Let $d \geq 2$ be a positive integer. Define the sequence $a_1,a_2,\dots$ by $$a_1=1 \ \text{and} \ a_{n+1}=a_n^d+1 \ \text{for all }n\geq 1.$$ Determine all pairs of positive integers $(p, q)$ such that $a_p$ divides $a_q$.

Let M = 7!.8!.9!.10!.11!.12!. How many factors of M are perfect squares ?

Let $n$ be a natural number. A tiling of a $2n \times 2n$ board is a placing of $2n^2$ dominos (of size $2 \times 1$ or $1 \times 2$) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two [i]sepearate tilings[/i] of a $2n \times 2n$ board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours. Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that $n$ is divisible by $3$ [i] Proposed by Tejaswi Navilarekallu [/i]

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. [i](IMO Problem 3)[/i] [i]Proposed by Germany, FR[/i]

If the base of a rectangle is increased by $ 10\%$ and the area is unchanged, then the altitude is decreased by: $ \textbf{(A)}\ 9\% \qquad\textbf{(B)}\ 10\% \qquad\textbf{(C)}\ 11\% \qquad\textbf{(D)}\ 11\frac {1}{9}\% \qquad\textbf{(E)}\ 9\frac {1}{11}\%$

Convex pentagon $ABCDE$ has the property that $\angle ADB = 20^o$, $\angle BEC = 16^o$, $\angle CAD = 3^o$,and $\angle DBE = 12^o$. What is the measure of $\angle ECA$?

Let $n$ be a positive integer. From the set $A=\{1,2,3,...,n\}$ an element is eliminated. What's the smallest possible cardinal of $A$ and the eliminated element, since the arithmetic mean of left elements in $A$ is $\frac{439}{13}$.

Given $n$ nonintersecting segments in the plane. Not a pair of those belong to the same straight line. We want to add several segments, connecting the ends of given ones, to obtain one nonselfintersecting broken line. Is it always possible?

Let there be a triangle $\triangle ABC$ with $BC=a$, $CA=b$, $AB=c$. Let $T$ be a point not inside $\triangle ABC$ and on the same side of $A$ with respect to $BC$, such that $BT-CT=c-b$. Let $n=BT$ and $m=CT$. Find the point $P$ that minimizes $f(P)=-a \cdot AP + m \cdot BP + n \cdot CP$.

Compare the number of distinct prime divisors of $200^2 \cdot 201^2 \cdot ... \cdot 900^2$ and $(200^2 -1)(201^2 -1)\cdot ... \cdot (900^2 -1) .$

Given a convex pentagon $ ABCDE. $ Let $ A_1 $ be the intersection of $ BD $ with $ CE $ and define $ B_1, C_1, D_1, E_1 $ similarly, $ A_2 $ be the second intersection of $ \odot (ABD_1),\odot (AEC_1) $ and define $ B_2, C_2, D_2, E_2 $ similarly. Prove that $ AA_2, BB_2, CC_2, DD_2, EE_2 $ are concurrent. [i]Proposed by Telv Cohl[/i]

Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.

$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$

Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$.

Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.

Among all possible representations of the positive integer $ n$ as $ n\equal{}\sum_{i\equal{}1}^k a_i$ with positive integers $ k, a_1 < a_2 < ...<a_k$, when will the product $ \prod_{i\equal{}1}^k a_i$ be maximum?

If $p (x) = c_0 + c_1x + c_2x^2 + c_3x^3$ is a polynomial with integer coefficients with $a, b,c$ integers and different from each other, prove that it cannot happen simultaneously that $p (a) = b$, $p (b) = c$ and $p (c) = a$.

The famous conjecture of Goldbach is the assertion that every even integer greater than $2$ is the sum of two primes. Except $2$, $4$, and $6$, every even integer is a sum of two positive composite integers: $n=4+(n-4)$. What is the largest positive even integer that is not a sum of two odd composite integers?

On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.

A honeybug crawls along the honeycombs with the unite length of their hexagons. He has moved from the node $A$ to the node $B$ along the shortest possible trajectory. Prove that the half of his way he moved in one direction.

Is it possible that by using the following transformations: \[ f(x) \mapsto x^2 \cdot f \left(\frac{1}{x}+1 \right) \ \ \ \text{or} \ \ \ f(x) \mapsto (x-1)^2 \cdot f\left(\frac{1}{x-1} \right)\] the function $f(x)=x^2+5x+4$ is sent to the function $g(x)=x^2+10x+8$ ?

In English class, you have discovered a mysterious phenomenon -- if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$, but if you spend $35$ minutes on the essay you somehow do not earn any points. It is 4AM, your English class starts at 8:05AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores? [i]Proposed by Evan Chen[/i]

Georgina calls a $992$-element subset $A$ of the set $S = \{1, 2, 3, \ldots , 1984\}$ a [b]halfthink set[/b] if [list] [*] the sum of the elements in $A$ is equal to half of the sum of the elements in $S$, and [*] exactly one pair of elements in $A$ differs by $1$. [/list] She notices that for some values of $n$, with $n$ a positive integer between $1$ and $1983$, inclusive, there are no halfthink sets containing both $n$ and $n+1$. Find the last three digits of the product of all possible values of $n$. [i]Proposed by Andrew Wu and Jason Wang[/i] (Note: wording changed from original to specify what $n$ can be.)

In a circle with centre at $O$ and diameter $AB$, two chords $BD$ and $AC$ intersect at $E$. $F$ is a point on $AB$ such that $EF \perp AB$. $FC$ intersects $BD$ in $G$. If $DE = 5$ and $EG =3$, determine $BG$.

Prove that for all non-negative numbers $x,y,z$ satisfying $x+y+z=1$, one has \[1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.\]