AoPS Community 2005 Cuba MO
www.artofproblemsolving.com/community/c4011324
by parmenides51
– Day 1
– Grades 10-12
1 Determine the smallest real number a such that there is a square of side a such that contains 5
unit circles inside it without common interior points in pairs.
2 There are n light bulbs in a circle and one of them is marked.
Let operation A:
Take a positive divisor d of the number n, starting with the light bulb marked and clockwise, we
count around the circumference from 1 to dn, changing the state (on or off) to those light bulbs
that correspond to the multiples of d.
Let operation B be:
Apply operationA to all positive divisors of n (to the first divider that is applied is with all the
light bulbs off and the remaining divisors is with the state resulting from the previous divisor).
Determine all the positive integers n, such that when applying the operation on B, all the light
bulbs are on.
3 There are two piles of cards, one with n cards and the other with m cards. A and B play alter-
nately, performing one of the following actions in each turn. following operations:
a) Remove a card from a pile.
b) Remove one card from each pile.
c) Move a card from one pile to the other.
Player A always starts the game and whoever takes the last one letter wins . Determine if there
is a winning strategy based on m and n, so that one of the players following her can win always.
– Day 2
– Grade 10
1 Determine all the quadrilaterals that can be divided by a diagonal into two triangles of equal
area and equal perimeter.
2 Determine the quadratic functions f (x) = ax2 + bx + c for which there exists an interval (h, k)
such that for all x ∈ (h, k) it holds that f (x)f (x + 1) < 0 and f (x)f (x − 1) < 0.
© 2024 AoPS Incorporated 1
AoPS Community 2005 Cuba MO
3 Determine all the quadruples of real numbers that satisfy the following:
The product of any three of these numbers plus the fourth is constant.
– Grade 11
4 Determine all functions f : R+ → R such that:
1 1
f (x)f (y) = f (xy) + +
x y
for all x, y positive reals.
5 On the circumcircle of triangle ABC, point P is taken in such a way that the perpendicular drawn
by the point P to the line AC cuts the circle also at the point Q, the perpendicular drawn by the
point Q to the line AB cuts the circle also at point R and the perpendicular drawn by point R
to the line BC cuts the circle also at the point P . Let O be the center of this circle. Prove that
∠P OC = 90o .
6 All positive differences ai − aj of five different positive integers a1 , a2 , a3 , a4 and a5 are all dif-
ferent. Let A be the set formed by the largest elements of each group of 5 elements that meet
said condition. Determine the minimum element of A.
– Grade 12
7 Determine all triples of positive integers (x, y, z) that satisfy
x < y < z, gcd(x, y) = 6, gcd(y, z) = 10, gcd(z, x) = 8 and lcm(x, y, z) = 2400.
8 Find the smallest real number A, such that there are two different triangles, with integer side-
lengths and so that the area of each be A.
9 Let x1 , x2 , . . . , xn and y1 , y2 , . . . , yn be positive reals such that
x1 + x2 + .. + xn ≥ yi ≥ x2i
for all i = 1, 2, . . . , n. Prove that
x1 x2 xn 1
++ + ... + > .
x1 y1 + x2 x2 y2 + x3 xn yn + x1 2n
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