Determine all integers $k$ such that the equation: $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{k}{xyz}$$ has an infinite number of integer solutions $(x,y,z)$ with gcd$(k,xyz)=1$.

Find the sum of all positive integers $n$ where the mean and median of $\{20, 42, 69, n\}$ are both integers. [i]Proposed by bissue[/i]

On sides $BC,CA,AB$ of a triangle $ABC$ points $D,E,F$ respectively are chosen so that $AD,BE,CF$ have a common point, say $G$. Suppose that one can inscribe circles in the quadrilaterals $AFGE,BDGF,CEGD$ so that each two of them have a common point. Prove that triangle $ABC$ is equilateral.

Circles $\omega$ and $\Omega$ are inscribed into the same angle. Line $\ell$ meets the sides of angles, $\omega$ and $\Omega$ in points $A$ and $F, B$ and $C, D$ and $E$ respectively (the order of points on the line is $A,B,C,D,E, F$). It is known that$ BC = DE$. Prove that $AB = EF$.

Find all values of real parameter $a$ for which equation $2{\sin}^4(x)+{\cos}^4(x)=a$ has real solutions

Let $k\geqslant 2$ be an integer, and $a,b$ be real numbers. prove that $a-b$ is an integer divisible by $k$ if and only if for every positive integer $n$ $$\lfloor an \rfloor \equiv \lfloor bn \rfloor \ (mod \ k)$$ Proposed by Navid Safaei

Let $n$ be an even positive integer. An $n$-degree monic polynomial $P(x)$ has $n$ real roots (not necessarily distinct). Suppose $y$ is a positive real number such that for any real number $t<y$, we have $P(t)>0$. Prove that \[P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.\]

In a triangle $ABC$, let $I$ denote the incenter. Let the lines $AI,BI$ and $CI$ intersect the incircle at $P,Q$ and $R$, respectively. If $\angle BAC = 40^o$, what is the value of $\angle QPR$ in degrees ?

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

Lamps of the hall switch by only five keys. Every key is connected to one or more lamp(s). By switching every key, all connected lamps will be switched too. We know that no two keys have same set of connected lamps with each other. At first all of the lamps are off. Prove that someone can switch just three keys to turn at least two lamps on.

Let ABCD be a convex quadrilateral; P,Q, R,S are the midpoints of AB, BC, CD, DA respectively such that triangles AQR, CSP are equilateral. Prove that ABCD is a rhombus. Find its angles.

Given a list $A$ of $n$ real numbers, the following algorithm, known as $\textit{insertion sort}$, sorts the elements of $A$ from least to greatest. \begin{tabular}{l} 1: \textbf{FUNCTION} $IS(A)$ \\ 2: $\quad$ \textbf{FOR} $i=0,\ldots, n-1$: \\ 3: $\quad\quad$ $j \leftarrow i$\\ 4: $\quad\quad$ \textbf{WHILE} $j>0$ \& $A[j-1]>A[j]:$\\ 5: $\quad\quad\quad$ \textbf{SWAP} $A[j], A[j-1]$\\ 6: $\quad\quad\quad$ $j \leftarrow j-1$\\ 7: \textbf{RETURN} $A$ \end{tabular} As $A$ ranges over all permutations of $\{1, 2, \ldots, n\}$, let $f(n)$ denote the expected number of comparisons (i.e., checking which of two elements is greater) that need to be made when sorting $A$ with insertion sort. Evaluate $f(13) - f(12)$.

It is given that $x, y, z$ are $3$ real numbers such that $\frac{x-y}{2+xy}+\frac{y-z}{2+yz}+\frac{z-x}{2+zx}=0$ Is it true that at least two of the three numbers must be equal? Justify your answer.

Consider $\triangle ABC$ with circumcenter $O$ and $\angle ABC$ obtuse. Construct $A'$ as the reflection of $A$ over $O$, and let $P$ be the intersection of $\overline{A'B}$ and $\overline{AC}$. Let $P'$ be the intersection of the circumcircle of $(OPA)$ with $\overline{AB}$. Given that the circumdiameter of $\triangle ABC$ is $25$, $\overline{AB} = 7$, and $\overline{BC} = 15$, find the length of $PP'$. [i]Proposed by Kevin Wu[/i]

An acute triangle $ ABC$ is given. Points $ A_1$ and $ A_2$ are taken on the side $ BC$ (with $ A_2$ between $ A_1$ and $ C$), $ B_1$ and $ B_2$ on the side $ AC$ (with $ B_2$ between $ B_1$ and $ A$), and $ C_1$ and $ C_2$ on the side $ AB$ (with $ C_2$ between $ C_1$ and $ B$) so that \[ \angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.\] The lines $ AA_1,BB_1,$ and $ CC_1$ bound a triangle, and the lines $ AA_2,BB_2,$ and $ CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

[b]Problem Section #2 b) Find the maximal value of $(x^3+1)(y^3+1)$, where $x,y \in \mathbb{R}$, $x+y=1$.

A function $g$ is such that for all integer $n$: $$g(n)=\begin{cases} 1\hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\geq 1 & \\ 0 \hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\leq 0 & \end{cases}$$ A function $f$ is such that for all integers $n\geq 0$ and $m\geq 0$: $$f(0,m)=0 \hspace{0.5cm} \textrm{and}$$ $$f(n+1,m)=\Bigl(1-g(m)+g(m)\cdot g(m-1-f(n,m))\Bigr)\cdot\Bigl(1+f(n,m)\Bigr)$$ Find all the possible functions $f(m,n)$ that satisfies the above for all integers $n\geq0$ and $m\geq 0$

Let $a$ and $b$ be integers and $p$ a prime. For each positive integer k, define$ A_k = \{n \in Z^+ |p^k$ divides $a^n - b^n\}$. Show that if $A_1$ is nonempty then $A_k$ is nonempty for all positive integers $k$

Prove that there exists a positive integer $n$, such that the remainder of $3^n$ when divided by $2^n$ is greater than $10^{2021} $.

Let $\triangle ABC$ be a triangle with $AB=5, BC=6, CA=7$. Suppose $P$ is a point inside $\triangle ABC$ such that $\triangle BPA\sim \triangle APC$. If $AP$ intersects $BC$ at $X$, find $\frac{BX}{CX}$. [i]Proposed by Nathan Ramesh

There are $n$ kids. From each two at least one of them has sent an SMS to the other. For each kid $A$, among the kids on which $A$ has sent an SMS, exactly 10% of them have sent an SMS to $A$. Determine the number of possible three-digit values of $n$.

Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate \[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]

There are a buch of 2000 stones. Two players play alternatively, following the next rules: ($a$)On each turn, the player can take 1, 2, 3, 4 or 5 stones [b]of[/b] the bunch. ($b$) On each turn, the player has forbidden to take the exact same amount of stones that the other player took just before of him in the last play. The loser is the player who can't make a valid play. Determine which player has winning strategy and give such strategy.

From a set of consecutive natural numbers one number is excluded so that the aritmetic mean of the remaining numbers is $ 50.55$. Find the initial set of numbers and the excluded number.

The sum of the first $n$ terms of the sequence $$1,1+2,1+2+2^2,\ldots,1+2+\cdots+2^{k-1},\ldots$$ is of the form $2^{n+R}+Sn^2+Tn+U$ for all $n>0.$ Find $R,S,T,$ and $U.$