2004 Canada National Olympiad

Part of Canada National Olympiad

Subcontests

(5)

1

set of positive integer divisors

T

be the set of all positive integer divisors of

2004^{100}

. What is the largest possible number of elements of a subset

S

T

such that no element in

S

divides any other element in

S

1

congruence

p

be an odd prime. Prove that: \displaystyle\sum_{k\equal{}1}^{p\minus{}1}k^{2p\minus{}1} \equiv \frac{p(p\plus{}1)}{2} \pmod{p^2}

1

circle

A,B,C,D

be four points on a circle (occurring in clockwise order), with

AB<AD

BC>CD

. The bisectors of angles

BAD

BCD

meet the circle at

X

Y

, respectively. Consider the hexagon formed by these six points on the circle. If four of the six sides of the hexagon have equal length, prove that

BD

must be a diameter of the circle.

1

Rooks on a 9x9 Board

How many ways can

8

mutually non-attacking rooks be placed on the

9\times9

chessboard (shown here) so that all

8

rooks are on squares of the same color? (Two rooks are said to be attacking each other if they are placed in the same row or column of the board.) [asy]unitsize(3mm); defaultpen(white); fill(scale(9)*unitsquare,black); fill(shift(1,0)*unitsquare); fill(shift(3,0)*unitsquare); fill(shift(5,0)*unitsquare); fill(shift(7,0)*unitsquare);
fill(shift(0,1)*unitsquare); fill(shift(2,1)*unitsquare); fill(shift(4,1)*unitsquare); fill(shift(6,1)*unitsquare); fill(shift(8,1)*unitsquare);
fill(shift(1,2)*unitsquare); fill(shift(3,2)*unitsquare); fill(shift(5,2)*unitsquare); fill(shift(7,2)*unitsquare);
fill(shift(0,3)*unitsquare); fill(shift(2,3)*unitsquare); fill(shift(4,3)*unitsquare); fill(shift(6,3)*unitsquare); fill(shift(8,3)*unitsquare);
fill(shift(1,4)*unitsquare); fill(shift(3,4)*unitsquare); fill(shift(5,4)*unitsquare); fill(shift(7,4)*unitsquare);
fill(shift(0,5)*unitsquare); fill(shift(2,5)*unitsquare); fill(shift(4,5)*unitsquare); fill(shift(6,5)*unitsquare); fill(shift(8,5)*unitsquare);
fill(shift(1,6)*unitsquare); fill(shift(3,6)*unitsquare); fill(shift(5,6)*unitsquare); fill(shift(7,6)*unitsquare);
fill(shift(0,7)*unitsquare); fill(shift(2,7)*unitsquare); fill(shift(4,7)*unitsquare); fill(shift(6,7)*unitsquare); fill(shift(8,7)*unitsquare);
fill(shift(1,8)*unitsquare); fill(shift(3,8)*unitsquare); fill(shift(5,8)*unitsquare); fill(shift(7,8)*unitsquare);
draw(scale(9)*unitsquare,black);[/asy]

1

Simple System

Find all ordered triples

(x,y,z)

of real numbers which satisfy the following system of equations: \left\{\begin{array}{rcl} xy & \equal{} & z \minus{} x \minus{} y \\ xz & \equal{} & y \minus{} x \minus{} z \\ yz & \equal{} & x \minus{} y \minus{} z \end{array} \right.