2018 Puerto Rico Team Selection Test

OMCC - CentroAmericam TST

1

Find all pairs (a,b) of positive integers that satisfy the equation a232b=1.

2

Let A={a1,a2,a3,a4,a5} be a set of 5 positive integers. Show that for any rearrangement of A, ai1, ai2, ai3, ai4, ai5, the product (ai1a1)(ai2a2)(ai3a3)(ai4a4)(ai5a5)is always even.

3

Let M be the point of intersection of diagonals AC and BD of the convex quadrilateral ABCD. Let K be the point of intersection of the extension of side AB (beyondA) with the bisector of the angle ACD. Let L be the intersection of KC and BD. If MACD=MBLD, prove that the angle BKC is equal to the angle CDB.

4

On a circumference of a circle, seven points are selected, at which different positive integers are assigned to each of them. Then fit simultaneously, each number is replaced by the least common multiple of the two neighboring numbers to it. If the same number n is obtained in each of the seven points, determine the smallest possible value for n. original wordingSobre una circunferencia de un círculo, se seleccionan siete puntos, a los cuales se le asignan enteros positivos distintos a cada uno de ellos. Luego, en forma simultánea, cada número se reemplaza por el mínimo común múltiplo de los dos números vecinos a él. Si se obtiene el mismo número n en cada uno de los siete puntos, determine el menor valor posible para n.[/url]

IberoAmerican IMO TST

1

Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to 2 and the sum of its terms is 200. How many progressions does Omar's list have?

2

Let ABC be an acute triangle and let P,Q be points on BC such that QAC=ABC and PAB=ACB. We extend AP to M so that P is the midpoint of AM and we extend AQ to N so that Q is the midpoint of AN. If T is the intersection point of BM and CN, show that quadrilateral ABTC is cyclic.

3

Let A be a set of m positive integers where m1. Show that there exists a nonempty subset B of A such that the sum of all the elements of B is divisible by m.

4

There are 4 piles of stones with the following quantities: 1004, 1005, 2009 and 2010. A legitimate move is to remove a stone from each from 3 different piles. Two players A and B play in turns. A begins the game . The player who, on his turn, cannot make a legitimate move, loses. Determine which of the players has a winning strategy and give a strategy for that player.

5

In the square shown in the figure, find the value of x.

6

Starting from an equilateral triangle with perimeter P0, we carry out the following iterations: the first iteration consists of dividing each side of the triangle into three segments of equal length, construct an exterior equilateral triangle on each of the middle segments, and then remove these segments (bases of each new equilateral triangle formed). The second iteration consists of apply the same process of the first iteration on each segment of the resulting figure after the first iteration. Successively, follow the other iterations. Let An be the area of the figure after the n- th iteration, and let Pn the perimeter of the same figure. If An=Pn, find the value of P0 (in its simplest form).