Let $\displaystyle{N = \prod_{k=1}^{1000} (4^k - 1)}$. Determine the largest positive integer $n$ such that $5^n$ divides evenly into $N$.

Let $p$ be a prime integer greater than $10^k$. Pete took some multiple of $p$ and inserted a $k-$digit integer $A$ between two of its neighbouring digits. The resulting integer C was again a multiple of $p$. Pete inserted a $k-$digit integer $B$ between two of neighbouring digits of $C$ belonging to the inserted integer $A$, and the result was again a multiple of $p$. Prove that the integer $B$ can be obtained from the integer $A$ by a permutation of its digits. [i](8 points)[/i] [i]Ilya Bogdanov[/i]

Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

A point in the plane is called [b]integral[/b] if both its $x$ and $y$ coordinates are integers. We are given a triangle whose vertices are integral. Its sides do not contain any other integral points. Inside the triangle, there are exactly 4 integral points. Must those 4 points lie on one line?

A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1,6,\dots,96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?

David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? $\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$

$20$ numbers are written on the board $1, 2, ... ,20$. Two players are putting signs before the numbers in turn ($+$ or $-$). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?

Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?

Let $P(x)$ denote the polynomial \[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$. [i]Victor Wang.[/i]

Determine all positive integers $d,$ such that there exists an integer $k\geq 3,$ such that One can arrange the numbers $d,2d,\ldots,kd$ in a row, such that the sum of every two consecutive of them is a perfect square.

Erick stands in the square in the 2nd row and 2nd column of a 5 by 5 chessboard. There are \$1 bills in the top left and bottom right squares, and there are \$5 bills in the top right and bottom left squares, as shown below. \[\begin{tabular}{|p{1em}|p{1em}|p{1em}|p{1em}|p{1em}|} \hline \$1 & & & & \$5 \\ \hline & E & & &\\ \hline & & & &\\ \hline & & & &\\ \hline \$5 & & & & \$1 \\ \hline \end{tabular}\] Every second, Erick randomly chooses a square adjacent to the one he currently stands in (that is, a square sharing an edge with the one he currently stands in) and moves to that square. When Erick reaches a square with money on it, he takes it and quits. The expected value of Erick's winnings in dollars is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $a \ge1$ be a real number and $z$ be a complex number such that $| z + a | \le a$ and $|z^2+ a | \le a$. Show that $| z | \le a$.

In triangle $ABC, \angle B = 60^o$, $O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$.The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.

A polynomial $A(x)$ is said to be [i]simple[/i] if $A(x)$ is divisible by $x$ but not divisible by $x^2$. Suppose that a polynomial $P(x)$ has a simple polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists a simple polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{2023}$.

The year $1978$ was “peculiar” in that the sum of the numbers formed with the first two digits and the last two digits is equal to the number formed with the middle two digits, i.e., $19+78=97$. What was the last previous peculiar year, and when will the next one occur?

Find the remainder when $(1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)$ is divided by $43$. Your answer should be an integer between $0$ and $42$.

We are given $13$ apparently equal balls, all but one having the same weight (the remaining one has a different weight). Is it posible to determine the ball with the different weight in $3$ weighings?

Let $ABC$ be a triangle with sides $BC = 13$, $CA = 14$ and $AB = 15$. We denote by $I$ the intersection point of the angle bisectors and $M$ to the midpoint of $AB$. The line $IM$ cuts at $P$ at the altitude drawn from $C$. Find the length of $CP$.

Find all triples $(x,y,z)$ of natural numbers that verify the equation $$2x^2y^2+2y^2z^2+2z^2x^2-x^4-y^4-z^4=576.$$

Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$, $b\ge 2$, $c\ge 2$. Adam and Boris play the following game. They alternately take turns with Adam first. On Adam’s turn, he can choose one of the polynomial’s coefficients and replace it with the sum of the other two coefficients. On Boris’s turn, he can choose one of the polynomial’s coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$, $b$ and $c$, determine who has a winning strategy.

Let $n$ be a natural number. Find all real numbers $x$ satisfying the equation $$\sum^n_{k=1}\frac{kx^k}{1+x^{2k}}=\frac{n(n+1)}4.$$

In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach $k$ vertices using at most two edges. Find the maximum value of $k$.

Let $AA_1 , BB_1$, and $CC_1$ be the altitudes of the non-isosceles acute-angled triangle $ABC$. The circles circumscibred around the triangles $ABC$ and $A_1 B_1 C$ intersect again at the point $P , Z$ is the intersection point of the tangents to the circumscribed circle of the triangle $ABC$ conducted at points $A$ and $B$ . Prove that lines $AP , BC$ and $ZC_1$ are concurrent.

Given the set of numbers $ \{1, 2, 3, \ldots, 100\} $. From this set, create 10 pairwise disjoint subsets $ N_i = \{a_{i,1}, a_{i,2}, ... a_{i,10} $ ($ i = 1, 2, \ldots, 10 $ ) so that the sum of the products $$ \sum_{i=10}^{10}\prod_{j=1}^{10} a_{i,j} $$ was the biggest.

suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)