Consider an urn containing $51$ white and $50$ black balls. Every turn, we randomly pick a ball, record the color of the ball, and then we put the ball back into the urn. We stop picking when we have recorded $n$ black balls, where $n$ is an integer randomly chosen from $\{1, 2,... , 100\}$. What is the expected number of turns?

Let $ABC$ be a triangle with $\angle BAC=40^\circ $, $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$. If $AD\parallel OG$, find $\angle ACB$.

It is known that $\cos 157^o = a$, where $a$ is given. Calculate $1^o$ in terms of $a$.

For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$

For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge\frac{8y\sqrt{xy}}{3\sqrt{3}}.$

Kathryn has a crush on Joe. Dressed as Catwoman, she attends the same school Halloween party as Joe, hoping he will be there. If Joe gets beat up, Kathryn will be able to help Joe, and will be able to tell him how much she likes him. Otherwise, Kathryn will need to get her hipster friend, Max, who is DJing the event, to play Joe’s favorite song, “Pieces of Me” by Ashlee Simpson, to get him out on the dance floor, where she’ll also be able to tell him how much she likes him. Since playing the song would be in flagrant violation of Max’s musical integrity as a DJ, Kathryn will have to bribe him to play the song. For every $\$10$ she gives Max, the probability of him playing the song goes up $10\%$ (from $0\%$ to $10\%$ for the first $\$10$, from $10\%$ to $20\%$ for the next $\$10$, all the way up to $100\%$ if she gives him $\$100$). Max only accepts money in increments of $\$10$. How much money should Kathryn give to Max to give herself at least a $65\%$ chance of securing enough time to tell Joe how much she likes him?

A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$.

$H$ is the orthocentre of an acute–angled triangle $ABC$. A point $E$ is taken on the line segment $CH$ such that $ABE$ is a right–angled triangle. Prove that the area of the triangle $ABE$ is the geometric mean of the areas of triangles $ABC$ and $ABH$.

Determine all real solutions of the equation: \[{ \frac{x^{2}}{x-1}+\sqrt{x-1}+\frac{\sqrt{x-1}}{x^{2}}}=\frac{x-1}{x^{2}}+\frac{1}{\sqrt{x-1}}+\frac{x^{2}}{\sqrt{x-1}} . \]

It is known that $x^2_1+ x^2_2+...+ x^2_6= 6$ and $x_1 + x_2 +....+ x_6 = 0.$ Prove that $ x_1x_2....x_6 \le \frac12$ . .

Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

Find all real numbers $c$ for which there exists a nonconstant two-variable polynomial $P(x, y)$ with real coefficients satisfying \[[P(x, y)]^2 = P(cxy, x^2 + y^2)\] for all real $x$ and $y$. [i]Nikolai Beluhov and Konstantin Garov[/i]

Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.

Let $(\Omega, \mathcal{A},\mathbb{P})$ be a standard probability space, and $\mathcal{X}$ be the set of all bounded random variables. For $t>0$, defined the mapping $R_t$ by \[R_t(X)=t\log \mathbb{E}[\exp(X/t)], \quad X \in \mathcal{X},\] and for $t \in (0,1)$ define the mapping $Q_t$ by \[Q_t(X)=\inf\{x \in \mathbb{R}: \mathbb{P}(X>x) \le t\}, \quad X \in \mathcal{X}.\] For two mappings $f,g:\mathcal{X} \to [-\infty,\infty)$, defined the operator $\square$ by \[f\square g(X)=\inf\{f(Y)+g(X-Y): Y \in \mathcal{X}\}, \quad X \in \mathcal{X}.\] (a) Show that, for $t,s>0$, \[R_t \square R_s=R_{t+s}.\] (b) Show that, for $t,s>0$ with $t+s<1$, \[Q_t \square Q_s=Q_{t+s}.\]

Let $ A,B,C,D,E$ be five distinct points, such that no three of them lie on the same line. Prove that \[ AB\plus{}BC\plus{}CA \plus{} DE < AD \plus{} AE \plus{} BD\plus{}BE \plus{} CD\plus{}CE .\]

A circle is tangent to both branches of the hyperbola $x^2 - 20y^2 = 24$ as well as the $x$-axis. Compute the area of this circle.

How many solutions does \[ x^2 - [x^2] = \left(x - [x]\right)^2 \] have satisfying $1 \leq x \leq n$?

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

Let $O$ be the center of the circle circumscribed to triangle $ABC$ and let $ S_ {A} $, $ S_ {B} $, $ S_ {C} $ be the circles centered on $O$ that are tangent to the sides $BC, CA, AB$ respectively. Show that the sum of the angle between the two tangents $ S_ {A} $ from $A$ plus the angle between the two tangents $ S_ {B} $ from $B$ plus the angle between the two tangents $ S_ {C} $ from $C$ is $180$ degrees.

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

You are given a square $n \times n$. The centers of some of some $m$ of its $1\times 1$ cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer $n \geq 3$, find the largest value of $m$ for which it is possible. [i]Proposed by Oleksiy Masalitin, Fedir Yudin[/i]

Let $ABC$ be an equilateral triangle and $\Gamma$ its incircle. If $D$ and $E$ are points on the segments $AB$ and $AC$ such that $DE$ is tangent to $\Gamma$, show that $\frac{AD}{DB}+\frac{AE}{EC}=1$.

Find a subset of $\{1,2, ...,2022\}$ with maximum number of elements such that it does not have two elements $a$ and $b$ such that $a = b + d$ for some divisor $d$ of $b$.

In a triangle, one excircle touches side $AB$ at point $C_1$ and the other touches side $BC$ at point $A_1$. Prove that on the straight line $A_1C_1$ the constructed excircles cut out equal segments.

We can conduct the following moves to a real number $x$: choose a positive integer $n$, and positives reals $a_1,a_2,\cdots, a_n$ whose reciprocals sum up to $1$. Let $x_0=x$, and $x_k=\sqrt{x_{k-1}a_k}$ for all $1\leq k\leq n$. Finally, let $y=x_n$. We said $M>0$ is "tremendous" if for any $x\in \mathbb{R}^+$, we can always choose $n,a_1,a_2,\cdots, a_n$ to make the resulting $y$ smaller than $M$. Find all tremendous numbers. [i]Proposed by ckliao914.[/i]