A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$ $\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601$

Let $n$ be even positive integer. There are $n$ real positive numbers written on the blackboard. In every step, we choose two numbers, erase them, and replace [i]each[/i] of then by their product. Show that for any initial $n$-tuple it is possible to obtain $n$ equal numbers on the blackboard after a finite number of steps. [i]Proposed by Peter Novotný[/i]

Let $\triangle ABC$ be triangle with incenter $I$ and circumcircle $\Gamma$. Let $M=BI\cap \Gamma$ and $N=CI\cap \Gamma$, the line parallel to $MN$ through $I$ cuts $AB$, $AC$ in $P$ and $Q$. Prove that the circumradius of $\odot (BNP)$ and $\odot (CMQ)$ are equal.

The point $M$ is inside the convex quadrilateral $ABCD$, such that \[ MA = MC, \hspace{0,2cm} \widehat{AMB} = \widehat{MAD} + \widehat{MCD} \quad \textnormal{and} \quad \widehat{CMD} = \widehat{MCB} + \widehat{MAB}. \] Prove that $AB \cdot CM = BC \cdot MD$ and $BM \cdot AD = MA \cdot CD.$

Sixteen points are placed in the centers of a $4 \times 4$ chess table in the following way: • • • • • • • • • • • • • • • • (a) Prove that one may choose $6$ points such that no isoceles triangle can be drawn with the vertices at these points. (b) Prove that one cannot choose $7$ points with the above property.

4. Let $\mathbb{N}$ denote the strictly positive integers. A function $f$ : $\mathbb{N}$ $\to$ $\mathbb{N}$ has the following properties which hold for all $n \in$ $\mathbb{N}$: a) $f(n)$ < $f(n+1)$; b) $f(f(f(n)))$ = 4$n$ Find $f(2022)$.

A natural number is called [i]bright [/i] if it is the sum of a perfect square and a perfect cube. Prove that if $r$ and $s$ are any two positive integers, then (a) there exist infinitely many positive integers $n$ such that both $r+n$ and $s+n$ are [i]bright[/i], (b) there exist infinitely many positive integers $m$ such that both rm and sm are [i]bright[/i].

The sequence $\{a_n\}$ is defined as follows: \begin{align*} a_n = \sqrt{1 + \left(1 + \frac 1n \right)^2} + \sqrt{1 + \left(1 - \frac 1n \right)^2}. \end{align*} What is the value of the expression given below? \begin{align*} \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}.\end{align*} $\textbf{(A)}\ \sqrt{18241} \qquad\textbf{(B)}\ \sqrt{18625} - 1 \qquad\textbf{(C)}\ \sqrt{18625} \qquad\textbf{(D)}\ \sqrt{19013} - 1\qquad\textbf{(E)}\ \sqrt{19013}$

Can the numbers from \( 1 \) to \( 2025 \) be arranged in a circle such that the difference between any two adjacent numbers has the form \( 2^k \) for some non-negative integer \( k \)? For different adjacent pairs of numbers, the values of \( k \) may be different. [i]Proposed by Anton Trygub[/i]

For what real values of $ k$, other than $ k \equal{} 0$, does the equation $ x^2 \plus{} kx \plus{} k^2 \equal{} 0$ have real roots? $ \textbf{(A)}\ {k < 0}\qquad \textbf{(B)}\ {k > 0} \qquad \textbf{(C)}\ {k \ge 1} \qquad \textbf{(D)}\ \text{all values of }{k}\qquad \textbf{(E)}\ \text{no values of }{k}$

Alex, Bill, and Charlie want to play a game of DotA. They each come online at a uniformly random time between $8:00$ and $8:05\text{ }\text{PM}$, and each person queues for $2$ minutes. However, if any of them sees any other of them online while queuing, they merge parties and restart the queue, again waiting for $2$ minutes starting from the merger time. For example, suppose that Alex logs in at $8:00\text{ PM}$, Bill logs in at $8:01\text{ PM}$, and Charlie logs in at $8:02:30\text{ PM}$ ($30$ seconds past $8:02\text{ PM}$). At $8:01\text{ PM}$, Alex and Bill would merge parties and queue for $2$ minutes starting at $8:01\text{ PM}$. At $8:02:30\text{ PM}$, Charlie would merge with Alex and Bill’s party, since Alex and Bill have waited together for only $1.5$ minutes. What is the probability that they will play as a party of $3$?

Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$. [color=#FF0000]Moderator says: Use the search before posting contest problems [url]http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=530783[/url][/color]

Prove that if $ a, b, c $ are non-negative numbers, then $$ a^3 + b^3 + c^3 + 3abc \geq a^2(b + c) + b^2(a + c) + c^2(a + b).$$

Prove that for any acute triangle, there is a point in space such that every line segment from a vertex of the triangle to a point on the line joining the other two vertices subtends a right angle at this point.

Solve in the set of real numbers: \[ 3\left(x^2 \plus{} y^2 \plus{} z^2\right) \equal{} 1, \] \[ x^2y^2 \plus{} y^2z^2 \plus{} z^2x^2 \equal{} xyz\left(x \plus{} y \plus{} z\right)^3. \]

Given a quadrilateral $ABCD$. $A ', B', C'$ and $D'$ are the midpoints of the sides $BC, CB, BA$ and $AB$, respectively. It is known that $AA'= CC'$, $BB'= DD'$. Is it true that $ABCD$ is a parallelogram? (M. Volchkevich)

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.

$D$ is an arbitrary point inside triangle $ABC$, and $E$ is inside triangle $BDC$. Prove that \[\frac{S_{DBC}}{(P_{DBC})^{2}}\geq\frac{S_{EBC}}{(P_{EBC})^{2}}\]

Determine the sum of the real roots of the equation $$x^2-8x+20=2\sqrt{x^2-8x+30}$$

The polynomial $$ Q(x_1,x_2,\ldots,x_4)=4(x_1^2+x_2^2+x_3^2+x_4^2)-(x_1+x_2+x_3+x_4)^2 $$ is represented as the sum of squares of four polynomials of four variables with integer coefficients. [b]a)[/b] Find at least one such representation [b]b)[/b] Prove that for any such representation at least one of the four polynomials isidentically zero. [i](A. Yuran)[/i]

How many integers $n$ with $10 \le n \le 500$ have the property that the hundreds digit of $17n$ and $17n+17$ are different? [i]Proposed by Evan Chen[/i]

Let $L_1$ and $L_2$ be two parallel lines and $L_3$ a line perpendicular to $L_1$ and $L_2$ at $H$ and $P$, respectively. Points $Q$ and $R$ lie on $L_1$ such that $QR = PR$ ($Q \ne H$). Let $d$ be the diameter of the circle inscribed in the triangle $PQR$. Point $T$ lies $L_2$ in the same semiplane as $Q$ with respect to line $L_3$ such that $\frac{1}{TH}= \frac{1}{d}- \frac{1}{PH}$ . Let $X$ be the intersection point of $PQ$ and $TH$. Find the locus of the points $X$ as $Q$ varies on $L_1$.

Let $A = \left\{ 1,2,3,4,5,6,7 \right\}$ and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.

Find all non-zero real numbers $a, b, c$ such that the following polynomial has four (not necessarily distinct) positive real roots. $$P(x) = ax^4 - 8ax^3 + bx^2 - 32cx + 16c$$

We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.