in the name of allah mathematical olympiad problems around the world

China(Team Selection Test) 180-274
China(Western Mathematical Olympiad) 275-291
China(North and South) 292-296
USA(AIME) 895-973
APMO 1989-2009
APMO 1989
1 Let x1, x2, · · · , xn be positive real numbers, and let
S = x1 + x2 + · · ·+ xn.
2! +
S3
2 Prove that the equation 6(6a2 + 3b2 + c2) = 5n2
has no solutions in integers except a = b = c = n = 0.
3 Let A1, A2, A3 be three points in the plane, and for convenience, let A4 = A1, A5 = A2. For n = 1, 2, and 3, suppose that Bn is the midpoint of AnAn+1, and suppose that Cn is the midpoint of AnBn. Suppose that AnCn+1 and BnAn+2 meet at Dn, and that AnBn+1
and CnAn+2 meet at En. Calculate the ratio of the area of triangle D1D2D3 to the area of triangle E1E2E3.
4 Let S be a set consisting of m pairs (a, b) of positive integers with the property that 1 ≤ a < b ≤ n. Show that there are at least
4m · (m− n2
4 ) 3n
triples (a, b, c) such that (a, b), (a, c), and (b, c) belong to S.
5 Determine all functions f from the reals to the reals for which
(1) f(x) is strictly increasing and (2) f(x) + g(x) = 2x for all real x,
where g(x) is the composition inverse function to f(x). (Note: f and g are said to be composition inverses if f(g(x)) = x and g(f(x)) = x for all real x.)
http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page
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APMO 1990
1 Given triangle ABC, let D, E, F be the midpoints of BC, AC, AB respectively and let G be the centroid of the triangle. For each value of ∠BAC, how many non-similar triangles are there in which AEGF is a cyclic quadrilateral?
2 Let a1, a2, · · · , an be positive real numbers, and let Sk be the sum of the products of a1, a2, · · · , an taken k at a time. Show that
SkSn−k ≥ (
for k = 1, 2, · · · , n− 1.
3 Consider all the triangles ABC which have a fixed base AB and whose altitude from C is a constant h. For which of these triangles is the product of its altitudes a maximum?
4 A set of 1990 persons is divided into non-intersecting subsets in such a way that
1. No one in a subset knows all the others in the subset,
2. Among any three persons in a subset, there are always at least two who do not know each other, and
3. For any two persons in a subset who do not know each other, there is exactly one person in the same subset knowing both of them.
(a) Prove that within each subset, every person has the same number of acquaintances.
(b) Determine the maximum possible number of subsets.
Note: It is understood that if a person A knows person B, then person B will know person A; an acquaintance is someone who is known. Every person is assumed to know one’s self.
5 Show that for every integer n ≥ 6, there exists a convex hexagon which can be dissected into exactly n congruent triangles.
APMO 1991
1 Let G be the centroid of a triangle ABC, and M be the midpoint of BC. Let X be on AB and Y on AC such that the points X, Y , and G are collinear and XY and BC are parallel. Suppose that XC and GB intersect at Q and Y B and GC intersect at P . Show that triangle MPQ is similar to triangle ABC.
2 Suppose there are 997 points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least 1991 red points in the plane. Can you find a special case with exactly 1991 red points?
3 Let a1, a2, · · · , an, b1, b2, · · · , bn be positive real numbers such that a1 + a2 + · · · + an = b1 + b2 + · · ·+ bn. Show that
a2 1
a1 + b1 +
a2 2
2
4 During a break, n children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule:
He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on.
Determine the values of n for which eventually, perhaps after many rounds, all children will have at least one candy each.
5 Given are two tangent circles and a point P on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point P .
APMO 1992
1 A triangle with sides a, b, and c is given. Denote by s the semiperimeter, that is s = a + b + c
2 .
Construct a triangle with sides s−a, s− b, and s−c. This process is repeated until a triangle can no longer be constructed with the side lengths given.
For which original triangles can this process be repeated indefinitely?
2 In a circle C with centre O and radius r, let C1, C2 be two circles with centres O1, O2 and radii r1, r2 respectively, so that each circle Ci is internally tangent to C at Ai and so that C1, C2 are externally tangent to each other at A.
Prove that the three lines OA, O1A2, and O2A1 are concurrent.
3 Let n be an integer such that n > 3. Suppose that we choose three numbers from the set {1, 2, . . . , n}. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations. (a) Show that if we choose all three numbers greater than
n
2 , then the values of these combinations are all distinct. (b) Let p be
a prime number such that p ≤ √
n. Show that the number of ways of choosing three numbers so that the smallest one is p and the values of the combinations are not all distinct is precisely the number of positive divisors of p− 1.
4 Determine all pairs (h, s) of positive integers with the following property:
If one draws h horizontal lines and another s lines which satisfy (i) they are not horizontal, (ii) no two of them are parallel, (iii) no three of the h + s lines are concurrent,
then the number of regions formed by these h + s lines is 1992.
5 Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.
APMO 1993
1 Let ABCD be a quadrilateral such that all sides have equal length and ∠ABC = 60o. Let l be a line passing through D and not intersecting the quadrilateral (except at D). Let E and F be the points of intersection of l with AB and BC respectively. Let M be the point of intersection of CE and AF .
Prove that CA2 = CM × CE.
2 Find the total number of different integer values the function
f(x) = [x] + [2x] + [ 5x
3 Let
g(x) = cn+1x n+1 + cnxn + · · ·+ c0
be non-zero polynomials with real coefficients such that g(x) = (x + r)f(x) for some real number r. If a = max(|an|, . . . , |a0|) and c = max(|cn+1|, . . . , |c0|), prove that
a
4 Determine all positive integers n for which the equation
xn + (2 + x)n + (2− x)n = 0
has an integer as a solution.
5 Let P1, P2, . . ., P1993 = P0 be distinct points in the xy-plane with the following properties: (i) both coordinates of Pi are integers, for i = 1, 2, . . . , 1993; (ii) there is no point other than Pi and Pi+1 on the line segment joining Pi with Pi+1 whose coordinates are both integers, for i = 0, 1, . . . , 1992.
Prove that for some i, 0 ≤ i ≤ 1992, there exists a point Q with coordinates (qx, qy) on the line segment joining Pi with Pi+1 such that both 2qx and 2qy are odd integers.
APMO 1994
1 Let f : R → R be a function such that (i) For all x, y ∈ R,
f(x) + f(y) + 1 ≥ f(x + y) ≥ f(x) + f(y)
(ii) For all x ∈ [0, 1), f(0) ≥ f(x), (iii) −f(−1) = f(1) = 1.
Find all such functions f .
2 Given a nondegenerate triangle ABC, with circumcentre O, orthocentre H, and circumradius R, prove that |OH| < 3R.
3 Let n be an integer of the form a2 + b2, where a and b are relatively prime integers and such that if p is a prime, p ≤
√ n, then p divides ab. Determine all such n.
4 Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?
5 You are given three lists A, B, and C. List A contains the numbers of the form 10k in base 10, with k any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively:
... ...
Prove that for every integer n > 1, there is exactly one number in exactly one of the lists B or C that has exactly n digits.
APMO 1995
1 Determine all sequences of real numbers a1, a2, . . ., a1995 which satisfy:
2 √
an − (n− 1) ≥ an+1 − (n− 1), for n = 1, 2, . . . 1994,
and 2 √
a1995 − 1994 ≥ a1 + 1.
2 Let a1, a2, . . ., an be a sequence of integers with values between 2 and 1995 such that: (i) Any two of the ai’s are realtively prime, (ii) Each ai is either a prime or a product of primes. Determine the smallest possible values of n to make sure that the sequence will contain a prime number.
3 Let PQRS be a cyclic quadrilateral such that the segments PQ and RS are not parallel. Consider the set of circles through P and Q, and the set of circles through R and S. Determine the set A of points of tangency of circles in these two sets.
4 Let C be a circle with radius R and centre O, and S a fixed point in the interior of C. Let AA′ and BB′ be perpendicular chords through S. Consider the rectangles SAMB, SBN ′A′, SA′M ′B′, and SB′NA. Find the set of all points M , N ′, M ′, and N when A moves around the whole circle.
5 Find the minimum positive integer k such that there exists a function f from the set Z of all integers to {1, 2, . . . k} with the property that f(x) 6= f(y) whenever |x− y| ∈ {5, 7, 12}.
APMO 1996
1 Let ABCD be a quadrilateral AB = BC = CD = DA. Let MN and PQ be two segments
perpendicular to the diagonal BD and such that the distance between them is d > BD
2 , with
M ∈ AD, N ∈ DC, P ∈ AB, and Q ∈ BC. Show that the perimeter of hexagon AMNCQP does not depend on the position of MN and PQ so long as the distance between them remains constant.
2 Let m and n be positive integers such that n ≤ m. Prove that
2nn! ≤ (m + n)! (m− n)!
≤ (m2 + m)n
3 If ABCD is a cyclic quadrilateral, then prove that the incenters of the triangles ABC, BCD, CDA, DAB are the vertices of a rectangle.
4 The National Marriage Council wishes to invite n couples to form 17 discussion groups under the following conditions:
(1) All members of a group must be of the same sex; i.e. they are either all male or all female.
(2) The difference in the size of any two groups is 0 or 1.
(3) All groups have at least 1 member.
(4) Each person must belong to one and only one group.
Find all values of n, n ≤ 1996, for which this is possible. Justify your answer.
5 Let a, b, c be the lengths of the sides of a triangle. Prove that
√ a + b− c +
√ b + c− a +
√ c + a− b ≤
APMO 1998
1 Let F be the set of all n-tuples (A1, . . . , An) such that each Ai is a subset of {1, 2, . . . , 1998}. Let |A| denote the number of elements of the set A. Find∑
(A1,...,An)∈F
|A1 ∪A2 ∪ · · · ∪An|
2 Show that for any positive integers a and b, (36a + b)(a + 36b) cannot be a power of 2.
3 Let a, b, c be positive real numbers. Prove that( 1 +
a
b
)( 1 +
b
c
)( 1 +
c
a
) ≥ 2
( 1 +
) .
4 Let ABC be a triangle and D the foot of the altitude from A. Let E and F lie on a line passing through D such that AE is perpendicular to BE, AF is perpendicular to CF , and E and F are different from D. Let M and N be the midpoints of the segments BC and EF , respectively. Prove that AN is perpendicular to NM .
5 Find the largest integer n such that n is divisible by all positive integers less than 3 √
n.
APMO 1999
1 Find the smallest positive integer n with the following property: there does not exist an arithmetic progression of 1999 real numbers containing exactly n integers.
2 Let a1, a2, . . . be a sequence of real numbers satisfying ai+j ≤ ai + aj for all i, j = 1, 2, . . . . Prove that
a1 + a2
for each positive integer n.
3 Let Γ1 and Γ2 be two circles intersecting at P and Q. The common tangent, closer to P , of Γ1 and Γ2 touches Γ1 at A and Γ2 at B. The tangent of Γ1 at P meets Γ2 at C, which is different from P , and the extension of AP meets BC at R. Prove that the circumcircle of triangle PQR is tangent to BP and BR.
4 Determine all pairs (a, b) of integers with the property that the numbers a2 + 4b and b2 + 4a are both perfect squares.
5 Let S be a set of 2n + 1 points in the plane such that no three are collinear and no four concyclic. A circle will be called Good if it has 3 points of S on its circumference, n−1 points in its interior and n− 1 points in its exterior. Prove that the number of good circles has the same parity as n.
x3 i
101 .
2 Find all permutations a1, a2, . . . , a9 of 1, 2, . . . , 9 such that
a1 + a2 + a3 + a4 = a4 + a5 + a6 + a7 = a7 + a8 + a9 + a1
and a2
9 + a2 1
3 Let ABC be a triangle. Let M and N be the points in which the median and the angle bisector, respectively, at A meet the side BC. Let Q and P be the points in which the perpendicular at N to NA meets MA and BA, respectively. And O the point in which the perpendicular at P to BA meets AN produced.
Prove that QO is perpendicular to BC.
4 Let n, k be given positive integers with n > k. Prove that:
1 n + 1
kk(n − k)n−k
5 Given a permutation (a0, a1, . . . , an) of the sequence 0, 1, . . . , n. A transportation of ai with aj is called legal if ai = 0 for i > 0, and ai−1 + 1 = aj . The permutation (a0, a1, . . . , an) is called regular if after a number of legal transportations it becomes (1, 2, . . . , n). For which numbers n is the permutation (1, n, n − 1, . . . , 3, 2, 0) regular?
APMO 2001
1 For a positive integer n let S(n) be the sum of digits in the decimal representation of n. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of n is called a stump of n. Let T (n) be the sum of all stumps of n. Prove that n = S(n) + 9T (n).
2 Find the largest positive integer N so that the number of integers in the set {1, 2, . . . , N} which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).
3 Let two equal regular n-gons S and T be located in the plane such that their intersection is a 2n-gon (n ≥ 3). The sides of the polygon S are coloured in red and the sides of T in blue.
Prove that the sum of the lengths of the blue sides of the polygon S ∩ T is equal to the sum of the lengths of its red sides.
4 A point in the plane with a cartesian coordinate system is called a mixed point if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point.
5 Find the greatest integer n, such that there are n + 4 points A, B, C, D, X1, . . . , Xn in the plane with AB 6= CD that satisfy the following condition: for each i = 1, 2, . . . , n triangles ABXi and CDXi are equal.
APMO 2002
1 Let a1, a2, a3, . . . , an be a sequence of non-negative integers, where n is a positive integer. Let
An = a1 + a2 + · · ·+ an
Prove that a1!a2! . . . an! ≥ (bAnc!)n
where bAnc is the greatest integer less than or equal to An, and a! = 1 × 2 × · · · × a for a ≥ 1(and 0! = 1). When does equality hold?
2 Find all positive integers a and b such that
a2 + b
are both integers.
3 Let ABC be an equilateral triangle. Let P be a point on the side AC and Q be a point on the side AB so that both triangles ABP and ACQ are acute. Let R be the orthocentre of triangle ABP and S be the orthocentre of triangle ACQ. Let T be the point common to the segments BP and CQ. Find all possible values of ∠CBP and ∠BCQ such that the triangle TRS is equilateral.
4 Let x, y, z be positive numbers such that
1 x
+ 1 y
+ 1 z
xyz + √
x + √
y + √
z
5 Let R denote the set of all real numbers. Find all functions f from R to R satisfying:
(i) there are only finitely many s in R such that f(s) = 0, and
(ii) f(x4 + y) = x3f(x) + f(f(y)) for all x, y in R.
APMO 2003
1 Let a, b, c, d, e, f be real numbers such that the polynomial
p(x) = x8 − 4x7 + 7x6 + ax5 + bx4 + cx3 + dx2 + ex + f
factorises into eight linear factors x−xi, with xi > 0 for i = 1, 2, . . . , 8. Determine all possible values of f .
2 Suppose ABCD is a square piece of cardboard with side length a. On a plane are two parallel lines `1 and `2, which are also a units apart. The square ABCD is placed on the plane so that sides AB and AD intersect `1 at E and F respectively. Also, sides CB and CD intersect `2 at G and H respectively. Let the perimeters of 4AEF and 4CGH be m1 and m2 respectively.
Prove that no matter how the square was placed, m1 + m2 remains constant.
3 Let k ≥ 14 be an integer, and let pk be the largest prime number which is strictly less than k. You may assume that pk ≥ 3k/4. Let n be a composite integer. Prove: (a) if n = 2pk, then n does not divide (n− k)!; (b) if n > 2pk, then n divides (n− k)!.
4 Let a, b, c be the sides of a triangle, with a + b + c = 1, and let n ≥ 2 be an integer. Show that
n √
2 2
5 Given two positive integers m and n, find the smallest positive integer k such that among any k people, either there are 2m of them who form m pairs of mutually acquainted people or there are 2n of them forming n pairs of mutually unacquainted people.
1 Determine all finite nonempty sets S of positive integers satisfying
i + j
(i, j) is an element of S for all i,j in S,
where (i, j) is the greatest common divisor of i and j.
2 Let O be the circumcenter and H the orthocenter of an acute triangle ABC. Prove that the area of one of the triangles AOH, BOH and COH is equal to the sum of the areas of the other two.
3 Let a set S of 2004 points in the plane be given, no three of which are collinear. Let L denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of S with at most two colours, such that for any points p, q of S, the number of lines in L which separate p from q is odd if and only if p and q have the same colour.
Note: A line ` separates two points p and q if p and q lie on opposite sides of ` with neither point on `.
4 For a real number x, let bxc stand for the largest integer that is less than or equal to x. Prove that ⌊
(n− 1)! n(n + 1)
) ( b2 + 2
) ( c2 + 2
a, b, c.
APMO 2005
1 Prove that for every irrational real number a, there are irrational real numbers b and b′ so that a + b and ab′ are both rational while ab and a + b′ are both irrational.
2 Let a, b, c be positive real numbers such that abc = 8. Prove that
a2√ (1 + a3)(1 + b3)
≥ 4 3
3 Prove that there exists a triangle which can be cut into 2005 congruent triangles.
4 In a small town, there are n×n houses indexed by (i, j) for 1 ≤ i, j ≤ n with (1, 1) being the house at the top left corner, where i and j are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by (1, c), where c ≤ n
2 . During each subsequent
time interval [t, t + 1], the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended neighbors of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters? A house indexed by (i, j) is a neighbor of a house indexed by (k, l) if |i− k|+ |j − l| = 1.
5 In a triangle ABC, points M and N are on sides AB and AC, respectively, such that MB = BC = CN . Let R and r denote the circumradius and the inradius of the triangle ABC, respectively. Express the ration MN/BC in terms of R and r.
APMO 2006
1 Let n be a positive integer. Find the largest nonnegative real number f(n) (depending on n) with the following property: whenever a1, a2, ..., an are real numbers such that a1+a2+· · ·+an
is an integer, there exists some i such that ai −
1 2
≥ f(n).
2 Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden ratio.
3 Let p ≥ 5 be a prime and let r be the number of ways of placing p checkers on a p × p checkerboard so that not all checkers are in the same row (but they may all be in the same column). Show that r is divisible by p5. Here, we assume that all the checkers are identical.
4 Let A,B be two distinct points on a given circle O and let P be the midpoint of the line segment AB. Let O1 be the circle tangent to the line AB at P and tangent to the circle O. Let l be the tangent line, different from the line AB, to O1 passing through A. Let C be the intersection point, different from A, of l and O. Let Q be the midpoint of the line segment BC and O2 be the circle tangent to the line BC at Q and tangent to the line segment AC. Prove that the circle O2 is tangent to the circle O.
5 In a circus, there are n clowns who dress and paint themselves up using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set of colours and no more than 20 clowns may use any one particular colour. Find the largest number n of clowns so as to make the ringmaster’s order possible.
APMO 2007
1 Let S be a set of 9 distinct integers all of whose prime factors are at most 3. Prove that S contains 3 distinct integers such that their product is a perfect cube.
P.S:It from http://www.kms.or.kr/competitions/apmo/
Now I see ”
{ The contest problems are to be kept confidential until they are posted on the offcial APMO website. Please do not disclose nor discuss the problems over the internet until that date. No calculators are to be used during the contest. ”
Am I wrong? If so, Please Mods locked topics of mine on this contest. :) Thanks!
2 Let ABC be an acute angled triangle with ∠BAC = 600 and AB > AC. Let I be the incenter, and H the orthocenter of the triangle ABC . Prove that 2∠AHI = 3∠ABC.
Now I see ”
3 Consider n disks C1;C2; ...;Cn in a plane such that for each 1 ≤ i < n, the center of Ci is on the circumference of Ci+1, and the center of Cn is on the circumference of C1. Define the score of such an arrangement of n disks to be the number of pairs (i; j) for which Ci properly contains Cj . Determine the maximum possible score.
Now I see ”
4 Let x; y and z be positive real numbers such that √
x+ √
y+ √
+
APMO 2007
5 A regular (5× 5)-array of lights is defective, so that toggling the switch for one light causes each adjacent light in the same row and in the same column as well as the light itself to change state, from on to off, or from off to on. Initially all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Find all the possible positions of this light.
APMO 2008
1 Let ABC be a triangle with ∠A < 60. Let X and Y be the points on the sides AB and AC, respectively, such that CA + AX = CB + BX and BA + AY = BC + CY . Let P be the point in the plane such that the lines PX and PY are perpendicular to AB and AC, respectively. Prove that ∠BPC < 120.
See all problems from APMO 2008 here, http://www.mathlinks.ro/viewtopic.php?p=10739771073977
2 Students in a class form groups each of which contains exactly three members such that any two distinct groups have at most one member in common. Prove that, when the class size is 46, there is a set of 10 students in which no group is properly contained.
3 Let Γ be the circumcircle of a triangle ABC. A circle passing through points A and C meets the sides BC and BA at D and E, respectively. The lines AD and CE meet Γ again at G and H, respectively. The tangent lines of Γ at A and C meet the line DE at L and M , respectively. Prove that the lines LH and MG meet at Γ.
4 Consider the function f : N0 → N0, where N0 is the set of all non-negative integers, defined by the following conditions :
(i) f(0) = 0; (ii) f(2n) = 2f(n) and (iii) f(2n + 1) = n + 2f(n) for all n ≥ 0.
(a) Determine the three sets L = {n|f(n) < f(n + 1)}, E = {n|f(n) = f(n + 1)}, and G = {n|f(n) > f(n+1)}. (b) For each k ≥ 0, find a formula for ak = max{f(n) : 0 ≤ n ≤ 2k} in terms of k.
5 Let a, b, c be integers satisfying 0 < a < c − 1 and 1 < b < c. For each k, 0 ≤ k ≤ a, Let rk, 0 ≤ rk < c be the remainder of kb when divided by c. Prove that the two sets {r0, r1, r2, · · · , ra} and {0, 1, 2, · · · , a} are different.
APMO 2009
1 Consider the following operation on positive real numbers written on a blackboard: Choose a number r written on the blackboard, erase that number, and then write a pair of positive real numbers a and b satisfying the condition 2r2 = ab on the board.
Assume that you start out with just one positive real number r on the blackboard, and apply this operation k2 − 1 times to end up with k2 positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed kr.
2 Let a1, a2, a3, a4, a5 be real numbers satisfying the following equations: a1
k2 + 1 +
Find the value of a1
37 +
a2
38 +
a3
39 +
a4
40 +
a5
41 (Express the value in a single fraction.)
3 Let three circles Γ1,Γ2,Γ3, which are non-overlapping and mutually external, be given in the plane. For each point P in the plane, outside the three circles, construct six points A1, B1, A2, B2, A3, B3 as follows: For each i = 1, 2, 3, Ai, Bi are distinct points on the circle Γi such that the lines PAi and PBi are both tangents to Γi. Call the point P exceptional if, from the construction, three lines A1B1, A2B2, A3B3 are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle.
4 Prove that for any positive integer k, there exists an arithmetic sequence a1
b1 , a2
b2 , a3
b3 , ...,
ak
bk of rational numbers, where ai, bi are relatively prime positive integers for each i = 1, 2, ..., k such that the positive integers a1, b1, a2, b2, ..., ak, bk are all distinct.
5 Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a 90 degrees left turn after every ` kilometer driving from start, Rob makes a 90 degrees right turn after every r kilometer driving from start, where ` and r are relatively prime positive integers.
In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair (`, r) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?
2004
1 Determine all integers a and b, so that (a3 + b)(a + b3) = (a + b)4
2 Solve the following equation for real numbers:
√ 4− x
2 (all square roots are non negative)
3 Given is a convex quadrilateral ABCD with ∠ADC = ∠BCD > 90. Let E be the point of intersection of the line AC with the parallel line to AD through B and F be the point of intersection of the line BD with the parallel line to BC through A. Show that EF is parallel to CD
4 The sequence < xn > is defined through: xn+1 = (
n
2004 +
2004 + 1 for n > 0
Let x1 be a non-negative integer smaller than 204 so that all members of the sequence are non-negative integers. Show that there exist infinitely many prime numbers in this sequence.
2005
1 Show for all integers n ≥ 2005 the following chaine of inequalities: (n + 830)2005 < n(n + 1) . . . (n + 2004) < (n + 1002)2005
2 Construct the semicircle h with the diameter AB and the midpoint M . Now construct the semicircle k with the diameter MB on the same side as h. Let X and Y be points on k, such
that the arc BX is 3 2
times the arc BY . The line MY intersects the line BX in D and the semicircle h in C. Show that Y ist he midpoint of CD.
3 For which values of k and d has the system x3 + y3 = 2 and y = kx + d no real solutions (x, y)?
4 Prove: if an infinte arithmetic sequence (an = a0 + nd) of positive real numbers contains two different powers of an integer a > 1, then the sequence contains an infinite geometric sequence (bn = b0q
n) of real numbers.
2006
1 Let 0 < x < y be real numbers. Let H = 2xy
x + y , G =
√ xy , A =
x + y
2 , Q =
√ x2 + y2
2 be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of x and y. As generally known H < G < A < Q. Arrange the intervals [H,G] , [G, A] and [A,Q] in ascending order by their length.
2 Let n > 1 be a positive integer an a a real number. Determine all real solutions (x1, x2, . . . , xn) to following system of equations: x1 +ax2 = 0 x2 +a2x3 = 0 xk +akxk+1 = 0 xn +anx1 = 0
3 In a non isosceles triangle ABC let w be the angle bisector of the exterior angle at C. Let D be the point of intersection of w with the extension of AB. Let kA be the circumcircle of the triangle ADC and analogy kB the circumcircle of the triangle BDC. Let tA be the tangent line to kA in A and tB the tangent line to kB in B. Let P be the point of intersection of tA and tB. Given are the points A and B. Determine the set of points P = P (C) over all points C, so that ABC is a non isosceles, acute-angled triangle.
4 Let < hn > n ∈ N a harmonic sequence of positive real numbers (that means that every hn
is the harmonic mean of its two neighbours hn−1 and hn+1 : hn = 2hn−1hn+1
hn−1 + hn+1 ) Show that:
if the sequence includes a member hj , which is the square of a rational number, it includes infinitely many members hk, which are squares of rational numbers.
2007
1 Let 0 < x0, x1, . . . , x669 < 1 be pairwise distinct real numbers. Show that there exists a pair
(xi, xj) with 0 < xixj(xj − xi) < 1
2007
2 Find all tuples (x1, x2, x3, x4, x5) of positive integers with x1 > x2 > x3 > x4 > x5 > 0 and⌊ x1 + x2
3
⌋2
3
⌋2
= 38.
3 Let a be a positive real number and n a non-negative integer. Determine S − T , where
S = 2n+1∑
and T = 2n+1∑
a|b k 2 c|
4 Let M be the intersection of the diagonals of a convex quadrilateral ABCD. Determine all such quadrilaterals for which there exists a line g that passes through M and intersects the side AB in P and the side CD in Q, such that the four triangles APM , BPM , CQM , DQM are similar.
2008
1 Show: For all real numbers a, b, c with 0 < a, b, c < 1 is: √
a2bc + ab2c + abc2+ √
(1− a)2(1− b)(1− c) + (1− a)(1− b)2(1− c) + (1− a)(1− b)(1− c)2 <√ 3
2 For a real number x is [x] the next smaller integer to x, that is the integer g with g < g + 1, and {x} = x − [x] is the decimal part of x. Determine all triples (a, b, c) of real numbers, which fulfil the following system of equations: {a} + [b] + {c} = 2, 9 {b} + [c] + {a} = 5, 3 {c}+ [a] + {b} = 4, 0
3 Given is an acute angled triangle ABC. Determine all points P inside the triangle with
1 ≤ ∠APB
∠ACB , ∠BPC
∠BAC , ∠CPA
∠CBA ≤ 2
k=n
(2k + 1)n
k Show that there exists no n, for which an
is a non-negative integer.
Athens, Greece
1 Let a, b, c be positive real numbers. Find all real solutions (x, y, z) of the sistem: ax + by = (x− y)2 by + cz = (y − z)2 cz + ax = (z − x)2
2 Let ABCD be a cyclic quadrilateral and let HA,HB,HC ,HD be the orthocenters of the triangles BCD, CDA, DAB and ABC respectively. Show that the quadrilaterals ABCD and HAHBHCHD are congruent.
3 Show that for any positive integer m, there exists a positive integer n so that in the decimal representations of the numbers 5m and 5n, the representation of 5n ends in the representation of 5m.
4 Let a, b, c be positive real numbers. Find all real solutions (x, y, z) of the sistem: ax + by = (x− y)2 by + cz = (y − z)2 cz + ax = (z − x)2
Balkan MO 1985
1 In a given triangle ABC, O is its circumcenter, D is the midpoint of AB and E is the centroid of the triangle ACD. Show that the lines CD and OE are perpendicular if and only if AB = AC.
2 Let a, b, c, d ∈ [−π
2 , π
2 ] be real numbers such that sin a + sin b + sin c + sin d = 1 and cos 2a +
cos 2b + cos 2c + cos 2d ≥ 10 3
. Prove that a, b, c, d ∈ [0, π
6 ]
3 Let S be the set of all positive integers of the form 19a+85b, where a, b are arbitrary positive integers. On the real axis, the points of S are colored in red and the remaining integer numbers are colored in green. Find, with proof, whether or not there exists a point A on the real axis such that any two points with integer coordinates which are symmetrical with respect to A have necessarily distinct colors.
4 There are 1985 participants to an international meeting. In any group of three participants there are at least two who speak the same language. It is known that each participant speaks at most five languages. Prove that there exist at least 200 participans who speak the same language.
Balkan MO 1986
1 A line passing through the incenter I of the triangle ABC intersect its incircle at D and E and its circumcircle at F and G, in such a way that the point D lies between I and F . Prove that: DF · EG ≥ r2.
2 Let ABCD be a tetrahedron and let E,F, G,H, K,L be points lying on the edges AB,BC, CD,DA,DB,DC respectively, in such a way that AE ·BE = BF · CF = CG ·AG = DH ·AH = DK ·BK = DL · CL. Prove that the points E,F, G,H, K,L lie all on a sphere.
3 Let a, b, c be real numbers such that ab is not 0, c > 0 and let (an)n≥1 be the sequence of real
numbers defined by: a1 = a, a2 = b and an+1 = a2
n + c
an−1 , ∀n ≥ 2. Show that all the sequence’s
terms are integer numbers if and only if the numbers a, b and a2 + b2 + c
ab are integers.
Remark : as Valentin mentions here [url]http://www.mathlinks.ro/Forum/viewtopic.php?p=492872searchid = 51674358492872[/url], the5−thromaniantstproblemfrom2006, followsimmediatlyfromthisbmoproblem.HereisaniceproblemproposedbyT ituAndreescuforBMOin1985.
Let ABC a triangle and P a point such that the triangles PAB,PBC, PCA have the same area and the same perimeter. Prove that if:
a) P is in the interior of the trinagle ABC then ABC is equilateral. b) P is in the exterior of the trinagle ABC then ABC is right angled triangle. ;)
Balkan MO 1987
1 Let a be a real number and let f : R → R be a function satisfying: f(0) = 1 2
and f(x + y) =
f(x)f(a− y) + f(y)f(a− x), ∀x, y ∈ R. Prove that f is constant.
2 Find all real numbers x, y greater than 1, satisfying the condition that the numbers √
x− 1+√ y − 1 and
√ x + 1 +
√ y + 1 are nonconsecutive integers.
3 In the triangle ABC the following equality holds: sin23 A
2 cos48
BC .
4 Two circles K1 and K2, centered at O1 and O2 with radii 1 and √
2 respectively, intersect at A and B. Let C be a point on K2 such that the midpoint of AC lies on K1. Find the lenght of the segment AC if O1O2 = 2
Balkan MO 1988
1 Let ABC be a triangle and let M,N,P be points on the line BC such that AM,AN,AP are the altitude, the angle bisector and the median of the triangle, respectively. It is known that
[AMP ] [ABC]
= 1 4
. Find the angles of triangle ABC.
2 Find all polynomials of two variables P (x, y) which satisfy P (a, b)P (c, d) = P (ac + bd, ad + bc),∀a, b, c, d ∈ R
3 Let ABCD be a tetrahedron and let d be the sum of squares of its edges’ lengths. Prove that the tetrahedron can be included in a region bounded by two parallel planes, the distances
between the planes being at most
√ d
2 √
3
4 Let (an)n≥1 be a sequence defined by an = 2n + 49. Find all values of n such that an = pg, an+1 = rs, where p, q, r, s are prime numbers with p < q, r < s and q − p = s− r.
Nicosia, Cyprus
1 Let a, b, c be positive real numbers. Prove the inequality
1 a (b + 1)
.
2 Let ABC be a triangle and m a line which intersects the sides AB and AC at interior points D and F , respectively, and intersects the line BC at a point E such that C lies between B and E. The parallel lines from the points A, B, C to the line m intersect the circumcircle of triangle ABC at the points A1, B1 and C1, respectively (apart from A, B, C). Prove that the lines A1E , B1F and C1D pass through the same point.
Greece
3 Find all triplets of positive rational numbers (m,n, p) such that the numbers m+ 1 np
, n+ 1
mn are integers.
Valentin Vornicu, Romania
4 Let m be a positive integer and {an}n≥0 be a sequence given by a0 = a ∈ N, and
an+1 =
an + m otherwise.
Find all values of a such that the sequence is periodical (starting from the beginning).
3x − 5y = z2
in positive integers.
2 Let MN be a line parallel to the side BC of a triangle ABC, with M on the side AB and N on the side AC. The lines BN and CM meet at point P . The circumcircles of triangles BMP and CNP meet at two distinct points P and Q. Prove that ∠BAQ = ∠CAP .
3 A 9×12 rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres C1, C2..., C96 in such way that the following to conditions are both fulfilled
(i) the distances C1C2, ...C95C96, C96C1 are all equal to √
13
(ii) the closed broken line C1C2...C96C1 has a centre of symmetry?
{Bulgaria.
4 Denote by S the set of all positive integers. Find all functions f : S → S such that
f
{Bulgaria.
Vilnius, Lithuania
1 Given a sequence a1, a2, ... of non-negative real numbers satisfying the conditions:
1. an + a2n ≥ 3n 2. an+1 + n ≤ 2 √
an(n + 1)
for all n = 1, 2, ... indices
(1) Prove that the inequality an ≥ n holds for evere n ∈ N (2) Give an example of such a sequence
2 Let P (x) be a polynomial with a non-negative coefficients. Prove that if the inequality
P
( 1 x
) P (x) ≥ 1 holds for x = 1, then this inequality holds for each positive x.
3 Let p, q, r be positive real numbers and n a natural number. Show that if pqr = 1, then
1 pn + qn + 1
≤ 1.
4 Let x1, x2, ..., xn be real numbers with arithmetic mean X. Prove that there is a positive
integer K such that for any natural number i satisfying 1 ≤ i < K, we have 1
K − i
xj ≤
X. (In other words, the arithmetic mean of each of the lists {x1, x2, ..., xK}, {x2, x3, ..., xK}, {x3, ..., xK}, ..., {xK−1, xK}, {xK} is not greater than X.)
5 Determine the range of the following function defined for integer k,
f(k) = (k)3 + (2k)5 + (3k)7 − 6k
where (k)2n+1 denotes the multiple of 2n + 1 closest to k
6 A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is 1001. What is the sum of the six numbers on the faces?
7 Find all sets X consisting of at least two positive integers such that for every two elements m and n of the set X, where n ¿ m, there exists an element k of X such that n = mk2.
8 Let f (x) be a non-constant polynomial with integer coefficients, and let u be an arbitrary positive integer. Prove that there is an integer n such that f (n) has at least u distinct prime factors and f (n) 6= 0.
9 A set S of n−1 natural numbers is given (n ≥ 3). There exist at least at least two elements in this set whose difference is not divisible by n. Prove that it is possible to choose a non-empty subset of S so that the sum of its elements is divisible by n.
Vilnius, Lithuania
10 Is there an infinite sequence of prime numbers p1, p2, . . ., pn, pn+1, . . . such that |pn+1−2pn| = 1 for each n ∈ N?
11 Given a table m x n, in each cell of which a number +1 or -1 is written. It is known that initially exactly one -1 is in the table, all the other numbers being +1. During a move it is allowed to cell containing -1, replace this -1 by 0, and simultaneously multiply all the numbers in the neighbouring cells by -1 (we say that two cells are neighbouring if they have a common side). Find all (m,n) for which using such moves one can obtain the table containing zeros only, regardless of the cell in which the initial -1 stands.
12 There are 2n different numbers in a row. Bo one move we can onterchange any two numbers or interchange any 3 numbers cyclically (choose a, b, c and place a instead of b, b instead of c, c instead of a). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing order ?
13 The 25 member states of the European Union set up a committee with the following rules: 1) the committee should meet daily; 2) at each meeting, at least one member should be represented; 3) at any two different meetings, a different set of member states should be represented; 4) at nth meeting, for every k < n, the set of states represented should include at least one state that was represented at the kth meeting.
For how many days can the committee have its meetings ?
14 We say that a pile is a set of four or more nuts. Two persons play the following game. They start with one pile of n ≥ 4 nuts. During a move a player takes one of the piles that they have and split it into two nonempty sets (these sets are not necessarily piles, they can contain arbitrary number of nuts). If the player cannot move, he loses. For which values of n does the first player have a winning strategy?
15 A circle is divided into 13 segments, numbered consecutively from 1 to 13. Five fleas called A,B,C,D and E are sitting in the segments 1,2,3,4 and 5. A flea is allowed to jump to an empty segment five positions away in either direction around the circle. Only one flea jumps at the same time, and two fleas cannot be in the same segment. After some jumps, the fleas are back in the segments 1,2,3,4,5, but possibly in some other order than they started. Which orders are possible ?
16 Through a point P exterior to a given circle pass a secant and a tangent to the circle. The secant intersects the circle at A and B, and the tangent touches the circle at C on the same side of the diameter through P as the points A and B. The projection of the point C on the diameter is called Q. Prove that the line QC bisects the angle ∠AQB.
17 Consider a rectangle with sidelengths 3 and 4, pick an arbitrary inner point on each side of this rectangle. Let x, y, z and u denote the side lengths of the quadrilateral spanned by these four points. Prove that 25 ≤ x2 + y2 + z2 + u2 ≤ 50.
Vilnius, Lithuania
18 A ray emanating from the vertex A of the triangle ABC intersects the side BC at X and the
circumcircle of triangle ABC at Y . Prove that 1
AX +
.
19 Let D be the midpoint of the side BC of a triangle ABC. Let M be a point on the side BC such that ∠BAM = ∠DAC. Further, let L be the second intersection point of the circumcircle of the triangle CAM with the side AB, and let K be the second intersection point of the circumcircle of the triangle BAM with the side AC. Prove that KL BC.
20 Three fixed circles pass through the points A and B. Let X be a variable point on the first circle different from A and B. The line AX intersects the other two circles at Y and Z (with
Y between X and Z). Show that the ratio XY
Y Z is independent of the position of X.
Baltic Way 2005
1 Let a0 be a positive integer. Define the sequence {an}n≥0 as follows: if
an = j∑
i=0
an+1 = c2005 0 + c2005
1 + · · ·+ c2005 j .
Is it possible to choose a0 such that all terms in the sequence are distinct?
2 Let α, β and γ be three acute angles such that sinα + sinβ + sin γ = 1. Show that
tan2 α + tan2 β + tan2 γ ≥ 3 8 .
3 Consider the sequence {ak}k≥1 defined by a1 = 1, a2 = 1 2
and
1 4akak+1
for k ≥ 1.
Prove that 1
a3a5 + · · ·+ 1
a98a100 < 4.
4 Find three different polynomials P (x) with real coefficients such that P ( x2 + 1
) = P (x)2 +1
for all real x.
5 Let a, b, c be positive real numbers such that abc = 1. Proove that
a
c2 + 2 ≤ 1
6 Let N and K be positive integers satisfying 1 ≤ K ≤ N . A deck of N different playing cards is shuffled by repeating the operation of reversing the order of K topmost cards and moving these to the bottom of the deck. Prove that the deck will be back in its initial order after a number of operations not greater than (2N/K)2.
7 A rectangular array has n rows and 6 columns, where n > 2. In each cell there is written either 0 or 1. All rows in the array are different from each other. For each two rows (x1, x2, x3, x4, x5, x6) and (y1, y2, y3, y4, y5, y6), the row (x1y1, x2y2, x3y3, x4y4, x5y5, x6y6) can be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.
Baltic Way 2005
8 Consider a 25× 25 grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid?
9 A rectangle is divided into 200× 3 unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size 1× 2 is divisible by 3.
10 Let m = 30030 and let M be the set of positive divisors of m which have exactly 2 prime factors. Determine the smallest positive integer n with the following property: for any choice of n numbers from M , there exist 3 numbers a, b, c among them satisfying abc = m.
13 What the smallest number of circles of radius √
2 that are nedeed to cover a rectangle . (a)- of size 6 ∗ 3 ? (b)- of size 5 ∗ 3 ?
16 Let n be a positive integer, let p be prime and let q be a divisor of (n + 1)p − np. Show that p divides q − 1.
19 Is it possible to find 2005 different positive square numbers such that their sum is also a square number ?
20 Find all positive integers n = p1p2 · · · pk which divide (p1 + 1)(p2 + 1) · · · (pk + 1) where p1p2 · · · pk is the factorization of n into prime factors (not necessarily all distinct).
Baltic Way 2008
1 Problem 1 Determine all polynomials p(x) with real coefficients such that p((x + 1)3) = (p(x) + 1)3 and p(0) = 0.
2 Problem 2 Prove that if the real numbers a, b and c satisfy a2 + b2 + c2 = 3 then∑ a2
2 + b + c2 ≥ (a + b + c)2
12 . When does the inequality hold?
3 Does there exist an angle α ∈ (0, π/2) such that sinα, cos α, tanα and cotα, taken in some order, are consecutive terms of an arithmetic progression?
4 The polyminal P has integer coefficients and P(x)=5 for five different integers x.Show that there is no integer x such that -7¡P(x)¡5 or 5¡P(x)¡17
5 Suppose that Romeo and Juliet each have a regular tetrahedron to the vertices of which some positive real numbers are assigned. They associate each edge of their tetrahedra with the product of the two numbers assigned to its end points. Then they write on each face of their tetrahedra the sum of the three numbers associated to its three edges. The four numbers written on the faces of Romeo’s tetrahedron turn out to coincide with the four numbers written on Juliet’s tetrahedron. Does it follow that the four numbers assigned to the vertices of Romeo’s tetrahedron are identical to the four numbers assigned to the vertices of Juliet’s tetrahedron?
6 Find all finite sets of positive integers with at least two elements such that for any two numbers
a, b (a > b) belonging to the set, the number b2
a− b belongs to the set, too.
7 How many pairs (m,n) of positive integers with m < n fulfill the equation 3
2008 =
?
8 Consider a set A of positive integers such that the least element of A equals 1001 and the product of all elements of A is a perfect square. What is the least possible value of the greatest element of A?
9 Suppose that the positive integers a and b satisfy the equation ab − ba = 1008 Prove that a and b are congruent modulo 1008.
10 For a positive integer n, let S(n) denote the sum of its digits. Find the largest possible value
of the expression S(n)
S(16n) .
11 Consider a subset A of 84 elements of the set {1, 2, . . . , 169} such that no two elements in the set add up to 169. Show that A contains a perfect square.
Baltic Way 2008
12 In a school class with 3n children, any two children make a common present to exactly one other child. Prove that for all odd n it is possible that the following holds: For any three children A, B and C in the class, if A and B make a present to C then A and C make a present to B.
13 For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened: i) Every country voted for exactly three problems. ii) Any two countries voted for different sets of problems. iii) Given any three countries, there was a problem none of them voted for. Find the maximal possible number of participating countries.
14 Is it possible to build a 4× 4× 4 cube from blocks of the following shape consisting of 4 unit cubes?
15 Some 1× 2 dominoes, each covering two adjacent unit squares, are placed on a board of size n×n such that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is 2008, find the least possible value of n.
16 Problem 16 Let ABCD be a parallelogram. The circle with diameter AC intersects the line BD at points P and Q. The perpendicular to the line AC passing through the point C intersects the lines AB and AD at points X and Y , respectively. Prove that the points P, Q, X and Y lie on the same circle.
Click: I proved that XYKL is cyclic (where K,L are intersection points of circle with diameter AC and AB, AD) and I tried to show that KL,XY,PQ intersect in one point but I failed...
17 Assume that a, b, c and d are the sides of a quadrilateral inscribed in a given circle. Prove that the product (ab + cd)(ac + bd)(ad + bc) acquires its maximum when the quadrilateral is a square.
18 Let AB be a diameter of a circle S, and let L be the tangent at A. Furthermore, let c be a fixed, positive real, and consider all pairs of points X and Y lying on L, on opposite sides of A, such that |AX| · |AY | = c. The lines BX and BY intersect S at points P and Q, respectively. Show that all the lines PQ pass through a common point.
19 In a circle of diameter 1, some chords are drawn. The sum of their lengths is greater than 19. Prove that there is a diameter intersecting at least 7 chords.
20 Let M be a point on BC and N be a point on AB such that AM and CN are angle bisectors of
the triangle ABC. Given that ∠BNM
∠MNC =
∠BMN
2002
1 Prove that for all a, b, c ∈ R+ 0 we have
a
bc +
b
ac +
c
and determine when equality occurs.
2 Prove that there are no perfect squares in the array below:
11 111 1111 ... 22 222 2222 ... 33 333 3333 ... 44 444 4444 ... 55 555 5555 ... 66 666 6666 ... 77 777 7777 ... 88 888 8888 ... 99 999 9999 ...
3 Is it possible to number the 8 vertices of a cube from 1 to 8 in such a way that the value of the sum on every edge is different?
4 Two congruent right-angled isosceles triangles (with baselength 1) slide on a line as on the picture. What is the maximal area of overlap?
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 287[/img]
2003
1 Playing soccer with 3 goes as follows: 2 field players try to make a goal past the goalkeeper, the one who makes the goal stands goalman for next game, etc.
Arne, Bart and Cauchy played this game. Later, they tell their math teacher that A stood 12 times on the field, B 21 times on the field, C 8 times in the goal. Their teacher knows who made the 6th goal.
Who made it?
2 Through an internal point O of ABC one draws 3 lines, parallel to each of the sides, intersecting in the points shown on the picture.
+ |BE| |BC|
+ |CN | |CA|
.
3 Yesterday (=April 22, 2003) was Gittes birthday. She notices that her age equals the sum of the 4 digits of the year she was born in. How old is she?
4 The points in the plane with integer coordinates are numbered as below.
2004
1 Two 5× 1 rectangles have 2 vertices in common as on the picture. (a) Determine the area of overlap (b) Determine the length of the segment between the other 2 points of intersection, A and B.
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 290[/img]Howcanyougofromthenumber11to25byonlymultiplyingwith2ordecreasingwith3inaminimumnumberofsteps?
23 A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably)
While the salesmen isn’t watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?
4 How many pairs of positive integers (a, b) satisfy 1 a
+ 1 b
2005
1 [we’re 2005 while writing] According to a legend there is a monster that awakes every now and then to swallow everyone who is solving this problem, and then falls back asleep for as many years as the sum of the digits of that year. The monster first hit mathlinks/aops in the year +234.
But guys, don’t worry! Get your hopes up, and prove you’re safe this year, as well as for the coming 10 years! :D
[wording slightly adapted from original wording]
2 Starting with two points A and B, some circles and points are constructed as shown in the figure:the circle with centre A through B the circle with centre B through A the circle with centre C through A the circle with centre D through B the circle with centre E through A the circle with centre F through A the circle with centre G through A (I think the wording is not very rigorous, you should assume intersections from the drawing)
Show that M is the midpoint of AB.
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 291[/img]Provethat20052 can be written in at least 4 ways as the sum of 2 perfect (non-zero) squares.
34 (a) Be M an internal point of the convex quadrilateral ABCD. Prove that |MA| + |MB| < |AD| + |DC| + |CB|. (b) Be M an internal point of the triangle ABC. Note k = min(|MA|, |MB|, |MC|). Prove k + |MA| + |MB| + |MC| < |AB| + |BC| + |CA|.
n! ≤ (
)n
(please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)
3 Let {ak}k≥0 be a sequence given by a0 = 0, ak+1 = 3 · ak + 1 for k ∈ N.
Prove that 11 | a155
4 Given a cube in which you can put two massive spheres of radius 1. What’s the smallest possible value of the side - length of the cube? Prove that your answer is the best possible.
3 Find all continuous functions f : R → R such that
f(x)3 = − x
) , ∀x ∈ R.
lim n→+∞
n2 = +∞
1988
1 show that the polynomial x4 + 3x3 + 6x2 + 9x + 12 cannot be written as the product of 2 polynomials of degree 2 with integer coefficients.
2 A 3-dimensional cross is made up of 7 cubes, one central cube and 6 cubes that share a face with it. The cross is inscribed in a circle with radius 1. What’s its volume?
3 Work base 3. (so each digit is 0,1,2)
A good number of size n is a number in which there are no consecutive 1’s and no consecutive 2’s. How many good 10-digit numbers are there?
4 Be R a positive real number. If R, 1, R + 1 2
are triangle sides, call θ the angle between R and
R + 1 2
1989
1 Show that every subset of 1,2,...,99,100 with 55 elements contains at least 2 numbers with a difference of 9.
2 When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What’s the ratio of the areas of those pentagons?
3 Show that: α = ± π
12 + k · π
2 (k ∈ Z) ⇐⇒ |tanα|+ |cot α| = 4
4 Let D be the set of positive reals different from 1 and let n be a positive integer. If for
f : D → R we have xnf(x) = f(x2), and if f(x) = xn for 0 < x < 1
1989 and for x > 1989,
then prove that f(x) = xn for all x ∈ D.
1990
1 On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0]. You get a figure as below, find the area of the colored part.
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 277[/img]Leta and b be two primes having at least two digits, such that a > b.
Show that 240|
( a4 − b4
) and show that 240 is the greatest positive integer having this property.
23 We form a decimal code of 21 digits. the code may start with 0. Determine the probability that the fragment 0123456789 appears in the code.
4 Let f : R+ 0 → R+
0 be a strictly decreasing function.
(a) Be an a sequence of strictly positive reals so that ∀k ∈ N0 : k · f(ak) ≥ (k + 1) · f(ak+1) Prove that an is ascending, that lim
k→+∞ f(ak) = 0and that lim
k→+∞ ak = +∞
(b) Prove that there exist such a sequence (an) in R+ 0 if you know lim
x→+∞ f(x) = 0.
1991
1 Show that the number 111...111 with 1991 times the number 1, is not prime.
2 (a) Show that for every n ∈ N there is exactly one x ∈ R+ so that xn + xn+1 = 1. Call this xn. (b) Find lim
n→+∞ xn.
3 Given ABC equilateral, with X ∈ [A,B]. Then we define unique points Y,Z so that Y ∈ [B,C], Z ∈ [A,C], XY Z equilateral.
If Area (ABC) = 2 ·Area (XY Z), find the ratio of AX
XB , BY
ZA .
4 A word of length n that consists only of the digits 0 and 1, is called a bit-string of length n. (For example, 000 and 01101 are bit-strings of length 3 and 5.) Consider the sequence s(1), s(2), ... of bit-strings of length n > 1 which is obtained as follows : (1) s(1) is the bit- string 00...01, consisting of n − 1 zeros and a 1 ; (2) s(k + 1) is obtained as follows : (a) Remove the digit on the left of s(k). This gives a bit-string t of length n − 1. (b) Examine whether the bit-string t1 (length n, adding a 1 after t) is already in {s(1), s(2), ..., s(k)}. If this is the not case, then s(k + 1) = t1. If this is the case then s(k + 1) = t0.
For example, if n = 3 we get : s(1) = 001 → s(2) = 011 → s(3) = 111 → s(4) = 110 → s(5) = 101 → s(6) = 010 → s(7) = 100 → s(8) = 000 → s(9) = 001 → ...
Suppose N = 2n. Prove that the bit-strings s(1), s(2), ..., s(N) of length n are all different.
1992
1 For every positive integer n, determine the biggest positive integer k so that 2k| 3n + 1
2 It has come to a policeman’s ears that 5 gangsters (all of different height) are meeting, one of them is the clan leader, he’s the tallest of the 5. He knows the members will leave the building one by one, with a 10-minute break between them, and too bad for him Belgium has not enough policemen to follow all gangsters, so he’s on his own to spot the clanleader, and he can only follow one member.
So he decides to let go the first 2 people, and then follow the first one that is taller than those two. What’s the chance he actually catches the clan leader like this?
3 a conic with apotheme 1 slides (varying height and radius, with r < 1 2 ) so that the conic’s
area is 9 times that of its inscribed sphere. What’s the height of that conic?
4 Let A,B, P positive reals with P ≤ A+B. (a) Choose reals θ1, θ2 with A cos θ1+B cos θ2 = P and prove that
A sin θ1 + B sin θ2 ≤ √
(A + B − P )(A + B + P )
(b) Prove equality is attained when θ1 = θ2 = arccos (
P
and P = 1 4
( x2 + y2 − z2 − w2
) with 0 < x ≤ y ≤ x + z + w, z, w > 0 and z2 + w2 < x2 + y2.
Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts (x, y, z, w), the cyclical one has the greatest area.
1993
1 The 20 pupils in a class each send 10 cards to 10 (different) class members. [note: you cannot send a card to yourself.] (a) Show at least 2 pupils sent each other a card. (b) Now suppose we had n pupils sending m cards each. For which (m,n) is the above true? (That is, find minimal m(n) or maximal n(m))
2 A jeweler covers the diagonal of a unit square with small golden squares in the following way: - the sides of all squares are parallel to the sides of the unit square - for each neighbour is their sidelength either half or double of that square (squares are neighbour if they share a vertex) - each midpoint of a square has distance to the vertex of the unit square equal to 1 2 , 1 4 , 1 8 , ... of the diagonal. (so real length: ×
√ 2) - all midpoints are on the diagonal
(a) What is the side length of the middle square? (b) What is the total gold-plated area?
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 281[/img]Fora, b, c > 0 we have:
−1 <
)1993
< 1
34 Define the sequence oan as follows: oa0 = 1, oan = oan−1 · cos ( π
2n+1
1994
1 Let a, b, c > 0 the sides of a right triangle. Find all real x for which ax > bx + cx, with a is the longest side.
2 Determine all integer solutions (a,b,c) with c ≤ 94 for which: (a+ √
c)2+(b+ √
c
3 Two regular tetrahedrons A and B are made with the 8 vertices of a unit cube. (this way is unique)
What’s the volume of A ∪B?
4 Let (fi) be a sequence of functions defined by: f1(x) = x, fn(x) = √
fn−1(x)− 1 4 . (n ∈ N, n ≥
2) (a) Prove that fn(x) ≤ fn−1(x) for all x where both functions are defined. (b) Find for each n the points of x inside the domain for which fn(x) = x.
1995
1 Four couples play chess together. For the match, they’re paired as follows: (”man Clara” indicates the husband of Clara, etc.)
Bea⇐⇒ Eddy
Who is Hubert married to?
2 How many values of x ∈ [1, 3] are there, for which x2 has the same decimal part as x?
3 Points A,B, C, D are on a circle with radius R. |AC| = |AB| = 500, while the ratio between |DC|, |DA|, |DB| is 1, 5, 7. Find R.
4 Given a regular n-gon inscribed in a circle of radius 1, where n > 3. Define G(n) as the average length of the diagonals of this n-gon.
Prove that if n→∞, G(n)→ 4 π
.
1996
1 In triangle ADC we got AD = DC and D = 100. In triangle CAB we got CA = AB and A = 20.
Prove that AB = BC + CD.
2 Determine the gcd of all numbers of the form p8 − 1, with p a prime above 5.
3 Consider the points 1, 1 2 , 1 3 , ... on the real axis. Find the smallest value k ∈ N0 for which all
points above can be covered with 5 closed intervals of length 1 k .
4 Consider a real poylnomial p(x) = anxn + ... + a1x + a0. (a) If deg(p(x)) > 2 prove that deg(p(x)) = 2 + deg(p(x + 1) + p(x− 1)− 2p(x)). (b) Let p(x) a polynomial for which there are real constants r, s so that for all real x we have
p(x + 1) + p(x− 1)− rp(x)− s = 0
Prove deg(p(x)) ≤ 2. (c) Show, in (b) that s = 0 implies a2 = 0.
1997
1 Write the number 1997 as the sum of positive integers for which the product is maximal, and prove there’s no better solution.
2 In the cartesian plane, consider the curves x2 + y2 = r2 and (xy)2 = 1. Call Fr the convex polygon with vertices the points of intersection of these 2 curves. (if they exist)
(a) Find the area of the polygon as a function of r. (b) For which values of r do we have a regular polygon?
3 oa1b1 is isosceles with ∠a1ob1 = 36. Construct a2, b2, a3, b3, ... as below, with |oai+1| = |aibi| and ∠aiobi = 36, Call the summed area of the first k triangles Ak. Let S be the area of the isocseles triangle, drawn in - - -, with top angle 108 and |oc| = |od| = |oa1|, going through the points b2 and a2 as shown on the picture. (yes, cd is parallel to a1b1 there)
Show Ak < S for every positive integer k.
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 284[/img]Thirteenbirdsarriveandsitdowninaplane.It′sknownthatfromeach5− tupleofbirds, atleastfourbirdssitonacircle.DeterminethegreatestM ∈ {1, 2, ..., 13} such that from these 13 birds, at least M birds sit on a circle, but not necessarily M + 1 birds sit on a circle. (prove that your M is optimal)
1998
1 Prove there exist positive integers a,b,c for which a+ b+ c = 1998, the gcd is maximized, and 0 < a < b ≤ c < 2a. Find those numbers. Are they unique?
2 Given a cube with edges of length 1, e the midpoint of [bc], and m midpoint of the face cdc1d1, as on the figure. Find the area of intersection of the cube with the plane through the points a,m, e.
[img]http://www.mathlinks.ro/Forum/albumpic.php?picid = 279[/img]amagical3× 3 square is a 3× 3 matrix containing all number from 1 to 9, and of which the sum of every row, every column, every diagonal, are all equal.
Determine all magical 3× 3 square
34 A billiard table. (see picture) A white ball is on p1 and a red ball is on p2. The white ball is shot towards the red ball as shown on the pic, hitting 3 sides first. Find the minimal distance the ball must travel.
1999
1 Determine all 6-digit numbers (abcdef) so that (abcdef) = (def)2 where (x1x2...xn) is no multiplication but an n-digit number.
2 Let [mn] be a diameter of the circle C and [AB] a chord with given length on this circle. [AB] neither coincides nor is perpendicular to [MN ]. Let C,D be the orthogonal projections of A and B on [MN ] and P the midpoint of [AB]. Prove that ∠CPD does not depend on the chord [AB].
3 Determine all f : R → R for which
2 · f(x)− g(x) = f(y)− y and f(x) · g(x) ≥ x + 1.
4 Let a, b, m, n integers greater than 1. If an− 1 and bm + 1 are both primes, give as much info as possible on a, b, m, n.
2000
1 An integer consists of 7 different digits, and is a multiple of each of its digits.
What digits are in this nubmer?
2 Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.
3 Let pn be the n-th prime. (p1 = 2) Define the sequence (fj) as follows: - f1 = 1, f2 = 2 - ∀j ≥ 2: if fj = kpn for k < pn then fj+1 = (k + 1)pn - ∀j ≥ 2: if fj = p2
n then fj+1 = pn+1
(a) Show that all fi are different (b) from which index onwards are all fi at least 3 digits? (c) which integers do not appear in the sequence? (d) how many numbers with less than 3 digits appear in the sequence?
4 Solve for x ∈ [0, 2π[: sinx < cos x < tanx < cot x
2001
1 may be challenge for beginner section, but anyone is able to solve it if you really try.
show that for every natural n > 1 we have: (n− 1)2| nn−1 − 1
2 Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment.
Find the ”?”
3 In a circle we enscribe a regular 2001-gon and inside it a regular 667-gon with shared vertices.
Prove that the surface in the 2001-gon but not in the 667-gon is of the form k.sin3 ( π
2001
) .cos3
( π
2001
) with k a positive integer. Find k.
4 A student concentrates on solving quadratic equations in R. He starts with a first quadratic equation x2 + ax + b = 0 where a and b are both different from 0. If this first equation has solutions p and q with p ≤ q, he forms a second quadratic equation x2 + px + q = 0. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.
2002
1 Is it possible to number the 8 vertices of a cube from 1 to 8 in such a way that the value of the sum on every edge is different?
2 Determine all functions f : R → R so that ∀x : x · f( x
2 )− f(
2 x
100 <
1 10
4 A lamp is situated at point A and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What’s the area of its shadow?
2003
11-12
1 Playing soccer with 3 goes as follows: 2 field players try to make a goal past the goalkeeper, the one who makes the goal stands goalman for next game, etc.
Arne, Bart and Cauchy played this game. Later, they tell their math teacher that A stood 12 times on the field, B 21 times on the field, C 8 times in the goal. Their teacher knows who made the 6th goal.
Who made it?
2 Two circles C1 and C2 intersect at S. The tangent in S to C1 intersects C2 in A different from S. The tangent in S to C2 intersects C1 in B different from S. Another circle C3 goes through A,B, S. The tangent in S to C3 intersects C1 in P different from S and C2 in Q different from S. Prove that the distance PS is equal to the distance QS.
3 A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.
4 Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with r = 2
√ 2 goes through 4 points)
Prove that ∀n ∈ N,∃r so that the circle with midpoint 0,0 and radius r goes through at least n points.
2004
11-12
1 Consider a triangle with side lengths 501m, 668m, 835m. How many lines can be drawn with the property that such a line halves both area and perimeter?
2 Two bags contain some numbers, and the total number of numbers is prime.
When we tranfer the number 170 from 1 bag to bag 2, the average in both bags increases by one.
If the total sum of all numbers is 2004, find the number of numbers.
3 A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably)
While the salesmen isn’t watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?
4 Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex T is on the perpendicular line through the center O of the base of the prism (see figure). Let s denote the side of the base, h the height of the cell and θ the angle between the line TO and TV .
(a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent
rhombi. (b) the total surface area of the cell is given by the formula 6sh− 9s2
2 tan θ +
2005
1 For all positive integers n, find the remainder of (7n)! 7n · n!
upon division by 7.
2 We can obviously put 100 unit balls in a 10× 10× 1 box. How can one put 105 unit balls in? How can we put 106 unit balls in?
3 Prove that 20052 can be written in at least 4 ways as the sum of 2 perfect (non-zero) squares.
4 If n is an integer, then find all values of n for which √
n + √
(b) cos (
( 6π
7
) are the roots of an equation of the form ax3 + bx2 +
cx + d = 0 where a, b, c, d are integers. Determine a, b, c and d.
2 Let 4ABC be an equilateral triangle and let P be a point on [AB]. Q is the point on BC such that PQ is perpendicular to AB. R is the point on AC such that QR is perpendicular to BC. And S is the point on AB such that RS is perpendicular to AC. Q′ is the point on BC such that PQ′ is perpendicular to BC. R′ is the point on AC such that Q′R′ is perpendicular
to AC. And S′ is the point on AB such that R′S′ is perpendicular to AB. Determine |PB| |AB|
if S = S′.
3 Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?
4 Find all functions f : R\{0, 1} → R such that
f(x) + f
2008
First Grades
1 Squares BCA1A2 , CAB1B2 , ABC1C2 are outwardly drawn on sides of triangle 4ABC. If AB1A
′C2 , BC1B ′A2 , CA1C
(i) Lines BC and AA′ are orthogonal.
(ii)Triangles 4ABC and 4A′B′C ′ have common centroid
2 For arbitrary reals x, y and z prove the following inequality:
x2 + y2 + z2 − xy − yz − zx ≥ max{3(x− y)2
4 , 3(y − z)2
4 , 3(y − z)2
4 }
3 Let b be an even positive integer. Assume that there exist integer n > 1 such that bn − 1 b− 1
is
perfect square. Prove that b is divisible by 8.
4 Given are two disjoint sets A and B such that their union is N. Prove that for all positive integers n there exist different numbers a and b, both greater than n, such that either {a, b, a+ b} is contained in A or {a, b, a + b} is contained in B.
2008
Fourth Grades
1 Given are three pairwise externally tangent circles K1 , K2 and K3. denote by P1 tangent point of K2 and K3 and by P2 tangent point of K1 and K3.
Let AB (A and B are different from tangency points) be a diameter of circle K3. Line AP2
intersects circle K1 (for second time) at point X and line BP1 intersects circle K2(for second time) at Y .
If Z is intersection point of lines AP1 and BP2 prove that points X, Y and Z are collinear.
2 Find all positive integers a and b such that a4 + a3 + 1
a2b2 + ab2 + 1 is an integer.
3 A rectangular table 9 rows × 2008 columns is fulfilled with numbers 1, 2, ...,2008 in a such way that each number appears exactly 9 times in table and difference between any two numbers from same column is not greater than 3. What is maximum value of minimum sum in column (with minimal sum)?
2008
Second Grades
1 Given is an acute angled triangle 4ABC with side lengths a, b and c (in an usual way) and circumcenter O. Angle bisector of angle ∠BAC intersects circumcircle at points A and A1. Let D be projection of point A1 onto line AB, L and M be midpoints of AC and AB , respectively.
(i) Prove that AD = 1 2 (b + c)
(ii) If triangle 4ABC is an acute angled prove that A1D = OM + OL
2 IF a, b and c are positive reals such that a2 + b2 + c2 = 1 prove the inequality:
a5 + b5
ab(a + b) +
b5 + c5
bc(b + c) +
c5 + a5
ca(a + b) ≥ 3(ab + bc + ca)− 2
3 Prove that equation p4 + q4 = r4 does not have solu

in the name of allah mathematical olympiad problems around the world

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