F - G - 1

1. Find the number of positive integral solutions of the equation

2. (Picture) Find the value of

.

3. If ( ( )) , find the value of

.

4. If and , find the value of .

5. If ( ) and ( ) , discuss

( )

6. (Picture) If the graph of is tangent to √ , find

the value of .

7. Find ( ) ( )

for ( ) ( ) using limit methods.

8. Given equation has roots , and develop

formulae for

, , and .

9. For equation compute values for

(a)

(b)

D - F - 1

1. If is divisible by then it is also divisible by ……

2. If (

)

, find the value of

.

3. How many integers satisfy the equation ( ) ?

4. If , find the value of .

5. If ( ) where are constants and if ( ) , find

the value of ( ) .

6. If and are positive numbers such that and , find .

7. Find the units digit of .

8. If , and are distinct primes, find the smallest positive perfect cube that has

as a factor.

9. (SHME 87)

ABCD is a square. M and N are mid points of sides BC and CD respectively.

Find .

10. Observe {

State a general law suggested by these examples and prove it.

M

N

A

B

D

C

D

D - F - 2

1. has a right angle at . If

find .

2. If ( )

, then for , find the value of ( ) in terms of ( ) .

3, 4. An unknown polynomial of degree 37 yields:

(a) a reminder of 1 when divided by

(b) a reminder of 3 when divided by

(c) a reminder of 21 when divided by

(A) Write a sentence for 3, 4 (a)

(B) Write a sentence for 3, 4 (b)

(C) Write a sentence for 3, 4 (c)

(D) Write “a” sentence for 3, 4 (a), (b), (c) together.

5. (Symmetry) If and then in terms

of find .

6. (gr. 12 or strong 11) If then find the least value of

. (With proof)

7. The sum of infinite geometric series is 8 and the sum of the 2nd

and 3rd

terms is 3. Determine

8. The sides of a triangle are and √ . Find its greatest

angle.

9. Prove that the square of an odd integer is always of form

where .

10. If one root of equation is √ , find the sum

of the other two roots.

F - 2

1. Solve for x:

.

2. If and ( )

, find in terms of and .

3. Prove

, ( ).

4. How many integers with 4 different digits are there between 1000 and

9999, such that the absolute value of the difference between the 1st digit

and last digit is 2?

5. Given a function of real variable such that

( ) ( ) ( ). Find ( )

6. Given and are the roots of the equation find

and , such that and are roots of .

7. The lengths of the sides of a triangle are consecutive integers and the

largest angle is twice the smallest angle. Find the cosine of the smallest

angle.

8. (AHSME) Evaluate:

( ) ( ) ( ) ( ) ( ) .

9. (AHSME) Given the (area >0), ( ) ( ) and ( ),

and , find the minimum area of ?

10. Find the slope of tangent to at the point where the tangent at

( ) on curve meets curve again. (From Calculus text)

F - 2

1. Given equation has roots , , and develop

formulae for

, , and .

2. For equation compute values for

(a)

(b)

3. Given a function of real variable such that

( ) ( ) ( ). Find ( ) .

4. Solve for x:

5. Find the slope of tangent to at the point where the tangent at

( ) on curve meets curve again. (From Calculus text)

6. The sum of infinite geometric series is 8 and, the sum of . Find all values

of common ratio.

7. If and , find the value of .

8. If ( ( )) , find the value of

.

9. If the graph of is tangent to √ , find the

value of .

F - G - 3 - 1991

1. (1996 D) Solve the system:

{

2. (1990 D) All the digits 1, 2, 3… 8 are permutated. For any permutation,

the eight digits occupy the digits 1 to 8 in some order.

(a) In how many of these is the product of every 5 consecutive sets of

digits divisible by 5?

(b) In how many is the product of all pairs of adjacent digits divisible by 2?

3. (1995 D) The circle ( ) passes through ( ) and

intersects -axis at and . Find the coordinates of and and the

measure of .

4. (1995 D) are consecutive vertices of a regular polygon

with 12 sides. If and

, find in terms of and the

following: ,

, and .

5. (1994 D) A sequence of squares (infinite) have sides length

and . Let be the area of all squares and the perimeter of the

figure.

(a) Find and in terms of .

(b) Find such that

1

1

s

s

6. (1992 D) Ann and Bob are playing a game in which Ann rolls a die and

Bob tosses a coin until someone wins. Ann wins in a “six” and Bob a

“head”.

(a) What is the probability that Ann wins.

(b) Find , the expected number of times Ann rolls.

∑ ( ( ) ( ))

7. (1991 D) Let ( ) ( ) ( ) ( ) ( )

(a) What is the coefficient of in ( ) ?

(b) What is the coefficient of in ( ) ?

(c) Evaluate ( )

(d) Evaluate ( )

8. (1991 D) Evaluate where {

9. (1993 D) How many ordered triples of real numbers are, such that

are the roots of ?

10. If , find the least value of .

D - F - 3

1. Evaluate:

(√ √ √ )(√ √ √ )(√ √ √ )( √ √ √ )

2. If

( )( ) is true for all , solve for

3. The polynomial ( ) ( ) has factors

and .

(a) What are the values of and ?

(b) What are other factors of the polynomial?

4. The difference of the squares of 2 positive integers which differ by 2 is a

perfect square . Find the sum of the 4 smallest values of n.

5. Find the values of integer for which | | is prime.

6. If √ √

√ √

, find .

7. If and ( )

, find the value of .

8. Solve for

, which satisfy the equation

.

9. Find the perimeter of the plane region enclosed by all points ( )

satisfying | | | | .

10. If , find the value of .

D - F - 4

1. If

, find

.

2. Find the sum of the squares of all the real numbers satisfying

3. (AHSME 1963) If and and none is zero, find

4. Find the number of distinct pairs ( ) of real numbers satisfying the

equations

{

5. Opposite sides of regular hexagon are 12 cm apart. Find the length of

each side.

6. Find | √( ) | when .

7. If an integer is a solution of nd “a” is in base ,

find in base .

8. For all ( ) find the product

( ) ( )( ) (( ) ( ) ( ) )

9. Find the sum of

( )( )

10. Find the number of ordered triples ( ) of positive integers which

satisfy

{

F - 4

1. (A) Find the sum of

√

2. (A) Find √ ( )

3. (A) Find

4. (A) Find ( ) ( )

( ) ( )

5. (*) If and , find the value of .

6. (*) If has roots and , find the value of

.

7. (*) If and are successive terms in a geometric

sequence, find the value of .

8. (*) If , find the value of .

9. (*) The vertices of a triangle are ( ) ( ) and ( ). Find the

coordinates of the centroid.

10. (AHSME 1967) If and is positive and grows beyond all bounds, then

what ( ) ( ) approach?

(*) - University of Waterloo freshman preparedness test

(A) - Old “White Tests”

Week 4 XF

Problems from 1996 Honsberger Lecture

1. If 10-equally spaced points around a circle are joined consecutively, a

convex regular inscribed decagon P is obtained (Figure 4a); if every third

point s joined, a self-intersecting regular decagon Q is formed (Figure 4b).

Prove that the difference between the length of a side of Q and the length

of a side of P is equal to the radius of the circle.

Figure 4a - Decagon P

Figure 4b - Decagon Q

2. Without actually evaluating the integer prove that some digit

occurs at least four times in its decimal representation.

3. If is a positive integer and is a divisor of , prove that

cannot be a perfect square.

4. (Math Orchard) In a storeroom are 200 granite slabs, 120 of which weigh 7

tonnes and the rest 9 tonnes. On a railway flatcar, up to 40 tonnes can be

loaded. What is the least number of flatcars that will accommodate the

slabs?

5. (Math Horizons) Solve for :

6. (CMC Math Enrichment)

Form the sum of the squares of any two distincte positive integers. Double

this sum. Is it possible to express the resulting number as a sum of two

squares?

Try this process with another pair of integers. Will this always be true?

7. Which of the numbers or is equal

to . (Note: )

8. In her last game, Mary bowled 199 and this raised her average from 177

to 178. To raise her average to 179 with the next game, what must she

bowl?

9. (WMCS TQ 1998)

In √ and is the median to side .

is the altitude to side and intersect at . What is the lengh of

?

10. varies as the sum af and , and varies as the positive difference

of and . If and when , find and when .

Week 5 XF

1. (From a Bulgarian contest for special 12-year-old pupils)

Peter was camping at the foot of a mountain and left camp at 10 o’clock one morning

to walk to the summit. The path was horizontal for a distance and then rose to the

summit. He didn’t spend any time at the top but turned right around and returned to

camp by the same route, arriving back at 4 p.m. If Peter walked at 4 km/h (kilometers

per hour) on level ground, ascended at 3 km/h, and descended at 6 km/h, determine

within a half-hour the time when Peter was at the summit.

2. In Fig. 1, is tangent to a given circle. Prove that, for all choices of secant ,

the bisector of cuts an isosceles triangle from .

Fig. 1

T

3. (1998 Waterloo Math Seminar)

The coefficient of the term of a quadratic equation is 1. If the constant term in the

original equation is increased by 25 the roots are equal. If the constant term in the

original is increased by 9, one root is three time the other. Find the roots of the

original equation.

4. A freighter sailing due north at 12 km/h sights a cruiser straight ahead at an

unknown distance and speeding due east at unknown speed. After 15 minutes the

vessels are 10 km apart and then, 15 minutes later they are 13 km apart. How far

apart are the vessels when the cruiser is due east of the freighter?

5. (AMC 12 - 2001) Points ( ) ( ) ( ) and ( ) lie in the

first quadrant and are the vertices of quadrilateral . The quadrilateral formed by

joining the midpoints of and is a square. What is the sum of the

coordinates of point ?

x

x

A

C B

X Y

6. Four positive integers and have a product of and satisfy

{

What is ?

7. In rectangle , points and lie on so that and is the

midpoint of . Also, intersects at and at . The area of rectangle

is . Find the area of triangle .

D E C

A F G B

8. A polynomial of degree four with leading coefficient 1 and integer coefficients has

two real zeros, both of which are integers. Which of the following can also be a zero

of the polynomial?

9. In triangle . Point is on so that and

. Find . C

D

B A

10. Consider sequences of positive real numbers of the form in which

every term after the first is 1 less than the product of its two immediate neighbors. For

how many different values of does the term appear somewhere in the

sequence?

H J

D - F - 5

1. Find the last digit in the decimal numeral for ( ) .

2. At exactly 12 noon, the hour hand of a clock starts to move at twice the normal

speed and the minute hand moves at

the normal speed. Find the correct time when

the two hands meet again.

3. Given . Find the number of solutions ( ).

4. and are points on the lines and respectively. If ( ) is the

midpoint of and | | , find the equation of the locus of .

5. Chords and in the circle intersect at and . If and

, find the diameter of the circle.

6. Solve for , | | | | .

7. Put √ √ √ √ √ √ in order.

8. A woman walks from to at 4 km/h, from to at 3 km/h, then from to at 6

km/h and then from to at 4 km/h. If the total time is 6 hours and , find the

total distance walked.

9. (AHSME 1987) A ball was floating in a lake when the lake froze. The ball was

removed, leaving a hole 24 cm across the top and 8 cm deep. What was the radius of

the ball?

10. (AHSME 1987) If is prime and both roots of are integers,

then: (a) (d)

(b) (e)

(c)

A

B

D C E

F - 5

1. The centre of a circle is ( ) and its radius is 2. If its equation is

expressed in the form , find .

2. Two points and are 1000 m apart on a level plane lie due west of a

mountain. If the angles of elevation of the mountain are 30° and 60° from

and , find the height of the mountain.

3. Find the perimeter and area of the plane region enclosed by all points

( ) that satisfy | | | | .

4. Find

.

5. Find √

√ .

6. Find the equation of the normal to at .

7. Find the equation of the tangent to

( ) ( ) at .

8. If (

), replace each in by

.

The new expression is equal to:

(a) (b) (c) (d) (e)

9. Express

in the form , where , and .

10. Find

.

More Problems

1. Find √ .

2. How high must an observer be above the pole in order to have a line of

sight of a satellite in orbit at height H above the equator?

3. Find the digit in the 7000th decimal place of the expansion of

.

4. Simplify √ √

√ √

.

5. Express the element, , of the sequence defined by:

, as a function of .

6. Find a number less than 100 which is increased by 20% when its digits

are reversed.

7. What is the smallest positive multiple of 225 consisting solely of 0’s and

1’s?

8. Find all natural numbers , for which ( ) .

9. Let and be positive integers such that

{

How many values of the expression are possible?

10. What is the greatest common divisor of all numbers , for a

positive integer?