Navvye/Olympiad-Problems-CSV huggingface dataset:
2013 India IMO Training Camp
Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$.  Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively.  Prove that $KL = MN$.
2013 India IMO Training Camp
We define an operation $\oplus$ on the set $\{0, 1\}$ by

$$ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.$$

For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$.  For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$.



For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$.  Prove that $f$ is a bijection on the set of natural numbers.
2013 India IMO Training Camp
Let $a, b, c$ be positive real numbers such that $a + b + c = 1$.  If $n$ is a positive integer then prove that

$$ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. $$
2013 India IMO Training Camp
In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal.  If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.
2013 India IMO Training Camp
A marker is placed at the origin of an integer lattice.  Calvin and Hobbes play the following game.  Calvin starts the game and each of them takes turns alternatively.  At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction.  Hobbes wins if the marker is back at the origin any time after the first turn.  Prove or disprove that Calvin can prevent Hobbes from winning.



Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).
2013 India IMO Training Camp
Let $n \ge 2$ be an integer.  There are $n$ beads numbered $1, 2, \ldots, n$.  Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed).  For example, with $n \ge 5$, the necklace with four beads $1, 5, 3, 2$ in the clockwise order is same as the one with $5, 3, 2, 1$ in the clockwise order, but is different from the one with $1, 2, 3, 5$ in the clockwise order.



We denote by $D_0(n)$ (respectively $D_1(n)$) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least $3$.  Prove that $n - 1$ divides $D_1(n) - D_0(n)$.
2013 India IMO Training Camp
In a triangle $ABC$, with $\widehat{A} > 90^\circ$, let $O$ and $H$ denote its circumcenter and orthocenter, respectively.  Let $K$ be the reflection of $H$ with respect to $A$.  Prove that $K, O$ and $C$ are collinear if and only if $\widehat{A} - \widehat{B} = 90^\circ$.
2013 India IMO Training Camp
For a positive integer $n$, a cubic polynomial $p(x)$ is said to be $n$-good if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$.  Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.
2013 India IMO Training Camp
Find all functions $f$ from the set of real numbers to itself satisfying

$$ f(x(1+y)) = f(x)(1 + f(y)) $$

for all real numbers $x, y$.
2013 India IMO Training Camp
An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers.

a) Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$.

b) Decide whether $a=2$ is friendly.
2013 India IMO Training Camp
Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules:

(a) On every move of his $B$ passes $1$ coin from every box to an adjacent box.

(b) On every move of hers $A$ chooses several coins that were not involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box.

Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.
2013 India IMO Training Camp
For a positive integer $n$, a sum-friendly odd partition of $n$ is a sequence $(a_1, a_2, \ldots, a_k)$ of odd positive integers with $a_1 \le a_2 \le \cdots \le a_k$ and $a_1 + a_2 + \cdots + a_k = n$ such that for all positive integers $m \le n$, $m$ can be uniquely written as a subsum $m = a_{i_1} + a_{i_2} + \cdots + a_{i_r}$.  (Two subsums $a_{i_1} + a_{i_2} + \cdots + a_{i_r}$ and $a_{j_1} + a_{j_2} + \cdots + a_{j_s}$ with $i_1 < i_2 < \cdots < i_r$ and $j_1 < j_2 < \cdots < j_s$ are considered the same if $r = s$ and $a_{i_l} = a_{j_l}$ for $1 \le l \le r$.)  For example, $(1, 1, 3, 3)$ is a sum-friendly odd partition of $8$.  Find the number of sum-friendly odd partitions of $9999$.
2013 India IMO Training Camp
In a triangle $ABC$, let $I$ denote its incenter.  Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$.  The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively.  Prove that $K, L, M, I$ are concyclic.
2013 India IMO Training Camp
Let $h \ge 3$ be an integer and $X$ the set of all positive integers that are greater than or equal to $2h$.  Let $S$ be a nonempty subset of $X$ such that the following two conditions hold:



if $a + b \in S$ with $a \ge h, b \ge h$, then $ab \in S$;



if $ab \in S$ with $a \ge h, b \ge h$, then $a + b \in S$.



Prove that $S = X$.
2013 India IMO Training Camp
A positive integer $a$ is called a double number if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$.  For example, $283283$ is a double number.  Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.
2013 India IMO Training Camp
Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients.  One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence  with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$.  The process stops when all the elements of the sequence are of degree $1$.  If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that  the process stops with a sequence of $n$ identical polynomials of degree 1.
2013 India IMO Training Camp
In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$.  A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$.  Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$.  Let $Y$ be the point of intersection of the lines $CP$ and $AQ$.  Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.
2012 India IMO Training Camp
Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.
2012 India IMO Training Camp
Let $a\ge b$ and $c\ge d$ be real numbers. Prove that the equation

$$(x+a)(x+d)+(x+b)(x+c)=0$$

has real roots.
2012 India IMO Training Camp
How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true?
2012 India IMO Training Camp
Let $ABCD$ be a trapezium with $AB\parallel CD$. Let $P$ be a point on $AC$ such that $C$ is between $A$ and $P$; and let $X, Y$ be the midpoints of $AB, CD$ respectively. Let $PX$ intersect $BC$ in $N$ and $PY$ intersect $AD$ in $M$. Prove that $MN\parallel AB$.
2012 India IMO Training Camp
Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent:

$(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$

$(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$
2012 India IMO Training Camp
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a function such that $f(x+y+xy)=f(x)+f(y)+f(xy)$ for all $x, y\in\mathbb{R}$. Prove that $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x, y\in\mathbb{R}$.
2012 India IMO Training Camp
The cirumcentre of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP=\angle OBR$ if and only if $\angle PDQ=\angle RCO$.
2012 India IMO Training Camp
Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that

$$\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}$$
2012 India IMO Training Camp
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$



Proposed by Igor Voronovich, Belarus
2012 India IMO Training Camp
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with $$\sum^{2011}_{j=1} j  x^n_j = a^{n+1} + 1$$



Proposed by Warut Suksompong, Thailand
2012 India IMO Training Camp
Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$
2012 India IMO Training Camp
Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half.



Proposed by Gerhard Wöginger, Austria
2012 India IMO Training Camp
Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$  and $C$ at the point $E$  inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$.



Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea
2012 India IMO Training Camp
Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.
2012 India IMO Training Camp
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.
2012 India IMO Training Camp
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying

$$f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},$$

for all $x, y\in \mathbb{R}^{+}$.
2011 India IMO Training Camp
Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that:



$a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area.



$b) a\cdot AP=b\cdot BP=c\cdot PC.$
2011 India IMO Training Camp
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that

$$36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.$$

Proposed by  Nazar Serdyuk, Ukraine
2011 India IMO Training Camp
A set of $n$ distinct integer weights $w_1,w_2,\ldots, w_n$ is said to be balanced if after removing any one of weights, the remaining $(n-1)$ weights can be split into two subcollections (not necessarily with equal size)with equal sum.



$a)$ Prove that if there exist balanced sets of sizes $k,j$ then also a balanced set of size $k+j-1$.

$b)$ Prove that for all odd $n\geq 7$ there exist a balanced set of size $n$.
2011 India IMO Training Camp
Find all positive integer $n$ satisfying the conditions



$a)n^2=(a+1)^3-a^3$



$b)2n+119$ is a perfect square.
2011 India IMO Training Camp
Suppose $a_1,\ldots,a_n$ are non-integral real numbers for $n\geq 2$  such that ${a_1}^k+\ldots+{a_n}^k$ is an integer for all integers $1\leq k\leq n$. Prove that none of $a_1,\ldots,a_n$ is rational.
2011 India IMO Training Camp
Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called good  if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let



$$A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for  all} x\in X, y\in Y}$$



Prove that :

$a)$ If $X_0$ is not good then the number of pairs $(X_0,Y)$ in $A$ is even.

$b)$ the number of good subsets of $T$ is odd.
2011 India IMO Training Camp
Prove that for no integer $ n$ is $ n^7 + 7$ a perfect square.
2011 India IMO Training Camp
Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.
2011 India IMO Training Camp
Let $ABC$ be an acute-angled triangle. Let $AD,BE,CF$ be internal bisectors with $D, E, F$ on $BC, CA, AB$ respectively. Prove that

$$\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}$$
2011 India IMO Training Camp
Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that

$$(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0$$

for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that

$$a_{k+2011}=a_{k+2011^{2011}}.$$
2010 India IMO Training Camp
Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.
2010 India IMO Training Camp
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:



(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;

(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.



Determine $N(n)$ for all $n\ge 2$.
2010 India IMO Training Camp
Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that

$$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})$$
2010 India IMO Training Camp
Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers.

(Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)
2010 India IMO Training Camp
Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to

$$2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}$$
2010 India IMO Training Camp
Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively.  If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.
2010 India IMO Training Camp
Call a positive integer good if either $N=1$ or $N$ can be written as product of even number of prime numbers, not necessarily distinct.

Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers.



(a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers.

(b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.
2010 India IMO Training Camp
Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$.

Show that there exists $j<k$ and $l<m$ such that

$$a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}$$
2010 India IMO Training Camp
Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$
2010 India IMO Training Camp
Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$
2010 India IMO Training Camp
Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$
2009 India IMO Training Camp
Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$,

$ r$ being inradius.
2009 India IMO Training Camp
Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) = 1$, are elements of V.

$ (a,b)$ is connected to $ (a,b + kab)$ by an edge and to $ (a + kab,b)$ by another edge for all integer k.

Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.
2009 India IMO Training Camp
Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following:

$ a_1 = a \\
a_2 = b \\
a_n = \text{largest odd divisor of }(a_{n - 1} + a_{n - 2})$.

Prove that there exists a natural number $ N$ such that $ a_n = gcd(a,b) \forall n\ge N$.
2009 India IMO Training Camp
Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly.

Prove That $ \frac {1}{aR_a} + \frac {1}{bR_b} + \frac {1}{cR_c} = \frac {s^2}{rabc}$.
2009 India IMO Training Camp
Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients.

We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that

$ f(x)-g(y)=\prod_{i=1}^n{(a_ix+b_iy+c_i)}$.

Prove that there exists $ a,b,c\in\mathbb{C}$ such that

$ f(x)=(x+a)^n+c\text{ and }g(y)=(y+b)^n+c$.
2009 India IMO Training Camp
Prove The Following identity:

$ \sum_{j = 0}^n \left ({3n + 2 - j \choose j}2^j - {3n + 1 - j \choose j - 1}2^{j - 1}\right ) = 2^{3n}$.

The Second term on left hand side is to be regarded zero for j=0.
2009 India IMO Training Camp
Let $ P$ be any  point in the interior of a $ \triangle ABC$.Prove That

$ \frac{PA}{a}+\frac{PB}{b}+\frac{PC}{c}\ge \sqrt{3}$.
2009 India IMO Training Camp
Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n+4^n$.Prove That $ 7\mid n$.
2009 India IMO Training Camp
Let

$ f(x)=\sum_{k=1}^n a_k x^k$ and $ g(x)=\sum_{k=1}^n \frac{a_k x^k}{2^k -1}$ be two polynomials with real coefficients.

Let g(x) have $ 0,2^{n+1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.
2009 India IMO Training Camp
For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.
2009 India IMO Training Camp
Find all integers $ n\ge 2$ with the following property:



There exists three distinct primes $p,q,r$ such that

whenever $ a_1,a_2,a_3,\cdots,a_n$ are $ n$ distinct positive integers with the property that at least one of $ p,q,r$ divides $ a_j - a_k \ \forall 1\le j\le k\le n$,

one of $ p,q,r$ divides all of these differences.
2009 India IMO Training Camp
Let $ G$ be a simple graph with vertex set $ V=\{0,1,2,3,\cdots ,n+1\}$ .$ j$and$ j+1$ are connected by an edge for $ 0\le j\le n$. Let $ A$ be a subset of $ V$ and $ G(A)$ be the induced subgraph associated with $ A$. Let $ O(G(A))$ be number of components of $ G(A)$ having an odd number of vertices.

Let

$ T(p,r)=\{A\subset V \mid 0.n+1 \notin A,|A|=p,O(G(A))=2r\}$ for $ r\le p \le 2r$.

Prove That $ |T(p,r)|={n-r \choose{p-r}}{n-p+1 \choose{2r-p}}$.
2007 India IMO Training Camp
Show that in a non-equilateral triangle, the following statements are equivalent:

$(a)$ The angles of the triangle are in arithmetic progression.

$(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.
2007 India IMO Training Camp
Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define

$$T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \  f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.$$

Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$

(Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)
2007 India IMO Training Camp
Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that

$$a^2+b^2+c^2\leq 2abc+1.$$
2007 India IMO Training Camp
Given a finite string $S$ of symbols $X$ and $O$, we denote $\Delta(s)$ as the number of$X'$s in $S$ minus the number of $O'$s (For example, $\Delta(XOOXOOX)=-1$). We call a string $S$ balanced if every sub-string $T$ of (consecutive symbols) $S$ has the property $-1\leq \Delta(T)\leq 2.$ (Thus $XOOXOOX$ is not balanced, since it contains the sub-string $OOXOO$ whose $\Delta$ value is $-3.$ Find, with proof, the number of balanced strings of length $n$.
2007 India IMO Training Camp
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$
2007 India IMO Training Camp
Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation

$$f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);$$

For all $x,y\in\mathbb R.$
2006 India IMO Training Camp
Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.