The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80$

In all triangles $ABC$ does it hold that: $$\sum\sin^2\frac A2\cos^2A\ge\frac{3\left(s^2-(2R+r)^2\right)}{8R^2}$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

Consider 2024 distinct prime numbers $p_1, p_2, \dots, p_{2024}$ such that \[p_1+p_2+\dots+p_{1012}=p_{1013}+p_{1014}+\dots+p_{2024}.\] Let $A=p_1p_2\dots p_{1012}$ and $B=p_{1013}p_{1014}\dots p_{2024}$. Prove that $|A-B|\geq 4$.

Humberto and Luciano use the break between classes to have fun with the following game: Humberto writes a list of distinct positive integers on a green sheet of paper and hands it to Luciano. Luciano then writes on a board all the possible sums, without repetitions, of one or more different numbers written on the green sheet. For example, if Humberto writes $1$, $3$ and $4$ on the green sheet, Luciano will write $1$, $3$, $4$, $5$, $7$ and $8$ on the board. (a) Let $n$ be a positive integer. Determine all positive integers $k$ such that Humberto can write a list of $n$ numbers on the green sheet in order to guarantee that Luciano will write exactly $k$ numbers on the board. (b) Luciano now decides to write a list of $m$ distinct positive integers on a yellow sheet of paper. Determine the smallest positive integer $m$ such that it is possible for Luciano to write this list so that, for any list that Humberto writes on the green sheet, with a maximum of $2023$ numbers, not all the numbers on the yellow sheet will be written on the board.

In how many ways can four married couples sit in a merry-go-round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?

Let $a_1, a_2, a_3, \ldots$ be an infinite sequence of positive integers such that $a_1=4$, $a_2=12$, and for all positive integers $n$, \[a_{n+2}=\gcd\left(a_{n+1}^2-4,a_n^2+3a_n \right).\] Find, with proof, a formula for $a_n$ in terms of $n$.

The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $F_{2n-1}^{2}+F_{2n+1}^{2}+1=3F_{2n-1}F_{2n+1}$ for all $n \ge 1$.

Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 + |(a - b)(b - c)(c - a)|.$$

[b]p10 [/b]Kathy has two positive real numbers, $a$ and $b$. She mistakenly writes $$\log (a + b) = \log (a) + \log( b),$$ but miraculously, she finds that for her combination of $a$ and $b$, the equality holds. If $a = 2022b$, then $b = \frac{p}{q}$ , for positive integers $p, q$ where $gcd(p, q) = 1$. Find $p + q$. [b]p11[/b] Points $X$ and $Y$ lie on sides $AB$ and $BC$ of triangle $ABC$, respectively. Ray $\overrightarrow{XY}$ is extended to point $Z$ such that $A, C$, and $Z$ are collinear, in that order. If triangle$ ABZ$ is isosceles and triangle $CYZ$ is equilateral, then the possible values of $\angle ZXB$ lie in the interval $I = (a^o, b^o)$, such that $0 \le a, b \le 360$ and such that $a$ is as large as possible and $b$ is as small as possible. Find $a + b$. [b]p12[/b] Let $S = \{(a, b) | 0 \le a, b \le 3, a$ and $b$ are integers $\}$. In other words, $S$ is the set of points in the coordinate plane with integer coordinates between $0$ and $3$, inclusive. Prair selects four distinct points in $S$, for each selected point, she draws lines with slope $1$ and slope $-1$ passing through that point. Given that each point in $S$ lies on at least one line Prair drew, how many ways could she have selected those four points?

All $n+3$ offices of University of Somewhere are numbered with numbers $0,1,2, \ldots ,$ $n+1,$ $n+2$ for some $n \in \mathbb{N}$. One day, Professor $D$ came up with a polynomial with real coefficients and power $n$. Then, on the door of every office he wrote the value of that polynomial evaluated in the number assigned to that office. On the $i$th office, for $i$ $\in$ $\{0,1, \ldots, n+1 \}$ he wrote $2^i$ and on the $(n+2)$th office he wrote $2^{n+2}$ $-n-3$. [list=a] [*] Prove that Professor D made a calculation error [*] Assuming that Professor D made a calculation error, what is the smallest number of errors he made? Prove that in this case the errors are uniquely determined, find them and correct them. [/list]

Let $m, n$ be positive integers with $m<n$ and consider an $n\times n$ board from which its upper left $ m\times m$ part has been removed. An example of such board for $n=5$ and $m=2$ is shown below. Determine for which pairs $(m, n)$ this board can be tiled with $3\times 1$ tiles. Each tile can be positioned either horizontally or vertically so that it covers exactly three squares of the board. The tiles should not overlap and should not cover squares outside of the board.

All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection? $\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textbf{(D) } 27\qquad\textbf{(E) } 28$

Let the three sides of $\triangle ABC$ be $a,b,c$. Prove that $\displaystyle \frac{\sin^2A}{a}+\frac{\sin^2B}{b}+\frac{\sin^2C}{c} \le \frac{S^2}{abc}$ where $\displaystyle S=\frac{a+b+c}{2}$. Find the case where equality holds.

Let $J\subsetneq I\subseteq \mathbb R$ be non-empty open intervals, and let $f_1, f_2,\ldots$ be real polynomials satisfying the following conditions: [list] [*] $f_i(x)\ge 0$ for all $i\ge 1$ and $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)$ is finite for all $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)=1$ for all $x\in J$. [/list] Do these conditions imply that $\sum\limits_{i=1}^\infty f_i(x)=1$ also for all $x\in I$? [i]Proposed by András Imolay, Budapest[/i]

Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI=3$ and the distance from $I$ to $BC$ is $2$, what is the square of length on $BC$?

On an $8\times 8$ chessboard, 6 black rooks and k white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of $k$. (Two rooks attack each other if they are in the same row or column and no rooks are between them.)

Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$ $\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$

The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely: $ \textbf{(A)}\ 0, 12, -12\qquad\textbf{(B)}\ 0, 12\qquad\textbf{(C)}\ 12, -12\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 0 $

Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$. [i]Note:[/i] The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$.

There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$.

Quasi-rounding is a substitution one of the two closest integers instead of the given number. Given $n$ numbers. Prove that you can quasi-round them in such a way, that a sum of every subset of quasi-rounded numbers will deviate from the sum of the same subset of initial numbers not greater than $(n+1)/4$ .

A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{None of the preceding} $

Prove that for each positive integer $n$ there exist odd positive integers $x_n$ and $y_n$ such that ${x_{n}}^2 +7{y_{n}}^2 =2^n$.

The line passing through the center of the equilateral triangle $ ABC $ intersects the lines $ AB $, $ BC $ and $ CA $ at the points $ {{C} _ {1}} $, $ {{A} _ {1}} $ and $ {{B} _ {1}} $, respectively. Let $ {{A} _ {2}} $ be a point that is symmetric $ {{A} _ {1}} $ with respect to the midpoint of $ BC $; the points $ {{B} _ {2}} $ and $ {{C} _ {2}} $ are defined similarly. Prove that the points $ {{A} _ {2}} $, $ {{B} _ {2}} $ and $ {{C} _ {2}} $ lie on the same line tangent to the inscribed circle of the triangle $ ABC $. (Serdyuk Nazar)

Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]