Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 India IMO Training Camp
Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that

$$a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019$$Prove that

$$5m+12n\le 581.$$
2019 India IMO Training Camp
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$
2019 India IMO Training Camp
Let $n\ge 2$ be an integer. Solve in reals:

$$|a_1-a_2|=2|a_2-a_3|=3|a_3-a_4|=\cdots=n|a_n-a_1|.$$
2019 India IMO Training Camp
Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.
2019 India IMO Training Camp
Determine all positive integers $m$ satisfying the condition that there exists a unique positive integer $n$ such that there exists a rectangle which can be decomposed into $n$ congruent squares and can also be decomposed into $m+n$ congruent squares.
2019 India IMO Training Camp
There are $2019$ coins on a table. Some are placed with head up and others tail up. A group of $2019$ persons perform the following operations: the first person chooses any one coin and then turns it over, the second person choses any two coins and turns them over and so on and the $2019$-th person turns over all the coins. Prove that no matter which sides the coins are up initially, the $2019$ persons can come up with a procedure for turning the coins such that all the coins have smae side up at the end of the operations.
2018 India IMO Training Camp
Let $\Delta ABC$ be an acute triangle. $D,E,F$ are the touch points of incircle with $BC,CA,AB$ respectively. $AD,BE,CF$ intersect incircle at $K,L,M$ respectively. If,$$\sigma = \frac{AK}{KD} + \frac{BL}{LE} + \frac{CM}{MF}$$$$\tau = \frac{AK}{KD}.\frac{BL}{LE}.\frac{CM}{MF}$$Then prove that $\tau = \frac{R}{16r}$. Also prove that there exists integers $u,v,w$ such that, $uvw \neq 0$, $u\sigma + v\tau +w=0$.
2018 India IMO Training Camp
A $10$ digit number is called interesting if its digits are distinct and is divisible by $11111$. Then find the number of interesting numbers.
2018 India IMO Training Camp
Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$$$b_{n+1}= b_n + \frac{1}{2a_n}$$for all $n\in\mathbb N$.

Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.
2018 India IMO Training Camp
Let $ABCD$ be a convex quadrilateral inscribed in a circle with center $O$ which does not lie on either diagonal. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.
2018 India IMO Training Camp
For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that

(a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$

(b) $\sum_{k=1}^n (a_k+b_k)=2n;$

(c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$
2018 India IMO Training Camp
A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the interior of the polygon, in such a way that all the resulting triangles have vertices of all three colours.
2018 India IMO Training Camp
Let $A,B,C$ be three points in that order on a line $\ell$ in the plane, and suppose $AB>BC$. Draw semicircles $\Gamma_1$ and $\Gamma_2$ respectively with $AB$ and $BC$ as diameters, both on the same side of $\ell$. Let the common tangent to $\Gamma_1$ and $\Gamma_2$ touch them respectively at $P$ and $Q$, $P\ne Q$. Let $D$ and $E$ be points on the segment $PQ$ such that the semicircle $\Gamma_3$ with $DE$ as diameter touches $\Gamma_2$ in $S$ and $\Gamma_1$ in $T$.



Prove that $A,C,S,T$ are concyclic.

Prove that $A,C,D,E$ are concyclic.
2018 India IMO Training Camp
Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.
2018 India IMO Training Camp
For a natural number $k>1$, define $S_k$ to be the set of all triplets $(n,a,b)$ of natural numbers, with $n$ odd and $\gcd (a,b)=1$, such that $a+b=k$ and $n$ divides $a^n+b^n$. Find all values of $k$ for which $S_k$ is finite.
2018 India IMO Training Camp
Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.
2018 India IMO Training Camp
Let $ABC$ be a triangle and $AD,BE,CF$ be cevians concurrent at a point $P$. Suppose each of the quadrilaterals $PDCE,PEAF$ and $PFBD$ has both circumcircle and incircle. Prove that $ABC$ is equilateral and $P$ coincides with the center of the triangle.
2018 India IMO Training Camp
Let $n\ge 2$ be a natural number. Let $a_1\le a_2\le a_3\le \cdots \le a_n$ be real numbers such that $a_1+a_2+\cdots +a_n>0$ and $n(a_1^2+a_2^2+\cdots +a_n^2)=2(a_1+a_2+\cdots +a_n)^2.$ If $m=\lfloor n/2\rfloor+1$, the smallest integer larger than $n/2$, then show that $a_m>0.$
2017 India IMO Training Camp
Let $P_c(x)=x^4+ax^3+bx^2+cx+1$ and $Q_c(x)=x^4+cx^3+bx^2+ax+1$ with $a,b$ real numbers, $c \in \{1,2, \dots, 2017\}$ an integer and $a \ne c$. Define $A_c=\{\alpha | P_c(\alpha)=0\}$ and $B_c=\{\beta | P(\beta)=0\}$.



(a) Find the number of unordered pairs of polynomials $P_c(x), Q_c(x)$ with exactly two common roots.



(b) For any $1 \le c \le 2017$, find the sum of the elements of $A_c \Delta B_c$.
2017 India IMO Training Camp
Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$
2017 India IMO Training Camp
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively.



(a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic.



(b) If $R$ is the radius of $\Omega$ and $d$ is the distance between its centre and $P$, then find the radius of the circle in (a) in terms of $R$ and $d$.
2017 India IMO Training Camp
In an acute triangle $ABC$, points $D$ and $E$ lie on side $BC$ with $BD<BE$. Let $O_1, O_2, O_3, O_4, O_5, O_6$ be the circumcenters of triangles $ABD, ADE, AEC, ABE, ADC, ABC$, respectively. Prove that $O_1, O_3, O_4, O_5$ are con-cyclic if and only if $A, O_2, O_6$ are collinear.
2017 India IMO Training Camp
Let $a,b,c,d$ be pairwise distinct positive integers such that $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$is an integer. Prove that $a+b+c+d$ is not a prime number.
2017 India IMO Training Camp
There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either on or off. Every second, each lamp undergoes a change according to the following rule:



(a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is off right now. (Indices taken mod $n$.)



(b) Otherwise, $L_i$ is on right now.



Initially, all the lamps are off, except for $L_1$ which is on. Prove that for infinitely many integers $n$ all the lamps will be off eventually, after a finite amount of time.
2017 India IMO Training Camp
Let $a,b,c$ be distinct positive real numbers with $abc=1$. Prove that $$\sum_{\text{cyc}} \frac{a^6}{(a-b)(a-c)}>15.$$
2017 India IMO Training Camp
Define a sequence of integers $a_0=m, a_1=n$ and $a_{k+1}=4a_k-5a_{k-1}$ for all $k \ge 1$. Suppose $p>5$ is a prime with $p \equiv 1 \pmod{4}$. Prove that it is possible to choose $m,n$ such that $p \nmid a_k$ for any $k \ge 0$.
2017 India IMO Training Camp
Let $n \ge 1$ be a positive integer. An $n \times n$ matrix is called good if each entry is a non-negative integer, the sum of entries in each row and each column is equal. A permutation matrix is an $n \times n$ matrix consisting of $n$ ones and $n(n-1)$ zeroes such that each row and each column has exactly one non-zero entry.



Prove that any good matrix is a sum of finitely many permutation matrices.
2017 India IMO Training Camp
Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.
2017 India IMO Training Camp
Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$
2017 India IMO Training Camp
Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.
2017 India IMO Training Camp
For each $n \ge 2$ define the polynomial $$f_n(x)=x^n-x^{n-1}-\dots-x-1.$$Prove that



(a) For each $n \ge 2$, $f_n(x)=0$ has a unique positive real root $\alpha_n$;



(b) $(\alpha_n)_n$ is a strictly increasing sequence;



(c) $\lim_{n \rightarrow \infty} \alpha_n=2.$
2017 India IMO Training Camp
Let $a$ be a positive integer which is not a perfect square, and consider the equation $$k = \frac{x^2-a}{x^2-y^2}.$$Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
2016 India IMO Training Camp
Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit.
2016 India IMO Training Camp
We say a natural number $n$ is perfect if the sum of all the positive divisors of $n$ is equal to $2n$. For example, $6$ is perfect since its positive divisors $1,2,3,6$ add up to $12=2\times 6$. Show that an odd perfect number has at least $3$ distinct prime divisors.



Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result.
2016 India IMO Training Camp
Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$for all reals $x,y$.
2016 India IMO Training Camp
An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A move consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order.

[asy] size(3cm);

pair A=(0,0),D=(1,0),B,C,E,F,G,H,I;

G=rotate(60,A)*D;

B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A;

draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]
2016 India IMO Training Camp
Let $n$ be a natural number. A sequence $x_1,x_2, \cdots ,x_{n^2}$ of $n^2$ numbers is called $n-\textit{good}$ if each $x_i$ is an element of the set $\{1,2,\cdots ,n\}$ and the ordered pairs $\left(x_i,x_{i+1}\right)$ are all different for $i=1,2,3,\cdots ,n^2$ (here we consider the subscripts modulo $n^2$). Two $n-$good sequences $x_1,x_2,\cdots ,x_{n^2}$ and $y_1,y_2,\cdots ,y_{n^2}$ are called $\textit{similar}$ if there exists an integer $k$ such that $y_i=x_{i+k}$ for all $i=1,2,\cdots,n^2$ (again taking subscripts modulo $n^2$). Suppose that there exists a non-trivial permutation (i.e., a permutation which is different from the identity permutation) $\sigma$ of $\{1,2,\cdots ,n\}$ and an $n-$ good sequence $x_1,x_2,\cdots,x_{n^2}$ which is similar to $\sigma\left(x_1\right),\sigma\left(x_2\right),\cdots ,\sigma\left(x_{n^2}\right)$. Show that $n\equiv 2\pmod{4}$.
2016 India IMO Training Camp
Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.
2016 India IMO Training Camp
Let $n$ be a natural number. We define sequences $\langle a_i\rangle$ and $\langle b_i\rangle$ of integers as follows. We let $a_0=1$ and $b_0=n$. For $i>0$, we let $$\left( a_i,b_i\right)=\begin{cases} \left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\right) & \text{if } a_{i-1}<b_{i-1},\\
\left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\right) & \text{if } a_{i-1}>b_{i-1},\\
\left(a_{i-1},b_{i-1}\right) & \text{if } a_{i-1}=b_{i-1}.\end{cases}$$Given that $a_k=b_k$ for some natural number $k$, prove that $n+3$ is a power of two.
2016 India IMO Training Camp
Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.
2016 India IMO Training Camp
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.
2016 India IMO Training Camp
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)
2016 India IMO Training Camp
Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that



$A\cap B=\{1\};$

every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$;

each prime number is a divisor of some number in $A$ and also some number in $B$;

one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$.

Each set has infinitely many composite numbers.
2015 India IMO Training Camp
Find all positive integers $a,b$ such that $\frac{a^2+b}{b^2-a}$ and $\frac{b^2+a}{a^2-b}$ are also integers.
2015 India IMO Training Camp
A $10$-digit number is called a $\textit{cute}$ number if its digits belong to the set $\{1,2,3\}$ and the difference of every pair of consecutive digits is $1$.

a) Find the total number of cute numbers.

b) Prove that the sum of all cute numbers is divisibel by $1408$.
2015 India IMO Training Camp
Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.
2015 India IMO Training Camp
Let $ABC$ be a triangle in which $CA>BC>AB$. Let $H$ be its orthocentre and $O$ its circumcentre. Let $D$ and $E$ be respectively the midpoints of the arc $AB$ not containing $C$ and arc $AC$ not containing $B$. Let $D'$ and $E'$ be respectively the reflections of $D$ in $AB$ and $E$ in $AC$. Prove that $O, H, D', E'$ lie on a circle if and only if $A, D', E'$ are collinear.
2015 India IMO Training Camp
For a composite number $n$, let $d_n$ denote its largest proper divisor. Show that there are infinitely many $n$ for which $d_n +d_{n+1}$ is a perfect square.
2015 India IMO Training Camp
Every cell of a $3\times 3$ board is coloured either by red or blue. Find the number of all colorings in which there are no $2\times 2$ squares in which all cells are red.
2015 India IMO Training Camp
Let $ABCD$ be a convex quadrilateral and let the diagonals $AC$ and $BD$ intersect at $O$. Let $I_1, I_2, I_3, I_4$ be respectively the incentres of triangles $AOB, BOC, COD, DOA$. Let $J_1, J_2, J_3, J_4$ be respectively the excentres of triangles $AOB, BOC, COD, DOA$ opposite $O$. Show that $I_1, I_2, I_3, I_4$ lie on a circle if and only if $J_1, J_2, J_3, J_4$ lie on a circle.
2015 India IMO Training Camp
Let $f$ and $g$ be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose $\deg(f)$ is odd and the sets $\{f(a)\mid a\in \mathbb{Z}\}$ and $\{g(a)\mid a\in \mathbb{Z}\}$ are the same. Prove that there exists an integer $k$ such that $g(x)=f(x+k)$.
2015 India IMO Training Camp
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.
2015 India IMO Training Camp
Let $G$ be a simple graph on the infinite vertex set $V=\{v_1, v_2, v_3,\ldots\}$. Suppose every subgraph of $G$ on a finite vertex subset is $10$-colorable, Prove that $G$ itself is $10$-colorable.
2015 India IMO Training Camp
In a triangle $ABC$, a point $D$ is on the segment $BC$, Let $X$ and $Y$ be the incentres of triangles $ACD$ and $ABD$ respectively. The lines $BY$ and $CX$ intersect the circumcircle of triangle $AXY$ at $P\ne Y$ and $Q\ne X$, respectively. Let $K$ be the point of intersection of lines $PX$ and $QY$. Suppose $K$ is also the reflection of $I$ in $BC$ where $I$ is the incentre of triangle $ABC$. Prove that $\angle BAC=\angle ADC=90^{\circ}$.
2015 India IMO Training Camp
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.



Proposed by Belgium
2015 India IMO Training Camp
There are $n\ge 2$ lamps, each with two states: $\textbf{on}$ or $\textbf{off}$. For each non-empty subset $A$ of the set of these lamps, there is a $\textit{soft-button}$ which operates on the lamps in $A$; that is, upon $\textit{operating}$ this button each of the lamps in $A$ changes its state(on to off and off to on). The buttons are identical and it is not known which button corresponds to which subset of lamps. Suppose all the lamps are off initially. Show that one can always switch all the lamps on by performing at most $2^{n-1}+1$ operations.
2015 India IMO Training Camp
Let $A$ be a finite set of pairs of real numbers such that for any pairs $(a,b)$ in $A$ we have $a>0$. Let $X_0=(x_0, y_0)$ be a pair of real numbers(not necessarily from $A$). We define $X_{j+1}=(x_{j+1}, y_{j+1})$ for all $j\ge 0$ as follows: for all $(a,b)\in A$, if $ax_j+by_j>0$ we let $X_{j+1}=X_j$; otherwise we choose a pair $(a,b)$ in $A$ for which $ax_j+by_j\le 0$ and set $X_{j+1}=(x_j+a, y_j+b)$. Show that there exists an integer $N\ge 0$ such that $X_{N+1}=X_N$.
2014 India IMO Training Camp
Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$
2014 India IMO Training Camp
Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.
2014 India IMO Training Camp
In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.
2014 India IMO Training Camp
Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.
2014 India IMO Training Camp
Let $n$ be a natural number.A triangulation of a convex n-gon is a division of the polygon into $n-2$ triangles by drawing $n-3$ diagonals no two of which intersect at an interior point of the polygon.Let $f(n)$ denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine $f(n)$ in terms of $n$.
2014 India IMO Training Camp
For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$.

Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where

$f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.
2014 India IMO Training Camp
Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$.
2014 India IMO Training Camp
Starting with the triple $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$, define a sequence of triples $(x_{n},y_{n},z_{n})$ by

$x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$

$y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$

$ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$

for $n\geq 0$.Show that each of the sequences $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$  converges to a limit and find these limits.
2014 India IMO Training Camp
In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.
2014 India IMO Training Camp
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold:



$x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$



$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$



$v_{1},v_{2},v_{3}$ satisfy triangle inequality



$v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality.



Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.
2014 India IMO Training Camp
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that

$$ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} $$

Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
2014 India IMO Training Camp
In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.
2014 India IMO Training Camp
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
2014 India IMO Training Camp
In how many ways rooks can be placed on a $8$ by $8$ chess board such that every row and every column has at least one rook?

(Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)
2014 India IMO Training Camp
Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that  $$ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. $$
2014 India IMO Training Camp
Find all positive integers $x$ and $y$ such that $x^{x+y}=y^{3x}$.
2014 India IMO Training Camp
Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.
2013 India IMO Training Camp
For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$.  For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$.  Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.