International Competitions 2019

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a) + 2f(b) = f(f(a + b)).$$

In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C = \angle BAC$. Similarly, let $Q_1$ be a point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q = \angle CBA$.

Prove that points $P$, $Q$, $P_1$, and $Q_1$ are concyclic.

A social network has 2019 users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time:

Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and no longer friends with $C$. All other friendship statuses are unchanged.

Initially, 1010 users have 1009 friends each, and 1009 users have 1010 friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

Find all pairs $(k, n)$ of positive integers such that $$k! = (2^n - 1)(2^n - 2)(2^n - 4) \cdots (2^n - 2^{n-1}).$$

The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k > 0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n = 3$ the process starting with the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$, which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$.

Let $I$ be the incentre of acute triangle $ABC$ with $AB \neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ again at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangles $PCE$ and $PBF$ meet again at $Q$.

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(xf(y) + 2y) = f(xy) + xf(y) + f(f(y))$$ holds for all real numbers $x$ and $y$.

Let $n \geq 3$ be an integer. We say that a vertex $A_i$ ($1 \leq i \leq n$) of a convex polygon $A_1A_2\ldots A_n$ is Bohemian if its reflection with respect to the midpoint of the segment $A_{i-1}A_{i+1}$ (with $A_0 = A_n$ and $A_{n+1} = A_1$) lies inside or on the boundary of the polygon $A_1A_2\ldots A_n$. Determine the smallest possible number of Bohemian vertices a convex $n$-gon can have (depending on $n$).

(A convex polygon $A_1A_2\ldots A_n$ has $n$ vertices with all inner angles smaller than $180°$.)

Let $ABC$ be an acute-angled triangle with $AC > BC$ and circumcircle $\omega$. Suppose that $P$ is a point on $\omega$ such that $AP = AC$ and that $P$ is an interior point of the shorter arc $BC$ of $\omega$. Let $Q$ be the point of intersection of the lines $AP$ and $BC$. Furthermore, suppose that $R$ is a point on $\omega$ such that $QA = QR$ and that $R$ is an interior point of the shorter arc $AC$ of $\omega$. Finally, let $S$ be the point of intersection of the line $BC$ with the perpendicular bisector of the side $AB$. Prove that the points $P$, $Q$, $R$, and $S$ are concyclic.

Determine the smallest positive integer $n$ for which the following statement holds true: From any $n$ consecutive integers one can select a non-empty set of consecutive integers such that their sum is divisible by $2019$.

Determine the smallest and the greatest possible values of the expression $$\left(\frac{1}{a^2 + 1} + \frac{1}{b^2 + 1} + \frac{1}{c^2 + 1}\right)\left(\frac{a^2}{a^2 + 1} + \frac{b^2}{b^2 + 1} + \frac{c^2}{c^2 + 1}\right)$$ provided $a$, $b$, and $c$ are non-negative real numbers satisfying $ab + bc + ca = 1$.

Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x + \alpha) \leq (x^{20} + x^{19})P(x)$$ holds for all real numbers $x$.

There are $n$ boys and $n$ girls in a school class, where $n$ is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper. Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation).

Prove that every integer from $1$ to $2019$ can be represented as an arithmetic expression consisting of up to $17$ symbols $2$ and an arbitrary number of additions, subtractions, multiplications, divisions and brackets. The $2$'s may not be used for any other operation, for example to form multi-digit numbers (such as $222$) or powers (such as $2^2$).

Valid examples: $$\left((2 \times 2 + 2) \times 2 - \frac{2}{2}\right) \times 2 = 22, \quad (2 \times 2 \times 2 - 2) \times \left(2 \times 2 + \frac{2 + 2 + 2}{2}\right) = 42.$$

Let $ABC$ be an acute-angled triangle such that $AB < AC$. Let $D$ be the point of intersection of the perpendicular bisector of the side $BC$ with the side $AC$. Let $P$ be a point on the shorter arc $AC$ of the circumcircle of the triangle $ABC$ such that $DP \parallel BC$. Finally, let $M$ be the midpoint of the side $AB$. Prove that $\angle APD = \angle MPB$.

Let $ABC$ be a right-angled triangle with its right angle at $B$ and circumcircle $c$. Denote by $D$ the midpoint of the shorter arc $AB$ of $c$. Let $P$ be the point on the side $AB$ such that $CP = CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX = AY = PD$. Prove that the points $X$, $Y$, and $P$ are collinear.

Let $a$, $b$ and $c$ be positive integers satisfying $a < b < c < a + b$. Prove that $c(a - 1) + b$ does not divide $c(b - 1) + a$.

Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N + 3)$. Prove that there exist two indices $i$ and $j$ such that $N = F_i \cdot F_j$, where $(F_n)_{n=1}^{\infty}$ is the Fibonacci sequence defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n \geq 3$.