Express $$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^2 n +m n^2 +2mn }$$ as a rational number.

Two unit circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. $M$ is an arbitrary point of $\omega_1$, $N$ is an arbitrary point of $\omega_2$. Two unit circles $\omega_3$ and $\omega_4$ pass through both points $M$ and $N$. Let $C$ be the second common point of $\omega_1$ and $\omega_3$, and $D$ be the second common point of $\omega_2$ and $\omega_4$. Prove that $ACBD$ is a parallelogram.

We are given convex quadrilateral $ABCD$. Each of its sides is divided into $N$ line segments of equal length. The points of division of side $AB$ are connected with the points of division of side $CD$ by straight lines (which we call the first set of straight lines), and the points of division of side BC are connected with the points of division of side $DA$ by straight lines (which we call the second set of straight lines) as shown in the diagram, which illustrates the case $N = 4$. This forms $N^2$ smaller quadrilaterals. From these we choose $N$ quadrilaterals in such a way that any two are at least divided by one line from the first set and one line from the second set. Prove that the sum of the areas of these chosen quadrilaterals is equal to the area of $ABCD$ divided by $N$. (A Andjans, Riga) [img]http://4.bp.blogspot.com/-8Qqk4r68nhU/XVco29-HzzI/AAAAAAAAKgo/UY8mXxg7tD0OrS6bEnoAw7Vuf31BuOE8wCK4BGAYYCw/s1600/TOT%2B1980%2BSpring%2BJ4.png[/img]

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

Which of the following represents the force corresponding to the given potential? [asy] // Code by riben size(400); picture pic; // Rectangle draw(pic,(0,0)--(22,0)--(22,12)--(0,12)--cycle); label(pic,"-15",(2,0),S); label(pic,"-10",(5,0),S); label(pic,"-5",(8,0),S); label(pic,"0",(11,0),S); label(pic,"5",(14,0),S); label(pic,"10",(17,0),S); label(pic,"15",(20,0),S); label(pic,"-2",(0,2),W); label(pic,"-1",(0,4),W); label(pic,"0",(0,6),W); label(pic,"1",(0,8),W); label(pic,"2",(0,10),W); label(pic,rotate(90)*"F (N)",(-2,6),W); label(pic,"x (m)",(11,-2),S); // Tick Marks draw(pic,(2,0)--(2,0.3)); draw(pic,(5,0)--(5,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(11,0)--(11,0.3)); draw(pic,(14,0)--(14,0.3)); draw(pic,(17,0)--(17,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(2,12)--(2,11.7)); draw(pic,(5,12)--(5,11.7)); draw(pic,(8,12)--(8,11.7)); draw(pic,(11,12)--(11,11.7)); draw(pic,(14,12)--(14,11.7)); draw(pic,(17,12)--(17,11.7)); draw(pic,(20,12)--(20,11.7)); draw(pic,(22,2)--(21.7,2)); draw(pic,(22,4)--(21.7,4)); draw(pic,(22,6)--(21.7,6)); draw(pic,(22,8)--(21.7,8)); draw(pic,(22,10)--(21.7,10)); // Paths path A=(0,6)--(5,6)--(5,4)--(11,4)--(11,8)--(17,8)--(17,6)--(22,6); path B=(0,6)--(5,6)--(5,2)--(11,2)--(11,10)--(17,10)--(17,6)--(22,6); path C=(0,6)--(5,6)--(5,5)--(11,5)--(11,7)--(17,7)--(17,6)--(22,6); path D=(0,6)--(5,6)--(5,7)--(11,7)--(11,5)--(17,5)--(17,6)--(22,6); path E=(0,6)--(5,6)--(5,8)--(11,8)--(11,4)--(17,4)--(17,6)--(22,6); draw(A); label("(A)",(9.5,-3),4*S); draw(shift(35*right)*B); label("(B)",(45.5,-3),4*S); draw(shift(20*down)*C); label("(C)",(9.5,-23),4*S); draw(shift(35*right)*shift(20*down)*D); label("(D)",(45.5,-23),4*S); draw(shift(40*down)*E); label("(E)",(9.5,-43),4*S); add(pic); picture pic2=shift(35*right)*pic; picture pic3=shift(20*down)*pic; picture pic4=shift(35*right)*shift(20*down)*pic; picture pic5=shift(40*down)*pic; add(pic2); add(pic3); add(pic4); add(pic5); [/asy]

In the convex pentagon $ABCDE$, the following equalities hold: $\angle CDE=90^\circ$, $AC=AD$, and $BD=BE$. Prove that triangle $ABD$ and quadrilateral $ABCE$ have the same area.

Compute the sum of all integers $n$ such that $n^2-3000$ is a perfect square.

Quadrilateral $ABCD$ is given with $AD \nparallel BC$. The midpoints of $AD$ and $BC$ are denoted by $M$ and $N$, respectively. The line $MN$ intersects the diagonals $AC$ and $BD$ in points $K$ and $L$, respectively. Prove that the circumcircles of the triangles $AKM$ and $BNL$ have common point on the line $AB$.( Proposed by Emil Stoyanov ) [img]http://estoyanov.net/wp-content/uploads/2015/09/est.png[/img]

Let $\vartriangle ABC$ be an equilateral triangle with side length $1$ and let $\Gamma$ the circle tangent to $AB$ and $AC$ at $B$ and $C$, respectively. Let $P$ be on side $AB$ and $Q$ be on side $AC$ so that $PQ // BC$, and the circle through $A, P$, and $Q$ is tangent to $\Gamma$ . If the area of $\vartriangle APQ$ can be written in the form $\frac{\sqrt{a}}{b}$ for positive integers $a$ and $b$, where $a$ is not divisible by the square of any prime, find $a + b$.

Find all functions $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ such that the conditions $\quad a) \quad a-b \mid f(a)-f(b)$ for all $a\neq b$ and $a,b \in \mathbb{Z}^{+}$ $\quad b) \quad f(\varphi(a))=\varphi(f(a))$ for all $a \in \mathbb{Z}^{+}$ where $\varphi$ is the Euler's totient function. holds

Let $ AB$ be the diameter of semicircle $O$ , $C, D $ be points on the arc $AB$, $P, Q$ be respectively the circumcenter of $\triangle OAC $ and $\triangle OBD $ . Prove that:$CP\cdot CQ=DP \cdot DQ$.[asy] import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black; real h=sqrt(55/64); pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B); D(arc(O,1,0,180),darkgreen); D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue); D(O); [/asy]

Let $A$ be a set of $N$ residues $\pmod{N^2}$. Prove that there exists a set $B$ of $N$ residues $\pmod{N^2}$ such that the set $A+B=\{a+b \vert a \in A, b \in B \}$ contains at least half of all the residues $\pmod{N^2}$.

Find all he positive integers $x, y, z, t, w$, such as: $x+\frac{1}{y+\frac{1}{z+\frac{1}{t+\frac{1}{w}}}}=\frac{1998}{115}$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$ satisfy, $$f\left(x+yf(x+y)\right)=y^2+f(x)f(y)$$ [i]Proposed by Dorlir Ahmeti, Kosovo[/i]

Consider a triangle $ABC$. The tangent line to its circumcircle at point $C$ meets line $AB$ at point $D$. The tangent lines to the circumcircle of triangle $ACD$ at points $A$ and $C$ meet at point $K$. Prove that line $DK$ bisects segment $BC$. (F.Ivlev)

For $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n)$. Evaluate $$ 4\sum^{30}_{n=1} n \cdot T(n).$$

If $r$ is a number such that $r^2-6r+5=0$, find $(r-3)^2$

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

What is the largest possible value of the expression $$gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )$$ for naturals $n$? [hide]original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n? [/hide]

Let $ f (n) $ be a function that fulfills the following properties: $\bullet$ For each natural $ n $, $ f (n) $ is an integer greater than or equal to $ 0 $. $\bullet$ $f (n) = 2010 $, if $ n $ ends in $ 7 $. For example, $ f (137) = 2010 $. $\bullet$ If $ a $ is a divisor of $ b $, then: $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $. Find $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $ and justify your answer.

On a blank piece of paper, two points with distance $1$ is given. Prove that one can use (only) straightedge and compass to construct on this paper a straight line, and two points on it whose distance is $\sqrt{2021}$ such that, in the process of constructing it, the total number of circles or straight lines drawn is at most $10.$ Remark: Explicit steps of the construction should be given. Label the circles and straight lines in the order that they appear. Partial credit may be awarded depending on the total number of circles/lines.

Kongol rolls two fair 6-sided die. The probability that one roll is a divisor of the other can be expressed as $\frac{p}{q}$. Determine $p+q$. [i]2022 CCA Math Bonanza Lightning Round 3.1[/i]

Arnaldo selects a nonnegative integer $a$ and Bernaldo selects a nonnegative integer $b$. Both of them secretly tell their number to Cernaldo, who writes the numbers $5$, $8$, and $15$ on the board, one of them being the sum $a+b$. Cernaldo rings a bell and Arnaldo and Bernaldo, individually, write on different slips of paper whether they know or not which of the numbers on the board is the sum $a+b$ and they turn them in to Cernaldo. If both of the papers say NO, Cernaldo rings the bell again and the process is repeated. It is known that both Arnaldo and Bernaldo are honest and intelligent. What is the maximum number of times that the bell can be rung until one of them knows the sum? Personal note: They really phoned it in with the names there…

Given are the non zero natural numbers $a,b,c$ such that the number $\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}$ is rational. Prove that the number $\frac{a^2+b^2+c^2}{a+b+c}$ is an integer .

Determine the smallest integer $n \ge 4$ for which one can choose four different numbers $a, b, c, $ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$ .