Navvye/Olympiad-Problems-CSV huggingface dataset:
2001 India National Olympiad
If $a,b,c$ are positive real numbers such that $abc= 1$, Prove that $$ a^{b+c} b^{c+a} c^{a+b} \leq 1 . $$
2001 India National Olympiad
Show that given any nine integers, we can find four, $a, b, c, d$ such that $a + b - c - d$is divisible by $20$. Show that this is not always true for eight integers.
2001 India National Olympiad
$ABC$ is a triangle. $M$ is the midpoint of $BC$. $\angle MAB = \angle C$, and $\angle MAC = 15^{\circ}$. Show that $\angle AMC$ is obtuse. If $O$ is the circumcenter of $ADC$, show that $AOD$ is equilateral.
2001 India National Olympiad
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(x +y) = f(x) f(y) f(xy)$ for all $x, y \in \mathbb{R}.$
2000 India National Olympiad
The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. The line through $A$ parallel to $LK$ meets $MK$ at $P$, and the line through $A$ parallel to $MK$ meets $LK$ at $Q$. Show that the line $PQ$ bisects $AB$ and bisects $AC$.
2000 India National Olympiad
Solve for integers $x,y,z$: $$ \{ \begin{array}{ccc} x + y &=& 1 - z \\ x^3 + y^3 &=& 1 - z^2 . \end{array}  $$
2000 India National Olympiad
If $a,b,c,x$ are real numbers such that $abc \not= 0$ and $$ \frac{xb + (1-x)c}{a} = \frac{xc + (1-x)a}{b} = \frac{xa + (1-x) b }{c},  $$ then prove that $a = b = c$.
2000 India National Olympiad
In a convex quadrilateral $PQRS$, $PQ =RS$, $(\sqrt{3} +1 )QR = SP$ and $\angle RSP - \angle SQP = 30^{\circ}$. Prove that $\angle PQR - \angle QRS = 90^{\circ}.$
2000 India National Olympiad
Let $a,b,c$ be three real numbers such that $1 \geq a \geq b \geq c \geq 0$. prove that if $\lambda$ is a root of the cubic equation $x^3 + ax^2 + bx + c = 0$ (real or complex), then $| \lambda | \leq 1.$
2000 India National Olympiad
For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that



(i) $f(1999) > f (1996)$;



(ii) $f(2000) = f(1997)$.
1999 India National Olympiad
Let $ABC$ be an acute-angled triangle in which $D,E,F$ are points on $BC,CA,AB$ respectively such that $AD \perp BC$;$AE = BC$;  and $CF$ bisects $\angle C$ internally, Suppose $CF$ meets $AD$ and $DE$ in $M$ and $N$ respectively. If $FM$$= 2$, $MN =1$, $NC=3$, find the perimeter of $\Delta ABC$.
1999 India National Olympiad
In a village $1998$ persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equla to $3996$ feet. For this purpose, the field was divided into $1998$ equal parts. If each part had an integer area, find the length and breadth of the field.
1999 India National Olympiad
Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that $$ p(x)q(x) = x^5 +2x +1 . $$
1999 India National Olympiad
Let $\Gamma$ and $\Gamma'$ be two concentric circles. Let $ABC$ and $A'B'C'$ be any two equilateral triangles inscribed in $\Gamma$ and $\Gamma'$ respectively. If $P$ and $P'$ are any two points on $\Gamma$ and $\Gamma'$ respectively, show that $$ P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2. $$
1999 India National Olympiad
Given any four distinct positive real numbers, show that one can choose three numbers $A,B,C$ from among them, such that all three quadratic equations \begin{eqnarray*} Bx^2 + x + C &=& 0\\ Cx^2 + x + A &=& 0 \\ Ax^2 + x +B &=& 0 \end{eqnarray*} have only real roots, or all three equations have only imaginary roots.
1999 India National Olympiad
For which positive integer values of $n$ can the set $\{ 1, 2, 3, \ldots, 4n \}$ be split into $n$ disjoint $4$-element subsets $\{ a,b,c,d \}$ such that in each of these sets $a = \dfrac{b +c +d} {3}$.
1998 India National Olympiad
In a circle $C_1$ with centre $O$, let $AB$ be a chord that is not a diameter. Let $M$ be the midpoint of this chord $AB$. Take a point $T$ on the circle $C_2$ with $OM$ as diameter. Let the tangent to $C_2$ at $T$ meet $C_1$ at  $P$. Show that $PA^2 + PB^2 = 4 \cdot PT^2$.
1998 India National Olympiad
Let $a$ and $b$ be two positive rational numbers such that $\sqrt[3] {a} + \sqrt[3]{b}$ is also a rational number. Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers.
1998 India National Olympiad
Let $p , q, r , s$ be four integers such that $s$ is not divisible by $5$. If there is an integer $a$ such that $pa^3 + qa^2+ ra +s$ is divisible be 5, prove that there is an integer $b$ such that $sb^3 + rb^2 + qb + p$ is also divisible by 5.
1998 India National Olympiad
Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.
1998 India National Olympiad
Suppose $a,b,c$ are three rela numbers such that the quadratic equation $$ x^2 - (a +b +c )x + (ab +bc +ca) = 0 $$ has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that

(i) The numbers $a,b,c$ are all positive.

(ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.
1998 India National Olympiad
It is desired to choose $n$ integers from the collection of $2n$ integers, namely,   $0,0,1,1,2,2,\ldots,n-1,n-1$ such that the average of these $n$ chosen integers is itself an integer and as minimum as possible. Show that this can be done for each positive integer $n$ and find this minimum value for each $n$.
1997 India National Olympiad
Let $ABCD$ be a parallelogram. Suppose a line passing through $C$ and lying outside the parallelogram meets $AB$ and $AD$ produced at $E$ and $F$ respectively. Show that $$ AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF . $$
1997 India National Olympiad
Show that there do not exist positive integers $m$ and $n$ such that $$ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . $$
1997 India National Olympiad
If $a,b,c$ are three real numbers and $$ a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} = t $$ for some real number $t$, prove that $abc + t = 0 .$
1997 India National Olympiad
In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr $p$, show that $\dfrac{14}{10} < p < \dfrac{15}{10}$.
1997 India National Olympiad
Find the number of $4 \times 4$ array whose entries are from the set $\{ 0 , 1, 2, 3 \}$ and which are such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by $4$.
1997 India National Olympiad
Suppose $a$ and $b$ are two positive real numbers such that the roots of the cubic equation $x^3 - ax + b = 0$ are all real. If $\alpha$ is a root of this cubic with minimal absolute value, prove that $$ \dfrac{b}{a} < \alpha < \dfrac{3b}{2a}.  $$
1996 India National Olympiad
a) Given any positive integer $n$, show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \ldots, n$;



b) If for some positive integers $x$ and $y$, $x+j$ divides $y+j$ for all positive integers $j$, prove that $x = y$.
1996 India National Olympiad
Let $C_1$ and $C_2$ be two concentric circles in the plane with radii $R$ and $3R$ respectively. Show that the orthocenter of any triangle inscribed in circle $C_1$ lies in the interior of circle $C_2$. Conversely, show that every point in the interior of $C_2$ is the orthocenter of some triangle inscribed in $C_1$.
1996 India National Olympiad
Solve the following system for real $a , b, c, d, e$: $$ \left\{ \begin{array}{ccc} 3a & = & ( b + c+ d)^3 \\ 3b & = & ( c + d +e ) ^3 \\ 3c & = & ( d + e +a )^3 \\ 3d & = & ( e + a +b )^3 \\ 3e &=& ( a + b +c)^3. \end{array}\right.  $$
1996 India National Olympiad
Let $X$ be a set containing $n$ elements. Find the number of ordered triples $(A,B, C)$ of subsets of $X$ such that $A$ is a subset of $B$ and $B$ is a proper subset of $C$.
1996 India National Olympiad
Define a sequence $(a_n)_{n \geq 1}$ by $a_1 =1$ and $a_2 =2$ and $a_{n+2} = 2 a_{n+1} - a_n + 2$ for $n \geq 1$. prove that for any $m$ , $a_m a_{m+1}$ is also a term in this sequence.
1996 India National Olympiad
There is a $2n \times 2n$ array (matrix)  consisting of $0's$ and $1's$ and there are exactly $3n$ zeroes. Show that it is possible to remove all the zeroes by deleting some $n$ rows and some $n$ columns.
1995 India National Olympiad
In an acute angled triangle $ABC$, $\angle A = 30^{\circ}$, $H$ is the orthocenter, and $M$ is the midpoint of $BC$. On the line $HM$, take a point $T$ such that $HM = MT$. Show that $AT = 2 BC$.
1995 India National Olympiad
Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.
1995 India National Olympiad
Show that the number of $3-$element subsets $\{ a , b, c \}$ of $\{ 1 , 2, 3, \ldots, 63 \}$ with $a+b +c < 95$ is less than the number of those with $a + b +c \geq 95.$
1995 India National Olympiad
Let $ABC$ be a triangle and a circle $\Gamma'$ be drawn lying outside the triangle, touching its incircle $\Gamma$  externally, and also the two sides $AB$ and $AC$. Show that the ratio of the radii of the circles $\Gamma'$ and $\Gamma$ is equal to $\tan^ 2 { \left( \dfrac{ \pi - A }{4} \right) }.$
1995 India National Olympiad
Let $n \geq 2$. Let $a_1 , a_2 , a_3 , \ldots a_n$ be $n$ real numbers all less than $1$ and such that $|a_k - a_{k+1} | < 1$ for $1 \leq k \leq n-1$. Show that $$ \dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 .  $$
1995 India National Olympiad
Find all primes $p$ for which the quotient $$ \dfrac{2^{p-1} - 1 }{p} $$ is a square.
1994 India National Olympiad
Let $G$ be the centroid of the triangle $ABC$ in which the angle at $C$ is obtuse and $AD$ and $CF$ be the medians from $A$ and $C$ respectively onto the sides $BC$ and $AB$. If the points $\ B,\ D, \ G$ and $\ F$ are concyclic, show that $\dfrac{AC}{BC} \geq \sqrt{2}$. If further $P$ is a point on the line $BG$ extended such that $AGCP$ is a parallelogram, show that triangle $ABC$ and $GAP$ are similar.
1994 India National Olympiad
If $x^5 - x ^3 + x = a,$  prove that $x^6 \geq 2a - 1$.
1994 India National Olympiad
In any set of $181$ square integers, prove that one can always find a subset of $19$ numbers, sum of whose elements is divisible by $19$.
1994 India National Olympiad
Find the number of nondegenerate triangles whose vertices lie in the set of points $(s,t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, $s$ and $t$ are integers.
1994 India National Olympiad
A circle passes through the vertex of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at $M$ and $N$ respectively. If the distance from $C$ to the line segment $MN$ is equal to $5$ units, find the area of rectangle $ABCD$.
1994 India National Olympiad
Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.
1993 India National Olympiad
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ intersect at $P$. Let $O$ be the circumcenter of triangle $APB$ and $H$ be the orthocenter of triangle $CPD$. Show that the points $H,P,O$ are collinear.
1993 India National Olympiad
Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$. Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$.
1993 India National Olympiad
If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$, show that $$ ab +bc +cd \leq \dfrac{1}{4}.  $$
1993 India National Olympiad
Let $ABC$ be a triangle in a plane $\pi$. Find the set of all points $P$ (distinct from $A,B,C$ ) in the plane $\pi$ such that the circumcircles of triangles $ABP$, $BCP$, $CAP$ have the same radii.
1993 India National Olympiad
Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.
1993 India National Olympiad
Let $ABC$ be a triangle right-angled at $A$ and $S$ be its circumcircle. Let $S_1$ be the circle touching the lines $AB$ and $AC$, and the circle $S$ internally. Further, let $S_2$ be the circle touching the lines $AB$ and $AC$ and the circle $S$ externally. If $r_1, r_2$ be the radii of $S_1, S_2$ prove that $r_1 \cdot r_2 = 4 A[ABC]$.
1993 India National Olympiad
Let $A = \{ 1,2, 3 , \ldots, 100 \}$ and $B$ be a subset of $A$ having $53$ elements. Show that $B$ has 2 distinct elements $x$ and $y$ whose sum is divisible by $11$.
1993 India National Olympiad
Let $f$ be a bijective function from $A = \{ 1, 2, \ldots, n \}$ to itself. Show that there is a positive integer $M$ such that $f^{M}(i) = f(i)$ for each $i$ in $A$, where $f^{M}$ denotes the composition $f \circ f \circ \cdots \circ f$ $M$ times.
1993 India National Olympiad
Show that there exists a convex hexagon in the plane such that



(i) all its interior angles are equal;



(ii) its sides are $1,2,3,4,5,6$ in some order.
1992 India National Olympiad
In a triangle $ABC$, $\angle A = 2 \cdot \angle B$. Prove that $a^2 = b (b+c)$.
1992 India National Olympiad
If $x , y, z \in \mathbb{R}$ such that $x+y +z =4$ and $x^2 + y^2 +z^2 = 6$, then show that each of $x, y, z$ lies in the closed interval $\left[ \dfrac{2}{3} , 2 \right]$. Can $x$ attain the extreme value $\dfrac{2}{3}$ or $2$?
1992 India National Olympiad
Find the remainder when $19^{92}$ is divided by 92.
1992 India National Olympiad
Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$, $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$.
1992 India National Olympiad
Two circles $C_1$ and $C_2$ intersect at two distinct points $P, Q$ in a plane. Let a line passing through $P$ meet the circles $C_1$ and $C_2$ in $A$ and $B$ respectively. Let $Y$ be the midpoint of $AB$ and let $QY$ meet the cirlces $C_1$ and $C_2$ in $X$ and $Z$ respectively. Show that $Y$ is also the midpoint of $XZ$.
1992 India National Olympiad
Let $f(x)$ be a polynomial in $x$ with integer coefficients and suppose that for five distinct integers $a_1, \ldots, a_5$ one has $f(a_1) = f(a_2) = \ldots = f(a_5) = 2$. Show that there does not exist an integer $b$ such that $f(b) = 9$.
1992 India National Olympiad
Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.
1992 India National Olympiad
Determine all pairs $(m,n)$ of positive integers for which $2^{m} + 3^{n}$ is a perfect square.
1992 India National Olympiad
Let $A_1, A_2, \ldots, A_n$ be an $n$ -sided regular polygon. If $\frac{1}{A_1 A_2} = \frac{1}{A_1 A_3} + \frac{1}{A_1A_4}$, find $n$.
1992 India National Olympiad
Determine all functions $f : \mathbb{R} - [0,1] \to \mathbb{R}$ such that $$ f(x) + f \left( \dfrac{1}{1-x} \right) = \dfrac{2(1-2x)}{x(1-x)} .  $$
1991 India National Olympiad
Find the number of positive integers $n$ for which



(i) $n \leq 1991$;



(ii) 6 is a factor of $(n^2 + 3n +2)$.
1991 India National Olympiad
Given an acute-angled triangle $ABC$, let points $A' , B' , C'$ be located as follows:  $A'$ is the point where altitude from $A$ on $BC$ meets the outwards-facing semicircle on $BC$ as diameter. Points $B', C'$ are located similarly.

Prove that $A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2$ where $A[ABC]$ is the area of triangle $ABC$.
1991 India National Olympiad
Given a triangle $ABC$ let  \begin{eqnarray*} x &=& \tan\left(\dfrac{B-C}{2}\right) \tan \left(\dfrac{A}{2}\right) \\ y &=& \tan\left(\dfrac{C-A}{2}\right) \tan \left(\dfrac{B}{2}\right) \\ z &=& \tan\left(\dfrac{A-B}{2}\right) \tan \left(\dfrac{C}{2}\right). \end{eqnarray*} Prove that $x+ y + z + xyz = 0$.
1991 India National Olympiad
Let $a,b,c$ be real numbers with $0 < a< 1$, $0 < b < 1$, $0 < c < 1$, and $a+b + c = 2$.

Prove that $\dfrac{a}{1-a} \cdot \dfrac{b}{1-b} \cdot \dfrac{c}{1-c} \geq 8$.
1991 India National Olympiad
Triangle $ABC$ has an incenter $I$. Let points $X$, $Y$ be located on the line segments $AB$, $AC$ respectively, so that $BX \cdot AB = IB^2$ and $CY \cdot AC = IC^2$. Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$.
1991 India National Olympiad
(i) Determine the set of all positive integers $n$ for which $3^{n+1}$ divides $2^{3^n} + 1$;



(ii) Prove that $3^{n+2}$ does not divide $2^{3^n} + 1$ for any positive integer $n$.
1991 India National Olympiad
Solve the following system for real $x,y,z$

$$ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} $$
1991 India National Olympiad
There are $10$ objects of total weight $20$, each of the weights being a positive integers. Given that none of the weights exceeds $10$ , prove that the ten objects can be divided into two groups that balance each other when placed on 2 pans of a balance.
1991 India National Olympiad
Triangle $ABC$ has an incenter $I$ l its incircle touches the side $BC$ at $T$. The line through $T$ parallel to $IA$ meets the incircle again at $S$ and the tangent to the incircle at $S$ meets $AB , AC$ at points $C' , B'$ respectively. Prove that triangle $AB'C'$ is similar to triangle $ABC$.
1991 India National Olympiad
For any positive integer $n$ , let $s(n)$ denote the number of ordered pairs $(x,y)$ of positive integers for which $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$ . Determine the set of positive integers for which $s(n) = 5$
1990 India National Olympiad
Given the equation

$$ x^4 + px^3 + qx^2 + rx + s = 0$$

has four real, positive roots, prove that

(a) $ pr - 16s \geq 0$

(b) $ q^2 - 36s \geq 0$

with equality in each case holding if and only if the four roots are equal.
1990 India National Olympiad
Determine all non-negative integral pairs $ (x, y)$ for which

$$ (xy - 7)^2 = x^2 + y^2.$$
1990 India National Olympiad
Let $ f$ be a function defined on the set of non-negative integers and taking values in the same

set. Given that



(a) $ \displaystyle x - f(x) = 19\left[\frac{x}{19}\right] - 90\left[\frac{f(x)}{90}\right]$ for all non-negative integers $ x$;



(b) $ 1900 < f(1990) < 2000$,



find the possible values that $ f(1990)$ can take.

(Notation : here $ [z]$ refers to largest integer that is $ \leq z$, e.g. $ [3.1415] = 3$).
1990 India National Olympiad
Consider the collection of all three-element subsets drawn from the set $ \{1,2,3,4,\dots,299,300\}$.

Determine the number of those subsets for which the sum of the elements is a multiple of 3.
1990 India National Olympiad
Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity

$$ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$$

must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?
1990 India National Olympiad
Triangle $ ABC$ is scalene with angle $ A$ having a measure greater than 90 degrees. Determine

the set of points $ D$ that lie on the extended line $ BC$, for which

$$ |AD|=\sqrt{|BD| \cdot |CD|}$$

where $ |BD|$ refers to the (positive) distance between $ B$ and $ D$.
1990 India National Olympiad
Let $ ABC$ be an arbitrary acute angled triangle. For any point $ P$ lying within the triangle, let

$ D$, $ E$, $ F$ denote the feet of the perpendiculars from $ P$ onto the sides $ AB$, $ BC$, $ CA$ respectively.

Determine the set of all possible positions of the point $ P$ for which the triangle $ DEF$ is isosceles.

For which position of $ P$ will the triangle $ DEF$ become equilateral?
1989 India National Olympiad
Prove that the Polynomial $ f(x) = x^{4} + 26x^{3} + 56x^{2} + 78x + 1989$ can't be expressed as a product $ f(x) = p(x)q(x)$ , where $ p(x)$ and $ q(x)$ are both polynomial with integral coefficients and with degree at least $ 1$.
1989 India National Olympiad
Let $ a,b,c$ and $ d$ be any four real numbers, not all equal to zero. Prove that the roots of the polynomial $ f(x) = x^{6} + ax^{3} + bx^{2} + cx + d$ can't all be real.
1989 India National Olympiad
Let $ A$ denote a subset of the set $ \{ 1,11,21,31, \dots ,541,551 \}$ having the property that no two elements of $ A$ add up to $ 552$. Prove that $ A$ can't have more than $ 28$ elements.
1989 India National Olympiad
Determine all $n \in \mathbb{N}$ for which



$n$ is not the square of any integer,

$\lfloor \sqrt{n}\rfloor ^3$ divides $n^2$.
1989 India National Olympiad
For positive integers $ n$, define $ A(n)$ to be $ \frac {(2n)!}{(n!)^{2}}$. Determine the sets of positive integers $ n$ for which



(a) $ A(n)$ is an even number,



(b) $ A(n)$ is a multiple of $ 4$.
1989 India National Olympiad
Triangle $ ABC$ has incentre $ I$ and the incircle touches $ BC, CA$ at $ D, E$ respectively. Let $ BI$ meet $ DE$ at $ G$. Show that $ AG$ is perpendicular to $ BG$.
1988 India National Olympiad
Let $ m_1,m_2,m_3,\dots,m_n$ be a rearrangement of the numbers $ 1,2,\dots,n$. Suppose that $ n$ is odd. Prove that the product

$$ \left(m_1-1\right)\left(m_2-2\right)\cdots \left(m_n-n\right)$$

is an even integer.
1988 India National Olympiad
Prove that the product of 4 consecutive natural numbers cannot be a perfect cube.
1988 India National Olympiad
Five men, $ A$, $ B$, $ C$, $ D$, $ E$ are wearing caps of black or white colour without each knowing the colour of his cap. It is known that a man wearing black cap always speaks the truth while the ones wearing white always tell lies. If they make the following statements, find the colour worn by each of them:

$ A$ : I see three black caps and one white cap.

$ B$ : I see four white caps

$ C$ : I see one black cap and three white caps

$ D$ : I see your four black caps.
1988 India National Olympiad
If $ a$ and $ b$ are positive and $ a + b = 1$, prove that

$$ \left(a+\frac{1}{a}\right)^2+\left(b+\frac{1}{b}\right)^2 \geq \frac{25}{2}$$
1988 India National Olympiad
Show that there do not exist any distinct natural numbers $ a$, $ b$, $ c$, $ d$ such that $ a^3+b^3=c^3+d^3$ and $ a+b=c+d$.
1988 India National Olympiad
If $ a_0,a_1,\dots,a_{50}$ are the coefficients of the polynomial

$$ \left(1+x+x^2\right)^{25}$$

show that $ a_0+a_2+a_4+\cdots+a_{50}$ is even.
1988 India National Olympiad
Given an angle $ \angle QBP$ and a point $ L$ outside the angle $ \angle QBP$. Draw a straight line through $ L$ meeting $ BQ$ in $ A$ and $ BP$ in $ C$ such that the triangle $ \triangle ABC$ has a given perimeter.
1988 India National Olympiad
A river flows between two houses $ A$ and $ B$, the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from $ A$ to $ B$, using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks.
1988 India National Olympiad
Show that for a triangle with radii of circumcircle and incircle equal to $ R$, $ r$ respectively, the inequality $ R \geq 2r$ holds.
1987 India National Olympiad
Given $ m$ and $ n$ as relatively prime positive integers greater than one, show that

$$ \frac{\log_{10} m}{\log_{10} n}$$

is not a rational number.
1987 India National Olympiad
Determine the largest number in the infinite sequence

$$ 1, \sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4}, \dots, \sqrt[n]{n},\dots$$
1987 India National Olympiad
Let $ T$ be the set of all triplets $ (a,b,c)$ of integers such that $ 1 \leq a < b < c \leq 6$ For each triplet $ (a,b,c)$ in $ T$, take number $ a\cdot b \cdot c$. Add all these numbers corresponding to all the triplets in $ T$. Prove that the answer is divisible by 7.
1987 India National Olympiad
If $ x$, $ y$, $ z$, and $ n$ are natural numbers, and $ n\geq z$ then prove that the relation $ x^n + y^n = z^n$ does not hold.
1987 India National Olympiad
Find a finite sequence of 16 numbers such that:

(a) it reads same from left to right as from right to left.

(b) the sum of any 7 consecutive terms is $ -1$,

(c) the sum of any 11 consecutive terms is $ +1$.
1987 India National Olympiad
Prove that if coefficients of the quadratic equation $ ax^2+bx+c=0$ are odd integers, then the roots of the equation cannot be rational numbers.