The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$?

For integers $0 \le m,n \le 2^{2017}-1$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor m/2^k \right\rfloor$ and $\left\lfloor n/2^k \right\rfloor$ are both odd integers. Consider a $2^{2017} \times 2^{2017}$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 2^{2017}$) is \[ (-1)^{\alpha(i-1, j-1)}. \] For $1 \le i, j \le 2^{2017}$, let $M_{i,j}$ be the matrix with the same entries as $M$ except for the $(i,j)$th entry, denoted by $a_{i,j}$, and such that $\det M_{i,j}=0$. Suppose that $A$ is the $2^{2017} \times 2^{2017}$ matrix whose $(i,j)$th entry is $a_{i,j}$ for all $1 \le i, j \le 2^{2017}$. Compute the remainder when $\det A$ is divided by $2017$. [i]Proposed by Michael Ren and Ashwin Sah[/i]

Suppose that $n > m \geq 1$ are integers such that the string of digits $143$ occurs somewhere in the decimal representation of the fraction $\frac{m}{n}$. Prove that $n > 125.$

A contest has six problems worth seven points each. On any given problem, a contestant can score either $0$, $1$, or $7$ points. How many possible total scores can a contestant achieve over all six problems?

Let $S = {1, 2, 2^2, 2^3, ... , 2^{2021}}$. Compute the difference between the number of even digits and the number of odd digits across all numbers in $S$ (written as integers in base $10$ with no leading zeros). If E is the exact answer to this question and A is your answer, your score is given by $\max \, \left(0, \left\lfloor 25 - \frac{1}{2 \cdot 10^8}|E - A|^4\right\rfloor \right)$.

Solve for integers $a,b,c$ \[ (a+b+c)^3 + \frac{1}{2} (b+c)(c+a)(a+b) = 1 - abc \]

(a) On a blackboard are written $100$ different numbers. Prove that you can choose $8$ of them so that their average value is not equal to that of any $9$ of the numbers on the blackboard. (b) On a blackboard are written $100$ integers. For any $8$ of them, you can find $9$ numbers on the blackboard so that the average value of the $8$ numbers is equal to that of the $9$. Prove that all the numbers on the blackboard are equal. (A Shapovalov)

An amusement park has a collection of scale models, with ratio $1 : 20$, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number? $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }20$

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this? [i]Proposed by Oleksiy Masalitin[/i]

Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$ are divisible by $p$. Prove that $p$ divides each of $a,b,c$. $\quad$ Proposed by Navilarekallu Tejaswi

Let $ABCD$ be a convex circumscribed quadrilateral such that $\angle ABC+\angle ADC<180^{\circ}$ and $\angle ABD+\angle ACB=\angle ACD+\angle ADB$. Prove that one of the diagonals of quadrilateral $ABCD$ passes through the other diagonals midpoint.

The LHS Math Team is doing Karaoke. William sings every song, David sings every other song, Peter sings every third song, and Muztaba sings every fourth song. If they sing $600$ songs, find the average number of people singing each song.

Let $ABCD$ be a quadrilateral inscribed in circle $(O)$. Let $M,N$ be the midpoints of $AD,BC$. A line through the intersection $P$ of the two diagonals $AC,BD$ meets $AD,BC$ at $S, T$ respectively. Let $BS$ meet $AT$ at $Q$. Prove that three lines $AD,BC,PQ$ are concurrent if and only if $M, S, T,N$ are on the same circle. Đỗ Thanh Sơn

A group of tourists get on $10$ buses in the outgoing trip. The same group of tourists get on $8$ buses in the return trip. Assuming each bus carries at least $1$ tourist, prove that there are at least $3$ tourists such that each of them has taken a bus in the return trip that has more people than the bus he has taken in the outgoing trip.

For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$

Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions: (a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and (b) $\lim_{x\to \infty} f(x) = 0$.

There's a large pile of cards. On each card a number from $1,2,\ldots n$ is written. It is known that sum of all numbers on all of the cards is equal to $k\cdot n!$ for some $k$. Prove that it is possible to arrange cards into $k$ stacks so that sum of numbers written on the cards in each stack is equal to $n!$.

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.

Let $G$ be a simple, undirected, connected graph with $100$ vertices and $2013$ edges. It is given that there exist two vertices $A$ and $B$ such that it is not possible to reach $A$ from $B$ using one or two edges. We color all edges using $n$ colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of $n$.

Find the sum of (decimal) digits of the number $(10^{2021} + 2021)^2$?

Evaluate $\sum_{n=1}^{1994}{\left((-1)^{n}\cdot\left(\frac{n^2 + n + 1}{n!}\right)\right)}$ .

Let $m$ be a positive integer and $a$ and $b$ be distinct positive integers strictly greater than $m^2$ and strictly less than $m^2+m$. Find all integers $d$ such that $m^2 < d < m^2+m$ and $d$ divides $ab$.

Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times). a) Find the number of distinct real roots of the equation $f^{3}(x) = x$ b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$

A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set $\{5,6,7,8,9,10\}$