2013 CentroAmerican

Part of CentroAmerican

Subcontests

(3)

2

Passing Coins Around a Round Table

Around a round table the people

P_1, P_2,..., P_{2013}

are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none); there are a total of

10000

coins. Starting with

P_1

and proceeding in clockwise order, each person does the following on their turn:
[*]If they have an even number of coins, they give all of their coins to their neighbor to the left.
[*]If they have an odd number of coins, they give their neighbor to the left an odd number of coins (at least

1

and at most all of their coins) and keep the rest.
Prove that, repeating this procedure, there will necessarily be a point where one person has all of the coins.

ABC is an Acute Triangle

ABC

be an acute triangle and let

\Gamma

be its circumcircle. The bisector of

\angle{A}

BC

D

\Gamma

K

(different from

A

), and the line through

B

\Gamma

X

K

is the midpoint of

AX

\frac{AD}{DC}=\sqrt{2}

2

ABCD is a Convex Quadrilateral

ABCD

be a convex quadrilateral and let

M

be the midpoint of side

AB

. The circle passing through

D

AB

A

intersects the segment

DM

E

. The circle passing through

C

AB

B

intersects the segment

CM

F

. Suppose that the lines

AF

BE

intersect at a point which belongs to the perpendicular bisector of side

AB

A

E

C

are collinear if and only if

B

F

D

p(x)-q(x)=1

Determine all pairs of non-constant polynomials

p(x)

q(x)

, each with leading coefficient

1

n

n

roots which are non-negative integers, that satisfy

p(x)-q(x)=1

2

Drawing Squares on an Infinite Grid

Ana and Beatriz take turns in a game that starts with a square of side

1

drawn on an infinite grid. Each turn consists of drawing a square that does not overlap with the rectangle already drawn, in such a way that one of its sides is a (complete) side of the figure already drawn. A player wins if she completes a rectangle whose area is a multiple of

5

. If Ana goes first, does either player have a winning strategy?

Writing (n, 3^n) On a Board

Juan writes the list of pairs

(n, 3^n)

n=1, 2, 3,...

on a chalkboard. As he writes the list, he underlines the pairs

(n, 3^n)

n

3^n

have the same units digit. What is the

2013^{th}

underlined pair?