Let $n \in \mathbb{N}$ be such that $gcd(n, 6) = 1$. Let $a_1 < a_2 < \cdots < a_n$ and $b_1 < b_2 < \cdots < b_n$ be two collection of positive integers such that $a_j + a_k + a_l = b_j + b_k + b_l$ for all integers $1 \le j < k < l \le n$. Prove that $a_j = b_j$ for all $1 \le j \le n$.

Find all functions $f: \mathbb{Q} \to \mathbb{R}$ such that $f(xy)=f(x)f(y)+f(x+y)-1$ for all rationals $x,y$

Let $a,b,c \in \mathbb{R^+}$ such that $abc=1$. Prove that $$\sum_{a,b,c} \sqrt{\frac{a}{a+8}} \ge 1$$

Let $ABCD$ be a convex quadrilateral. Construct equilateral triangles $AQB$, $BRC$, $CSD$ and $DPA$ externally on the sides $AB$, $BC$, $CD$ and $DA$ respectively. Let $K, L, M, N$ be the mid-points of $P Q, QR, RS, SP$. Find the maximum value of $$\frac{KM + LN}{AC + BD}$$ .

For each point $X$ in the plane, a real number $r_X > 0$ is assigned such that $2|r_X - r_Y | \le |XY |$, for any two points $X, Y$ . (Here $|XY |$ denotes the distance between $X$ and $Y$) A frog can jump from $X$ to $Y$ if $r_X = |XY |$. Show that for any two points $X$ and $Y$ , the frog can jump from $X$ to $Y$ in a finite number of steps.

$O$ is the centre of the circumcircle of triangle $ABC$, and $M$ is its orthocentre. Point $A$ is reflected in the perpendicular bisector of the side $BC$,$ B$ is reflected in the perpendicular bisector of the side $CA$, and finally $C$ is reflected in the perpendicular bisector of the side $AB$. The images are denoted by $A_1, B_1, C_1$ respectively. Let $K$ be the centre of the inscribed circle of triangle $A_1B_1C_1$. Prove that $O$ bisects the line segment $MK$.

Given $2015$ points in the plane, show that if every four of them form a convex quadrilateral then the points are the vertices of a convex $2015-$sided polygon.

Let $a$ and $n$ denote positive integers such that $n|a^n-1$. Prove that the numbers $a+1,a^2+2, \cdots a^n+n$ all leave different remainders when divided by $n$.

For every positive integer$ n$, let $P(n)$ be the greatest prime divisor of $n^2+1$. Show that there are infinitely many quadruples $(a, b, c, d)$ of positive integers that satisfy $a < b < c < d$ and $P(a) = P(b) = P(c) = P(d)$.

Let $ABCD$ be a convex quadrilateral. In the triangle $ABC$ let $I$ and $J$ be the incenter and the excenter opposite to vertex $A$, respectively. In the triangle $ACD$ let $K$ and $L$ be the incenter and the excenter opposite to vertex $A$, respectively. Show that the lines $IL$ and $JK$, and the bisector of the angle $BCD$ are concurrent.

Let $f:\mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}$ be defined by $f(0)=0$, $$f(2n+1)=2f(n)$$for $n \ge 0$ and $$f(2n)=2f(n)+1$$for $n \ge 1$ If $g(n)=f(f(n))$, prove that $g(n-g(n))=0$ for all $n \ge 0$.

Find all pairs of cubic equations $x^3+ax^2+bx+c=0$ and $x^3+bx^2+ax+c=0$ where $a, b$ are positive integers, $c\neq 0$ is an integer, such that each equation has three integer roots and exactly one of these three roots is common to both the equations.

Show that there are no positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ such that $$(1+a_1 \omega)(1+a_2 \omega)(1+a_3 \omega)(1+a_4 \omega)(1+a_5 \omega)(1+a_6 \omega)$$is an integer where $\omega$ is an imaginary $5$th root of unity.

For $ n \in \mathbb{N}$, let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n > 1$, the product $s(n - 1)s(n)s(n + 1)$ is an even number.

Suppose a $m \times m$ square can be divided into $7$ rectangles such that no two rectangles have a common interior point and the side-lengths of the rectangles form the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \}$. Find the maximum value of $m$.

Let $k \in \mathbb{N}$, let $x_k$ denote the nearest integer to $\sqrt k$. Show that for each $m \in \mathbb {N}$, $$\sum_{k=1}^{m} \frac{1}{x_k} = f(m)+ \frac{m}{f(m)+1}$$, where $f(m)$ is the integer part of $\frac{\sqrt{4m-3}-1}{2}$

A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and $WY$ meet $AB$ at points $N$ and $M$. Prove that the length of segment $NM$ doesn’t depend on point $C$.

Suppose $a,b,c\in[0,2]$ and $a+b+c=3$. Find the maximal and minimal value of the expression $$\sqrt{a(b+1)}+\sqrt{b(c+1)}+\sqrt{c(a+1)}.$$

Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$

Find all positive integer $n$ such that $$\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1$$holds for all $\theta$ which are not integral multiples of $\frac{\pi}{2}$

Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) = (x^2 + x + 1)^n. \]

Let $A$ be a non empty subset of positive reals such that for every $a,b,c \in A$, the number $ab+bc+ca$ is rational. Prove that $\frac{a}{b}$ is a rational for every $a,b \in A$.

Let $ABC$ be at triangle with incircle $\Gamma$. Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three circles inside $\triangle ABC$ each of which is tangent to $\Gamma$ and two sides of the triangle and their radii are $1,4,9$. Find the radius of $\Gamma$.

Let $S$ be a set of in $3-$ space such that each of the points in $S$ has integer coordinates $(x,y,z)$ with $1 \le x,y,z \le n $. Suppose the pairwise distances between these points are all distinct. Prove that $$|S| \le min \{(n+2)\sqrt{\frac{n}{3}},n\sqrt{6} \}$$

Show that there are infinitely many natural numbers which are simultaneously a sum of two squares and a sum of two cubes but which are not a sum of two $6-$th powers.

Find all real polynomials $P(x)$ that satisfy $$P(x^3-2)=P(x)^3-2$$

Prove that there exists a real number $C > 1$ with the following property. Whenever $n > 1$ and $a_0 < a_1 < a_2 <\cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1} \cdots \frac{1}{a_n}$ form an arithmetic progression, then $a_0 > C^n$.

Let $n\ge2$ and let $p(x)=x^n+a_{n-1}x^{n-1} \cdots a_1x+a_0$ be a polynomial with real coefficients. Prove that if for some positive integer $k(<n)$ the polynomial $(x-1)^{k+1}$ divides $p(x)$ then $$\sum_{i=0}^{n-1}|a_i| \ge 1 +\frac{2k^2}{n}$$

The sequence $<a_n>$ is defined as follows, $a_1=a_2=1$, $a_3=2$, $$a_{n+3}=\frac{a_{n+2}a_{n+1}+n!}{a_n},$$$n \ge 1$. Prove that all the terms in the sequence are integers.

Prove that there exists a set of infinitely many positive integers such that the elements of no finite subset of this set add up to a perfect square.