We play a game of musical chairs with $n$ chairs numbered $1$ to $n$. You attach $n$ leaves, numbered $1$ to $n$, to the chairs in such a way that the number on a leaf does not match the number on the chair it is attached to. One player sits on each chair. Every time you clap, each player looks at the number on the leaf attached to his current seat and moves to sit on the seat with that number. Prove that, for any $m$ that is not a prime power with$ 1 < m \leq n$, it is possible to attach the leaves to the seats in such a way that after $m$ claps everyone has returned to the chair they started on for the first time.

Let $ \Omega$ is circle with radius $ R$ and center $ O$. Let $ \omega$ is a circle inside of the $ \Omega$ with center $ I$ radius $ r$. $ X$ is variable point of $ \omega$ and tangent line of $ \omega$ pass through $ X$ intersect the circle $ \Omega$ at points $ A,B$. A line pass through $ X$ perpendicular with $ AI$ intersect $ \omega$ at $ Y$ distinct with $ X$.Let point $ C$ is symmetric to the point $ I$ with respect to the line $ XY$.Find the locus of circumcenter of triangle $ ABC$ when $ X$ varies on $ \omega$

The binomial coefficient $\tbinom{n}{k}$ can be defined as the coefficient of $x^k$ in the expansion of $(1+x)^n.$ Similarly, define the trinomial coefficient $\tbinom{n}{k}_3$ as the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n.$ Determine the number of integers $k$ with $0 \le k \le 4048$ such that $\tbinom{2024}{k}_3 \equiv 1 \pmod{3}.$

In triangle $ ABC$ we have $ \angle{ACB} \equal{} 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD \equal{} AC$ and $ DB \equal{} DC$. Prove that $ \angle{BAC} \equal{} 3\angle{BAD}$.

There are $2022$ equally spaced points on a circular track $\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle. Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labellings of the $2022$ points. [i]Kevin Cong[/i]

Find all pairs $(p,q)$ of prime numbers which $p>q$ and $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer.

In triangle \( ABC \), let \( I \) be the \( A \)-excenter. Points \( X \) and \( Y \) are placed on line \( BC \) such that \( B \) is between \( X \) and \( C \), and \( C \) is between \( Y \) and \( B \). Moreover, \( B \) and \( C \) are the contact points of \( BC \) with the \( A \)-excircle of triangles \( BAY \) and \( AXC \), respectively. Let \( J \) be the \( A \)-excenter of triangle \( AXY \), and let \( H' \) be the reflection of the orthocenter of triangle \( ABC \) with respect to its circumcenter. Prove that \( I \), \( J \), and \( H' \) are collinear. Proposed by Ali Nazarboland

Find all positive integer $n$ for which there exist real number $a_1,a_2,\ldots,a_n$ such that $$\{a_j-a_i|1\le i<j\le n\}=\left\{1,2,\ldots,\frac{n(n-1)}2\right\}.$$

Let $m, n$ be positive integers, define: $f(x)=(x-1)(x^2-1)\cdots(x^m-1)$, $g(x)=(x^{n+1}-1)(x^{n+2}-1)\cdots(x^{n+m}-1)$. Show that there exists a polynomial $h(x)$ of degree $mn$ such that $f(x)h(x)=g(x)$, and its $mn+1$ coefficients are all positive integers.

Given triangle $ABC$. Point $B_1$ bisects the length of the broken line $ABC$ (composed of segments $AB$ and $BC$), point $C_1$ bisects the length of the broken line$ACB$, point $A_1$ bisects the length of of the broken line $CAB$. Through points $A_1$, $B_1$ and $C_1$ straight lines $\ell_A$ ,$\ell_B$, $\ell_C$ are drawn parallel to the bisectors angles $BAC$, $ABC$ and $ACB$ respectively. Prove that the lines $\ell_A$, $\ell_B$ and $\ell_C$ intersect at one point.

In a right triangle $ABC$ with $\angle BAC =90^o $and $\angle ABC= 54^o$, point $M$ is the midpoint of the hypotenuse $[BC]$ , point $D$ is the foot of the angle bisector drawn from the vertex $C$ and $AM \cap CD = \{E\}$. Prove that $AB= CE$.

Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size $2 \times 1$ or $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)

The Fibonacci sequence is dened by $f_1=f_2=1$ and $f_n=f_{n-1}+f_{n-2}$ for $n\ge 3$. A Pythagorean triangle is a right-angled triangle with integer side lengths. Prove that $f_{2k+1}$ is the hypotenuse of a Pythagorean triangle for every positive integer $k$ with $k\ge 2$

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

Let $\Gamma$ be the maximum possible value of $a+3b+9c$ among all triples $(a,b,c)$ of positive real numbers such that \[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\] If $\Gamma = \frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, then find $p+q$.

Let $M=\{3,4,5,6,7,8,9,10,11,12,13,15,17,19,21,23\}$. Prove that for any $a_i>0,i=\overline{1,n},n\in M$ the inequality holds: $$\frac{a_1^2}{(a_2+a_3)^4}+\frac{a_2^2}{(a_3+a_4)^4}+\ldots+\frac{a_{n-1}^2}{(a_n+a_1)^4}+\frac{a_n^2}{(a_1+a_2)^4}\ge\frac{n^3}{16s^2},$$ where $s=\sum_{i=1}^na_i$. [i]Proposed by Marius Olteanu[/i]

Inside triangle \( ABC \), points \( D \) and \( E \) are chosen such that \( \angle ABD = \angle CBE \) and \( \angle ACD = \angle BCE \). Point \( F \) on side \( AB \) is such that \( DF \parallel AC \), and point \( G \) on side \( AC \) is such that \( EG \parallel AB \). Prove that \( \angle BFG = \angle BDC \). [i]Proposed by Anton Trygub[/i]

If $a+\log_2 3,a+\log_4 3,a+\log_8 3$ are a geometric series, then the common ratio is________.

Let $\{a_n\}$ be a sequence satisfying: $a_1=1$ and $a_{n+1}=2a_n+n\cdot (1+2^n),(n=1,2,3,\cdots)$. Determine the general term formula of $\{a_n\}$.

Suppose that $ f: [a,b]\rightarrow \mathbb{R} $ be a monotonic function and for every $ x_1,x_2\in [a,b] $ that $ x_1<x_2 $ ,there exist $ c\in (a,b) $ such that $ \int _{x_1}^{x_2}f(x)dx=f(c)(x_1-x_2) $ a) Show that $ f $ be the continuous function on interval $ (a,b) $ b) Suppose that $ f $ is integrable function on interval $ [a,b] $ but $ f $ isn't a monotonic function then ,is it the result of part a) right?

Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.

The bisector of acute angle $\alpha$ of the right triangle $ABC$ splits the side $a$ in two segments with lengths $m$ and $n$. Find the acute angle $\beta$ and the lenths of the other two sides by knowing $m$ and $n$.

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

A convex polygon $P$ can be partitioned into $27$ parallelograms. Prove that it can also be partitioned into $21$ parallelograms.

Two circles with radii $3$ and $4$ are externally tangent at $P$. Let $A \neq P$ be on the first circle and $B \neq P$ be on the second circle, and let the tangents at $A$ and $B$ to the respective circles intersect at $Q$. Given that $QA = QB$ and $AB$ bisects $PQ$, compute the area of $QAB$.