Given $a$ and $b$ distinct positive integers, show that the system of equations $x y +zw = a$ $xz + yw = b$ has only finitely many solutions in integers $x, y, z,w$.

We construct similar isosceles triangles on the sides of the triangle $ ABC $: triangle $ APB $ outside the triangle $ ABC $ ($ AP = PB $), triangle $ CQA $ outside the triangle $ ABC $ ($ CQ = QA $), triangle $ CRB $ inside the triangle $ ABC $ ($ CR = RB $). Prove that $ APRQ $ is a parallelogram or that the points $ A, P, R, Q $ lie on a straight line.

Let $P = \{p_1,p_2,\ldots, p_{10}\}$ be a set of $10$ different prime numbers and let $A$ be the set of all the integers greater than $1$ so that their prime decomposition only contains primes of $P$. The elements of $A$ are colored in such a way that: [list] [*] each element of $P$ has a different color, [*] if $m,n \in A$, then $mn$ is the same color of $m$ or $n$, [*] for any pair of different colors $\mathcal{R}$ and $\mathcal{S}$, there are no $j,k,m,n\in A$ (not necessarily distinct from one another), with $j,k$ colored $\mathcal{R}$ and $m,n$ colored $\mathcal{S}$, so that $j$ is a divisor of $m$ and $n$ is a divisor of $k$, simultaneously. [/list] Prove that there exists a prime of $P$ so that all its multiples in $A$ are the same color.

Let $k$ be a circle with centre $O$. Let $AB$ be a chord of this circle with midpoint $M\neq O$. The tangents of $k$ at the points $A$ and $B$ intersect at $T$. A line goes through $T$ and intersects $k$ in $C$ and $D$ with $CT < DT$ and $BC = BM$. Prove that the circumcentre of the triangle $ADM$ is the reflection of $O$ across the line $AD$.

In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$[i]-clique[/i]. If a vertex is connected with all other vertices in the graph, we call it a [i]central[/i] vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that [b](1)[/b] $G$ does not contain a $(k+1)$-[i]clique[/i]; [b](2)[/b] if we add an arbitrary edge to $G$, that creates a $(k+1)$-[i]clique[/i]. Find the least possible number of [i]central[/i] vertices in $G$.

Let $AB$ be a triangle and $\Gamma$ the excircle, relative to the vertex $A$, with center $D$. The circle $\Gamma$ is tangent to the lines $AB$ and $AC$ at $E$ and $F$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with $BD$ and $CD$, respectively. If $O$ is the point of intersection of $BQ$ and $CP$, show that the distance from $O$ to the line $BC$ is equal to the radius of the inscribed circle in the triangle $ABC$.

the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide. find the number of all possible codes(in terms of $n$).

What is the least possible value of $$(x+1)(x+2)(x+3)(x+4)+2019$$where $x$ is a real number? $\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$

Suppose in a triangle $\triangle ABC$, $A$ , $B$ , $C$ are the three angles and $a$ , $b$ , $c$ are the lengths of the sides opposite to the angles respectively. Then prove that if $sin(A-B)= \frac{a}{a+b}\sin A \cos B - \frac{b}{a+b}\sin B \cos A$ then the triangle $\triangle ABC$ is isoscelos.

A scalene triangle $ABC$ is inscribed in a circle $\Gamma$. The bisector of angle $A$ meets $BC$ at $E$. Let $M$ be the midpoint of the arc $BAC$. The line $ME$ intersects $\Gamma$ again at $D$. Show that the circumcentre of triangle $AED$ coincides with the intersection point of the tangent to $\Gamma$ at $D$ and the line $BC$.

Let $\omega$ be a circle with center $O$ and a non-diameter chord $BC$ of $\omega$. A point $A$ varies on $\omega$ such that $\angle BAC<90^{\circ}$. Let $S$ be the reflection of $O$ through $BC$. Let $T$ be a point on $OS$ such that the bisector of $\angle BAC$ also bisects $\angle TAS$. 1. Prove that $TB=TC=TO$. 2. $TB, TC$ cut $\omega$ the second times at points $E, F$, respectively. $AE, AF$ cut $BC$ at $M, N$, respectively. Let $SM$ intersects the tangent line at $C$ of $\omega$ at $X$, $SN$ intersects the tangent line at $B$ of $\omega$ at $Y$. Prove that the bisector of $\angle BAC$ also bisects $\angle XAY$.

Over a period of $k$ consecutive days, a total of $2014$ babies were born in a certain city, with at least one baby being born each day. Show that: (a) If $1014 < k \le 2014$, there must be a period of consecutive days during which exactly $100$ babies were born. (b) By contrast, if $k = 1014$, such a period might not exist.

A ladder is formed by removing some consecutive unit squares of a $10\times 10$ chessboard such that for each $k-$th row ($k\in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. How many rectangles formed by composition of unit squares does the ladder have? $ \textbf{(A)}\ 625 \qquad\textbf{(B)}\ 715 \qquad\textbf{(C)}\ 1024 \qquad\textbf{(D)}\ 1512 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $m,n\ge2$ be positive integers. On an $m\times n$ chessboard, some unit squares are occupied by rooks such that each rook attacked by odd number of other rooks. Determine the maximum number of rooks that can be placed on the chessboard.

A convex quadrilateral $ABCD$ has right angles at $A$ and $C$. A point $E$ lies on the extension of the side $AD$ beyond $D$ so that$\angle ABE =\angle ADC$. The point $K$ is symmetric to the point $C$ with respect to point $A$. Prove that$\angle ADB =\angle AKE$ .

Let be three complex numbers $ z,t,u, $ whose affixes in the complex plane form a triangle $ \triangle . $ [b]a)[/b] Let be three non-complex numbers $ a,b,c $ that sum up to $ 0. $ Prove that $$ |az+bt+cu|=|at+bu+cz|=|au+bz+ct| $$ if $ \triangle $ is equilateral. [b]b)[/b] Show that $ \triangle $ is equilateral if $$ |z+2t-3u|=|t+2u-3z|=|u+2z-3t| . $$

The triangle $ABC$ is given. On the arc $BC$ of its circumscribed circle, which does not contain point $A$, the variable point $X$ is selected, and on the rays $XB$ and $XC$, the variable points $Y$ and $Z$, respectively, so that $XA = XY = XZ$. Prove that the line $YZ$ passes through a fixed point. [i]Proposed by A. Kuznetsov[/i]

Given a triangle $ABC$. Construct a point $D$ on the side $AB$ and point $E$ on the side $AC$ so that $BD = CE$ and $\angle ADC = \angle BEC$

Let $a,b$ be positive real numbers satisfying $2ab=a-b$. Denote for any positive integer $k$ $x_k$ and $y_k$ to be the closest integer to $ak$ and $bk$, respectively (if there are two closest integers, choose the larger one). Prove that any positive integer $n$ appears in the sequence $(x_k)_{k\ge 1}$ if and only if it appears at least three times in the sequence $(y_k)_{k\ge 1}$.

On the planet Brick, which has the shape of a rectangular parallelepiped with edges of $1$ km,$ 2$ km and $4$ km, the Little Prince built a house in the center of the largest face. What is the distance from the house to the most remote point on the planet? (The distance between two points on the surface of a planet is defined as the length of the shortest path along the surface connecting these points.)

Calculate the following indefinite integrals. [1] $\int \frac{dx}{e^x-e^{-x}}$ [2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$ [3] $\int (3^x-2)^2 dx$ [4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$ [5] $\int \frac{dx}{1-\cos x}dx$

An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.

On a table there are 2004 boxes, and in each box a ball lies. I know that some the balls are white and that the number of white balls is even. In each case I may point to two arbitrary boxes and ask whether in the box contains at least a white ball lies. After which minimum number of questions I can indicate two boxes for sure, in which white balls lie?

Let $ABCD$ be a cyclic quadrilateral. The diagonals $AC$ and $BD$ meet at $P$, and $DA $ and $CB$ meet at $Q$. Suppose $PQ$ is perpendicular to $AC$. Let $E$ be the midpoint of $AB$. Prove that $PE$ is perpendicular to $BC$.

The water city of Platense consists of many platforms and bridges between them. Each bridge connects two platforms and there is not two bridges connecting the same two platforms. The mayor wants to switch some bridges by a series of moves in the following way: if there are three platforms $A,B,C$ and bridges $AB$ and $AC$ ([b]no[/b] bridge $BC$), he can switch bridge $AB$ to a bridge $BC$. A configuration of bridges is [i]good[/i] if it is possible to go to any platfom from any platform using only bridges. Starting in a good configuration, prove that the mayor can reach any other good configuration, whose the quantity of bridges is the same, using the allowed moves.