MathDB

1

Ten boxes in a circle

Ten boxes are arranged in a circle. Each box initially contains a positive number of golf balls. A move consists of taking all of the golf balls from one of the boxes and placing them into the boxes that follow it in a counterclockwise direction, putting one ball into each box. Prove that if the next move always starts with the box where the last ball of the previous move was placed, then after some number of moves, we get back to the initial distribution of golf balls in the boxes.

9

1

Determine the number of sequences

Suppose that

m

and

k

are positive integers. Determine the number of sequences

x_1, x_2, x_3, \dots , x_{m-1}, x_m

with
[*]

x_i

an integer for

i = 1, 2, 3, \dots , m

, [*]

1\le x_i \le k

for

i = 1, 2, 3, \dots , m

, [*]

x_1\neq x_m

, and [*]no two consecutive terms equal.

8

1

Determine an infinite family of quadruples (a, b, c, d)

Determine an infinite family of quadruples

(a, b, c, d)

of positive integers, each of which is a solution to

a^4+b^5+c^6=d^7

.

7

1

Folded sheet of paper

A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?

6

1

Prove that <ABC=3<PBC

Triangle

ABC

is right-angled at

C

.

AQ

is drawn parallel to

BC

with

Q

and

B

on opposite sides of

AC

so that when

BQ

is drawn, intersecting

AC

at

P

, we have

PQ = 2AB

. Prove that

\angle ABC = 3\angle PBC

.

5

1

Determine all positive integers "n"

Determine all positive integers

n

for which

n(n + 9)

is a perfect square.

4

1

Find the probability of sum 6

Three fair six-sided dice are thrown. Determine the probability that the sum of the numbers on the three top faces is

6

.

3

1

Prove that a polynomial does not exist

Prove that there does not exist a polynomial

f(x)

with integer coefficients for which

f(2008) = 0

and

f(2010) = 1867

.

2

1

Right triangle ABC with altitude d

Triangle

ABC

is right-angled at

C

with

AC = b

and

BC = a

. If

d

is the length of the altitude from

C

to

AB

, prove that

\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{d^2}

1

Determine all solutions

Determine all solutions to the system of equations \begin{align*}x+y+z=2 \\ x^2-y^2-z^2=2 \\ x-3y^2+z=0\end{align*}

2009 Canadian Mathematical Olympiad Qualification Repechage

Subcontests

Ten boxes in a circle

Determine the number of sequences

Determine an infinite family of quadruples (a, b, c, d)

Folded sheet of paper

Prove that <ABC=3<PBC

Determine all positive integers "n"

Find the probability of sum 6

Prove that a polynomial does not exist

Right triangle ABC with altitude d

Determine all solutions

2009 Canadian Mathematical Olympiad Qualification Repechage

Subcontests

Ten boxes in a circle

Determine the number of sequences

Determine an infinite family of quadruples (a, b, c, d)

Folded sheet of paper

Prove that &lt;ABC=3&lt;PBC

Determine all positive integers &quot;n&quot;

Find the probability of sum 6

Prove that a polynomial does not exist

Right triangle ABC with altitude d

Determine all solutions

Prove that <ABC=3<PBC

Determine all positive integers "n"