Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that points $A$, $X$, and $Y$ are collinear.
Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|,|b-d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1,1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_{0}, y_{0})=(1,1)$, $(x_{1}, y_{1})$, $(x_{2}, y_{2}), \ldots, (x_{n}, y_{n})$ such that, for all $i=1,2, \ldots, n$, $1 \leq x_{i}, y_{i} \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_{i}, y_{i})$. In this case, each cell $(x_{i}, y_{i})$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.
Let $n$ be a positive integer and $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers. Prove that
$$
\sum_{i=1}^{n} \frac{1}{2^{i}}\left(\frac{2}{1+a_{i}}\right)^{2^{i}} \geq \frac{2}{1+a_{1} a_{2} \ldots a_{n}}-\frac{1}{2^{n}} .
$$
Prove that for every positive integer $t$ there is a unique permutation $a_{0}, a_{1}, \ldots, a_{t-1}$ of $0,1, \ldots, t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2 a_{i}}$ is odd and $2 a_{i} \neq t+i$.
Line $\ell$ intersects sides $B C$ and $A D$ of cyclic quadrilateral $A B C D$ in its interior points $R$ and $S$ respectively, and intersects ray $D C$ beyond point $C$ at $Q$, and ray $B A$ beyond point $A$ at $P$. Circumcircles of the triangles $Q C R$ and $Q D S$ intersect at $N \neq Q$, while circumcircles of the triangles $P A S$ and $P B R$ intersect at $M \neq P$. Let lines $M P$ and $N Q$ meet at point $X$, lines $A B$ and $C D$ meet at point $K$ and lines $B C$ and $A D$ meet at point $L$. Prove that point $X$ lies on line $K L$.
Let $ABCD$ be a trapezoid with parallel sides $AB$ and $CD$, with $\angle BAD = 90^\circ$ and with $AB + CD = BC$. Furthermore, let $M$ be the mid-point of $AD$.
Prove that $\angle CMB = 90^\circ$.
Anna, Berta and Clara write the square numbers $1, 4, 9, \dots, 2025$ on a blackboard, compute their sum and observe that it is divisible by $3$. Then, they agree to the following game: In each round, Anna will cross out one number, then Berta will do the same, and then Clara will do the same. This continues until all numbers are crossed out. Clara has the goal that the sum of the remaining numbers after each round is divisible by $3$.
a) Prove that Anna cannot stop Clara from reaching her goal if Clara has Berta's help.
b) Prove that Berta can stop Clara from reaching her goal even if Clara has Anna's help.
Determine the smallest constant $C$ such that the inequality
$$
(X + Y)^2 (X^2 + Y^2 + C) + (1 - XY)^2 \geq 0
$$
holds for all real numbers $X$ and $Y$.
For which values of $X$ and $Y$ does equality hold for this smallest constant $C$?
Let $ABC$ be an acute triangle with $AB \gt AC$. Let $D$, $E$ and $F$ denote the feet of its altitudes on $BC$, $AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$.
Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$.
Initially, the numbers $1, 2, \ldots, 2024$ are written on a blackboard. Trixi and Nana play a game, taking alternate turns. Trixi plays first.
The player whose turn it is chooses two numbers $a$ and $b$, erases both, and writes their (possibly negative) difference $a - b$ on the blackboard. This is repeated until only one number remains on the blackboard after $2023$ moves. Trixi wins if this number is divisible by $3$, otherwise Nana wins.
Let $ABC$ be an obtuse triangle with orthocenter $H$ and centroid $S$. Let $D$, $E$ and $F$ be the midpoints of segments $BC$, $AC$, $AB$, respectively.
Show that the circumcircle of triangle $ABC$, the circumcircle of triangle $DEF$ and the circle with diameter $HS$ have two distinct points in common.
For each prime number $p$, determine the number of residue classes modulo $p$ which can be represented as $a^2 + b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers.
Let $\alpha$ and $\beta$ be real numbers with $\beta \neq 0$. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$
f(\alpha f(x) + f(y)) = \beta x + f(y)
$$
holds for all real $x$ and $y$.
Let $h$ be a semicircle with diameter $AB$. The two circles $k_1$ and $k_2$, $k_1 \neq k_2$, touch the segment $AB$ at the points $C$ and $D$, respectively, and the semicircle $h$ from the inside at the points $E$ and $F$, respectively. Prove that the four points $C$, $D$, $E$ and $F$ lie on a circle.
Let $n \ge 3$ be an integer. A *circle dance* is a dance that is performed according to the following rule: On the floor, $n$ points are marked at equal distances along a large circle. At each of these points is a sheet of paper with an arrow pointing either clockwise or counterclockwise. One of the points is labeled "Start". The dancer starts at this point. In each step, he first changes the direction of the arrow at his current position and then moves to the next point in the new direction of the arrow.
a) Show: Each circle dance visits each point infinitely often.
b) How many different circle dances are there? Two circle dances are considered to be the same if they differ only by a finite number of steps at the beginning and then always visit the same points in the same order. (The common sequence of steps may begin at different times in the two dances.)
A positive integer is called *powerful* if all exponents in its prime factorization are $\geq 2$.
Prove that there are infinitely many pairs of powerful consecutive positive integers.
Let $ABC$ be an acute triangle with orthocenter $H$. The circumcircle of the triangle $BHC$ intersects $AC$ a second time in point $P$ and $AB$ a second time in point $Q$.
Prove that $H$ is the circumcenter of the triangle $APQ$.