Let $n$ be an odd positive integer such that $\varphi (n)$ and $\varphi (n+1)$ are both powers of $2$ (here $\varphi(n)$ denotes Euler’s totient function). Prove that $n+1$ is a power of $2$ or $n = 5$.

Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$for all reals $x, y$.

Four points lie on a plane such that no three of them are collinear. Consider the four triangles formed by taking any three points at a time. If the inradii of these four triangles are all equal, prove that the four triangles are congruent.

Suppose $n$ is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least $n$. Express the $n-$th smallest number in this set in terms of $n$.

Let $I$ and $O$ be respectively the incentre and circumcentre of a triangle $ABC$. If $AB = 2$, $AC = 3$ and $\angle AIO = 90^{\circ}$, find the area of $\triangle ABC$.

Let $ABCD$ be a convex quadrilateral in which $$\angle BAC = 48^{\circ}, \angle CAD = 66^{\circ}, \angle CBD = \angle DBA.$$Prove that $\angle BDC = 24^{\circ}$.

Let $\pi (n)$ denote the largest prime divisor of $n$ for any positive integer $n > 1$. Let $q$ be an odd prime. Show that there exists a positive integer $k$ such that $$\pi \left(q^{2^k}-1\right)< \pi\left(q^{2^k}\right)<\pi \left( q^{2^k}+1\right).$$

Given a convex polygon, show that it has three consecutive vertices such that the circle through them contains the polygon.

Find a real function $f : [0,\infty)\to \mathbb R$ such that $f(2x+1) = 3f(x)+5$, for all $x$ in $[0,\infty)$.

Two triangles $ABC$ and $DEF$ have the same incircle. If a circle passes through $A,B,C,D,E$ prove that it also passes through $F$.

Consider a set of $2016$ distinct points in the plane, no four of which are collinear. Prove that there is a subset of $63$ points among them such that no three of these $63$ points are collinear.

If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x), f(x)g(x), f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3 - 3x^2 + 5$ and $x^2 - 4x$ are written on the blackboard. Can we write a nonzero polynomial of the form $x^n - 1$ after a finite number of steps? Justify your answer.

Determine all functions $f:\mathbb R\to\mathbb R$ such that for all $x, y \in \mathbb R$ $$f(xf(y) - yf(x)) = f(xy) - xy.$$

Five airlines operate in a country consisting of $36$ cities. Between any pair of cities exactly one airline operates two way flights. If some airlines operates between cities $A,B$ and $B,C$ we say that the ordered triple $A,B,C$ is properly-connected. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.

Let $f$ be a polynomial with real coefficients and suppose $f$ has no nonnegative real root. Prove that there exists a polynomial $h$ with real coefficients such that the coefficients of $fh$ are nonnegative.

Find all nonnegative integers $k, n$ which satisfy $2^{2k+1} + 9\cdot 2^k + 5 = n^2.$

The set of all positive real numbers is partitioned into three mutually disjoint non-empty subsets: $\mathbb R^+ = A \cup B\cup C$ and $A \cap B = B \cap C = C \cap A = \emptyset$ whereas none of $A, B, C$ is empty. Show that one can choose $a \in A, b \in B$ and $c \in C$ such that $a,b, c$ are the sides of a triangle. Is it always possible to choose three numbers from three different sets $A,B,C$ such that these three numbers are the sides of a right-angled triangle?

Find all $n \in \mathbb N$ such that $n = \varphi (n) + 402$, where $\varphi$ denotes the Euler phi function.

The diagonals $AD, BE$ and $CF$ of a convex hexagon concur at a point $M$. Suppose the six triangles $ABM, BCM, CDM, DEM, EFM$ and $FAM$ are all acute-angled and the circumcentre of all these triangles lie on a circle. Prove that the quadrilaterals $ABDE, BCEF$ and $CDFA$ have equal areas.

Let $n \in \mathbb N$. Prove that for each factor $m \ge n$ of $n(n + 1)/2$, one can partition the set $\{1,2, 3,\cdots , n\}$ into disjoint subsets such that the sum of elements in each subset is equal to $m$.

Is it possible to define an operation $\star$ on $\mathbb Z$ such that for any $a, b, c$ in $\mathbb Z, (a \star b) \star c = a \star (b \star c)$ holds; for any $x, y$ in $\mathbb Z, x \star x \star y = y \star x \star x=y$?

Show that there are infinitely many rational triples $(a, b, c)$ such that $$a + b + c = abc = 6.$$

Let $a$ and $k$ be positive integers. Prove that for every positive integer $d$ there exists a positive integer $n$ such that $d$ divides $ka^n + n.$

Call a non-constant polynomial real if all its coecients are real. Let $P$ and $Q$ be polynomials with complex coefficients such that the composition $P \circ Q$ is real. Show that if the leading coefficient of $Q$ and its constant term are both real, then $P$ and $Q$ are real.

Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$

For even positive integer $n$ we put all numbers $1, 2, \cdots , n^2$ into the squares of an $n \times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that it is possible to have $\frac{S_1}{S_2}=\frac{39}{64}$.

Let $A_1A_2A_3\cdots A_{10}$ be a regular decagon and $A=A_1A_4\cap A_2A_5, B=A_1A_6\cap A_2A_7, C=A_1A_9\cap A_2A_{10}.$ Find the angles of the triangle $ABC$.

Solve the equation for primes $p$ and $q$: $$p^3-q^3=pq^3-1.$$

Find all real numbers $a$ such that there exists a function $f:\mathbb R\to \mathbb R$ such that the following conditions are simultaneously satisfied: (a) $f(f(x))=xf(x)-ax,\;\forall x\in\mathbb{R};$ (b) $f$ is not a constant function; (c) $f$ takes the value $a$.

Consider a $2n\times 2n$ chessboard with all the $4n^2$ cells being white to start with. The following operation is allowed to be performed any number of times: "Three consecutive cells (in a row or column) are recolored (a white cell is colored black and a black cell is colored white)." Find all possible values of $n\ge 2$ for which using the above operation one can obtain the normal chess coloring of the given board.

A real polynomial of odd degree has all positive coefficients. Prove that there is a (possibly trivial) permutation of the coefficients such that the resulting polynomial has exactly one real zero.

Let $K$ and $L$ be the centers of the excircles of a non-isosceles triangle $ABC$ opposite $B$ and $C$ respectively. Let $M$ and $N$ be points in the plane of the triangle such that $BM$ bisects $AC$ and $CN$ bisects $AB$. Prove that the lines $KM$ and $NK$ meet on $BC$. NoteThe problem in its current formulation is trivially wrong. No possible rectification is known to OP / was sent to the participants.