For each integer $n \geq 2$, let $F(n)$ denote the greatest prime factor of $n$. A [i]strange pair[/i] is a pair of distinct primes $p$ and $q$ such that there is no integer $n \geq 2$ for which $F(n)F(n+1)=pq$. Prove that there exist infinitely many strange pairs.

A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?

In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$

Find all positive integers $(a,b,c)$ such that $$ab-c,\quad bc-a,\quad ca-b$$ are all powers of $2$. [i]Proposed by Serbia[/i]

Let $ABC$ be a triangle. A circle is tangent to sides $AB, AC$ and to the circumcircle of $ABC$ (internally) at points $P, Q, R$ respectively. Let $S$ be the point where $AR$ meets $PQ$. Show that \[\angle{SBA}\equiv \angle{SCA}\]

Let $100$ discs lie on the plane in such a way that each two of them have a common point. Prove that there exists a point lying inside at least $15$ of these discs. (M. Kharitonov, A. Polyansky)

Farmer John has a (flexible) fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he call enclose an maximum possible area of $A_0$, and when he uses one of the walls or both walls, he gets a maximum of area of $A_1$ and $A_2$ respectively. If $n=\frac{A_1}{A_0}+\frac{A_2}{A_1}$, find $\lfloor 1000n\rfloor$. [i]Proposed by Yannick Yao[/i]

There are $13$ marked points on the circumference of a circle with radius $13$. Prove that we can choose three of the marked points which form a triangle with area less than $13$.

Among a set of $2N$ coins, all identical in appearance, $2N - 2$ are real and $2$ are fake. Any two real coins have the same weight . The fake coins have the same weight, which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most $4$ times, if (a) $N = 16$, ( b ) $N = 11$ ? (A Shapovalov)

Let a simple polygon be defined as a polygon in which no consecutive sides are parallel and no two non-consecutive sides share a common point. Given that all vertices of a simple polygon $P$ are lattice points (in a Cartesian coordinate system, each vertex has integer coordinates), and each side of $P$ has integer length, prove that the perimeter must be even.

Let $ a$ and $ k$ be positive integers. Let $ a_i$ be the sequence defined by $ a_1 \equal{} a$ and $ a_{n \plus{} 1} \equal{} a_n \plus{} k\pi(a_n)$ where $ \pi(x)$ is the product of the digits of $ x$ (written in base ten) Prove that we can choose $ a$ and $ k$ such that the infinite sequence $ a_i$ contains exactly $ 100$ distinct terms

Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p \plus{} b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$.

The integer $n$, between 10000 and 99999, is $abcde$ when written in decimal notation. The digit $a$ is the remainder when $n$ is divided by 2, the digit $b$ is the remainder when $n$ is divided by 3, the digit $c$ is the remainder when $n$ is divided by 4, the digit $d$ is the remainder when $n$ is divied by 5, and the digit $e$ is the reminader when $n$ is divided by 6. Find $n$.

Let $A, B, C, D, E$ be five points on a circle such that $|AB| = |CD|$ and $|BC| = |DE|$. The segments $AD$ and $BE$ intersect at $F$. Let $M$ denote the midpoint of segment $CD$. Prove that the circle of center $M$ and radius $ME$ passes through the midpoint of segment $AF$.

Prove that $2^{147} - 1$ is divisible by $343.$

Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Let $n>1 \in \mathbb{N}$ and $a_1, a_2, ..., a_n$ be a sequence of $n$ natural integers. Let: $$b_1 = \left[\frac{a_2 + \cdots + a_n}{n-1}\right], b_i = \left[\frac{a_1 + \cdots + a_{i-1} + a_{i+1} + \cdots + a_n}{n-1}\right], b_n = \left[\frac{a_1 + \cdots + a_{n-1}}{n-1}\right]$$ Define a mapping $f$ by $f(a_1,a_2, \cdots a_n) = (b_1,b_2,\cdots,b_n)$. a) Let $g: \mathbb{N} \to \mathbb{N}$ be a function such that $g(1)$ is the number of different elements in $f(a_1,a_2, \cdots a_n)$ and $g(m)$ is the number od different elements in $f^m(a_1,a_2, \cdots a_n) = f(f^{m-1}(a_1,a_2, \cdots a_n)); m>1$. Prove that $\exists k_0 \in \mathbb{N}$ s.t. for $m \ge k_0$ the function $g(m)$ is periodic. b) Prove that $\sum_{m=1}^k \frac{g(m)}{m(m+1)} < C$ for all $k \in \mathbb{N}$, where $C$ is a function that doesn't depend on $k$.

The $x$ and $y$ figures satisfy a condition: for every $n\ge1$ the number $$xx...x6yy...y4$$ ($n$ times $x$ and $n$ times $y$) is a perfect square. Find all possible $x$ and $y$.

How many $4 \times 4$ arrays whose entries are $0$s and $1$s are there such that the row sums (the sum of the entries in each row) are $1,2,3,$ and $4,$ in some order, and the column sums (the sum of the entries in each column) are also $1,2,3,$ and $4,$ in some order? For example, the array $\begin{bmatrix} 1 & 1 & 1 & 0\\ 0 & 1 & 1 & 0\\ 1 & 1 & 1 & 1\\ 0 & 1 & 0 & 0 \end{bmatrix}$ satisfies the condition. $\textbf{(A)}144~\textbf{(B)}240~\textbf{(C)}336~\textbf{(D)}576~\textbf{(E)}624$

In a convex quadrilateral $ABCD$, angles $B$ and $D$ are right angles. On on sides $AB$, $BC$, $CD$, $DA$ points $K$, $L$, $M$, $N$ are taken respectively so that $KN \perp AC$ and $LM \perp AC$. Prove that $KM$, $LN$ and $AC$ intersect at one point.

Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number. (The triangular numbers are the $t_n = n(n+1)/2$ with $n$ in $\{0,1,2,\dots\}$.)

In a family album, there are ten photos. On each of them, three people are pictured: in the middle stands a man, to the right of him stands his brother, and to the left of him stands his son. What is the least possible total number of people pictured, if all ten of the people standing in the middle of the ten pictures are different.

Let $n \ge 2$ be a positive integer. Suppose $a_1, a_2, \dots, a_n$ are distinct integers. For $k = 1, 2, \dots, n$, let \[ s_k := \prod_{\substack{i \not= k, \\ 1 \le i \le n}} |a_k - a_i|, \] i.e. $s_k$ is the product of all terms of the form $|a_k - a_i|$, where $i \in \{ 1, 2, \dots, n \}$ and $i \not= k$. Find the largest positive integer $M$ such that $M$ divides the least common multiple of $s_1, s_2, \dots, s_n$ for any choices of $a_1, a_2, \dots, a_n$.

We have a machine that has an input and an output. The input is a letter from the finite set $I$ and the output is a lamp that at each moment has one of the colors of the set $C=\{c_1,\dots,c_p\}$. At each moment the machine has an inner state that is one of the $n$ members of finite set $S$. The function $o: S \rightarrow C$ is a surjective function defining that at each state, what color must the lamp be, and the function $t:S \times I \rightarrow S$ is a function defining how does giving each input at each state changes the state. We only shall see the lamp and we have no direct information from the state of the car at current moment. In other words a machine is $M=(S,I,C,o,t)$ such that $S,I,C$ are finite, $t:S \times I \rightarrow S$ , and $o:S \rightarrow C$ is surjective. It is guaranteed that for each two different inner states, there's a sequence of inputs such that the color of the lamp after giving the sequence to the machine at the first state is different from the color of the lamp after giving the sequence to the machine at the second state. (a) The machine $M$ has $n$ different inner states. Prove that for each two different inner states, there's a sequence of inputs of length no more than $n-p$ such that the color of the lamp after giving the sequence to the machine at the first state is different from the color of the lamp after giving the sequence to the machine at the second state. (b) Prove that for a machine $M$ with $n$ different inner states, there exists an algorithm with no more than $n^2$ inputs that starting at any unknown inner state, at the end of the algorithm the state of the machine at that moment is known. Can you prove the above claim for $\frac{n^2}{2}$?