The three integers $2000$, $19$ and $n$ are written on the board. Ana and Beto play the following game:
Ana starts and then they play in turns. Each move consists of erasing one of the numbers on the board and replacing it with the difference of the other two (the larger minus the smaller). Only moves in which one of the written numbers changes are allowed. The player who cannot move on their turn loses.
Prove that for every value of $n$ the game has a winner, and determine who wins if the numbers on the board are $2000$, $19$ and $2019$.
Let $n$ be a positive integer. There are $n$ balls numbered from 1 to $n$ and three boxes of different colours.
Find the smallest $n$ such that for every placement of the $n$ balls into the three boxes there are always, in one same box, two balls such that the difference of the numbers written on them (the larger minus the smaller) is equal to the square of an integer.
In triangle $ABC$, let $D$ and $E$ be points on the sides $AB$ and $AC$ respectively such that $BD = CE$. Let $M$ and $N$ be the midpoints of $BC$ and $DE$ respectively.
Prove that the bisector of angle $BAC$ is parallel to the line $MN$.
Bruno chooses a positive integer $X$. Then Flor chooses four integers $a, b, c, d$ and computes $N = (a-b)(b-c)(c-d)(d-a)(a-c)(b-d)$, the product of the six differences between these four numbers. Determine the largest value of $X$ for which Bruno can be certain that $N$ will be a multiple of $X$.
On a $9 \times 9$ board some cells must be coloured red, at least one. For each colouring, let $P$ be the number of cells, coloured or not, that have an even number of red neighbouring cells. (Two cells are neighbours if they share a side.) Give a colouring of the board that has the smallest possible value of $P$ and prove that there cannot be a smaller value.
Let $A, B, C$ be the colours of the three boxes, and assign to each number the colour of the box that contains it. Now the problem is to find the maximum value of $n$ such that it is possible to colour the numbers $1, 2, 3, \ldots, n$ with the colours $A, B, C$ so that no two numbers of the same colour differ by the square of an integer.
We say that three positive integers $a$, $b$, $c$ form a *family* if the following two conditions hold:
$$
\bullet\ a+b+c=900;
$$
• there exists an integer $n$, $n \ge 2$, such that $\frac{a}{n-1} = \frac{b}{n} = \frac{c}{n+1}$.
We have a $7 \times 7$ board. Julián colours 29 cells black. Then Pilar must place on the board an L-shaped piece (an "elbow") that covers exactly three cells, like those in the figure (oriented in any way). If the three cells covered by the elbow are black, Pilar wins.
Determine whether Julián can make the colouring in such a way that it is impossible for Pilar to win.
Let $\Gamma$ be a circle with centre $S$ and radius $r$, and let $A$ be a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not lie on the line $AS$, and consider the point $O$ where the perpendicular bisectors of triangle $ABC$ meet, that is, the circumcentre of $ABC$.
Determine all possible positions of the point $O$ as $B$ varies on the circle $\Gamma$.
We call *similar* positive integers that have exactly the same digits. For example, $1241$, $2114$, $4211$ are similar numbers, but $1424$ is not similar to the previous ones.
Decide whether there exist three similar numbers of $300$ digits each, with all their digits different from $0$, such that the sum of two of them is equal to the third. If the answer is yes, give an example, and if it is no, justify why.
In a club some pairs of members are friends. Given $k \ge 3$, we say that a club is *k-friendly* if in every group of $k$ members these can be seated at a round table so that every pair of neighbours are friends.
a) Prove that if a club is 6-friendly then it is 7-friendly.
b) Is it true that if a club is 9-friendly then it is 10-friendly?
Let $n$ be a natural number. We define $f(n)$ as the number of ways of writing $n$ as a sum of powers of 2, where the order in which each term appears is taken into account. For example, $f(4) = 6$ because $4$ can be written as $4$; $2+2$; $2+1+1$; $1+2+1$; $1+1+2$; $1+1+1+1$.
Find the smallest $n$ greater than $2019$ for which $f(n)$ is odd.
A set of distinct positive integers is called *singular* if, for each of its elements, after crossing out that element the remaining ones can be split into two sets with no common elements such that the sums of the elements of the two groups are equal. Find the smallest positive integer $n \gt 1$ such that there exists a singular set $A$ with $n$ elements.
Let $n \ge 1$ be an integer. We have two sequences, each of $n$ positive real numbers, $a_1, a_2, ..., a_n$ and $b_1, b_2, ..., b_n$, such that $a_1 + a_2 + ... + a_n = 1$ and $b_1 + b_2 + ... + b_n = 1$. Find the smallest possible value that the sum
$$
\frac{a_1^2}{a_1+b_1} + \frac{a_2^2}{a_2+b_2} + \dots + \frac{a_n^2}{a_n+b_n}
$$
can take.
In triangle $ABC$ we have $A\hat{C}B = 2 \cdot A\hat{B}C$. Moreover, $P$ is a point inside triangle $ABC$ such that $AP = AC$ and $PB = PC$. Prove that $B\hat{A}C = 3 \cdot B\hat{A}P$.
We have a set $M$ of 2019 real numbers such that for every pair $a, b$ of numbers of $M$ the number $a^2 + b\sqrt{2}$ is rational. Prove that for every $a$ in $M$ the number $a\sqrt{2}$ is rational.
We have an arithmetic progression of 7 terms in which all the terms are distinct prime numbers. Determine the smallest possible value of the last term of such a progression.
CLARIFICATION: In an arithmetic progression with difference $d$, each term is equal to the previous one plus $d$.
The natural numbers from $1$ to $300$ inclusive are placed around a circle. We say that such an arrangement is *alternating* if each number is smaller than both of its neighbours or larger than both of its neighbours. We call a pair of neighbouring numbers a *good pair* if, after removing that pair from the circle, the remaining numbers form an alternating arrangement.
Determine the smallest possible number of good pairs in an alternating arrangement of the numbers from $1$ to $300$ inclusive.
Let $\mathbb{Z}^{+}$ be the set of positive integers. Determine all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ such that $a^{2} + f(a) f(b)$ is divisible by $f(a) + b$ for all positive integers $a$ and $b$.
