Find all the pairs of integers $(m,n)$ satisfying the equality $3(m^2+n^2)-7(m+n)=-4$

Let $a, b, c$ be positive real numbers. Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}\geq\sqrt{\frac{2ab}{3a+b+2c}}+\sqrt{\frac{2bc}{3b+c+2a}}+\sqrt{\frac{2ca}{3c+a+2b}}.$$

A positive integer is called [i]good [/i] if it can be written as the sum of two distinct integer squares. A positive integer is called [i]better [/i]if it can be written in at least two was as the sum of two integer squares. A positive integer is called [i]best [/i] if it can be written in at least four ways as the sum of two distinct integer squares. a) Prove that the product of two good numbers is good. b) Prove that $ 5$ is good, $ 2005$ is better, and $ 2005^2$ is best.

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

Find positive integers $x, y, z$ such that $x > z > 1999 \cdot 2000 \cdot 2001 > y$ and $2000x^{2}+y^{2}= 2001z^{2}.$

For which pair $(A,B)$, \[ x^2+xy+y=A \\ \frac{y}{y-x}=B \] has no real roots? $ \textbf{(A)}\ (1/2,2) \qquad\textbf{(B)}\ (-1,1) \qquad\textbf{(C)}\ (\sqrt 2, \sqrt 2) \qquad\textbf{(D)}\ (1,1/2) \qquad\textbf{(E)}\ (2,2/3) $

Let $a_0=1$ and for all $n\ge 1$ let $a_n$ be the smaller root of the equation $$4^{-n}x^2-x+a_{n-1} = 0.$$ Given that $a_n$ approaches a value $L$ as $n$ goes to infinity, what is the value of $L$?

Show that for every convex polygon whose area is less than or equal to $1$, there exists a parallelogram with area $2$ containing the polygon.

Let $ABCD$ be a trapezoid such that $|AB|=9$, $|CD|=5$ and $BC\parallel AD$. Let the internal angle bisector of angle $D$ meet the internal angle bisectors of angles $A$ and $C$ at $M$ and $N$, respectively. Let the internal angle bisector of angle $B$ meet the internal angle bisectors of angles $A$ and $C$ at $L$ and $K$, respectively. If $K$ is on $[AD]$ and $\dfrac{|LM|}{|KN|} = \dfrac 37$, what is $\dfrac{|MN|}{|KL|}$? $ \textbf{(A)}\ \dfrac{62}{63} \qquad\textbf{(B)}\ \dfrac{27}{35} \qquad\textbf{(C)}\ \dfrac{2}{3} \qquad\textbf{(D)}\ \dfrac{5}{21} \qquad\textbf{(E)}\ \dfrac{24}{63} $

Higher Secondary P7 If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.

Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.

The pentagon $ABCDE$ is inscribed in the circle. Line segments $AC$ and $BD$ intersect at point $K$. Line segment $CE$ touches the circumcircle of triangle $ABK$ at point $N$. Find the angle $CNK$ if $\angle ECD = 40^o.$

Let $A,B,C$ be points on $[OX$ and $D,E,F$ be points on $[OY$ such that $|OA|=|AB|=|BC|$ and $|OD|=|DE|=|EF|$. If $|OA|>|OD|$, which one below is true? $\textbf{(A)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)>\text{Area}(DBF)$ $\textbf{(B)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)=\text{Area}(DBF)$ $\textbf{(C)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)<\text{Area}(DBF)$ $\textbf{(D)}$ If $m(\widehat{XOY})<45^\circ$ then $\text{Area}(AEC)<\text{Area}(DBF)$, and if $45^\circ < m(\widehat{XOY})<90^\circ$ then $\text{Area}(AEC)>\text{Area}(DBF)$ $\textbf{(E)}$ None of above

Let $\mathcal{C}$ be a family of subsets of $A=\{1,2,\dots,100\}$ satisfying the following two conditions: 1) Every $99$ element subset of $A$ is in $\mathcal{C}.$ 2) For any non empty subset $C\in\mathcal{C}$ there is $c\in C$ such that $C\setminus\{c\}\in \mathcal{C}.$ What is the least possible value of $|\mathcal{C}|$?

Let $x_1<x_2< \ldots <x_{2024}$ be positive integers and let $p_i=\prod_{k=1}^{i}(x_k-\frac{1}{x_k})$ for $i=1,2, \ldots, 2024$. What is the maximal number of positive integers among the $p_i$?

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

The sides of a triangle have lengths of $ 15$, $ 20$, and $ 25$. Find the length of the shortest altitude. $ \text{(A)}\ 6 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 12.5 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$

Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.

Define a sequence $(a_n)$ recursively by $a_1=0, a_2=2, a_3=3$ and $a_n=\max_{0<d<n} a_d \cdot a_{n-d}$ for $n \ge 4$. Determine the prime factorization of $a_{19702020}$.

Suppose that $p,p+d,p+2d,p+3d,p+4d$, and $p+5d$ are six prime numbers, where $p$ and $d$ are positive integers. Show that $d$ must be divisible by $2,3,$ and $5$.

[b]7.[/b] For the differential equation $ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}= 2\frac{\partial^2 u}{\partial x \partial y} $ find all solutions of the form $u(x,y)=f(x)g(y)$. [b](R. 14)[/b]

On $\triangle ABC$, let $M$ the midpoint of $AB$ and $N$ the midpoint of $CM$. Let $X$ a point such that $\angle XMC=\angle MBC$ and $\angle XCM=\angle MCB$ with $X,B$ in opposite sides of line $CM$. Let $\Omega$ the circumcircle of triangle $\triangle AMX$ [b]a)[/b] Show that $CM$ is tangent to $\Omega$ [b]b)[/b] Show that the lines $NX$ and $AC$ meet at $\Omega$

Find all pairs of integers $ (x,y)$ satisfying $ 1\plus{}1996x\plus{}1998y\equal{}xy.$

$ 100\times 19.98\times 1.998\times 1000\equal{} $ $ \text{(A)}\ (1.998)^{2}\qquad\text{(B)}\ (19.98)^{2}\qquad\text{(C)}\ (199.8)^{2}\qquad\text{(D)}\ (1998)^{2}\qquad\text{(E)}\ (19980)^{2} $

Consider a regular 2n-gon $ P$,$A_1,A_2,\cdots ,A_{2n}$ in the plane ,where $n$ is a positive integer . We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$ .We color the sides of $P$ in 3 different colors (ignore the vertices of $P$,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover ,from every point in the plane external to $P$ , points of most 2 different colors on $P$ can be seen .Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently). [i]Proposed by Viktor Simjanoski, Macedonia[/i] JBMO 2017, Q4