2000 CentroAmerican

Part of CentroAmerican

Subcontests

(3)

2

Area of a parallelogram

ABCDE

be a convex pentagon. If

P

Q

R

S

are the respective centroids of the triangles

ABE

BCE

CDE

DAE

PQRS

is a parallelogram and its area is

2/9

ABCD

"Nice" representations

Let's say we have a nice representation of the positive integer

n

if we write it as a sum of powers of 2 in such a way that there are at most two equal powers in the sum (representations differing only in the order of their summands are considered to be the same). a) Write down the 5 nice representations of 10. b) Find all positive integers with an even number of nice representations.

2

Tiling a 15xn board

Determine all positive integers

n

such that it is possible to tile a

15 \times n

board with pieces shaped like this: [asy]size(100); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,0)--(6,0)); draw((4,1)--(7,1)); draw((4,2)--(7,2)); draw((5,3)--(6,3)); draw((4,1)--(4,2)); draw((5,0)--(5,3)); draw((6,0)--(6,3)); draw((7,1)--(7,2));[/asy]

Circles with AB and AC as diameters

ABC

be an acute-angled triangle.

C_{1}

C_{2}

are two circles of diameters

AB

AC

, respectively.

C_{2}

AB

intersect again at

F

C_{1}

AC

intersect again at

E

BE

C_{2}

P

CF

C_{1}

Q

AP=AQ

2

Divisors of 26

Find all three-digit numbers

abc

a \neq 0

a^{2}+b^{2}+c^{2}

is a divisor of 26.

Writing numbers within an equilateral triangle

Write an integer on each of the 16 small triangles in such a way that every number having at least two neighbors is equal to the difference of two of its neighbors. Note: Two triangles are said to be neighbors if they have a common side. [asy]size(100); pair P=(0,0); pair Q=(2, 2*sqrt(3)); pair R=(4,0); draw(P--Q--R--cycle); pair B=midpoint(P--Q); pair A=midpoint(P--B); pair C=midpoint(B--Q); pair E=midpoint(Q--R); pair D=midpoint(Q--E); pair F=midpoint(E--R); pair H=midpoint(R--P); pair G=midpoint(R--H); pair I=midpoint(H--P); draw(A--I); draw(B--H); draw(C--G); draw(I--D); draw(H--E); draw(G--F); draw(C--D); draw(B--E); draw(A--F);[/asy]