Let $\triangle ABC$ be equilateral. Let $D$ be the midpoint of side $AC,$ and let $DEFG$ be a square such that $D, F, B$ are collinear and $E,G$ lie on $AB,CB$ respectively. What fraction of the area of $\triangle ABC$ is covered by square $DEFG?$ [i]Proposed by Lohith Tummala[/i]

Consider the triangle $ ABC$ with side-lenghts equal to $ a,b,c$. Let $ p\equal{}\frac{a\plus{}b\plus{}c}{2}$, $ R$-the radius of circumcircle of the triangle $ ABC$, $ r$-the radius of the incircle of the triangle $ ABC$ and let $ l_a,l_b,l_c$ be the lenghts of bisectors drawn from $ A,B$ and $ C$, respectively, in the triangle $ ABC$. Prove that: $ l_al_b\plus{}l_bl_c\plus{}l_cl_a\leq p\sqrt{3r^2\plus{}12Rr}$ [i]Proposer[/i]: [b]Baltag Valeriu[/b]

Let $M=(0,1)\cap \mathbb Q$. Determine, with proof, whether there exists a subset $A\subset M$ with the property that every number in $M$ can be uniquely written as the sum of finitely many distinct elements of $A$.

Let \(c\) be a positive real number. Alice wishes to pick an integer \(n\) and a sequence \(a_1\), \(a_2\), \(\ldots\) of distinct positive integers such that \(a_{i} \leq ci\) for all positive integers \(i\) and \[n, \qquad n + a_1, \qquad n + a_1 - a_2, \qquad n + a_1 - a_2 + a_3, \qquad \cdots\] is a sequence of distinct nonnegative numbers. Find all \(c\) such that Alice can fulfil her wish.

A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\frac{\sqrt{3}}{3}x$. The three lines create an equilateral triangle. What is the perimeter of the triangle? $ \textbf{(A) }2\sqrt{6}\qquad\textbf{(B) }2+2\sqrt{3}\qquad\textbf{(C) }6\qquad\textbf{(D) }3+2\sqrt{3}\qquad\textbf{(E) }6+\frac{\sqrt{3}}{3} $

Prove that for any real numbers $ x_1, x_2, \ldots, x_{1981} $, $ y_1, y_2, \ldots, y_{1981} $ such that $ \sum_{j=1}^{1981} x_j = 0 $, $ \sum_{j=1}^{1981} y_j = 0 $ the inequality occurs $$ \sqrt{\sum_{j=1}^{1981} (x_j^2+y_j^2)} \leq \frac{1}{\sqrt{2}} \sum_{j=1}^{1981} \sqrt{x_j^2+y_j^2}.$$

A positive integer is a non-trivial perfect power if it can be expressed as $n^k$ where $n$ and $k$ are positive integers and $k>1$. Show that there exist arbitrarily large consecutive square numbers with no other non-trivial perfect powers between them.

Let $ m \equal{} 2007^{2008}$, how many natural numbers n are there such that $ n < m$ and $ n(2n \plus{} 1)(5n \plus{} 2)$ is divisible by $ m$ (which means that $ m \mid n(2n \plus{} 1)(5n \plus{} 2)$) ?

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation $$x(x-y)+y(y-z)+z(z-x) = 1?$$$\textbf{(A)}\: x>y$ and $y=z$ $\textbf{(B)}\: x=y-1$ and $y=z-1$ $\textbf{(C)} \: x=z+1$ and $y=x+1$ $\textbf{(D)} \: x=z$ and $y-1=x$ $\textbf{(E)} \: x+y+z=1$

In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements: i) $P$ is the circumcentre of triangle $ABC$. ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$. iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$. Prove that $S$ is the incentre of triangle $ABC$.

66 dwarfs have a total of 111 hats. Each of the hats belongs to a dwarf and colored by 66 different colors. Festivities are organized where each of these dwarfs wears their own hat. There is no dwarf pair wearing the same colored hat in any of the festivities. For any two of the festivities, there exist a dwarf wearing a hat of a different color in these festivities. Find the maximum value of the number of festivities that can be organized.

In quadrilateral $ABCD$, $AC = 5$, $CD = 7$, and $AD = 3$. The angle bisector of $\angle CAD$ intersects $CD$ at $E$. If $\angle CBD = 60^o$ and $\angle AED = \angle BEC$, compute the value of $AE + BE$.

$3n-1$ points are given in the plane, no three are collinear. Prove that one can select $2n$ of them whose convex hull is not a triangle.

All of the digits of a seven-digit positive integer are either $7$ or $8.$ If this integer is divisible by $9,$ what is the sum of its digits?

We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. Prove that $F_{n-1}F_{n}F_{n+1}L_{n-1}L_{n}L_{n+1}(n \geq 2)$ is not a perfect square.

Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.

1. Baron Munchhausen was told that some polynomial $P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ is such that $P(x)+P(-x)$ has exactly 45 distinct real roots. Baron doesn't know the value of $n$. Nevertheless he claims that he can determine one of the coefficients $a_{n}, \ldots, a_{1}, a_{0}$ (indicating its position and value). Isn't Baron mistaken? Boris Frenkin

In $\triangle ABC$, $CA=1960\sqrt{2}$, $CB=6720$, and $\angle C = 45^{\circ}$. Let $K$, $L$, $M$ lie on $BC$, $CA$, and $AB$ such that $AK \perp BC$, $BL \perp CA$, and $AM=BM$. Let $N$, $O$, $P$ lie on $KL$, $BA$, and $BL$ such that $AN=KN$, $BO=CO$, and $A$ lies on line $NP$. If $H$ is the orthocenter of $\triangle MOP$, compute $HK^2$. [hide="Clarifications"] [list] [*] Without further qualification, ``$XY$'' denotes line $XY$.[/list][/hide] [i]Evan Chen[/i]

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB,$ or to their extensions. a) Prove that the tetrahedron $SABC$ is regular. b) Prove conversely that for every regular tetrahedron five such spheres exist.

A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly $1989$ minutes? [asy] draw((0,0)--(20,0), EndArrow); draw((0,0)--(0,25), EndArrow); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,10)--(10,10)--(10,0)--(15,0)--(15,15)--(0,15)--(0,20)--(10,20),linewidth(2)); draw((0,20)--(10,20), EndArrow); draw((3.5,.5)--(4,.5)--(4,2), EndArrow); draw((4,3.5)--(4,4)--(2.5,4), EndArrow); draw((2,5.5)--(1,5.5)--(1,7), EndArrow); draw((1,8)--(1,9)--(2.5,9), EndArrow); draw((8,9.5)--(9,9.5)--(9,8), EndArrow); draw((10.5,2)--(10.5,1)--(12,1), EndArrow); draw((13,.5)--(14,.5)--(14,2), EndArrow); draw((14.5,13)--(14.5,14)--(13,14), EndArrow); draw((2,15.5)--(1,15.5)--(1,17), EndArrow); draw((.5,18)--(.5,19)--(2,19), EndArrow); label("x", (21,0), E); label("y", (0,26), N); label("4", (0,20), W); label("3", (0,15), W); label("2", (0,10), W); label("1", (0,5), W); label("0", (0,0), SW); label("1", (5,0), S); label("2", (10,0), S); label("3", (15,0), S); [/asy] $\textbf{(A)}\ (35,44) \qquad\textbf{(B)}\ (36,45) \qquad\textbf{(C)}\ (37,45) \qquad\textbf{(D)}\ (44,35) \qquad\textbf{(E)}\ (45,36)$

There are several coins in a row, and the [i]allowed move[/i] is to remove exactly one coin from the row, which can either be the first or the last. In the initial distribution, there are $n$ coins with not necessarily equal values. Ana and María alternate turns. Ana starts, making two moves, then María makes one move, then Ana makes two moves, and so on until no coins remain: Ana makes two moves and María makes one. (Only in the last move can Ana take one coin if only one coin is left.) Ana's goal is to ensure she takes at least $\dfrac{2}{3}$ of the total value of the coins. Determine if Ana can achieve her goal with certainty if [list=a] [*]$n=2013$ [*]$n=2014$ [/list] If the answer is yes, provide a strategy to achieve it; if the answer is no, give a specific sequence of coins and explain how María prevents Ana from achieving her goal.

A positive integer $d$ and a permutation of positive integers $a_1,a_2,a_3,\dots$ is given such that for all indices $i\geq 10^{100}$, $|a_{i+1}-a_{i}|\leq 2d$ holds. Prove that there exists infinity many indices $j$ such that $|a_j -j|< d$.

The sequence $a_1, a_2, a_3, ...$ satisfies that $a_1 = 3$, $a_2 = -1$, $a_na_{n-2} +a_{n-1} = 2$ for all $n \ge 3$. Calculate $a_1 + a_2+ ... + a_{99}$.

It is known that the graph of the function $y = f(x)$ after a rotation of $45^o$ around a certain point turns into the graph of the function $y = x^3 + ax^2 + 19x + 97$. At what $a$ is this possible?

A $\textit{cubic sequence}$ is a sequence of integers given by $a_n =n^3 + bn^2 + cn + d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers. $\textbf{(a)}$ Show that there exists a cubic sequence such that the only terms of the sequence which are squares of integers are $a_{2015}$ and $a_{2016}$. $\textbf{(b)}$ Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part $\textbf{(a)}$.