Navvye/Olympiad-Problems-CSV huggingface dataset:
2006 India IMO Training Camp
Let $ABC$ be a triangle and let $P$ be a point in the plane of $ABC$ that is inside the region of the angle $BAC$ but outside triangle $ABC$.



(a) Prove that any two of the following statements imply the third.





(i) the circumcentre of triangle $PBC$ lies on the ray $\stackrel{\to}{PA}$.



(ii) the circumcentre of triangle $CPA$ lies on the ray $\stackrel{\to}{PB}$.



(iii) the circumcentre of triangle $APB$ lies on the ray $\stackrel{\to}{PC}$.



(b) Prove that if the conditions in (a) hold, then the circumcentres of triangles $BPC,CPA$ and $APB$ lie on the circumcircle of triangle $ABC$.
2006 India IMO Training Camp
Let $p$ be a prime number and let $X$ be a finite set containing at least $p$ elements. A collection of pairwise mutually disjoint $p$-element subsets of $X$ is called a $p$-family. (In particular, the empty collection is a $p$-family.) Let $A$(respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.
2006 India IMO Training Camp
Let $ABC$ be an equilateral triangle, and let $D,E$ and $F$ be points on $BC,BA$ and $AB$ respectively. Let $\angle BAD= \alpha, \angle CBE=\beta$ and $\angle ACF =\gamma$. Prove that if $\alpha+\beta+\gamma \geq 120^\circ$, then the union of the triangular regions $BAD,CBE,ACF$ covers the triangle $ABC$.
2006 India IMO Training Camp
Let $ABC$ be a triangle with inradius $r$, circumradius $R$, and with sides $a=BC,b=CA,c=AB$. Prove that

$$\frac{R}{2r} \ge \left(\frac{64a^2b^2c^2}{(4a^2-(b-c)^2)(4b^2-(c-a)^2)(4c^2-(a-b)^2)}\right)^2.$$
2006 India IMO Training Camp
the positive divisors $d_1,d_2,\cdots,d_k$ of a positive integer $n$ are ordered

$$1=d_1<d_2<\cdots<d_k=n$$

Suppose $d_7^2+d_{15}^2=d_{16}^2$. Find all possible values of $d_{17}$.
2006 India IMO Training Camp
Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that

$$b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.$$
2006 India IMO Training Camp
Find  all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.
2006 India IMO Training Camp
Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that

$$\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N$$

Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that

$$\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}$$
2006 India IMO Training Camp
Let $A_1,A_2,\ldots,A_n$ be subsets of a finite set $S$ such that $|A_j|=8$ for each $j$. For a subset $B$ of $S$ let $F(B)=\{j \mid 1\le j\le n \ \ \text{and} \  A_j \subset B\}$. Suppose for each subset $B$ of $S$ at least one of the following conditions holds





(a) $|B| > 25$,



(b) $F(B)={\O}$,



(c) $\bigcap_{j\in F(B)} A_j \neq {\O}$.



Prove that $A_1\cap A_2 \cap \cdots \cap A_n \neq {\O}$.
2005 India IMO Training Camp
Prove that one can find a $n_{0} \in \mathbb{N}$ such that $\forall m \geq n_{0}$, there exist three positive integers $a$, $b$ , $c$ such that



(i) $m^3 < a < b < c < (m+1)^3$;



(ii) $abc$ is the cube of an integer.
2005 India IMO Training Camp
Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using non-intersecting diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.
2005 India IMO Training Camp
Let $0 <a <b$ be two rational numbers. Let $M$ be a set of positive real numbers with the properties:



(i) $a \in M$ and $b \in M$;



(ii) if $x$ $\in M$ and $y \in M$, then $\sqrt{xy} \in M$.



Let $M^*$denote the set of all irrational numbers in $M$. prove that every $c,d$ such that $a <c <d<b$, $M^*$ contains an element $m$ with property $c<m<d$
2005 India IMO Training Camp
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying

$$ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}$$

for any two positive integers $ m$ and $ n$.



Remark. The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:

$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.

By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).



Proposed by Mohsen Jamali, Iran
2005 India IMO Training Camp
A merida path of order $2n$ is a lattice path in the first quadrant of $xy$- plane joining $(0,0)$ to $(2n,0)$ using three kinds of steps $U=(1,1)$, $D= (1,-1)$ and $L= (2,0)$, i.e. $U$ joins $x,y)$ to $(x+1,y+1)$ etc... An ascent in a merida path is a maximal string of consecutive steps of the form $U$. If $S(n,k)$ denotes the number of merdia paths of order $2n$ with exactly $k$ ascents, compute $S(n,1)$ and $S(n,n-1)$.
2005 India IMO Training Camp
Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$
2005 India IMO Training Camp
Given real numbers $a,\alpha,\beta, \sigma \ and \ \varrho$ s.t. $\sigma, \varrho > 0$ and $\sigma \varrho = \frac{1}{16}$, prove that there exist integers $x$ and $y$ s.t.



$$ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho  $$
2005 India IMO Training Camp
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.



Proposed by Marcin Kuczma, Poland
2005 India IMO Training Camp
For a given triangle ABC, let X be a variable point on the line BC such that the point C lies between the points B and X. Prove that the radical axis of the incircles of the triangles ABX and ACX passes through a point independent of X.



This is a slight extension of the IMO Shortlist 2004 geometry problem 7 and can be found, together with the proposed solution, among the files uploaded at  . Note that the problem was proposed by Russia. I could not find the names of the authors, but I have two particular persons under suspicion. Maybe somebody could shade some light on this...



Darij
2005 India IMO Training Camp
Determine all positive integers $n > 2$ , such that $$ \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6)  $$
2005 India IMO Training Camp
For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$
2004 India IMO Training Camp
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
2004 India IMO Training Camp
For $a,b,c$ positive reals find the minimum  value of $$ \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}. $$
2004 India IMO Training Camp
Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutations.
2004 India IMO Training Camp
Prove that in any triangle $ABC$,

$$ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }.  $$
2004 India IMO Training Camp
Find all triples $(x,y,n)$ of positive integers such that $$ (x+y)(1+xy) = 2^{n}  $$
2004 India IMO Training Camp
Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$
2004 India IMO Training Camp
Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$
2004 India IMO Training Camp
A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be Athenian set if there is a point $P$ of the plane satsifying



(*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$;

(**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct.



(a) Find all Athenian sets in the plane.

(b) For a given Athenian set, find the set of all points $P$ in the plane satisfying (*) and (**)
2004 India IMO Training Camp
Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$
2004 India IMO Training Camp
The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$
2004 India IMO Training Camp
Show that the only solutions of te equation $$ p^{k} + 1 = q^{m}  $$, in positive integers $k,q,m > 1$ and prime $p$ are

(i) $(p,k,q,m) = (2,3,3,2)$

(ii) $k=1 , q=2,$and  $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$
2004 India IMO Training Camp
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that



$$ f(x+y) = f(x)f(y) - c \sin{x} \sin{y}  $$ for all reals $x,y$ where $c> 1$ is a given constant.
2004 India IMO Training Camp
Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively.

(a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that $$ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)}  $$ where $B = \angle ABC$ and $C = \angle BCA$

(b) Prove that $$ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2}  $$
2004 India IMO Training Camp
Define a function $g: \mathbb{N} \mapsto \mathbb{N}$ by the following rule:

(a) $g$ is nondecrasing

(b) for each $n$, $g(n)$ i sthe number of times $n$ appears in the range of $g$,



Prove that $g(1) = 1$ and $g(n+1) = 1 + g( n +1 - g(g(n)))$ for all $n \in \mathbb{N}$
2004 India IMO Training Camp
Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds $v_1$ and $v_2$ respectively, where $v_1$ and $v_2$ are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that



(a) at any given time $t$, at least one of the runners is at a distance not more than $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units from the starting point.

(b) there is a time $t$ such that both the runners are at least $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units away from the starting point. (All disstances are measured along the track). $[x]$ is the greatest integer function.
2004 India IMO Training Camp
Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that $$ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n}  $$
2004 India IMO Training Camp
Find all primes $p \geq 3$ with the following property: for any prime $q<p$, the number

$$

p - \Big\lfloor \frac{p}{q} \Big\rfloor q

$$

is squarefree (i.e. is not divisible by the square of a prime).
2004 India IMO Training Camp
Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the  ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.
2004 India IMO Training Camp
Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ and $Q(x) = x^2 + px + q$be two real polynomials. Suppose that there exista an interval $(r,s)$ of length greater than $2$ SUCH THAT BOTH $P(x)$ AND $Q(x)$ ARE nEGATIVE FOR $X \in (r,s)$ and both are positive for $x > s$ and $x<r$. Show that there is a real $x_0$ such that $P(x_0) < Q(x_0)$
2004 India IMO Training Camp
An integer $n$ is said to be good if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.



Proposed by Hojoo Lee, Korea
2003 India IMO Training Camp
Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.
2003 India IMO Training Camp
Find all triples $(a,b,c)$ of positive integers such that

(i) $a \leq b \leq c$;

(ii) $\text{gcd}(a,b,c)=1$; and

(iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$.
2003 India IMO Training Camp
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,

$$f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).$$
2003 India IMO Training Camp
There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.
2003 India IMO Training Camp
On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.
2003 India IMO Training Camp
A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$, the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$(see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$.

[asy]

draw((30,0)--(-70,0), Arrow);

draw((30,0)--(-20,-40));

draw((-20,-40)--(80,-40), Arrow);

draw((0,-60)--(-40,20), dashed, Arrow);

draw((0,-60)--(0,15), dashed);

draw((0,15)--(40,-65),dashed, Arrow);

[/asy]
2003 India IMO Training Camp
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
2003 India IMO Training Camp
Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that

(a) triangle $ABC$ is acute,

(b) $a+b+c>r+r_1+r_2+r_3$.
2003 India IMO Training Camp
Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.
2003 India IMO Training Camp
Let $n$ be a positive integer greater than $1$, and let $p$ be a prime such that $n$ divides $p-1$ and $p$ divides $n^3-1$. Prove that $4p-3$ is a square.
2002 India IMO Training Camp
Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.
2002 India IMO Training Camp
Show that there is a set of $2002$ consecutive positive integers containing exactly $150$ primes. (You may use the fact that there are $168$ primes less than $1000$)
2002 India IMO Training Camp
Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$?
2002 India IMO Training Camp
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$.  Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that $$ OD + DH = OE + EH = OF + FH$$ and the lines $AD$, $BE$, and $CF$ are concurrent.
2002 India IMO Training Camp
Let $a,b,c$ be positive reals such that $a^2+b^2+c^2=3abc$. Prove that

$$\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}$$
2002 India IMO Training Camp
Determine the number of $n$-tuples of integers $(x_1,x_2,\cdots ,x_n)$ such that $|x_i| \le 10$ for each $1\le i \le n$ and $|x_i-x_j| \le 10$ for $1 \le i,j \le n$.
2002 India IMO Training Camp
Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.
2002 India IMO Training Camp
Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$.





(a) Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$



(b) Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.
2002 India IMO Training Camp
On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on.

If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?
2002 India IMO Training Camp
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying

$$ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
 + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}

$$

for all nonnegative integers $ p$, $ q$, $ r$.
2002 India IMO Training Camp
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.
2002 India IMO Training Camp
Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is olympic if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.
2002 India IMO Training Camp
Let $ABC$ and $PQR$ be two triangles such that





(a) $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.



(b) $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$





Prove that $AB+AC=PQ+PR$.
2002 India IMO Training Camp
Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$
2002 India IMO Training Camp
Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality



$$

\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +

\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.

$$
2002 India IMO Training Camp
Is it possible to find  $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?
2002 India IMO Training Camp
Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.
2002 India IMO Training Camp
Consider the square grid with $A=(0,0)$ and $C=(n,n)$ at its diagonal ends. Paths from $A$ to $C$ are composed of moves one unit to the right or one unit up. Let $C_n$ (n-th catalan number) be the number of paths from $A$ to $C$ which stay on or below the diagonal $AC$. Show that the number of paths from $A$ to $C$ which cross $AC$ from below at most twice is equal to $C_{n+2}-2C_{n+1}+C_n$
2002 India IMO Training Camp
Let $ABC$ be an acute triangle.  Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that

$$

\angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad

\angle CFB = 2 \angle ACB.

$$Let $D'$  be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$   be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum

$$

\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.

$$
2002 India IMO Training Camp
Let $a,b,c$ be positive real numbers. Prove that

$$\frac{a}b+\frac{b}c+\frac{c}a \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}$$
2002 India IMO Training Camp
Given a prime $p$, show that there exists a positive integer $n$ such that the decimal representation of $p^n$ has a block of $2002$ consecutive zeros.
2001 India IMO Training Camp
Let $x$ , $y$ , $z>0$. Prove that if $xyz\geq xy+yz+zx$, then $xyz \geq 3(x+

y+z)$.
2001 India IMO Training Camp
Two symbols $A$ and $B$ obey the rule $ABBB = B$. Given a word $x_1x_2\ldots x_{3n+1}$ consisting of $n$ letters $A$ and $2n+1$ letters $B$, show that there is a unique cyclic permutation of this word which reduces to $B$.
2001 India IMO Training Camp
In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that:

$$AL+BM+CN \leq 3(AD+BE+CF)$$

When does equality occur?
2001 India IMO Training Camp
For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.
2001 India IMO Training Camp
Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.
2001 India IMO Training Camp
Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A  \right | \neq \left |B  \right |$.