Let $ABC$ be a triangle in which $\angle BAC = 60^{\circ}$ . Let $P$ (similarly $Q$) be the point of intersection of the bisector of $\angle ABC$(similarly of $\angle ACB$) and the side $AC$(similarly $AB$). Let $r_1$ and $r_2$ be the in-radii of the triangles $ABC$ and $AP Q$, respectively. Determine the circum-radius of $APQ$ in terms of $r_1$ and $r_2$.

Let $\tau(n)$ be the number of positive divisors of a natural number $n$, and $\sigma(n)$ be their sum. Find the largest real number $\alpha$ such that \[\frac{\sigma(n)}{\tau(n)}\ge\alpha \sqrt{n}\] for all $n \ge 1$.

Suppose $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that \[2f (f (x)) = (x^2 - x)f (x) + 4 - 2x\] for all real $x$. Find $f (2)$ and all possible values of $f (1)$. For each value of $f (1)$, construct a function achieving it and satisfying the given equation.

In a lottery, a person must select six distinct numbers from $1, 2, 3,\dots, 36$ to put on a ticket. The lottery commitee will then draw six distinct numbers randomly from $1, 2, 3, \ldots, 36$. Any ticket with numbers not containing any of these $6$ numbers is a winning ticket. Show that there is a scheme of buying $9$ tickets guaranteeing at least one winning ticket, but $8$ tickets are not enough to guarantee a winning ticket in general.

Let $<a_n>$ be a sequence of non-negative real numbers such that $a_{m+n} \le a_m +a_n$ for all $m,n \in \mathbb{N}$. Prove that \[\sum_{k=1}^{N} \frac{a_k}{k^2}\ge \frac{a_N}{4N}\ln N\] for any $N \in \mathbb{N}$, where $\ln$ denotes the natural logarithm.

Let $T$ be an isosceles right triangle. Let $S$ be the circle such that the difference in the areas of $T \cup S$ and $T \cap S$ is the minimal. Prove that the centre of $S$ divides the altitude drawn on the hypotenuse of $T$ in the golden ratio (i.e., $\frac{(1 + \sqrt{5})}{2}$)

Let $X$ be the set of all positive real numbers. Find all functions $f : X \longrightarrow X$ such that \[f (x + y) \ge f (x) + yf (f (x))\] for all $x$ and $y$ in $X$.

Let $ABC$ be an acute triangle with $\angle BAC = 30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$ , respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$ , respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$ . Prove that $\angle BPC = 90^{\circ}$.

Let $C$ be a circle, $A_1 , A_2,\ldots ,A_n$ be distinct points inside $C$ and $B_1 , B_2 ,\ldots ,B_n$ be distinct points on $C$ such that no two of the segments $A_1B_1 , A_2 B_2 ,\ldots ,A_n B_n$ intersect. A grasshopper can jump from $A_r$ to $A_s$ if the line segment $A_r A_s$ does not intersect any line segment $A_t B_t (t \neq r, s)$. Prove that after a certain number of jumps, the grasshopper can jump from any $A_u$ to any $A_v$ .

For all $a, b, c > 0$ and $abc = 1$, prove that \[\frac{1}{a(a+1)+ab(ab+1)}+\frac{1}{b(b+1)+bc(bc+1)}+\frac{1}{c(c+1)+ca(ca+1)}\ge\frac{3}{4}\]

Let $(a_n )_{n\ge 1}$ be a sequence of integers that satisfies \[a_n = a_{n-1} -\text{min}(a_{n-2} , a_{n-3} )\] for all $n \ge 4$. Prove that for every positive integer $k$, there is an $n$ such that $a_n$ is divisible by $3^k$ .

A positive integer is called monotonic if when written in base $10$, the digits are weakly increasing. Thus $12226778$ is monotonic. Note that a positive integer cannot have first digit $0$. Prove that for every positive integer $n$, there is an $n$-digit monotonic number which is a perfect square.

Let $I$ be the incentre of a triangle $ABC$ and $\Gamma_a$ be the excircle opposite $A$ touching $BC$ at $D$. If $ID$ meets $\Gamma_a$ again at $S$, prove that $DS$ bisects $\angle BSC$.

For a positive integer $n$, consider the set \[S = \{0, 1, 1 + 2, 1 + 2 + 3, \ldots, 1 + 2 + 3 +\ldots + (n - 1)\}\] Prove that the elements of $S$ are mutually incongruent modulo $n$ if and only if $n$ is a power of $2$.

Let $P (x)$ be a polynomial with integer coefficients. Given that for some integer $a$ and some positive integer $n$, where \[\underbrace{P(P(\ldots P}_{\text{n times}}(a)\ldots)) = a,\] is it true that $P (P (a)) = a$?

Consider $2011^2$ points arranged in the form of a $2011 \times 2011$ grid. What is the maximum number of points that can be chosen among them so that no four of them form the vertices of either an isosceles trapezium or a rectangle whose parallel sides are parallel to the grid lines?

Let $P$ be a point inside a triangle $ABC$ such that \[\angle P AB = \angle P BC = \angle P CA\] Suppose $AP, BP, CP$ meet the circumcircles of triangles $P BC, P CA, P AB$ at $X, Y, Z$ respectively $(\neq P)$ . Prove that \[[XBC] + [Y CA] + [ZAB] \ge 3[ABC]\]

In a party among any four persons there are three people who are mutual acquaintances or mutual strangers. Prove that all the people can be separated into two groups $A$ and $B$ such that in $A$ everybody knows everybody else and in $B$ nobody knows anybody else.

Prove that, for any positive integer $n$, there exists a polynomial $p(x)$ of degree at most $n$ whose coefficients are all integers such that, $p(k)$ is divisible by $2^n$ for every even integer $k$, and $p(k) -1$ is divisible by $2^n$ for every odd integer $k$.

Let $x$ be a positive real number and let $k$ be a positive integer. Assume that $x^k+\frac{1}{x^k}$ and $x^{k+1}+\frac{1}{x^{k+1}}$ are both rational numbers. Prove that $x+\frac{1}{x}$ is also a rational number.

Construct a triangle, by straight edge and compass, if the three points where the extensions of the medians intersect the circumcircle of the triangle are given.

Let $a, b, c$ be positive integers for which \[ac = b^2 + b + 1\] Prove that the equation \[ax^2 - (2b + 1)xy + cy^2 = 1\] has an integer solution.

The seats in the Parliament of some country are arranged in a rectangle of $10$ rows of $10$ seats each. All the $100$ $MP$s have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. $4$ members at most) how much they earn. They feel a lot of envy towards each other: an $MP$ is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of $MP$s who are satisfied with their salaries?

On a circle there are $n$ red and $n$ blue arcs given in such a way that each red arc intersects each blue one. Prove that some point is contained by at least $n$ of the given coloured arcs.

Does the sequence \[11, 111, 1111, 11111, \ldots\] contain any fifth power of a positive integer? Justify your answer.

For which $n \ge 1$ is it possible to place the numbers $1, 2, \ldots, n$ in some order $(a)$ on a line segment, or $(b)$ on a circle so that for every $s$ from $1$ to $\frac{n(n+1)}{2}$, there is a connected subset of the segement or circle such that the sum of the numbers in that subset is $s$?

Let $ABC$ be a scalene triangle. Let $l_A$ be the tangent to the nine-point circle at the foot of the perpendicular from $A$ to $BC$, and let $l_A'$ be the tangent to the nine-point circle from the mid-point of $BC$. The lines $l_A$ and $l_A'$ intersect at $A'$ . Define $B'$ and $C'$ similarly. Show that the lines $AA' , BB'$ and $CC'$ are concurrent.

Let $n > 1$ be a positive integer. Find all $n$-tuples $(a_1 , a_2 ,\ldots, a_n )$ of positive integers which are pairwise distinct, pairwise coprime, and such that for each $i$ in the range $1 \le i \le n$, \[(a_1 + a_2 + \ldots + a_n )|(a_1^i + a_2^i + \ldots + a_n^i )\].

Let $a, b$ and $c$ be positive real numbers. Prove that \[\frac{\sqrt{a^2+bc}}{b+c}+\frac{\sqrt{b^2+ca}}{c+a}+\frac{\sqrt{c^2+ab}}{a+b}\ge\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\]

Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$. Let \[AO = 5, BO =6, CO = 7, DO = 8.\] If $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, determine $\frac{OM}{ON}$ .

Let $S(k)$ denote the digit-sum of a positive integer $k$(in base $10$). Determine the smallest positive integer $n$ such that \[S(n^2 ) = S(n) - 7\]

Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that $(x + y)f (x) \le x^2 + f (xy) + 110$, for all $x, y$ in $\mathbb{N}$. Determine the minimum and maximum values of $f (23) + f (2011)$.

Suppose there are $n$ boxes in a row and place $n$ balls in them one in each. The balls are colored red, blue or green. In how many ways can we place the balls subject to the condition that any box $B$ has at least one adjacent box having a ball of the same color as the ball in $B$? [Assume that balls in each color are available abundantly.]

Let $H$ be the orthocentre and $O$ be the circumcentre of an acute triangle $ABC$. Let $AD$ and $BE$ be the altitudes of the triangle with $D$ on $BC$ and $E$ on $CA$. Let $K =OD \cap BE, L = OE \cap AD$. Let $X$ be the second point of intersection of the circumcircles of triangles $HKD$ and $HLE$, and let $M$ be the midpoint of side $AB$. Prove that points $K, L, M$ are collinear if and only if $X$ is the circumcentre of triangle $EOD$.

Prove that there exist integers $a, b, c$ all greater than $2011$ such that \[(a+\sqrt{b})^c=\ldots 2010 \cdot 2011\ldots\] [Decimal point separates an integer ending in $2010$ and a decimal part beginning with $2011$.]