A sequence of integers $(a_n)$ satisfies $a_{n+1} = a_n^3 + 1999$ for $n = 1,2,....$ Prove that there exists at most one $n$ for which $a_n$ is a perfect square.

Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$. Prove that: a) $(AB)^2 = AB.$ b) $(AB - BA)^3 = O_n.$

For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$?

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

Given is a natural number $n \geq 3$. What is the smallest possible value of $k$ if the following statements are true? For every $n$ points $ A_i = (x_i, y_i) $ on a plane, where no three points are collinear, and for any real numbers $ c_i$ ($1 \le i \le n$) there exists such polynomial $P(x, y)$, the degree of which is no more than $k$, where $ P(x_i, y_i) = c_i $ for every $i = 1, \dots, n$. (The degree of a nonzero monomial $ a_{i,j} x^{i}y^{j} $ is $i+j$, while the degree of polynomial $P(x, y)$ is the greatest degree of the degrees of its monomials.)

Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$

Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.

Let $I$ be the center of inscribed circle of right triangle $\Delta ABC$ with $\angle A = 90$ and point $M$ is the midpoint of $(BC)$.The bisector of $\angle BAC$ intersects the circumcircle of $\Delta ABC $ in point $W$.Point $U$ is situated on the line $AB$ such that the lines $AB$ and $WU$ are perpendiculars.Point $P$ is situated on the line $WU$ such that the lines $PI$ and $WU$ are perpendiculars.Prove that the line $MP$ bisects the segment $CI$.

Let $n$ be a positive integer. Aavid has a card deck consisting of $ 2n$ cards, each colored with one of $n$ colors such that every color is on exactly two of the cards. The $2n$ cards are randomly ordered in a stack. Every second, he removes the top card from the stack and places the card into an area called the pit. If the other card of that color also happens to be in the pit, Aavid collects both cards of that color and discards them from the pit. Of the $(2n)!$ possible original orderings of the deck, determine how many have the following property: at every point, the pit contains cards of at most two distinct colors.

Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$. [i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]

Five men and seven women stand in a line in random order. Let m and n be relatively prime positive integers so that $\tfrac{m}{n}$ is the probability that each man stands next to at least one woman. Find $m + n.$

We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?

Find the value of \[+1+2+3-4-5-6+7+8+9-10-11-12+\cdots -2020,\] where the sign alternates between $+$ and $-$ after every three numbers.

Determine all positive integers $n \ge 2$ that satisfy the following condition; For all integers $a, b$ relatively prime to $n$, \[a \equiv b \; \pmod{n}\Longleftrightarrow ab \equiv 1 \; \pmod{n}.\]

A box contains answer $4032$ scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of the scripts and keep the other outside. If there is one script with even score and the other with odd score, we put back the script with the odd score and keep the other script outside. After following this procedure a number of times, there are 3 scripts left among which there is at least one script each with odd and even scores. Find, with proof, the number of scripts with odd scores among the three left.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

1.14. Let P and Q be interior points of triangle ABC such that \ACP = \BCQ and \CAP = \BAQ. Denote by D;E and F the feet of the perpendiculars from P to the lines BC, CA and AB, respectively. Prove that if \DEF = 90, then Q is the orthocenter of triangle BDF.

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.

Prove that any integer $n$ satisfying the inequality $n <(44 + \sqrt{1975})^100 <n + 1$ is odd.

Let $ABC$ be an acute angled triangle (with $AB<AC<BC$) inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ (with center $A$ and radius $AB$) intersects side $BC$ at point $D$ and the circumcircle $c(O,R)$ at point $E$. Prove that side $AC$ bisects angle $\angle DAE$.

Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

Let $a, b, c, d$ be real numbers. Prove that is $$(a^2 + b^2 + 1)(c^2 + d^2 + 1) \ge 2(a + c)(b + d).$$

In a $10\times 10$ table, positive numbers are written. It is known that, looking left-right, the numbers in each row form an arithmetic progression and, looking up-down, the numbers is each column form a geometric progression. Prove that all the ratios of the geometric progressions are equal.

Given an integer $n \geq 3$, determine the maximum value of product of $n$ non-negative real numbers $x_1,x_2, \ldots , x_n$ when subjected to the condition \begin{align*} \sum_{k=1}^n \frac{x_k}{1+x_k} =1 \end{align*}

A tournament is held among $n$ teams, following such rules: a) every team plays all others once at home and once away.(i.e. double round-robin schedule) b) each team may participate in several away games in a week(from Sunday to Saturday). c) there is no away game arrangement for a team, if it has a home game in the same week. If the tournament finishes in 4 weeks, determine the maximum value of $n$.