1976 Canada National Olympiad

Part of Canada National Olympiad

Subcontests

(8)

1

Weights

Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.

1

Four coplanar points

A,B,C,D

are four points in space, such that \angle ABC\equal{}\angle BCD\equal{}\angle CDA\equal{}\angle DAB\equal{}\pi/2, prove that

A,B,C,D

lie in a plane.

1

Points connected by red and blue line segments

Each of the 36 line segments joining 9 distinct points on a circle is coloured either red or blue. Suppose that each triangle determined by 3 of the 9 points contains at least one red side. Prove that there are four points such that the 6 segments connecting them are all red.

1

Polynomial

P(x,y)

be a polynomial in two variables

x,y

such that P(x,y)\equal{}P(y,x) for every

x,y

(for example, the polynomial x^2\minus{}2xy\plus{}y^2 satisfies this condition). Given that (x\minus{}y) is a factor of

P(x,y)

, show that (x\minus{}y)^2 is a factor of

P(x,y)

1

Sum of consecutive numbers

Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.

1

Locus

AB

be a diameter of a circle,

C

be any fixed point between

A

B

on this diameter, and

Q

be a variable point on the circumference of the circle. Let

P

be the point on the line determined by

Q

C

for which \frac{AC}{CB}\equal{}\frac{QC}{CP}. Describe, with proof, the locus of the point

P

1

Chess tournament

Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students for grade eight participated in the chess tournament? Is the solution unique?

1

Positive integers

Suppose n(n\plus{}1)a_{n\plus{}1}\equal{}n(n\minus{}1)a_n\minus{}(n\minus{}2)a_{n\minus{}1} for every positive integer

n\ge1

. Given that a_0\equal{}1,a_1\equal{}2, find \frac{a_0}{a_1}\plus{}\frac{a_1}{a_2}\plus{}\frac{a_2}{a_3}\plus{}\dots\plus{}\frac{a_{50}}{a_{51}}.