Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

A [i]semi-palindrome[/i] is a four-digit number whose first two digits and last two digits are identical. For instance, $2323$ and $5757$ are semi-palindromes, but $1001$ and $2324$ are not. What is the difference between the largest semi-palindrome and smallest semi-palindrome? $\textbf{(A) } 7979\qquad\textbf{(B) } 8080\qquad\textbf{(C) } 8181\qquad\textbf{(D) } 8484\qquad\textbf{(E) } 8989$

Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2023$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2023$ loses. Who wins if every player wants to win? [i]Proposed by Mykhailo Shtandenko[/i]

Find some natural number $a$ such that $2a$ is a perfect square, $3a$ is a perfect cube, $5a$ is the fifth power of some natural number.

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

Let $ABCD$ be a rectangle and $P$ be a point on the side$ AD$ such that $\angle BPC = 90^o$. The perpendicular from $A$ on $BP$ cuts $BP$ at $M$ and the perpendicular from $D$ on $CP$ cuts $CP$ in $N$. Show that the center of the rectangle lies in the $MN$ segment.

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$? $\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$

A man travels $ m$ feet due north at $ 2$ minutes per mile. He returns due south to his starting point at $ 2$ miles per minute. The average rate in miles per hour for the entire trip is: $ \textbf{(A)}\ 75\qquad \textbf{(B)}\ 48\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 24\qquad\\ \textbf{(E)}\ \text{impossible to determine without knowing the value of }{m}$

Let $n\geq 3$ be an odd integer. Determine how many real solutions there are to the set of $n$ equations \[\left\{\begin{array}{cc}x_1(x_1+1)=x_2(x_2-1)\\x_2(x_2+1)=x_3(x_3-1)\\ \vdots \\ x_n(x_n+1) = x_1(x_1-1)\end{array}\right.\]

Let \(ABC\) be a right-angled triangle with \(\angle B=90^{\circ}\). Let \(I\) be the incentre if \(ABC\). Extend \(AI\) and \(CI\); let them intersect \(BC\) in \(D\) and \(AB\) in \(E\) respectively. Draw a line perpendicular to \(AI\) at \(I\) to meet \(AC\) in \(J\), draw a line perpendicular to \(CI\) at \(I\) to meet \(AC\) at \(K\). Suppose \(DJ=EK\). Prove that \(BA=BC\).

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?

Given $\triangle ABC$ with circumcircle $\ell$. Point $M$ in $\triangle ABC$ such that $AM$ is the angle bisector of $\angle BAC$. Circle with center $M$ and radius $MB$ intersects $\ell$ and $BC$ at $D$ and $E$ respectively, $(B \not= D, B \not= E)$. Let $P$ be the midpoint of arc $BC$ in $\ell$ that didn't have $A$. Prove that $AP$ angle bisector of $\angle DPE$ if and only if $\angle B = 90^{\circ}$.

Let $O$ be the centre of the circumcircle of triangle $ABC$ and let $I$ be the centre of the incircle of triangle $ABC$. A line passing through the point $I$ is perpendicular to the line $IO$ and passes through the incircle at points $P$ and $Q$. Prove that the diameter of the circumcircle is equal to the perimeter of triangle $OPQ$.

The area of the rectangular region is [asy] draw((0,0)--(4,0)--(4,2.2)--(0,2.2)--cycle,linewidth(.5 mm)); label(".22 m",(4,1.1),E); label(".4 m",(2,0),S); [/asy] $\text{(A)}\ \text{.088 m}^2 \qquad \text{(B)}\ \text{.62 m}^2 \qquad \text{(C)}\ \text{.88 m}^2 \qquad \text{(D)}\ \text{1.24 m}^2 \qquad \text{(E)}\ \text{4.22 m}^2$

Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$.

Is there a described $n$-gon in which each side is longer than the diameter of the inscribed circle a) at $n = 4$? b) when $n = 7$? c) when $n = 6$? [i]P.A.Kozhevnikov[/i]

A group of $ 100$ students numbered $ 1$ through $ 100$ are playing the following game. The judge writes the numbers $ 1$, $ 2$, $ \ldots$, $ 100$ on $ 100$ cards, places them on the table in an arbitrary order and turns them over. The students $ 1$ to $ 100$ enter the room one by one, and each of them flips $ 50$ of the cards. If among the cards flipped by student $ j$ there is card $ j$, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than $ 1$ percent of chance to win?

$2000$ people are standing on a line. Each one of them is either a [i]liar[/i], who will always lie, or a [i]truth-teller[/i], who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line.

Given non-constant linear functions $p_1(x), p_2(x), \dots p_n(x)$. Prove that at least $n-2$ of polynomials $p_1p_2\dots p_{n-1}+p_n, p_1p_2\dots p_{n-2} p_n + p_{n-1},\dots p_2p_3\dots p_n+p_1$ have a real root.

Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$. [i]Proposed by Sharvil Kesarwani[/i]

For all positive integers $ n$, let $ f(n) \equal{} \log_{2002} n^2$. Let \[ N \equal{} f(11) \plus{} f(13) \plus{} f(14) \] Which of the following relations is true? $ \textbf{(A)}\ N < 1 \qquad \textbf{(B)}\ N \equal{} 1 \qquad \textbf{(C)}\ 1 < N < 2 \qquad \textbf{(D)}\ N \equal{} 2 \qquad \textbf{(E)}\ N > 2$

Let $n$ and $k$ be two positive integers such that $1\leq n \leq k$. Prove that, if $d^k+k$ is a prime number for each positive divisor $d$ of $n$, then $n+k$ is a prime number.

For each positive integer $k$ we denote by $S(k)$ the sum of its digits, for example $S(132)=6$ and $S(1000)=1$. A positive integer $n$ is said to be $\textbf{fascinating}$ if it holds that $n = \frac{k}{S(k)}$ for some positive integer $k$. For example, the number $11$ is $\textbf{fascinating}$ since $11 = \frac{198}{S(198)} ($since $\frac{198}{S(198)}=\frac{198}{1+9+8}=\frac{198}{18} = 11)$. Prove that there exists a positive integer less than $2021$ and that it is not $\textbf{fascinating}$.

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]