Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.

Let $ ABC$ be a triangle such that \[ \frac{BC}{AB \minus{} BC}\equal{}\frac{AB \plus{} BC}{AC}\] Determine the ratio $ \angle A : \angle C$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Solve in positive real numbers the following equation $x^{-y} + y^{-x} = 4 - x - y$.

Bob plays a game on the whiteboard. Initially, the numbers $\{1, 2, ...,n\}$ are shown. On each turn, Bob takes two numbers from the board $x$, $y$, erases them both, and writes down $2x + y$ onto the board. In terms of n, what is the maximum possible value that Bob can end up with?

Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter? [img]https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png[/img]

Let $n \ge 2$ be an integer. Let $x_1 \ge x_2 \ge ... \ge x_n$ and $y_1 \ge y_2 \ge ... \ge y_n$ be $2n$ real numbers such that $$0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n $$ $$\text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2.$$ Prove that $$\sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}.$$ [i]Proposed by David Speyer and Kiran Kedlaya[/i]

It is known that each two of the 12 competitors, that participated in the finals of the competition “Mathematical duels”, have a common friend among the other 10. Prove that there is one of them that has at least 5 friends among the group.

Let $\omega_1$, $\omega_2$, $\omega_3$ are three externally tangent circles, with $\omega_1$ and $\omega_2$ tangent at $A$. Choose points $B$ and $C$ on $\omega_1$ so that lines $AB$ and $AC$ are tangent to $\omega_3$. Suppose the line $BC$ intersect $\omega_3$ at two distinct points, and $X$ is the intersection further away to $B$ and $C$ than the other one. Prove that one of the tangent lines of $\omega_2$ passing through $X$, is also tangent to an excircle of triangle $ABC$. [i]Proposed by Ivan Chan Kai Chin[/i]

suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

Let $\mathcal P$ denote the set of planes in three-dimensional space with positive $x$, $y$, and $z$ intercepts summing to one. A point $(x,y,z)$ with $\min \{x,y,z\} > 0$ lies on exactly one plane in $\mathcal P$. What is the maximum possible integer value of $\left(\frac{1}{4} x^2 + 2y^2 + 16z^2\right)^{-1}$? [i]Proposed by Sammy Luo[/i]

Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$. Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$

Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

On a table lie $289$ coins that form a square array $17 \times 17$. All coins are facing with the crown up. In one move, it is possible to reverse any five coins lying in a row: vertical, horizontal or diagonal. Is it possible that after a number of such moves, all the coins to be arranged with tails up?

Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively. (a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$ (b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]

Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than $11$. P.s. the current year in the problem is $2010$.

Alan draws a convex $2020$-gon $\mathcal{A}=A_1A_2\dotsm A_{2020}$ with vertices in clockwise order and chooses $2020$ angles $\theta_1, \theta_2, \dotsc, \theta_{2020}\in (0, \pi)$ in radians with sum $1010\pi$. He then constructs isosceles triangles $\triangle A_iB_iA_{i+1}$ on the exterior of $\mathcal{A}$ with $B_iA_i=B_iA_{i+1}$ and $\angle A_iB_iA_{i+1}=\theta_i$. (Here, $A_{2021}=A_1$.) Finally, he erases $\mathcal{A}$ and the point $B_1$. He then tells Jason the angles $\theta_1, \theta_2, \dotsc, \theta_{2020}$ he chose. Show that Jason can determine where $B_1$ was from the remaining $2019$ points, i.e. show that $B_1$ is uniquely determined by the information Jason has. [i]Proposed by Andrew Gu.[/i]

A uniform horizontal circular plate of weight $ Q $ kG is supported at points $ A $, $ B $, $ C $ lying on the circumference of the plate, with $ AC = BC $ and $ ACB = 2\alpha $. What weight $ x $ kG must be placed on the plate at the other end $ D $ of the diameter drawn from point $ C $ so that the pressure of the plate on the support at $ C $G is equal to zero?

Let $ABC$ be an acute triangle with $| AB | > | AC |$. Let $D$ be the foot of the altitude from $A$ to $BC$, let $K$ be the intersection of $AD$ with the internal bisector of angle $B$, Let $M$ be the foot of the perpendicular from $B$ to $CK$ (it could be in the extension of segment $CK$) and$ N$ the intersection of $BM$ and $AK$ (it could be in the extension of the segments). Let $T$ be the intersection of$ AC$ with the line that passes through $N$ and parallel to $DM$. Prove that $BM$ is the internal bisector of the angle $\angle TBC$

A Pokemon Go player starts at $(0,0)$ and carries a pedometer that records the number of steps taken. He then takes steps with length $1$ unit in the north, south, east, or west direction, such that each move after the first is perpendicular to the move before it. Somehow, the player eventually returns to $(0, 0)$, but he had visited no point (except $(0, 0)$) twice. Let $n$ be the number on the pedometer when the player returns to $(0, 0)$. Of the numbers from $1$ to $2019$ inclusive, how many can be the value of $n$?

Find the number of nondegenerate triangles whose vertices lie in the set of points $(s,t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, $s$ and $t$ are integers.

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]

Let $a_1, a_2, \cdots, a_{2022}$ be nonnegative real numbers such that $a_1+a_2+\cdots +a_{2022}=1$. Find the maximum number of ordered pairs $(i, j)$, $1\leq i,j\leq 2022$, satisfying $$a_i^2+a_j\ge \frac 1{2021}.$$

1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer. 2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$.