ARML Practice ProblemsArvind Thiagarajan, 2005-2006

May 7, 2006

1 Geometry Problems

1. Find the coordinates of the point on the circle with equation (x 6)2 + (y 5)2 = 25that is nearest the point (2, 11). (TJ ARML 06 Practice 1)

2. In rectangle ABCD, AB = 6 and BC = 8. Equilateral Triangles ADE and DCF aredrawn on the exterior of the rectangle. If the area of triangle BEF is a

3 + b, find the

ordered pair of rational numbers (a, b). (TJ ARML 06 Practice 1)

3. The ratio of the width of a rectangle to its length equals the ratio of its length to halfits perimeter. If the area of the rectangle is 50(

5 1) and the width of the rectangle

is k(5 1), find the value of k. (TJ ARML 06 Practice 2)

4. In 4ABC, AB = 5, BC = 6, and AC = 7. Points P on AB, Q on BC, and R on ACare located so that AP = 2, BQ = 2, CR = 3. If the area of 4ABC is x, then the areaof 4PQR is kx

35. Find the value of k. (TJ ARML 06 Practice 2)

5. Point C lies on a circle one of whose diameters is AB. The bisector of CAB intersectsBC at D and intersects the circle at E. If BD = 25 and CD = 7, find BE.(TJ ARML 06 Practice 2)

6. In 4ABC, D is on AB so that AD:DB = 1 : 2, and G is on CD so that CG:GD = 3 : 2.If BG intersects AC at F, find BG:GF. (TJ ARML 06 Practice 4)

7. A chord of a triangle is a line segment whose endpoints lie on the sides of the triangle(but not at the vertices). For a triangle whose sides have lengths of 4, 5, and 6, findthe length of the shortest chord that divides the triangle into two regions of equal area.(TJ ARML 06 Practice 4)

8. In right triangle4ABC, AC = 4 and BC = 8. A square is drawn exterior to the trianglewith AB as one side. Find the distance from C to the intersection of the diagonals ofthe square. (TJ ARML 06 Practice 5)

9. In trapezoid ABCD, the ratio of base AB to base CD is 2 : 3. Diagonals AC and BDintersect at point X, and the line through X parallel to AB intersects AD at point P.Find the ratio of the area of 4PAX to the area of 4ABD. (TJ ARML 06 Practice 5)

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10. The longer base of an isosceles trapezoid is a chord of a circle, and the shorter base istangent to the circle. If the length of one leg is 5 and the lengths of the bases are 24and 18, find the area of the circle. (TJ ARML 06 Practice 5)

11. Acute triangle 4ABC is inscribed in a circle. Altitudes AM and CN are extended tomeet the circle again at P and Q respectively. If PQ:AC = 7 : 2, find the numericalvalue of sinB. (TJ ARML 06 Practice 5)

12. The length of each side of 4ABC is 6. If X is the trisection point of CA nearer C andif median AM intersects BX at U, then MU = k

3. Find k. (TJ ARML 06 Practice 6)

13. The length of a radius of a circle is 3 + 23. Three congruent circles are drawn in the

interior of the original circle, each internally tangent to the original circle and externallytangent to the others. Find the length of a radius of one of the three congruent circles.(TJ ARML 06 Practice 6)

14. In parallelogram ABCD, A is acute and AB = 5. Point E is on AD with AE = 4 andBE = 3. A line through B, perpendicular to CD, intersects CD at F. If BF = 5, findEF. (TJ ARML 06 Practice 6)

15. Find the numerical value of b for which the length of the path from A(0, 2) to B(b, 0)to C(c, 10) to D(5, 9) will be a minimum. (TJ ARML 06 Practice 6)

16. In equilateral triangle 4ABC, points D, E, and F are on AB, BC, and CA, respectively,with AD = BE = CF = 1 and DB = EC = FA =

3. The area of 4DEF is a+ b3.

Find the ordered pair of rational numbers (a, b). (TJ ARML 06 Practice 6)

17. Find the area of an equiangular octagon, the lengths of whose sides are alternately 1and

2. (TJ ARML 06 Practice 9)

18. The length of the base of an isosceles triangle is 20. In the plane of the triangle, a point4 units from this base is 10 units from each leg. Find the area of the triangle.(TJ ARML 06 Practice 9)

2 Number Theory Problems

1. Let [x] denote the greatest integer n such that n x. Let f(x) = [ x121

2

][121

2

x]. If

0 < x < 90, then the range of f consists of k elements. Find the value of k.(TJ ARML 06 Practice 1)

2. Find all triples of real numbers (x, y, z) such thatx+ yz = 6y + xz = 6z + xy = 6 (TJ ARML 06 Practice 1)

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3. In base eight, the four-digit numeral BBCC is the square of the two-digit numeral AA.Find the ordered triple of digits (A,B,C). (TJ ARML 06 Practice 1)

4. In arranging the ordered pairs of positive integers thusly:(1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), (1, 4) , such that if two ordered pairs have adifferent element-sum, the one with the smaller element-sum comes first and if theyhave the same element-sum, the one with the smaller first element comes first. Findthe 1978th ordered pair. (TJ ARML 06 Practice 2)

5. The positive integer n, when divided by 3, 4, 5, 6, and 7, leaves remainders of 2, 3, 4, 5,and 6 respectively. Find the least possible value of n. (TJ ARML 06 Practice 4)

6. In base fifty, the integer x is represented by the numeral CC and x3 is represented bythe numeral ABBA. If C > 0, express all possible values of B in base ten.(TJ ARML 06 Practice 4)

7. The integers between 1 and 1000 inclusive are written in a row. Sam started at at 1 andcircled every 24th number in red. Janet started at at 1 and circled every 15th numberin blue. What is the smallest possible positive difference between a red number and ablue number? (TJ ARML 06 Practice 4)

8. The sum of 5 positive integers x, y, z, w, and u is equal to their product. If x y z w u, find the product xyzwu. (TJ ARML 06 Practice 9)

9. The sum of 19 consecutive positive integers equals p3, where p is a prime number.Compute the smallest of the 19 integers. (TJ ARML 06 Practice 10)

10. Determine all positive primes p such that p1994 + p1995 is a perfect square.(TJ ARML 06 Practice 10)

11. Compute the number of ordered triples (A,B,C) with A,B,C (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)such that k is an integer if

k =.ABC + .ACB + .BAC + .BCA+ .CAB + .CAB + .CBA

.A+ .B + .C.

(TJ ARML 06 Practice 10)

3 Logarithms and Exponents Problems

1. Find all positive numbers x that satisfy(2 + log x)3 + (1 + log x)3 = (1 + log x2)3. (TJ ARML 06 Practice 4)

2. Of all ordered triples of positive integers (x, y, z) that satisfy 23x + 24y = 25z, find thesmallest value of z. (TJ ARML 06 Practice 6)

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3. Find all ordered pairs of real numbers (x, y) such that logx2

y3= 1 and log x2y3 = 7.

(TJ ARML 06 Practice 9)

4 Sequences and Series

1. Express the following sum as a quotient of two integers:15n=2

2

(n 1)(n+ 1).(TJ ARML 06 Practice 9)

2. The sum of 1999 positive numbers in an increasing arithmetic progression is 1. Computethe width of the smallest interval containing all possible values of the common difference.Do not leave your answer in factored form. (TJ ARML 06 Practice 10)

5 Combinatorics

1. A regular dodecahedron is a polyhedron whose 12 faces are congruent regular polygons.An interior diagonal of this polyhedron is a diagonal which is not wholly contained inone of the faces. A regular dodedecahedron has 30 edges and 20 vertices. How manyinterior diagonals does it have. (TJ ARML 06 Practice 9)

6 Trigonometry Problems

1. If sin6 + cos6 = 23, find all possible values of sin 2. (TJ ARML 06 Practice 1)

2. If 0 < x < pi, find all values of x that satisfy sin 5x+ sin 3x = 0.(TJ ARML 06 Practice 1)

3. Find the numerical value of sin 40 sin 80 + sin 80 sin 160 + sin 160 sin 320.(TJ ARML 06 Practice 2)

4. If cos x+cos y+cos z = sinx+sin y+sin z = 0, find the numerical value of cos (x y)+cos (y z) + cos (z x). (TJ ARML 06 Practice 4)

5. If x 6= kpi2, where k is an integer, then

cot2 x tan2 x2 + cot2 x+ tan2 x

+ 2 sin2 x has a minimum value of m and maximum value of M , for

all real number x. Find the ordered pair (m,M). (TJ ARML 06 Practice 6)

6. Find the numerical value ofsin 18 cos 12 + cos 162 cos 102

sin 22 cos 8 + cos 158 cos 98.

(TJ ARML 06 Practice 9)

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7 Polynomial Problems

1. a > 0, b > 0, a 6= b, and ab+ b

a

ab ba +

ab ba

ab+ b

a=ab

Write an equation expressing a explicitly in terms of b. (TJ ARML 06 Practice 1)

2. If a, b, and c are different numbers and if a3 + 3a+ 14 = 0, b3 + 3b+ 14 = 0 and

c3 + 3c+ 14 = 0, find the value of1

a+

1

b+

1

c. (TJ ARML 06 Practice 2)

3. Two drivers, A and B, were 225 km apart. They traveled towards each other at thesame constant speed of x km per hour, with A having had a head start of 30 minutes.Upon meeting, each continued to the others starting point at a constant speed of x10km per hour. If A completed the entire trip in 5 hours, find x.(TJ ARML 06 Practice 2)

4. If x, y, z, a, b, and c are nonzero real numbers satisfying(4x2 + 9y2 + z2)(a2 + b2 + c2) = (2ax+ 3by + cz)2, find the continued ratio x : y : z interms of a, b, and c. (TJ ARML 06 Practice 2)

5. One of the roots ofx2 + 1

x+

x

x2 + 1=

29

10is

1 +k

5, where k is a negative integer. Find k.

(TJ ARML 06 Practice 4)

6. If n > 1, find the two smallest integral values of n for which x2 + x + 1 is a factor of(x+ 1)n xn 1, over the set of polynomials with integral coeffecients. (TJ ARML 06Practice 5)

7. Every expression of the form a2b2+b2c2+c2d2+d2a2 can be expressed as the sum of twosquares in at least two different ways. Find any one of the three possible ordered pairs ofpositive integers (x, y), with x > y, that satisfies x2+y2 = 442102+102332+33252+52442.(TJ ARML 06 Practice 5)

8. Let A =xr

1 xr , B =xs

1 xs , and C =xr+s

1 xr+s , where (1 xr)(1 xs)(1 xr+s) 6= 0.

Write an equation expressing C explicitly in terms of A and B.(TJ ARML 06 Practice 6)

9. Find all ordered pairs of integers (x, y) which satisfy x2 + 4x+ y2 = 9.(TJ ARML 06 Practice 9)

10. The roots of ax2 + bx + c = 0 are irrational, but their calculator approximations are0.8430703308 and 0.5930703308. If a, b, and c are integers whose greatest common

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divisor is 1 and which satisfy a > 0, |b| 10 and |c| 10, compute the ordered triple(a, b, c). (TJ ARML 06 Practice 10)

8 Algorithmic Problems (Graph Theory, Greedy Algorithm, etc.)

1. One hundred pennies are arranged in seven stacks, of which no two stacks contain thesame number of pennies. A student counts the number of pennies in each stack andtakes 50 pennies in such a way as to disturb the fewest number of stacks. He ends uptaking pennies from N stacks. For all such arrangements of pennies, what is the largestpossible value of N that will be necessary. (TJ ARML 06 Practice 5)

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