Let \( A \) be a set of 2025 positive real numbers. For a subset \( T \subseteq A \), define \( M_T \) as the median of \( T \) when all elements of \( T \) are arranged in increasing order, with the convention that \( M_\emptyset = 0 \). Define \[ P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T. \] Find the smallest real number \( C \) such that for any set \( A \) of 2025 positive real numbers, the following inequality holds: \[ P(A) - Q(A) \leq C \cdot \max(A), \] where \(\max(A)\) denotes the largest element in \( A \).

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point $Q$ on the circular road (see diagram). [img]6763[/img] It is served by two buses, each of which requires 20 minutes for the round trip. Bus No. 1, upon leaving the terminus, travels along the straight road, once around the circle clockwise and returns along the straight road to the terminus. Bus No. 2, reaching the terminus 10 minutes after Bus No. 1, has a similar route except that it proceeds counterclockwise around the circle. Both buses run continuously and do not wait at any point on the route except for a negligible amount of time to pick up and discharge passengers. A man plans to wait at a point $P$ which is $x$ miles ($0\le x < 12$) from the terminus along the route of Bus No. 1 and travel to the terminus on one of the buses. Assuming that he chooses to board that bus which will bring him to his destination at the earliest moment, there is a maximum time $w(x)$ that his journey (waiting plus travel time) could take. Find $w(2)$; find $w(4)$. For what value of $x$ will the time $w(x)$ be the longest? Sketch a graph of $y = w(x)$ for $0\le x < 12$.

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

Find all functions $h: \mathbb{Z}\to \mathbb{Z}$ such that for all $x,y\in \mathbb{Z}$: \[h(x+y)+h(xy)=h(x)h(y)+1.\]

The center of the inscribed circle of triangle $ABC$ is $I$. Let $e$ be the perpendicular line on $CI$ passing through $I$. The line $e$ itnersects the side $AC$ at $A'$ and the side $BC$ at point $B'$. Let $A''$ be the symmetric point of $A$ wrt $A'$, $B''$ be the symmetric point of $B$ wrt $B'$. Prove that $A''B''$ is a line tangent to the incircle.

Let $k$ be a fixed odd integer, $k>3$. Prove: There exist infinitely many positive integers $n$, such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$, and $d_1+d_2=n+k$.

Let $x_1,x_2,\dots,x_{31}$ be real numbers. Then find the maximum value can $$\sum_{i,j=1,2,\dots,31, \; i\neq j}{\lceil x_ix_j \rceil }-30\left(\sum_{i=1,2,\dots,31}{\lfloor x_i^2 \rfloor } \right)$$ achieve. P.S.: For a real number $x$ we denote the smallest integer that does not subseed $x$ by $\lceil x \rceil$ and the biggest integer that does not exceed $x$ by $\lfloor x \rfloor$. For example $\lceil 2.7 \rceil=3$, $\lfloor 2.7 \rfloor=2$ and $\lfloor 4 \rfloor=\lceil 4 \rceil=4$

Let $n{}$ and $k{}$ be natural numbers, with $n > 2k$. In the deck of cards, each card contains a subset of the set $\{1, 2, \ldots , n\}$ consisting of at least $k+1$, but no more than $n-k$ elements. Each $m$-element set is written exactly on $m-k$ cards. Is it possible to split these cards into $n- 2k$ stacks so that in each stack all subsets on the cards are different, and any two of them intersect?

The circle is inscribed in $\vartriangle ABC$. Let $L, M, N$ be the tangent points of the circle with sides $AB, AC, BC$, respectively. Prove that $\angle MLN$ is always an acute angle.

Let $v$ and $w$ be real numbers such that, for all real numbers $a$ and $b$, the inequality \[(2^{a+b}+8)(3^a+3^b) \leq v(12^{a-1}+12^{b-1}-2^{a+b-1})+w\] holds. Compute the smallest possible value of $128v^2+w^2$. [i]Proposed by Luke Robitaille[/i]

An ordered pair \((k,n)\) of positive integers is [i]good[/i] if there exists an ordered quadruple \((a,b,c,d)\) of positive integers such that \(a^3+b^k=c^3+d^k\) and \(abcd=n\). Prove that there exist infinitely many positive integers \(n\) such that \((2022,n)\) is not good but \((2023,n)\) is good. [i]Proposed by Luke Robitaille[/i]

Prove that in a meeting of $285$ people, at least one of them has given an even number of handshakes ($0$ is considered an even number and corresponds to an assistant who does not shake any hand).

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that \[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\] where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.

For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \] Find all natural numbers $ n $ such that $ s(n) = 2010 $

Find all positive integers $n$ such that the number $\frac{(2n)!+1}{n!+1}$ is positive integer.

Find the least positive integer $n$ such that there are at least $1000$ unordered pairs of diagonals in a regular polygon with $n$ vertices that intersect at a right angle in the interior of the polygon.

Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$. (a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$. (b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have \[ p(x+1)+p(x-1)-rp(x)-s=0 \]Prove $\deg(p(x))\le 2$. (c) Show, in (b) that $s=0$ implies $a_2=0$.

Find the least positive integer $n$ such that $15$ divides the product \[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\] , for every positive integers $a_1, a_2, \dots, a_{15}$.

A polynomial $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such that $n\leq 1000$. Given that \[p(1001) = a\cdot 2^b - c,\] where $a$ is an odd integer, and $0 < c < 2007$, find $c-(a+b)$.

Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that: [list] [*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and [*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$. [/list] [i]Proposed by Ricky Liu[/i]

Through point $ P $ inside triangle $ ABC $, straight lines were drawn, parallel to the sides, until they intersect with the sides. In the three resulting parallelograms, diagonals that do not contain point $ P $, are drawn. Points $ A_1 $, $ B_1 $ and $ C_1 $ are the intersection points of the lines containing these diagonals such that $A_1$ and $A$ are in different sides of line $BC$ and $B_1$ and $C_1$ are similar. Prove that if hexagon $ AC_1BA_1CB_1 $ is inscribed and convex, then point $ P $ is the orthocenter of triangle $ A_1B_1C_1 $.

The non-isosceles triangle $ABC$ is inscribed in the circle $\omega$. The tangent line to this circle at the point $C$ intersects the line $AB$ at the point $D$. Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$, respectively. The point $M$ is on the side $AB$ such that $\frac{AK}{BL} = \frac{AM}{BM}$. Let the perpendiculars from the point $M$ to the straight lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$ respectively. Prove that $2\angle CQP=\angle ACB$

Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.\] Find the sum of all possible values of $a+b$.