2019 Argentina National Olympiad Level 2

Level 2

Day 1

1

We say that three positive integers a, b and c form a family if the following two conditions are satisfied: a+b+c=900. There exists an integer n, with n, such that \frac{a}{n-1}=\frac{b}{n}=\frac{c}{n+1}. Determine the number of such families.

2

A 7 \times 7 grid is given. Julián colors 29 cells black. Pilar must then place an L-shaped piece, covering exactly three cells (oriented in any direction, as shown in the figure). Pilar wins if all three cells covered by the L-shaped piece are black. Can Julián color the grid in such a way that it is impossible for Pilar to win? [asy][asy] size(1.5cm); draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0)); [/asy][/asy]

3

Let \Gamma be a circle of center S and radius r and A a point outside the circle. Let BC be a diameter of \Gamma such that B does not belong to the line AS, and we consider the point O where the perpendicular bisectors of triangle ABC intersect, that is, the circumcenter of ABC. Determine all possible locations of point O when B varies in circle \Gamma.

Day 2

4

We define similar numbers as positive integers that have exactly the same digits (but possibly in another order). For example, 1241, 2114 and 4211 are similar numbers, but 1424 is not similar to the other three. Determine whether there exist three similar numbers, each with 300 digits (all digits being non-zero), such that the sum of two of them equals the third. If the answer is yes, provide an example; if not, justify why it is impossible.

5

In a club, some pairs of members are friends. Given an integer k \geqslant 3, we say a club is k-friendly if, in any group of k members, they can be seated at a round table such that each pair of neighbors are friends. Prove that if a club is 6-friendly, then it is also 7-friendly. Is it true that if a club is 9-friendly, then it is also 10-friendly?

6

Let n be a natural number. We define f(n) as the number of ways to express n as a sum of powers of 2, where the order of the terms is taken into account. For example, f(4) = 6, because 4 can be written as: \begin{align*} 4;\\ 2 + 2;\\ 2 + 1 + 1;\\ 1 + 2 + 1;\\ 1 + 1 + 2;\\ 1 + 1 + 1 + 1. \end{align*}Find the smallest n greater than 2019 for which f(n) is odd.