For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds: For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, [u]all of the same size[/u], let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which \[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\] a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$. b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.

Define $f(n):$ the number of integral points of line segment $OA_n$ ($O$ and $A_n$ not included), where $A_n(n,n+3)$. Then, $f(1)+f(2)+\cdots+f(1990)=$________.

Let $ABCD$ be a tangential quadrilateral with $BC> BA$. The point $P$ is on the segment $BC$, such that $BP = BA$ . Show that the bisector of $\angle BCD$, the perpendicular on line $BC$ through $P$ and the perpendicular on $BD$ through $A$, intersect at one point.

A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.

You are participating in a virtual stock market, with many different stocks. For a stock $S$, there is a list of prices where the $i$th number is the price of the stock on day $i$. On each day $i$, you are given the stock's current price (in dollars), and you can either buy a share of stock $S$, sell your share of stock $S$, or do nothing, but you may only take one of these actions per day, and you may not have more than one share of stock $S$ at a time. Each stock is independent, so for example on the first day, you may buy a share of $S$ and a share of $T$, and on the second day you may sell your share of $T$. At USMCA Trading LLC, you are given $2021!$ different stocks, where each stock's list of prices corresponds to a unique permutation of the first $2021$ positive integers, to trade for $2021$ days. You start out with $M$ dollars, and at the end of $2021$ days, you end up with $N$ dollars. Assume $M$ is large enough so that you can never run out of money during the $2021$ days. What is the maximum possible value of $N - M$?

On an $n\times n$ chart, where $n \geq 4$, stand "$+$" signs in the cells of the main diagonal and "$-$" signs in all the other cells. You can change all the signs in one row or in one column, from $-$ to $+$ or from $+$ to $-$. Prove that you will always have $n$ or more $+$ signs after finitely many operations.

Find the value of $\frac12+\frac{4}{2^2} +\frac{9}{2^3} +\frac{16}{2^4} + ...$

Given are real $y>1$ and positive integer $n \leq y^{50}$ such that all prime divisors of $n$ do not exceed $y$. Prove that $n$ is a product of $99$ positive integer factors (not necessarily primes) not exceeding $y$.

In the space, there is a convex polyhedron $D$ such that for every vertex of $D$, there are an even number of edges passing through that vertex. We choose a face $F$ of $D$. Then we assign each edge of $D$ a positive integer such that for all faces of $D$ different from $F$, the sum of the numbers assigned on the edges of that face is a positive integer divisible by $2024$. Prove that the sum of the numbers assigned on the edges of $F$ is also a positive integer divisible by $2024$.

Peter has a wooden square stamp divided into a grid. He coated some $102$ cells of this grid with black ink. After that, he pressed this stamp $100$ times on a list of paper so that each time just those $102$ cells left a black imprint on the paper. Is it possible that after his actions the imprint on the list is a square $101 \times 101$ such that all the cells except one corner cell are black? (Alexsandr Gribalko)

Initially, the number $10$ is written on the board. In each subsequent moves, you can either (i) erase the number $1$ and replace it with a $10$, or (ii) erase the number $10$ and replace it with a $1$ and a $25$ or (iii) erase a $25$ and replace it with two $10$. After sometime, you notice that there are exactly one hundred copies of $1$ on the board. What is the least possible sum of all the numbers on the board at that moment?

Consider the set $A = \{1, 2, 3, ..., 2n - 1\}$, where $n \ge 2$ is a positive integer. We remove from the set $A$ at least $n - 1$ elements such that: • if $a \in A$ has been removed, and $2a \in A$, then $2a$ has also been removed, • if $a, b \in A (a \ne b)$ have been removed and $a + b \in A$, then $a + b$ has also been removed. Which numbers have to be removed such that the sum of the remaining numbers is maximum?

Given quadratic trinomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$, where $a>c$. It is known that for every real $t$ and $s$ with $t+s=1$ the polynomial $B(x)=tP(x)+sQ(x)$ has at least one real root. Prove that $bc \geq ad$.

Let $p$ be a prime number. Find all positive integers $a$ and $b$ such that: $\frac{4a + p}{b}+\frac{4b + p}{a}$ and $ \frac{a^2}{b}+\frac{b^2}{a}$ are integers.

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 $\lfloor 4 \rfloor=\lceil 4 \rceil=4$

Evan the ant lives on a right hexagonal pyramid $ABCDEFP$ whose base is regular hexagon $ABCDEF$, and $PA=PB=PC=PD=PE=PF=38\sqrt{3}$. Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$, respectively. Let $X$ be the point on segment $MP$ and $Y$ be the point on segment $NP$ such that $MX=NY=\sqrt{3}$. Given that $PM=37\sqrt{3}$, the length of the shortest path from $X$ to $Y$ that Evan can take by crawling along the surface of $ABCDEFP$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Lightning 4.4[/i]