In the figure, the triangles $ABC$ and $CDE$ are equilateral, with side lengths $1$ and $4$, respectively. Moreover, $B$, $C$ and $D$ are collinear and $F$ and $G$ are midpoints of $BC$ and $CD$, respectively. Let $P$ be the intersection point of $AF$ and $BE$. Determine the area of the shaded triangle $BPG$. [img]https://fv5-4.failiem.lv/thumb_show.php?i=qmpfykxcek&view&v=1&PHPSESSID=1f433228a75b4117c35f707722c547c423d3d671[/img]

Let $a, b, c$ be the lengths of the sides of a triangle, and $h_a, h_b, h_c$ the lengths of the heights corresponding to the sides $a, b, c,$ respectively. If $t \geq \frac{1} {2}$ is a real number, show that there is a triangle with sidelengths $$ t\cdot a + h_a, \ t\cdot b + h_b , \ t\cdot c + h_c.$$

Find the maximum value of $x+y$, given $ x^2 + y^2 - 3y - 1 = 0 $.

An Olympiad has 99 tasks. Several participants of the Olympiad are standing in a circle. They all solved different sets of tasks. Any two participants standing side by side do not have a common solved problem, but have a common unsolved one. Prove that the number of participants in the circle does not exceed \[2^{99}-\binom{99}{50}.\]

Let G, O, D, I, and T be digits that satisfy the following equation: \begin{tabular}{ccccc} &G&O&G&O\\ +&D&I&D&I\\ \hline G&O&D&O&T \end{tabular} (Note that G and D cannot be $0$, and that the five variables are not necessarily different.) Compute the value of GODOT.

Find all integers $n \geq 1$ such that for all $x_1,...,x_n \in \mathbb{R}$ the following inequality is satisfied $$\left(\frac{x_1^n+...+x_n^n}{n}-x_1....x_n\right)\left(x_1+...+x_n\right) \geq 0$$

Let $n$ be a positive integer. A pair of $n$-tuples $(a_1,\cdots{}, a_n)$ and $(b_1,\cdots{}, b_n)$ with integer entries is called an [i]exquisite pair[/i] if $$|a_1b_1+\cdots{}+a_nb_n|\le 1.$$ Determine the maximum number of distinct $n$-tuples with integer entries such that any two of them form an exquisite pair. [i]Pakawut Jiradilok and Warut Suksompong, Thailand[/i]

Let $x_1,x_2,\cdots,x_n$ be positive real numbers such that $x_1+x_2+\cdots+x_n=1$ $(n\ge 2)$. Prove that\[\sum_{i=1}^n\frac{x_i}{x_{i+1}-x^3_{i+1}}\ge \frac{n^3}{n^2-1}.\]here $x_{n+1}=x_1.$

Show that for real variables $1 \leq a, b \leq 2$ the following inequality holds. $$2(a+b)^2 \leq 9ab $$

An ant walks across the floor of a circular path of radius $r$ and moves in a straight line, but sometimes stops. Each time it stops, before resuming the march, it rotates $60^o$ alternating the direction (if the last time it turned $60^o$ to its right, the next one does it $60^o$ to its left, and vice versa). Find the maximum possible length of the path the ant goes through. Prove that the length found is, in fact, as long as possible. Figure: turn $60^o$ to the right .

Bokos tribus have $2021$ closed chests, we know that every chest have some amount of rupias and some amount of diamonts. They are going to do a deal with Link, that consits that Link will stay with a amount of chests and Bokos with the rest. Before opening the chests, Link has to say the amount of chest that he will stay with. After this the chests open and Link has to choose the chests with the amount that he previously said. Link doesn´t want to make Bokos angry so he wants to say the smallest number of chest that he will stay with, but guaranteeing that he stay with at least with the half of diamonts, and at least the half of the rupias. What number does Link needs to say?

Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$.

Kevin has a set $S$ of $2014$ points scattered on an infinitely large planar gameboard. Because he is bored, he asks Ashley to evaluate \[ x = 4f_4 + 6f_6 + 8f_8 + 10f_{10} + \cdots \] while he evaluates \[ y = 3f_3 + 5f_5+7f_7+9f_9 + \cdots, \] where $f_k$ denotes the number of convex $k$-gons whose vertices lie in $S$ but none of whose interior points lie in $S$. However, since Kevin wishes to one-up everything that Ashley does, he secretly positions the points so that $y-x$ is as large as possible, but in order to avoid suspicion, he makes sure no three points lie on a single line. Find $\left\lvert y-x \right\rvert$. [i]Proposed by Robin Park[/i]

Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$. (Here we always assume that an angle bisector is a ray.) [i]Proposed by Sergey Berlov, Russia[/i]

For each integer $n\geq0$, let $S(n)=n-m^2$, where $m$ is the greatest integer with $m^2\leq n$. Define a sequence by $a_0=A$ and $a_{k+1}=a_k+S(a_k)$ for $k\geq0$. For what positive integers $A$ is this sequence eventually constant?

Minimal distance of a finite set of different points in space is length of the shortest segment, whose both ends belong to this set and segment has length greater than $0$. a) Prove there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,15R$. b) Does there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,2R$?

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

We have a large supply of black, white, red and green hats. And we want to give $8$ of these hats to $8$ students that are sitting around a round table. Find the number of ways of doing that in each of these cases (assuming for the purposes of this problem that students will notchange their places, and that hats of the same color are identical) a) Each hat to be used must be either red or green. b) Exactly two hats of each color are to be used c) Exactly two hats of each color are to be used, and every two hats of the same color are to be given to two adjacent students. d) Exactly two hats of each color are to be used, and no two hats of the same color are to be given to two adjacent students. e) There are no restrictions on the number of hats of each color that are to be used, but no two hats of the same color are to be given to two adjacent students.

Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of \[AB^2 + 2AC^2 - 3AD^2.\]

There are $2019$ points given in the plane. A child wants to draw $k$ (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal $k$, such that for any initial configuration of points it is possible to draw $k$ discs with the above property?

Prove such $a$, that all the numbers $\cos a, \cos 2a, \cos 4a, ... , \cos (2^na)$ are negative.

Equilateral $\triangle ABC$ with side length $14$ is rotated about its center by angle $\theta$, where $0 < \theta < 60^{\circ}$, to form $\triangle DEF$. The area of hexagon $ADBECF$ is $91\sqrt{3}$. What is $\tan\theta$? [asy] defaultpen(fontsize(13)); size(200); pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C; draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F)); [/asy] $\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~\displaystyle\frac{5\sqrt{3}}{11}\qquad\textbf{(C)}~\displaystyle\frac{4}{5}\qquad\textbf{(D)}~\displaystyle\frac{11}{13}\qquad\textbf{(E)}~\displaystyle\frac{7\sqrt{3}}{13}$

A set $M$ consists of $n$ elements. Find the greatest $k$ for which there is a collection of $k$ subsets of $M$ such that for any subsets $A_{1},...,A_{j}$ from the collection, there is an element belonging to an odd number of them

Triangle $ABC$ has side length $AB = 5$, $BC = 12$, and $CA = 13$. Circle $\Gamma$ is centered at point $X$ exterior to triangle $ABC$ and passes through points $A$ and $C$. Let $D$ be the second intersection of $\Gamma$ with segment $\overline{BC}$. If $\angle BDA = \angle CAB$, the radius of $\Gamma$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]2016 CCA Math Bonanza Lightning #3.3[/i]

Let $A = \{1,2,3,4,5,6,7,8\}$, $B = \{9,10,11,12,13,14,15,16\}$ and $C =\{17,18,19,20,21,22,23,24\}$. Find the number of triples $(x, y, z)$ such that $x \in A, y \in B, z \in C $ and $x + y + z = 36$.