Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime

Chandler the Octopus is at a tentacle party! At this party, there is $1$ creature with $2$ tentacles, $2$ creatures with $3$ tentacles, $3$ creatures with $4$ tentacles, all the way up to $14$ creatures with $15$ tentacles. Each tentacle is distinguishable from all other tentacles. For some $2\le m < n \le 15$, a creature with m tentacles “meets” a creature with n tentacles; “meeting” another creature consists of shaking exactly 1 tentacle with each other. Find the number of ways there are to pick distinct $m < n$ between $2$ and $15$, inclusive, and then to pick a creature with $m$ tentacles to “meet” a selected creature with $n$ tentacles. [i]Proposed by Armaan Tipirneni, Richard Chen, and Denise the Octopus[/i]

In the figure below, we know that $AB = CD$ and $BC = 2AD$. Prove that $\angle BAD = 30^o$. [img]https://3.bp.blogspot.com/-IXi_8jSwzlU/W1R5IydV5uI/AAAAAAAAIzo/2sREnDEnLH8R9zmAZLCkVCGeMaeITX9YwCK4BGAYYCw/s400/IGO%2B2015.el3.png[/img]

For how many primes $ p$, there exits unique integers $ r$ and $ s$ such that for every integer $ x$ $ x^{3} \minus{} x \plus{} 2\equiv \left(x \minus{} r\right)^{2} \left(x \minus{} s\right)\pmod p$? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

What is the area of the figure in the complex plane enclosed by the origin and the set of all points $\tfrac{1}{z}$ such that $(1-2i)z+(-2i-1)\overline{z}=6i$?

A rectangle is divided into $9$ small rectangles if by parallel lines to its sides, as shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/e/d/1fd545862a3c7950249ec54a631c74e59fb9ed.png[/img] The four numbers written indicate the areas of the four corresponding rectangles. Prove that the total area of the rectangle is greater than or equal to $90$.

A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.

Two players $A$ and $B$ play the following game. Before the game starts, $A$ chooses 1000 not necessarily different odd primes, and then $B$ chooses half of them and writes them on a blackboard. In each turn a player chooses a positive integer $n$, erases some primes $p_1$, $p_2$, $\dots$, $p_n$ from the blackboard and writes all the prime factors of $p_1 p_2 \dotsm p_n - 2$ instead (if a prime occurs several times in the prime factorization of $p_1 p_2 \dotsm p_n - 2$, it is written as many times as it occurs). Player $A$ starts, and the player whose move leaves the blackboard empty loses the game. Prove that one of the two players has a winning strategy and determine who. Remark: Since 1 has no prime factors, erasing a single 3 is a legal move.

A design $X$ is an array of the digits $1,2,..., 9$ in the shape of an $X$, for example, [img]https://cdn.artofproblemsolving.com/attachments/8/e/d371a2cd442cb7a8784e1cc7635344df722e20.png[/img] We will say that a design $X$ is [i]balanced [/i] if the sum of the numbers of each of the diagonals match. Determine the number of designs $X$ that are balanced.

Let $n$ be a positive integer. On the blackboard all quadratic polynomials with positive integer coefficients, that do not exceed $n$, without real roots are written Find all $n$ for which the number of written polynomials is even [i]A. Voidelevich[/i]

An arbitrary point $M$ is chosen inside the triangle $ABC$. Prove that $MA + MB + MC \le max (AB + BC, BC + AC, AC + AB)$. (N. Sedrakyan)

In the accompanying figure, semicircles with centers$ A$ and $B$ have radii $4$ and $2$, respectively. Furthermore, they are internally tangent to the circle of diameter $PQ$. Also the semicircles with centers $ A$ and $ B$ are externally tangent to each other. The circle with center $C$ is internally tangent to the semicircle with diameter $PQ$ and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center $C$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/281b335f6a2d6230a5b79060e6d85d6ca6f06c.png[/img]

Compute $\sum_{k=0}^{2n} (-1)^k a_k^2$ where $a_k$ are the coefficients in the expansion \[(1- \sqrt 2 x +x^2)^n =\sum_{k=0}^{2n} a_k x^k.\]

Square ABCD has side length 7. Let $A_1$, $B_1$, $C_1$, and $D_1$ be points on rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, where each point is $3$ units from the end of the ray so that $A_1B_1C_1D_1$ forms a second square as shown. SImilarly, let $A_2$, $B_2$, $C_2$, and $D_2$ be points on segments $A_1B_1$, $B_1C_1$, $C_1D_1$, and $D_1A_1$, respectively, forming another square where $A_2$ divides segment $A_1B_1$ into two pieces whose lengths are in the same ratio as $AA_1$ is to $A_1B$. Continue this process to construct square $A_nB_nC_nD_n$ for each positive integer $n$. Find the total of all the perimeters of all the squares. [asy] size(180); pair[] A={(-1,-1),(-1,1),(1,1),(1,-1),(-1,-1)}; string[] X={"A","B","C","D"}; for(int k=0;k<10;++k) { for(int m=0;m<4;++m) { if(k==0) label("$"+X[m]+"$",A[m],A[m]); if(k==1) label("$"+X[m]+"_1$",A[m],A[m]); draw(A[m]--A[m+1]); A[m]+=3/7*(A[m+1]-A[m]); } A[4]=A[0]; }[/asy]

You are given \( 11 \) numbers with an arithmetic mean of \( 10 \). Each of the first \( 4 \) numbers is increased by \( 20 \), and each of the last \( 7 \) numbers is decreased by \( 24 \). What is the arithmetic mean of the new \( 11 \) numbers?

The angle bisectors $AA_1$ and $BB_1$ of the triangle ABC intersect at point $O$. Prove that when the angle $C$ is equal to $60^0$, then $OA_1=OB_1$

The square \[ \begin{tabular}{|c|c|c|} \hline 50&\textit{b}&\textit{c}\\ \hline \textit{d}&\textit{e}&\textit{f}\\ \hline \textit{g}&\textit{h}&2\\ \hline \end{tabular} \]is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of $ g$? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 136$

Denote by $\mathbb Z^2$ the set of all points $(x,y)$ in the plane with integer coordinates.  For each integer $n\geq 0$, let $P_n$ be the subset of $\mathbb Z^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k\leq n$.  Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.

Integers $x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1$ satisfy the inequality $(-1)^{x_k} x_{k-1}x_{k+1} >0$ for $k = 1,2,...,n$. Prove that the difference $\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k|$ is divisible by $4$.

This sequence lists the perfect squares in increasing order: \[0,1,4,9,16,\cdots ,a,10^8,b,\cdots\] Determine the value of $b-a$.

Given a real number $\alpha$, find all pairs $(f,g)$ of functions $f,g :\mathbb{R} \to \mathbb{R}$ such that $$xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) \;\;\;\;\;\;\;\;\;\;\; ,\forall x,y \in \mathbb{R}.$$

Each day John's mother sends him to the store with $\$1$ to buy widgets and gadgets, each of which cost a whole number of cents. On the first day John comes back with $4$ widgets, $5$ gadgets, and $35$ cents in change. On the second day, John comes back with $5$ widgets, $4$ gadgets, and $39$ cents in change. On the third day, John comes back with only $c$ cents in change. He hands his mother the change, telling her that he had tripped coming home and broken all the widgets and gadgets. His mother, thinking for a moment, begins yelling at him for lying, as she noticed that there was no way he could have received exactly $c$ cents in change given the price of widgets and gadgets. What is the sum of the digits of the least possible value of $c$? $\textbf{(A) }10\qquad\textbf{(B) }13\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }\text{impossible to determine}$

The triangle $ABC$ satisfies $0\le AB\le 1\le BC\le 2\le CA\le 3$. What is the maximum area it can have?

The real numbers $a, b, c, d$ satisfy $a > b > c > d$ and $$a + b + c + d = 2004 \,\,\, and \,\,\, a^2 - b^2 + c^2 - d^2 = 2004.$$ Answer, with proof, to the following questions: a) What is the smallest possible value of $a$? b) What is the number of possible values of $a$?

There are $2n$ points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose $n$ drawn segments having no common ends?