In acute-angled triangle $ABC, I$ is the center of the inscribed circle and holds $| AC | + | AI | = | BC |$. Prove that $\angle BAC = 2 \angle ABC$.

Determine all polynomials $P (x)$ with real coefficients that apply $P (x^2) + 2P (x) = P (x)^2 + 2$.

For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically? Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.

Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.

Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different. For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$. Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) + \phi (b) + gcd (a, b)$. Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.

Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$ at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$.

For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$. Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.

Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$for all $x, y \in Z$

Given are two positive integers $k$ and $n$ with $k \le n \le 2k - 1$. Julian has a large stack of rectangular $k \times 1$ tiles. Merlin calls a positive integer $m$ and receives $m$ tiles from Julian to place on an $n \times n$ board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number $m$ that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?