Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.

Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four

The ratio of $w$ to $x$ is $4 : 3$, the ratio of $y$ to $z$ is $3 : 2$, and the ratio of $z$ to $x$ is $1 : 6$. What is the ratio of $w$ to $y$? $\textbf{(A) }4:3 \qquad \textbf{(B) }3:2 \qquad \textbf{(C) } 8:3 \qquad \textbf{(D) } 4:1 \qquad \textbf{(E) } 16:3 $

For any positive integer n, evaluate $$\sum_{i=0}^{\lfloor \frac{n+1}{2} \rfloor} {n-i+1 \choose i}$$ , where $\lfloor n \rfloor$ is the greatest integer less than or equal to $n$ .

Find all functions $f: R \rightarrow R$ such that \[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\] for all $x,y \in R$

Given that the value of \[\sum_{k=1}^{2021} \frac{1}{1^2+2^2+3^2+\cdots+k^2}+\sum_{k=1}^{1010} \frac{6}{2k^2-k}+\sum_{k=1011}^{2021} \frac{24}{2k+1}\] can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Aidan Duncan[/i]

A trapezoid has bases with lengths equal to $5$ and $15$ and legs with lengths equal to $13$ and $13.$ Determine the area of the trapezoid.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.

Find the number of pairs $(m,n)$ of positive integers with $1\le m<n\le 30$ such that there exists a real number $x$ satisfying $$\sin(mx)+\sin(nx)=2.$$

In a graph with $n$ vertices every two vertices are connected by a unique path. For each two vertices $u$ and $v$, let $d(u, v)$ denote the distance between $u$ and $v$, i.e. the number of edges in the path connecting these two vertices, and $\deg(u)$ denote the degree of a vertex $u$. Let $W$ be the sum of pairwise distances between the vertices, and $D$ the sum of weighted pairwise distances: $\sum_{\{u, v\}}(\deg(u)+\deg(v))d(u, v)$. Prove that $D=4W-n(n-1)$.

$k$ marbles are placed onto the cells of a $2024 \times 2024$ grid such that each cell has at most one marble and there are no two marbles are placed onto two neighboring cells (neighboring cells are defined as cells having an edge in common). a) Assume that $k=2024$. Find a way to place the marbles satisfying the conditions above, such that moving any placed marble to any of its neighboring cells will give an arrangement that does not satisfy both the conditions. b) Determine the largest value of $k$ such that for all arrangements of $k$ marbles satisfying the conditions above, we can move one of the placed marble onto one of its neighboring cells and the new arrangement satisfies the conditions above.

An organization has $n$ members, each two of which know exactly one of the others. Prove that there is a member that knows everyone.

[b]Evaluate[/b] $\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$

Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

4.Prove that there exist only finitely many positive integers n such that $(\frac{n}{1}+1)(\frac{n}{2}+2)...(\frac{n}{n}+n)$ is an integer.

Does there exists a subset of positive integers with infinite members such that for every two members $a,b$ of this set \[a^2-ab+b^2|(ab)^2\]

Let $\mathbb R^\ast$ denote the set of nonzero reals. Find all functions $f: \mathbb R^\ast \to \mathbb R^\ast$ satisfying \[ f(x^2+y)+1=f(x^2+1)+\frac{f(xy)}{f(x)} \] for all $x,y \in \mathbb R^\ast$ with $x^2+y\neq 0$. [i]Proposed by Ryan Alweiss[/i]

Inside a triangle $ABC$ three circles with radius $1$ are drawn. (Circles can be tangent to each other and to the sides of the triangle, but can not have any common internal points.) Find the biggest value of $r$ for which one can state that he can always draw a fourth circle inside the triangle of radius $r$, which does not intersect three already drawn circles.

Find all positive integers $n$, for which there exist positive integers $a>b$, satisfying $n=\frac{4ab}{a-b}$.

Two locations A and B are connected by a 5-mile trail which features a lookout C. A group of 15 hikers started at A and walked along the trail to C. Another group of 10 hikers started at B and walked along the trail to C. The total distance travelled to C by all hikers from the group that started in A was equal to the total distance travelled to C by all hikers from the group that started in B. Find the distance (in miles) from A to C along the trail.

Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for evert integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times $111$ appears on this sequence?

Three standard six-sided dice are rolled. What is the probability that the product of the values on the top faces of the three dice is a perfect cube?

Let a quardilateral $ABCD$ with $AB=AD$ and $\widehat B=\widehat D=90$. At $CD$ we take point $E$ and at $BC$ we take point $Z$ such that $AE\bot DZ$. Prove that $AZ\bot BE$