Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 India National Olympiad
Let $ABC$ be a right-angle triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $CBD$ are equal. If this common value is $r^{\prime}$ and if $r$ is the inradius of triangle $ABC$, prove that

$$ \cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}. $$
2016 India National Olympiad
Consider a nonconstant arithmetic progression $a_1, a_2,\cdots, a_n,\cdots$. Suppose there exist relatively prime positive integers $p>1$ and $q>1$ such that $a_1^2, a_{p+1}^2$ and $a_{q+1}^2$ are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.
2015 India National Olympiad
Let $ABC$ be a right-angled triangle with $\angle{B}=90^{\circ}$. Let $BD$ is the altitude from $B$ on $AC$. Let $P,Q$ and $I $be the incenters of triangles $ABD,CBD$ and $ABC$ respectively.Show that circumcenter of triangle $PIQ$ lie on the hypotenuse $AC$.
2015 India National Olympiad
For any natural number $n > 1$ write the finite decimal expansion of $\frac{1}{n}$ (for example we write $\frac{1}{2}=0.4\overline{9}$ as its infinite decimal expansion not $0.5)$. Determine the length of non-periodic part of the (infinite) decimal expansion of $\frac{1}{n}$.
2015 India National Olympiad
Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$.
2015 India National Olympiad
There are four basketball players $A,B,C,D$. Initially the ball is with $A$. The ball is always passed from one person to a different person.

In how many ways can the ball come back to $A$ after $\textbf{seven}$ moves? (for example $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A$, or $A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A)$.
2015 India National Olympiad
Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$
2015 India National Olympiad
Show that from a set of $11$ square integers one can select six numbers $a^2,b^2,c^2,d^2,e^2,f^2$ such that $a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}$.
2014 India National Olympiad
In a triangle $ABC$, let $D$ be the point on the segment $BC$ such that $AB+BD=AC+CD$. Suppose that the points $B$, $C$ and the centroids  of triangles $ABD$ and $ACD$ lie on a circle. Prove that $AB=AC$.
2014 India National Olympiad
Let $n$ be a natural number. Prove that,

$$ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor $$

is even.
2014 India National Olympiad
Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$
2014 India National Olympiad
Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.
2014 India National Olympiad
In a acute-angled triangle $ABC$, a point $D$ lies on the segment $BC$. Let $O_1,O_2$ denote the circumcentres of triangles $ABD$ and $ACD$ respectively. Prove that the line joining the circumcentre of triangle $ABC$ and the orthocentre of triangle $O_1O_2D$ is parallel to $BC$.
2014 India National Olympiad
Let $n>1$ be a natural number. Let $U=\{1,2,...,n\}$, and define $A\Delta B$ to be the set of all those elements of $U$ which belong to exactly one of $A$ and $B$. Show that $|\mathcal{F}|\le 2^{n-1}$, where $\mathcal{F}$ is a collection of subsets of $U$ such that for any two distinct elements of $A,B$ of $\mathcal{F}$ we have $|A\Delta B|\ge 2$. Also find all such collections $\mathcal{F}$ for which the maximum is attained.
2013 India National Olympiad
Let $\Gamma_1$ and $\Gamma_2$ be two circles touching each other externally at $R.$ Let $O_1$ and $O_2$ be the centres of $\Gamma_1$ and $\Gamma_2,$ respectively. Let $\ell_1$ be a line which is tangent to $\Gamma_2$ at $P$ and passing through $O_1,$ and let $\ell_2$ be the line tangent to $\Gamma_1$ at $Q$ and passing through $O_2.$ Let $K=\ell_1\cap \ell_2.$ If $KP=KQ$ then prove that the triangle $PQR$ is equilateral.
2013 India National Olympiad
Find all $m,n\in\mathbb N$ and primes $p\geq 5$ satisfying

$$m(4m^2+m+12)=3(p^n-1).$$
2013 India National Olympiad
Let $a,b,c,d \in \mathbb{N}$ such that  $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.
2013 India National Olympiad
Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove that $T_n - n$ is always even.
2013 India National Olympiad
In an acute triangle $ABC,$ let $O,G,H$ be its circumcentre, centroid and orthocenter. Let $D\in BC, E\in CA$ and $OD\perp BC, HE\perp CA.$ Let $F$ be the midpoint of $AB.$ If the triangles $ODC, HEA, GFB$ have the same area, find all the possible values of $\angle C.$
2013 India National Olympiad
Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$
2012 India National Olympiad
Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.
2012 India National Olympiad
Let $p_1<p_2<p_3<p_4$ and $q_1<q_2<q_3<q_4$ be two sets of prime numbers, such that $p_4 - p_1 = 8$ and $q_4 - q_1= 8$. Suppose $p_1 > 5$ and $q_1>5$. Prove that $30$ divides $p_1 - q_1$.
2012 India National Olympiad
Define a sequence $<f_0 (x), f_1 (x), f_2 (x), \dots>$ of functions by $$f_0 (x) = 1$$$$f_1(x)=x$$$$(f_n(x))^2 - 1 = f_{n+1}(x) f_{n-1}(x)$$for $n \ge 1$. Prove that each $f_n (x)$ is a polynomial with integer coefficients.
2012 India National Olympiad
Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be good if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ smaller triangles of equal area. Determine the number of good points for a given triangle $ABC$.
2012 India National Olympiad
Let $ABC$ be an acute angled triangle. Let $D,E,F$ be points on $BC, CA, AB$ such that $AD$ is the median, $BE$ is the internal bisector and $CF$ is the altitude. Suppose that $\angle FDE=\angle C, \angle DEF=\angle A$ and $\angle EFD=\angle B.$ Show that $ABC$ is equilateral.
2012 India National Olympiad
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and



$(i) f(xy) + f(x)f(y) = f(x) + f(y)$



$(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0 $



for all $x,y \in \mathbb{Z}$, simultaneously.



$(a)$ Find the set of all possible values of the function $f$.



$(b)$ If $f(10) \ne 0$ and $f(2) = 0$, find the set of all integers $n$ such that $f(n) \ne 0$.
2011 India National Olympiad
Let $D,E,F$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE.$ Show that $\triangle ABC$ is equilateral.
2011 India National Olympiad
Call a natural number $n$ faithful if there exist natural numbers $a<b<c$ such that $a|b,$ and $b|c$ and $n=a+b+c.$

$(i)$ Show that all but a finite number of natural numbers are faithful.

$(ii)$ Find the sum of all natural numbers which are not faithful.
2011 India National Olympiad
Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0$ be two polynomials with integral coefficients such that $a_n-b_n$ is a prime and $a_nb_0-a_0b_n\neq 0,$ and $a_{n-1}=b_{n-1}.$ Suppose that there exists a rational number $r$ such that $P(r)=Q(r)=0.$ Prove that $r\in\mathbb Z.$
2011 India National Olympiad
Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
2011 India National Olympiad
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Gamma.$ Let $E,F,G,H$ be the midpoints of arcs $AB,BC,CD,AD$ of $\Gamma,$ respectively. Suppose that $AC\cdot BD=EG\cdot FH.$ Show that $AC,BD,EG,FH$ are all concurrent.
2011 India National Olympiad
Find all functions $f:\mathbb{R}\to \mathbb R$ satisfying

$$f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),$$For all $x,y\in\mathbb R$.
2010 India National Olympiad
Let $ ABC$ be a triangle with circum-circle $ \Gamma$. Let $ M$ be a point in the interior of triangle $ ABC$ which is also on the bisector of $ \angle A$. Let $ AM, BM, CM$ meet $ \Gamma$ in $ A_{1}, B_{1}, C_{1}$ respectively. Suppose $ P$ is the point of intersection of $ A_{1}C_{1}$ with $ AB$; and $ Q$ is the point of intersection of $ A_{1}B_{1}$ with $ AC$. Prove that $ PQ$ is parallel to $ BC$.
2010 India National Olympiad
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n - 2)!$.
2010 India National Olympiad
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:

$$ (x^2 + xy + y^2)(y^2 + yz + z^2)(z^2 + zx + x^2) = xyz$$

$$ (x^4 + x^2y^2 + y^4)(y^4 + y^2z^2 + z^4)(z^4 + z^2x^2 + x^4) = x^3y^3z^3$$
2010 India National Olympiad
How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions

$$ a_j^2 - a_ja_{j + 1} + a_{j + 1}^2$$

for $ j = 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?
2010 India National Olympiad
Let $ ABC$ be an acute-angled triangle with altitude $ AK$. Let $ H$ be its ortho-centre and $ O$ be its circum-centre. Suppose $ KOH$ is an acute-angled triangle and $ P$ its circum-centre. Let $ Q$ be the reflection of $ P$ in the line $ HO$. Show that $ Q$ lies on the line joining the mid-points of $ AB$ and $ AC$.
2010 India National Olympiad
Define a sequence $ < a_n > _{n\geq0}$ by $ a_0 = 0$, $ a_1 = 1$ and

$$ a_n = 2a_{n - 1} + a_{n - 2},$$

for $ n\geq2.$



$ (a)$ For every $ m > 0$ and $ 0\leq j\leq m,$ prove that $ 2a_m$ divides

$ a_{m + j} + ( - 1)^ja_{m - j}$.



$ (b)$ Suppose $ 2^k$ divides $ n$ for some natural numbers $ n$ and $ k$. Prove that $ 2^k$ divides $ a_n.$
2009 India National Olympiad
Let $ ABC$ be a tringle and let $ P$ be an interior point such that $ \angle BPC = 90 ,\angle BAP = \angle BCP$.Let $ M,N$ be the mid points of $ AC,BC$ respectively.Suppose $ BP = 2PM$.Prove that $ A,P,N$ are collinear.
2009 India National Olympiad
Define a a sequence $ {<{a_n}>}^{\infty}_{n=1}$ as follows



$ a_n=0$, if number of positive divisors of $ n$ is  odd

$ a_n=1$, if number of positive divisors of $ n$ is even



(The positive divisors of $ n$ include $ 1$ as well as $ n$.)Let $ x=0.a_1a_2a_3........$ be the real number whose decimal expansion contains $ a_n$ in the $ n$-th place,$ n\geq1$.Determine,with proof,whether $ x$ is rational or irrational.
2009 India National Olympiad
Find all real numbers $ x$ such that:

$ [x^2+2x]={[x]}^2+2[x]$



(Here $ [x]$ denotes the largest integer not exceeding $ x$.)
2009 India National Olympiad
All the points in the plane are colored using three colors.Prove that there exists a triangle with vertices having the same color such that either it is isosceles or its angles are in geometric progression.
2009 India National Olympiad
Let $ ABC$ be an acute angled triangle and let $ H$ be its ortho centre. Let $ h_{max}$ denote the largest altitude of the triangle $ ABC$. Prove that:



$AH + BH + CH\leq2h_{max}$
2009 India National Olympiad
Let $ a,b,c$ be positive real numbers such that $ a^3 + b^3 = c^3$.Prove that:



$ a^2 + b^2 - c^2 > 6(c - a)(c - b)$.
2008 India National Olympiad
Let $ ABC$ be triangle, $ I$ its in-center; $ A_1,B_1,C_1$ be the reflections of $ I$ in $ BC, CA, AB$ respectively. Suppose the circum-circle of triangle $ A_1B_1C_1$ passes through $ A$. Prove that $ B_1,C_1,I,I_1$ are concylic, where $ I_1$ is the in-center of triangle $ A_1,B_1,C_1$.
2008 India National Olympiad
Find all triples $ \left(p,x,y\right)$ such that $ p^x=y^4+4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers.
2008 India National Olympiad
Let $ A$ be a set of real numbers such that $ A$ has at least four elements. Suppose $ A$ has the property that $ a^2 + bc$ is a rational number for all distinct numbers $ a,b,c$ in $ A$. Prove that there exists a positive integer $ M$ such that $ a\sqrt{M}$ is a rational number for every $ a$ in $ A$.
2008 India National Olympiad
All the points with integer coordinates in the $ xy$-Plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point $ (0,0)$ is red and the point $ (0,1)$ is blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.
2008 India National Olympiad
Let $ ABC$ be a triangle; $ \Gamma_A,\Gamma_B,\Gamma_C$ be three equal, disjoint circles inside $ ABC$ such that $ \Gamma_A$ touches $ AB$ and $ AC$; $ \Gamma_B$ touches $ AB$ and $ BC$; and $ \Gamma_C$ touches $ BC$ and $ CA$. Let $ \Gamma$ be a circle touching circles $ \Gamma_A, \Gamma_B, \Gamma_C$ externally. Prove that the line joining the circum-centre $ O$ and the in-centre $ I$ of triangle $ ABC$ passes through the centre of $ \Gamma$.
2008 India National Olympiad
Let $ P(x)$ be a polynomial with integer coefficients. Prove that there exist two polynomials $ Q(x)$ and $ R(x)$, again with integer coefficients, such that

(i) $ P(x) \cdot Q(x)$ is a polynomial in $ x^2$ , and

(ii) $ P(x) \cdot R(x)$ is a polynomial in $ x^3$.
2007 India National Olympiad
In a triangle $ ABC$ right-angled at $ C$ , the median through $ B$ bisects the angle between $ BA$ and the bisector of $ \angle B$. Prove that

$$ \frac{5}{2} < \frac{AB}{BC} < 3$$
2007 India National Olympiad
Let $ n$ be a natural number such that $ n = a^2 + b^2 +c^2$ for some natural numbers $ a,b,c$. Prove that

$$ 9n = (p_1a+q_1b+r_1c)^2 + (p_2a+q_2b+r_2c)^2 + (p_3a+q_3b+r_3c)^2$$

where $ p_j$'s , $ q_j$'s , $ r_j$'s are all nonzero integers. Further, if $ 3$ does not divide at least one of $ a,b,c,$ prove that $ 9n$ can be expressed in the form $ x^2+y^2+z^2$, where $ x,y,z$ are natural numbers none of which is divisible by $ 3$.
2007 India National Olympiad
Let $ m$ and $ n$ be positive integers such that $ x^2 - mx +n = 0$ has real roots $ \alpha$ and $ \beta$.



Prove that $ \alpha$ and $ \beta$ are integers if and only if $ [m\alpha] + [m\beta]$ is the square of an integer.



(Here $ [x]$ denotes the largest integer not exceeding $ x$)
2007 India National Olympiad
Let $ \sigma = (a_1, a_2, \cdots , a_n)$ be permutation of $ (1, 2 ,\cdots, n)$. A pair $ (a_i, a_j)$ is said to correspond to an inversion of $\sigma$ if $ i<j$ but $ a_i>a_j$. How many permutations of $ (1,2,\cdots,n)$, $ n \ge 3$, have exactly two inversions?



For example, In the permutation $(2,4,5,3,1)$, there are 6 inversions corresponding to the pairs $ (2,1),(4,3),(4,1),(5,3),(5,1),(3,1)$.
2007 India National Olympiad
Let $ ABC$ be a triangle in which $ AB=AC$. Let $ D$ be the midpoint of $ BC$ and $ P$ be a point on $ AD$. Suppose $ E$ is the foot of perpendicular from $ P$ on $ AC$. Define

$$ \frac{AP}{PD}=\frac{BP}{PE}=\lambda , \ \ \ \frac{BD}{AD}=m , \ \ \ z=m^2(1+\lambda)$$

Prove that

$$ z^2 - (\lambda^3 - \lambda^2 - 2)z + 1 = 0$$



Hence show that $ \lambda \ge 2$ and $ \lambda = 2$ if and only if $ ABC$ is equilateral.
2007 India National Olympiad
If $ x$, $ y$, $ z$ are positive real numbers, prove that

$$ \left(x + y + z\right)^2 \left(yz + zx + xy\right)^2 \leq 3\left(y^2 + yz + z^2\right)\left(z^2 + zx + x^2\right)\left(x^2 + xy + y^2\right) .$$
2006 India National Olympiad
In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that



(1) $IO$ is perpendicular to $BI$;



(2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.
2006 India National Olympiad
Prove that for every positive integer $n$ there exists a unique ordered pair $(a,b)$ of positive integers such that

$$ n = \frac{1}{2}(a + b - 1)(a + b - 2) + a .  $$
2006 India National Olympiad
Let $X=\mathbb{Z}^3$ denote the set of all triples $(a,b,c)$ of integers. Define $f: X \to X$ by $$ f(a,b,c) = (a+b+c, ab+bc+ca, abc) .  $$

Find all triples $(a,b,c)$  such that $$ f(f(a,b,c)) = (a,b,c) . $$
2006 India National Olympiad
Some 46 squares are randomly chosen from a $9 \times 9$ chess board and colored in red. Show that there exists a $2\times 2$ block of 4 squares of which at least three are colored in red.
2006 India National Olympiad
In a cyclic quadrilateral $ABCD$, $AB=a$, $BC=b$, $CD=c$, $\angle ABC = 120^\circ$ and $\angle ABD = 30^\circ$. Prove that



(1) $c \ge a + b$;



(2) $|\sqrt{c + a} - \sqrt{c + b} | = \sqrt{c - a - b}$.
2006 India National Olympiad
(a) Prove that if $n$ is a integer such that $n \geq 4011^2$ then there exists an integer $l$ such that $$ n < l^2 < (1 + \frac{1}{{2005}})n .  $$



(b) Find the smallest positive integer $M$ for which whenever an integer $n$ is such that $n \geq M$

then there exists an integer $l$ such that $$ n < l^2 < (1 + \frac{1}{{2005}})n .  $$
2005 India National Olympiad
Let $M$ be the midpoint of side $BC$ of a triangle $ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L,K$ being nearer to $A$

than $L$. If $AK = KL = LM$, prove that the sides of triangle $ABC$ are in the ratio $5 : 10 : 13$ in some order.
2005 India National Olympiad
Let $\alpha$ and $\beta$ be positive integers such that  $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$. Find the minimum possible value of $\beta$.
2005 India National Olympiad
Let $p, q, r$  be positive real numbers, not all equal, such that some two of the equations \begin{eqnarray*} px^2 + 2qx + r &=& 0 \\ qx^2 + 2rx + p &=& 0 \\ rx^2 + 2px + q &=& 0 . \\ \end{eqnarray*} have a common root, say $\alpha$. Prove that



$a)$ $\alpha$ is real and negative;



$b)$ the remaining third quadratic equation has non-real roots.
2005 India National Olympiad
All possible $6$-digit numbers, in each of which the digits occur in nonincreasing order (from left to right, e.g. $877550$) are written as a sequence in increasing order. Find the $2005$-th number in this sequence.
2005 India National Olympiad
Let $x_1$ be a given positive integer. A sequence $\{x_n\}_ {n\geq 1}$ of positive integers is such that $x_n$, for $n \geq 2$, is obtained from $x_ {n-1}$ by adding some nonzero digit of $x_ {n-1}$. Prove that



a) the sequence contains an even term;



b) the sequence contains infinitely many even terms.
2005 India National Olympiad
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that $$ f(x^2 + yf(z)) = xf(x) + zf(y) , $$ for all $x, y, z \in \mathbb{R}$.
2004 India National Olympiad
$ABCD$ is a convex quadrilateral. $K$, $L$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $CD$, $DA$. $BD$ bisects $KM$ at $Q$. $QA = QB = QC = QD$ , and$\frac{LK}{LM} = \frac{CD}{CB}$. Prove that $ABCD$ is a square
2004 India National Olympiad
$p > 3$  is a prime. Find all integers $a$, $b$, such that $a^2 + 3ab + 2p(a+b) + p^2 = 0$.
2004 India National Olympiad
If $a$ is a real root of $x^5 - x^3 + x - 2 = 0$, show that $[a^6] =3$
2004 India National Olympiad
$ABC$ is a triangle, with sides $a$, $b$, $c$ , circumradius $R$, and exradii $r_a$, $r_b$, $r_c$If $2R\leq r_a$, show that $a > b$, $a > c$, $2R > r_b$, and $2R > r_c$.
2004 India National Olympiad
S is the set of all ($a$, $b$, $c$, $d$, $e$, $f$) where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides abcdef for all members of $S$.
2004 India National Olympiad
Show that the number of 5-tuples ($a$, $b$, $c$, $d$, $e$) such that $abcde = 5(bcde + acde + abde + abce + abcd)$ is odd
2003 India National Olympiad
Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.
2003 India National Olympiad
Find all primes $p,q$ and even $n>2$ such that $p^n+p^{n-1}+...+1=q^2+q+1$.
2003 India National Olympiad
Show that $8x^4 - 16x^3 + 16x^2 - 8x + k = 0$ has at least one real root for all real $k$. Find the sum of the non-real roots.
2003 India National Olympiad
Find all $7$-digit numbers which use only the digits $5$ and $7$ and are divisible by $35$.
2003 India National Olympiad
Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$, $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$.
2003 India National Olympiad
Each lottery ticket has a 9-digit numbers, which uses only the digits $1$, $2$, $3$. Each ticket is colored red, blue or green. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket $122222222$ is red, and ticket $222222222$ is green. What color is ticket $123123123$ ?
2002 India National Olympiad
For a convex hexagon $ ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements.



($ a_1$) $ AB$ is parallel to $ DE$. ($ a_2$)$ AE = BD$.



($ b_1$) $ BC$ is parallel to $ EF$. ($ b_2$)$ BF = CE$.



($ c_1$) $ CD$ is parallel to $ FA$. ($ c_2$) $ CA = DF$.



$ (a)$ Show that if all six of these statements are true then the hexagon is cyclic.



$ (b)$ Prove that, in fact, five of the six statements suffice.
2002 India National Olympiad
Find the smallest positive value taken by $a^3 + b^3 + c^3 - 3abc$ for positive integers $a$, $b$, $c$ .

Find all $a$, $b$, $c$ which give the smallest value
2002 India National Olympiad
If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.
2002 India National Olympiad
Is it true that there exist 100 lines in the plane, no three concurrent, such that they intersect in exactly 2002 points?
2002 India National Olympiad
Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).
2002 India National Olympiad
The numbers $1, 2, 3$, $\ldots$, $n^2$ are arranged in an $n\times n$ array, so that the numbers in each row increase from left to right, and the numbers in each column increase from top to bottom. Let $a_{ij}$ be the number in position $i, j$. Let $b_j$ be the number of possible values for $a_{jj}$. Show that $$ b_1 + b_2 + \cdots + b_n = \frac{ n(n^2-3n+5) }{3} . $$
2001 India National Olympiad
Let $ABC$ be a triangle in which no angle is $90^{\circ}$. For any point $P$ in the plane of the triangle, let $A_1, B_1, C_1$ denote the reflections of $P$ in the sides $BC,CA,AB$ respectively. Prove that



(i) If $P$ is the incenter or an excentre of $ABC$, then $P$ is the circumenter of $A_1B_1C_1$;



(ii)  If $P$ is the circumcentre of $ABC$, then $P$ is the orthocentre of $A_1B_1C_1$;



(iii) If $P$ is the orthocentre of $ABC$, then $P$ is either the incentre or an excentre of $A_1B_1C_1$.
2001 India National Olympiad
Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.