International Competitions 2021

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n + 1, \ldots, 2n$ each on different cards. He then shuffles these $n + 1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

Show that the inequality $$\sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{|x_i - x_j|} \leqslant \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{|x_i + x_j|}$$ holds for all real numbers $x_1, \ldots, x_n$.

Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD$. The point $E$ on the segment $AC$ satisfies $\angle ADE = \angle BCD$, the point $F$ on the segment $AB$ satisfies $\angle FDA = \angle DBC$, and the point $X$ on the line $AC$ satisfies $CX = BX$. Let $O_1$ and $O_2$ be the circumcentres of the triangles $ADC$ and $EXD$, respectively. Prove that the lines $BC$, $EF$, and $O_1O_2$ are concurrent.

Let $\Gamma$ be a circle with centre $I$, and $ABCD$ a convex quadrilateral such that each of the segments $AB$, $BC$, $CD$ and $DA$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $AIC$. The extension of $BA$ beyond $A$ meets $\Omega$ at $X$, and the extension of $BC$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $AD$ and $CD$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that $$AD + DT + TX + XA = CD + DY + YZ + ZC.$$

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.

Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a < k < b$.

Let $m \geqslant 2$ be an integer, $A$ be a finite set of (not necessarily positive) integers, and $B_1, B_2, B_3, \ldots, B_m$ be subsets of $A$. Assume that for each $k = 1, 2, \ldots, m$ the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $m/2$ elements.

Determine all real numbers $A$ such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying

$$x_{n+1} = A - \frac{1}{x_n}$$

for every integer $n \geqslant 1$, has only finitely many negative terms.

Let $m$ and $n$ be positive integers. Some squares of an $m \times n$ board are coloured red. A sequence $a_1, a_2, \ldots, a_{2r}$ of $2r \geqslant 4$ pairwise distinct red squares is called a bishop circuit if for every $k \in \{1, \ldots, 2r\}$, the squares $a_k$ and $a_{k+1}$ lie on a diagonal, but the squares $a_k$ and $a_{k+2}$ do not lie on a diagonal (here $a_{2r+1} = a_1$ and $a_{2r+2} = a_2$).

In terms of $m$ and $n$, determine the maximum possible number of red squares on an $m \times n$ board without a bishop circuit.

(Remark. Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of $45°$.)

Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A$, $D$, $E$ and $F$ are concyclic.

Let $n \geqslant 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$-gon. Zagi plans to make a journey of $n - 1$ jumps such that in the $i$-th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots, n - 1\}$. Prove that if after $\lceil \frac{n}{2} \rceil$ jumps Zagi has visited $\lceil \frac{n}{2} \rceil + 1$ distinct vertices, then after $n - 1$ jumps Zagi will have visited all of the vertices.

(Remark. For a real number $x$, we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$.)

Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that the inequality

$$f(x^2) - f(y^2) \leqslant (f(x) + y)(x - f(y))$$

holds for all real numbers $x$ and $y$.

Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor) = \lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$

(a) there exists an $n$-pretty polynomial;

(b) any $n$-pretty polynomial has a degree of at least $\frac{2n + 1}{3}$.

(Remark. For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)

Let $n$, $b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n - 1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n - 1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.

Let $n$ be a positive integer. Prove that in a regular $6n$-gon, we can draw $3n$ diagonals with pairwise distinct ends and partition the drawn diagonals into $n$ triplets so that:

• the diagonals in each triplet intersect in one interior point of the polygon and
• all these $n$ intersection points are distinct.

Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$. The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$, respectively. Lines $EF$ and $BC$ meet at $G$. Prove that $AD$ and $DG$ are perpendicular.

Let $ABC$ be a triangle and let $M$ be the midpoint of the segment $BC$. Let $X$ be a point on the ray $AB$ such that $2\angle CXA = \angle CMA$. Let $Y$ be a point on the ray $AC$ such that $2\angle AYB = \angle AMB$. The line $BC$ intersects the circumcircle of the triangle $AXY$ at $P$ and $Q$, such that the points $P$, $B$, $C$, and $Q$ lie in this order on the line $BC$. Prove that $PB = QC$.

Find all pairs $(n,p)$ of positive integers such that $p$ is prime and

$$1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdots + p^2).$$

Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.