Let $ABC$ be a triangle such that the altitude $CH$ and the sides $CA,CB$ are respectively equal to a side and two distinct diagonals of a regular heptagon. Prove that $\angle ACB<120^\circ$.

Determine the number of digits $1$ in the integer part of $\frac{10^{1992}}{10^{83}+7}$.

Given three distinct real numbers $a$, $b$, and $c$, show that at least two of the three following equations \[(x-a)(x-b)=x-c\] \[(x-c)(x-b)=x-a\] \[(x-c)(x-a)=x-b\] have real solutions.

Let $O_1$ and $O_2$ be concentric circles with radii 4 and 6, respectively. A chord $AB$ is drawn in $O_1$ with length $2$. Extend $AB$ to intersect $O_2$ in points $C$ and $D$. Find $CD$.

Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.

Given a circle of radius $R$. Find the ratio of the largest area of ​​the circumscribed quadrilateral to the smallest area of ​​the inscribed one.

Let $p$ be an odd prime. Suppose that positive integers $a,b,m,r$ satisfy $p\nmid ab$ and $ab > m^2$. Prove that there exists at most one pair of coprime positive integers $(x,y)$ such that $ax^2+by^2=mp^r$.

Two persons are playing the following game on a $n\times m$ table, with drawn lines: Person $\#1$ starts the game. Each person in their move, folds the table on one of its lines. The one that could not fold the table on their turn loses the game. Who has a winning strategy?

The sum of all terms of a finite arithmetical progression of integers is a power of two. Prove that the number of terms is also a power of two.

Given a positive integer $N$. Prove that \[\sum_{m=1}^N \sum_{n=1}^N \frac{1}{mn^2+m^2n+2mn}<\frac{7}{4}.\] [i]Proposed by tan-1[/i]

Determine the integers $k\geq 2$ for which the sequence $\Big\{ \binom{2n}{n} \pmod{k}\Big\}_{n\in \mathbb{Z}_{\geq 0}}$ is eventually periodic.

In triangle $ABC$ let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$. [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]Triangle $ABC$ is acute.[/list][/hide]

Let $ABC$ be an acute scalene triangle with altitudes $AE$ $(E \in BC)$ and $BD$ $(D \in AC)$. Point $M$ lies on $AC$, such that $AM = AE$ and $C,A$ and $M$ lie in this order. Point $L$ lies on $BC$, such that $BL=BD$ and $C,B$ and $L$ lie in this order. Let $P$ be the midpoint of $DE$. Prove that $EM,DL$ and the perpendicular from $P$ to $AB$ are concurrent.

Let $n$ be a positive integer; and $S(n)$ denote the sum of all digits in the decimal representation of $n$. A positive integer obtained by removing one or several digits from the right hand end of the decimal representation of $n$ is called the [i]truncation[/i] of $n$. The sum of all truncations of $n$ is denoted as $T(n)$. Prove that $S(n)+9T(n)=n$

Let $N$ be the number of $105$-digit positive integers that contain the digit 1 an odd number of times. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Harry Kim[/i]

For a triangle $ ABC $, let $ B_1 ,C_1 $ be the excenters of $ B, C $. Line $B_1 C_1 $ meets with the circumcircle of $ \triangle ABC $ at point $ D (\ne A) $. $ E $ is the point which satisfies $ B_1 E \bot CA $ and $ C_1 E \bot AB $. Let $ w $ be the circumcircle of $ \triangle ADE $. The tangent to the circle $ w $ at $ D $ meets $ AE $ at $ F $. $ G , H $ are the points on $ AE, w $ such that $ DGH \bot AE $. The circumcircle of $ \triangle HGF $ meets $ w $ at point $ I ( \ne H ) $, and $ J $ be the foot of perpendicular from $ D $ to $ AH $. Prove that $ AI $ passes the midpoint of $ DJ $.

Let ABC be a triangle with$|AB|< |AC|. $ Let $k$ be the circumcircle of $\triangle ABC$ and let $O$ be the center of $k$. Point $M$ is the midpoint of the arc $BC $ of $k$ not containing $A$. Let $D $ be the second intersection of the perpendicular line from $M$ to $AB$ with $ k$ and $E$ be the second intersection of the perpendicular line from $M$ to $AC $ with $k$. Points $X $and $Y $ are the intersections of $CD$ and $BE$ with $OM$ respectively. Denote by $k_b$ and $k_c$ circumcircles of triangles $BDX$ and $CEY$ respectively. Let $G$ and $H$ be the second intersections of $k_b$ and $k_c $ with $AB$ and $AC$ respectively. Denote by ka the circumcircle of triangle $AGH.$ Prove that $O$ is the circumcenter of $\triangle O_aO_bO_c, $where $O_a, O_b, O_c $ are the centers of $k_a, k_b, k_c$ respectively.

Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.

[b]3.[/b] Denoting by $E$ the class of trigonometric polynomials of the form $f(x)=c_{0}+c_{1}cos(x)+\dots +c_{n} cos(nx)$, where $c_{0} \geq c_{1} \geq \dots \geq c_{n}>0$, prove that $(1-\frac{2}{\pi})\frac{1}{n+1}\leq min_{{f\epsilon E}}( \frac{max_{\frac{\pi}{2}\leq x\leq \pi} \left | f(x) \right |}{max_{0\leq x\leq 2\pi} \left | f(x) \right |})\leq (\frac{1}{2}+\frac{1}{\sqrt{2}})\frac{1}{n+1}$. [b](S. 24)[/b]

Show that points $A, B, C$ and $D$ can be placed on the plane in such a way that the quadrilateral $ABCD$ has an area which is twice the area of the quadrilateral $ADBC.$

Show that for any four positive real numbers $ a,b,c,d $ and four negative real numbers $ e,f,g,h, $ the terms $ ae+bc,ef+cg,fd+gh,da+hb $ are not all positive.

A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared. Determine all possible first moves of the first player after which he has a winning strategy. [i]Proposed by Ilya Bogdanov & Vladimir Bragin, Russia[/i]

The director of a Zoo has bought eight elephants numbered by $1, 2, \ldots , 8$. He has forgotten their masses but he remembers that each elephant starting with the third one has the mass equal to the sum of the masses of two preceding ones. Suddenly the director hears a rumor that one of the elephants has lost his mass. How can the director perform two weightings on balancing scales without weights to either find this elephant or make sure that this was just a rumor? (It is known that no elephant gained mass and no more than one elephant lost mass.) [i]Alexandr Gribalko[/i]

For every natural number $n$, find all polynomials $x^2+ax+b$, where $a^2 \geq 4b$, that divide $x^{2n} + ax^n + b$.

In a $3$ by $3$ table, by a $k$-worm, we mean a path of different cells $(S_1,S_2,...,S_k)$ such that each two consecutive cells have one side in common. The $k$-worm at each steep can go one cell forward and turn to the $(S,S_1,...,S_{k-1})$ if $S$ is an unfilled cell which is adjacent (has one side in common) with $S_1$. Find the maximum number of $k$ such that there is a $k$-worm $(S_1,...,S_k)$ such that after finitly many steps can be turned to $(S_k,...,S_1)$.