2007 Canada National Olympiad

Part of Canada National Olympiad

Subcontests

(5)

1

Problem 2

You are given a pair of triangles for which two sides of one triangle are equal in length to two sides of the second triangle, and the triangles are similar, but not necessarily congruent. Prove that the ratio of the sides that correspond under the similarity is a number between \frac {1}{2}(\sqrt {5} \minus{} 1) and \frac {1}{2}(\sqrt {5} \plus{} 1).

1

Problem 1

What is the maximum number of non-overlapping

2\times 1

dominoes that can be placed on a

8\times 9

checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board.

1

Numbers

For two real numbers

a

b

ab\neq 1

\ast

a\ast b=\frac{a+b-2ab}{1-ab}.

Start with a list of

n\geq 2

real numbers whose entries

x

0<x<1

. Select any two numbers

a

b

in the list; remove them and put the number

a\ast b

at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains.

a.

Prove that this single number is the same regardless of the choice of pair at each stage.

b.

Suppose that the condition on the numbers

x

0<x\leq 1

. What happens if the list contains exactly one

1

1

Give a polynomial example for f if f(xy)+f(y-x)\geq f(y+x)

f

is a real-valued function for which

f(xy)+f(y-x)\geq f(y+x)

for all real numbers

x

y

. a) Give a non-constant polynomial that satisfies the condition. b) Prove that

f(x)\geq 0

x

1

Four circles

Let the incircle of triangle

ABC

BC,\, CA

AB

D,\, E

F,

respectively. Let

\omega,\,\omega_{1},\,\omega_{2}

\omega_{3}

denote the circumcircles of triangle

ABC,\, AEF,\, BDF

CDE

respectively. Let

\omega

\omega_{1}

A

P,\,\omega

\omega_{2}

B

Q,\,\omega

\omega_{3}

C

R.

a.

\omega_{1},\,\omega_{2}

\omega_{3}

intersect in a common point.

b.

PD,\, QE

RF

are concurrent.