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Indian National Math Olympiad

Click on the year to find problems (and possibly discussions).

—- —- INMO 1986 INMO 1987 INMO 1988 INMO 1989
INMO 1990 INMO 1991 INMO 1992 INMO 1993 INMO 1994 INMO 1995
INMO 1996 INMO 1997 INMO 1998 INMO 1999 INMO 2000 INMO 2001
INMO 2002 INMO 2003 INMO 2004 INMO 2005 INMO 2006 INMO 2007
INMO 2008 INMO 2009 INMO 2010 INMO 2011 INMO 2012 INMO 2013
INMO 2014 INMO 2015 INMO 2016 INMO 2017 INMO 2018 INMO 2019

===INMO 1986===

  1. A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ?
  2. Solve $$ \log_{2}x+\log_{4}y+\log_{4}z=2 \log_{3}y+\log_{9}z+\log_{9}x=2 \log_{4}z+\log_{16}x+\log_{16}y=2 $$
  3. Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that $$ \frac{1}{\sqrt{c}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}} $$
  4. Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number.
  5. If \( P(x) \) is a polynomial with integer coefficients and a, b, c , three distinct integers, then show that it is impossible to have \(P(a)=b\), \(P(b)=c\), \(P(c)=a\).
  6. Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal.
  7. If \( a \), \( b \), \( x \), \( y \) are integers greater than 1 such that \( a \) and \( b \) have no common factor except 1 and \( x^{a}= y^{b} \) show that , for some integer greater than 1.
  8. Suppose \( A_{1},dots, A_{6}\) are six sets each with four elements and \( B_{1},dots,B_{n}\) are sets each with two elements, Let \( S = A_{1}\cup A_{2}\cup \cdots \cup A_{6}= B_{1}\cup \cdots \cup B_{n} \). Given that each elements of belongs to exactly four of the \( A \)’s and to exactly three of the \( B \)’s, find \( n\).
  9. Show that among all quadrilaterals of a given perimeter the square has the largest area

===INMO 1987===

  1. Given and as relatively prime positive integers greater than one, show that \([\frac{\log_{10}m}{\log_{10}n} ]\) is not a rational number.
    Determine the largest number in the infinite sequence \([ 1,\sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4},dots,\sqrt[n]{n},dots ]\)
  2. 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 \( ac.bc.c \). Add all these numbers corresponding to all the triplets in \( T \). Prove that the answer is divisible by 7.
  3. 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.
  4. 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 \( +1 \).
  5. 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.
  6. Construct the \( \triangle ABC \), given \( h_{a} \), \( h_{b} \) (the altitudes from \( A \) and \( B \)) and \( m_{a} \), the median from the vertex \( A \).
  7. Three congruent circles have a common point \( O \) and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the common point \( O \) are collinear.
  8. Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles.

===INMO 1988===

  1. 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 $$[\left(m_{1}-1\right)\left(m_{2}-2\right)\cdot\left(m_{n}-n\right)]$$ is an even integer.
  2. Prove that the product of 4 consecutive natural numbers cannot be a perfect cube.
  3. 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.
  4. If \( a \) and \( b \) are positive and \( a+b = 1 \), prove that $$[\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2}\geq\frac{25}{2}]$$
  5. 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 \( a^{3}+b^{3}=c^{3}+d^{3} \).
  6. 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.
  7. 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.
  8. 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.
  9. Show that for a triangle with radii of circumcircle and incircle equal to \( R \), \( r \) respectively, the inequality \( R\geq 2r \) holds.

===INMO 1989===

  1. Prove that the polynomial \( f(x)=x^4+26x^3+56x^2+78x+1989 \) cannot be expressed as a product \( f(x) = p(x)q(x) \),where \( p(x), q(x) \) are both polynomials with integral coefficients and with degree less than 4.
  2. Let a, b, c and d be any four real numbers, not all equal to zero. Prove that the roots of the polynomial \( x^6+ax^3+bx^2+cx+d \) cannot all be real.
  3. Let A denote a subset of the set {1, 11, 21, 31, . . . 541, 551} having the property that no two elements of A add up to 552. Prove that A cannot have more than 28 elements.
  4. Determine with proof, all the positive integers n for which: (a) n is not the square of any integer; and (b) \( [\sqrt{n}] \) divides \( n^2 \). (Notation : [x] denotes the largest integer that is less than or equal to x).
  5. 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.
  6. Triangle ABC has incenter I and the incircle touches BC, CA at D, E respectively. Let BI meet DE at G. Show that AG is perpendicular to BC.
  7. Let A be one of the two points of intersection of two circles with centers X , Y respectively. The tangents at A to the two circles meet the circles again at B.C . Let a point P belocated so that PX AY is a parallelogram. Show that P is also the circum-center of triangle ABC .

===INMO 1990===

  1. Given the equation \( x^6+px^3+qx^2+rx+s=0 \) has four real, positive roots, prove that (a) \( pr-16s \ge 0 \), (b) \( q^2-16s \ge 0 \) with equality in each case holding if and only if the four roots are equal.
  2. Determine all non-negative integral pairs (x, y) for which \( (xy-7)^2=x^2+y^2 \).Let f be a function defined on the set of non-negative integers and taking values in the same set. Given that \( (a) x − f(x) = 19[x/19] − 90[f(x)/90] \) 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 z, e.g. [3.1415] = 3).
  3. Let f be a function defined on the set of non-negative integers and taking values in the same set. Given that \( (a) x − f(x) = 19[x/19] − 90[f(x)/90] \) 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 z, e.g. [3.1415] = 3).
  4. Consider the collection of all three-element subsets drawn from the set {1, 2, 3, 4, . . . , 299, 300}. Determine the number of those subsets for which the sum of the elements is a multiple of 3.
  5. Let a, b, c denote the sides of a triangle. Show that the quantity \( \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \) must lie between the limits 3/2 and 2. Can equality hold at either limits?
  6. 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 \( |AD|=\sqrt{|BD||CD|} \) where |BD| refers to the (positive) distance between B and D.
  7. 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?

===INMO 1991===

  1. Find the number of positive integers n for which (a) \( n \le 1991 \) and (b) 6 is a factor of \( n^2+3n+2 \).
  2. Given any 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 semi-circle drawn on BC as diameter. Points \( B^’,C^’ \) are located similarly. Prove that \( |BCA^’|^2+|CAB^’|^2+|ABC^’|^2=|ABC|^2 \), where [ABC] denotes the area of triangle ABC, etc.
  3. Given a triangle ABC, define the quantities x, y, z as follows: \( x = \tan((B − C)/2) \tan(A/2), y = \tan((C − A)/2) \tan(B/2), z = \tan((A − B)/2) \tan(C/2) \). Prove that : $$ x + y + z + xyz = 0. $$
  4. Let a, b, c be real numbers with 0 < a < 1, 0 < b < 1, 0 < c < 1 and a+b+c = 2. Prove that $$ \frac{a}{1-a}.\frac{b}{1-b}.\frac{c}{1-c} \ge 8 $$.
  5. Triangle ABC has incenter I. Let points X, Y be located on the line segment AB, AC respectively so that : \( BX.AB=IB^2 \) and \( CY.AC=IC^2 \) Given that the points X, I, Y lie on a straight line, find the possible values of the measure of angle A.
  6. (a) Determine the set of all positive integers n for which \( 3^{n+1} \) divides \( 2^{j+1} \). (b) Prove that \( 3^{n+2} \) does not divide \( 2^{3^{n}}+1 \) for any positive integer n.
  7. Solve the following system of equations for real x, y, z: \( x+y-z=4, x^2-y^2+z^2=4,xyz=6 \).
  8. There are 10 objects with total weight 20, each of the weights being a positive integer. Given that none of the weights exceeds 10, prove that the 10 objects can be divided into two groups that balance each other when placed on the two pans of a balance.

===INMO 1992===

  1. In a triangle ABC, angle A is twice angle B. Show that \( a^2=b.(b+c) \).
  2. If x, y and z are three real numbers such that x + y + z = 4 and \( x^2+y^2+z^2=6 \), then show that each of x, y and z lies in the closed interval [2/3, 2], that is \( \frac{2}{3} \le x \le 2 \)$, \( \frac{2}{3} \le y \le 2 \),and. Can x attain the extreme value 2/3 or 2 ?
  3. Find the remainder when \( 19^{92} \) is divided by 92.
  4. 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, \( 1 \le k \le 5 \). \( (P_1, P_2, . . . P_k) \) does not form a permutation of {1, 2, . . . k}. That is \( P_1 \neq 0 \); \( (P_1, P_2) \) is not permutation of {1, 2}; \( (P_1, P_2, P_3) \) is not a permutation of {1, 2, 3}, etc.
  5. Two circles C1 and C2 intersect at two distinct points P and Q in a plane. Let a line passing through P meet the circles C1 and C2 in A and B respectively. Let Y be the mid-point of AB and QY meet the circles C1 and C2 in X and Z respectively. Show that Y is also the mid-point of XZ.
  6. Let \( f(x) \) be a polynomial in x with integer coefficients and suppose that for 5 distinct integers \( a_1,a_2,a_3,a_4 \)  and \( a_5 \) one has \( f(a_1)=f(a_2)=f(a_3)=f(a_4)=f(a_5)=2 \). Show that there does not exist an integer b such that \( f(b) = 9 \).
  7. Find the number of ways in which one can place the numbers \( 1, 2, 3, . . ., n^2 \) on the \( n^2 \) squares of n × n chessboard, one on each, such that the numbers in each row and each column are in arithmetic progression. (Assume \( n \ge 3 ) \).
  8. Determine all pairs (m, n) of positive integers for which \( 2^n+3^n \) is a perfect square.
  9. Let \( A_1A_2A_3 . . .A_n \) be an n-sided regular polygon such that \( \frac{1}{A_1A_2}=\frac{A_1}{A_3}+\frac{A_1}{A_4} \). Determine n, the number of sides of the polynomial.
  10. Determine all functions \( f:R{0,1} \mapsto R \) satisfying the functional relation,where x is a real number different from 0 and 1.

(Here \( R \) denotes the set of all real numbers.)

===INMO 1993===

  1. 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.
  2. If a, b, c, d are 4 non-negative real numbers and a + b + c + d = 1, show that \( ab+bc+cd \le \frac{1}{4} \).
  3. Let ABC be a triangle in a plane \( \sum \). Find the set of all points P (distinct from A, B, C) in the plane \( \sum \) such that the circumcircles of triangles ABP, BCP and CAP have the same radii.
  4. Let \( P(x)=x^2+ax+b \)= be a quadratic polynomial in which a and b are integers. Given any integer n, show that there is an integer M such that P(n) · P(n + 1) = P(M).
  5. Show that there is a natural number n such that n! when written in decimal notation (that is, in base 10) ends exactly in 1993 zeros.
  6. Let ABC be 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 \( r_1 \) and \( r_2 \) be the radii of the circles \( S_1 \) and \( S_2 \) respectively, show that \( r_1·r_2=4(area \Delta ABC) \).
  7. Let A = {1, 2, 3, . . . , 100} and B be a subset of A having 53 elements. Show that B has two distinct elements x and y whose sum is divisible by 11.
  8. Let f be a bijective (1-1 and onto) function from A = {1, 2, 3 . . . , n} to itself. Show that there is positive number such that, for each i in A. denotes the composite function \( f \) m-times.
  9. Show that there exists a convex hexagon in the plane such that (a) all its interior angles are equal, (b) all its sides are 1, 2, 3, 4, 5, 6 in some order.

===INMO 1994===

  1. Let G be the centroid of a triangle ABC in which the angle C is obtuse and AD and CF be the medians from A and C respectively onto the sides BC and AB. If the four points B, D, G and F are concyclic, show that \( \frac{AC}{BC} > \sqrt{2} \). If further P is a point on the line BG extended such that AGCP is a parallelogram, show that the triangle ABC and GAP are similar.
  2. If \( x^5-x^3+x=a \), prove that \( x^6 \ge 2a-1 \).
  3. 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.
  4. Find the number of nondegenerate triangles whose vertices lie in the set of points (s, t) in the plane such that \( 0 \le s \le 4 \), \( 0 \le t \le 4 \) with s and t integers.
  5. A circle passes through a vertex C of a triangle 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 the rectangle ABCD.
  6. If \( f:R \mapsto R \) is a function satisfying the properties (a) f(−x) = −f(x), (b) f(x + 1) = f(x) + 1, (c) \( f(\frac{1}{x})=\frac{f(x)}{x^2} \), for \( x \neq 0) \), prove that\( f(x) = x \) for all real values of x. Here \( R \) denotes the set of all real numbers.

===INMO 1995===

  1. In an acute-angled triangle ABC,<A=\( 30^0 \), H is the orthocenter and M is the mid-point of BC. On the line HM, take a point T such that HM = MT. Show that AT = 2BC.
  2. Show that there are infinitely many pairs (a, b) of relatively prime integers (not necessarily positive) such that both quadratic functions \( x^2+ax+b=0 \) and \( x^2+2ax+b=0 \) have integer roots.
  3. Show that the number of 3−element subset {a, b, c} of {1, 2, 3, . . . , 63} with a + b + c < 95 is less than the number of those with a + b + c > 95.
  4. Let ABC be triangle and a circle \( \Gamma^’ \) be drawn inside the triangle, touching its incircle \( \Gamma \) externally and also touching the two sides AB and AC. Show the ratio of the radii of the circles \( \Gamma \) and \( \Gamma^’ \) is equal to \( \tan^2{\frac{\pi-A}{4}} \).
  5. Let \( a_1 ,a_2 ,a_3 , . . . ,a_n \) be n real numbers all greater than 1 and such that \( 1 \le k \le {n-1} \) for. Show that \( \frac{a_1}{a_2}+ \frac{a_2}{a_3} + . . . . .+ \frac{a_{n-1}}{a_n}+ \frac{a_n}{a_{1}} < 2n-1 \)
  6. Find all primes p for which the quotient \( 2^{p-1}-1 \) is a square.

===INMO 1996===

  1. (a) Given any positive integer n, show that there exist distinct positive integers x and y such that x + j divides y + j for j = 1, 2, 3, · · · , n. (b) If for some positive integers x and y, x + j divides y + j for all positive integers j, prove that x = y.
  2. Let C1 and C2 be two concentric circles in the plane with radii R and 3R respectively. Show that the orthocentre of any triangle inscribed in circle C1 lies in the interior of circle C2. Conversely, show that also every point in the interior of C2 is the orthocentre of some triangle inscribed in C1.
  3. Solve the following system of equations for real numbers a, b, c, d, e. \( 3a=(b+c+d)^2,3b=(c+d+e)^2,3c=(d+e+a)^2,3d=(e+a+b)^2,3e=(a+b+c)^2 \).
  4. Let X be a set containing n elements. Find the number of all 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.
  5. Define a sequence \( (a_n)_{n \ge 1} \) by \( a_1=1,a_2=2 \) and \( a_{n+2}=2a_{n+1}-a_{n}+2 \) for \( n \ge 1 \). Prove that for any m, \( a_{m}a_{m+1} \) is also a term in the sequence.
  6. There is a 2n × 2n array (matrix) consisting of 0’s and 1’s and there are exactly 3n zeros.Show that it is possible to remove all the zeros by deleting some n rows and some n columns. [Note: A m × n array is a rectangular arrangement of mn numbers in which there are m horizontal rows and n vertical columns.]

===INMO 1997===

  1. 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 \( AC^2+CE.CF=AB.AE+AD.AF \)
  2. Show that there do not exist positive integers m and n such that \( \frac{m}{n}+\frac{m+1}{n}=4 \).
  3. If a, b, c are three distinct real numbers and \( (a+ \frac{1}{b})+(b+\frac{1}{c})+(c+\frac{1}{a})=t \) for some real number t, prove that abc + t = 0.
  4. In a unit square one hundred segments are drawn from the center to to the sides dividing the square into one hundred parts (triangles and possibly quadrilaterals). If all the parts have equal perimeter p show that 1 · 4 < p < 1 · 5.
  5. Find the number of 4 × 4 arrays whose entries are from the set {0, 1, 2, 3} and which are such that the sum of numbers in each of the four rows and each of the four columns is divisible by 4. (An m × n array is an arrangement of mn numbers in m rows and n columns.)
  6. 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 is a root of this cubic with minimum absolute value, prove that \( \frac{b}{a} \le \alpha \le \frac{3b}{2a} \)

===INMO 1998===

  1. In a circle \( C_1 \) with centre O, let AB be a chord that is not a diameter. Let M be the midpoint of AB. Take a point T on the circle \( C_2 \) with OM as diameter. Let the tangent to \( C_2 \) at T meet \( C_1 \) in P. Show that \( PA^2+PB^2=4PT^2 \)
  2. Let a and b be two positive \( a^{\frac{1}{3}}+b^{\frac{1}{3}} \) rational numbers such that \( a^{\frac{1}{3}}+b^{\frac{1}{3}} \) is also a rational number. Prove that themselves are rational numbers.
  3. 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 by 5, prove that there is an integer b such that \( sb^3+rb^2+qb+p \) is also divisible by 5.
  4. Suppose ABCD is a cyclic quadrilateral inscribed in a circle of radius one unit. If \( AB.BC.CD.DA \ge 4 \),prove that ABCD is a square.
  5. Suppose a, b, c are three real 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 \( \beta \neq 0 \) are real numbers [here \( i=\sqrt{-1} \)]. 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.
  6. It is desired to choose n integers from the collection of 2n integers, namely, 0, 0, 1, 1, 2, 2, . . . , n−1, n−1 such that the average (that is, the arithmetic mean) 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 average for each n.

===INMO 1999===

  1. Let ABC be an acute angled triangle in which D, E, F are points on BC, CA, AB respectively such that AD is perpendicular to BC; AE = EC; and CF bisects <C internally. Suppose CF meets AD and DE in M and N respectively. If FM = 2, MN = 1, NC = 3, find the perimeter of the triangle ABC.
  2. In a village 1998 persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equal to 3996 feet. For this purpose, the field was divided into 1998 equal parts. If each part had an integer area ( measured in Sq. ft.), find the length and breadth of the field.
  3. 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 \(p(x)q(x)=x^5+2x+1 \).
  4. 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 \( {P^’A}^2+{P^’B}+{P^’C}={A^’P}^2+{B^’P}^2+{C^’P}^2 \)
  5. Given any four distinct real numbers, show that one can choose three numbers, say, A, B, C from among them such that all the three quadratic equations \( Bx^2+x+C=0, Cx^2+x+A=0, Ax^2+x+B=0 \) have only real roots or all the three equations have only imaginary roots.
  6. For which positive integer values of n can the set {1, 2, 3, 4, . . . ,4n} be split into n disjoint 4-element subsets {a, b, c, d} such that in each of these sets \( a=\frac{b+c+d}{3} \)?

===INMO 2000===

  1. The incircle of triangle ABC touches the sides BC,CA and AB in K,L and M respectively. The line through A and parallel to LK meets MK in P and the line through A and parallel to MK meets LK in Q. Show that the line PQ bisects the sides AB and AC of the triangle ABC.
  2. Solve for integers x, y, z: \(x+y=1-z,x^3+y^3=1-z^2 \).
  3. If a, b, c, x are real numbers such that \(abc \neq 0 \) 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.
  4. In a convex quadrilateral PQRS, PQ = RS, \((\sqrt{3}-1) \)QR = SP and < RSP − < SPQ = \(30^0 \). Prove that <PQR – <QRS = \(90^0 \)
  5. Let a, b, c be three real numbers such that \(1 \ge a \ge b \ge c \ge 0 \). Prove that if \(\lambda \) is a root of the cubic equation \(x^3+ax^2+bx+c=0 \) (real or complex), then \(|\lambda| \le 1 \).
  6. For any natural numbers n, \((n \ge 3) \), let f(n) denote the number of non congruent integer-sided triangles with perimeter n (e. g., f(3) = 1, f(4) = 0, f(7) = 2). Show that (a) f(1999) > f(1996);(b) f(2000) = f(1997).

===INMO 2001===

  1. Let ABC be a triangle in which no angle is \(90^0 \). 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 the following statements
    (a) If P is the incentre or an excentre of ABC, then P is the circumcentre of \(A_1B_1C_1 \);
    (b) If P is the circumcentre of ABC, then P is the orthocentre of \(A_1B_1C_1 \);
    (c) If P is the orthocentre of ABC, then P is either the incentre or an excentre of \(A_1B_1C_1 \).
  2. Show that the equation \( x^2 + y^2 + z^2 = (x − y)(y − z)(z − x) \) has infinitely many solutions in integers x, y, z.
  3. If a, b, c are positive real numbers such that abc = 1, prove that $$ a^{b+c} b^{c+a} c^{a+b} \le 1 $$.
  4. Given any nine integers show that it is possible to choose, from among them, four integers a, b, c, d such that a + b − c − d is divisible by 20. Further show that such a selection is not possible if we start with eight integers instead of nine.
  5. Let ABC be a triangle and D be the midpoint of side BC. Suppose <DAB = <BCA and <DAC = \(15^0 \). Show that ADC is obtuse. Further, if O is the circumcentre of ADC, prove that triangle AOD is equilateral.
  6. Let R denote the set of all real numbers. Find all functions \(f : R \mapsto R \) satisfying the condition f(x + y) = f(x)f(y)f(xy) for all x, y in R.

===INMO 2002===

  1. For a convex hexagon ABCDEF , consider the following six 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 the six statements are true , then the hexagon is cyclic ( i. e. , it can be inscribed in a circle ). (b) Prove that , in fact , any five of these six statements also imply that the hexagon is cyclic.
  2. Determine the least positive value taken by the expression a3 + b3 + c3 = 3abc as a, b, c vary over all positive integers . Find also all triples (a, b, c) for which this least value is attained .
  3. Let x, y be positive reals such that x + y = 2. Prove that \(x^3y^3(x^3 + y^3) le 2 \).
  4. Do there exist 100 lines in the plane , no three of them concurrent , such that they intersect exactly in 2002 points ?
  5. Do there exist three distinct positive real numbers a, b, c such that the numbers a, b, c, b + c −a, c + a − b, a + b − c and a + b + c form a 7-term arithmetic progression in some order ?
  6. Suppose the \(n^2 \) numbers 1, 2, 3, · · · , \(n^2 \) are arranged to form an n by n array consisting of n rows and n columns such that the numbers in each row ( from left to right ) and each column ( from top to bottom ) are in increasing order . Denote by \(a_{jk} \) the number in the j − th row and k − ”th” column . Suppose \(b_j \) is the maximum possible number of entries that can occur as \(a_{jj} \) , \(1 le j le n \) . Prove that $$ b_1 + b_2 + b_3 + · · · b_n \le \frac{n}{3}(n^2 − 3n + 5) $$.( Example : In the case n = 3 , the only numbers which can occur as \(a_{22} \) are 4, 5 or 6 so that \(b_2 \) = 3 . )

===INMO 2003===

  1. Consider an acute triangle ABC and let P be an interior point of ABC . Suppose the lines BP and CP , when produced , meet AC and AB in E and F respectively . Let D be the point where AP intersects the line segment EF and K be the foot of perpendicular from D on to BC . Show that DK bisects <EKF .
  2. Find all primes p and q ,and even numbers n > 2 , satisfying the equation \( p^n+ p^{n−1}+ · · · + p + 1 = q^2+ q + 1 .
  3. Show that for every real number a the equation \( 8x^4− 16x^3+ 16x^2− 8x + a = 0 \) has at least one non-real root and find the sum of all the non-real roots of the equation .
  4. Find all 7-digit numbers formed by using only the digits 5 and 7 , and divisible by both 5 and 7 .
  5. Let ABC be a triangle with sides \( a, b, c \) . Consider a triangle \( A_1B_1C_1 \) with sides equal to \( a +\frac{b}{2}, b +\frac{c}{2}, c +\frac{a}{2} \). Show that [\( A_1B_1C_1 \)] ≥ \( \frac{9}{4}[ABC] \), where [XY Z] denotes the area of the triangle XY Z .
  6. In a lottery tickets are given nine-digit numbers using only the digits 1, 2, 3 . They are also coloured red , blue or green in such a way that two tickets whose numbers differ in all the nine places get different colours . Suppose the ticket bearing the number 122222222 is red and that bearing the number 222222222 is green . Determine , with proof , the colour of the ticket bearing the number 123123123 .

===INMO 2004===

  1.  Consider a convex quadrilateral \( ABCD \), in which \( K, L, M, N \) are the midpoints of the sides \( AB, BC, CD, DA \) respectively . Suppose (a) BD bisects \( KM \) at \( Q \) ; (b) \( QA = QB = QC = QD \) ; and (c) \( LK/LM = CD/CB \) . Prove that ABCD is a square .
  2.  Suppose p is a prime greater than 3. Find all pairs of integers (a, b) satisfying the equation \( a^2+ 3ab + 2p(a + b) + p^2= 0 \).
  3. If \( \alpha \) is a real root of the equation \( x^5 − x^3 + x − 2 = 0 \) , prove that [\(\alpha_6\)] = 3  . (For any real number a , we denote by [a] the greatest integer not exceeding a . )
  4. Let R denote the circumradius of a triangle \( ABC \); \( a, b, c \) its sides \( BC, CA, AB \) ; and \( r_a, r_b, r_c \) its exradii opposite \( A, B, C \). If \( 2R ≤ r_a \) , prove that
    (i) \( a > b \) and \( a > c \) ;
    (ii) \( 2R > r_b \) and \( 2R > r_c \) .
  5. Let S denote the set of all 6-tuples \( (a, b, c, d, e, f) \) of positive integers such that \( a^2 + b^2 + c^2 +d^2 + e^2 = f^2 \). Consider the set \( T = {abcdef : (a, b, c, d, e, f) ∈ S} \). Find the greatest common divisor of all the members of T.
  6. Prove that the number of 5-tuples of positive integers \( (a,b,c,d,e) \) satisfying the equation \( abcde = 5(bcde + acde + abde + abce + abcd) \) is an odd integer .

===INMO 2005===

  1. 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.
  2.  Let \( \alpha \) and \( \beta \) be positive integers such that \( \frac{43}{197} < \frac{\alpha}{\beta} < \frac{17}{77} \).Find the minimum possible value of β.
  3. Let \( p, q, r \) be positive real numbers, not all equal, such that some two of the equations \( px^2+ 2qx + r = 0, qx^2+ 2rx + p = 0, rx^2 + 2px + q = 0 \), have a common root, say \( \alpha \). Prove that
    (a) \( \alpha \) is real and negative; and
    (b) the remaining third equation has non-real roots.
  4. All possible 6-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g.,877550) are written as a sequence in increasing order. Find the 2005-th number in this sequence.
  5. Let \( x_1 \) be a given positive integer. A sequence \( (x_n)_{n=1} ^{\infty} = (x_1, x_2, x_3, · · ·) \) of positive integers is such that \( x_n \), for \( n \ge 2 \), is obtained from \( {x_n−1} \) by adding some nonzero digit of \( x_{n−1} \).
    Prove that (a) the sequence has an even number;
    (b) the sequence has infinitely many even numbers.
  6. Find all functions \( f : R \mapsto R \) such that \( f(x^2 + yf(z)) = xf(x) + zf(y) \), for all \( x, y, z \) in \( R \). (Here \( R \) denotes the set of all real numbers.)

===INMO 2006===

  1. In a non equilateral triangle \(ABC \), the sides \(a, b, c \) form an arithmetic progression. Let \(I \) and \(O \) denote the incentre and circumcentre of the triangle respectively.
    (i) Prove that \(IO \) is perpendicular to \(BI \).
    (ii) Suppose \(BI \) extended meets \(AC \) in \(K \), and \(D, E \) are the midpoints of \(BC, BA \) respectively. Prove that \(I \) is the circumcentre of triangle \(DKE \).
  2. Prove that for every positive integer \(n \) there exists a unique ordered pair \((a, b) \) of positive integers such that \( n =\frac{1}{2}(a + b − 1)(a + b − 2) + a \).
  3. Let \(X \) denote the set of all triples \((a, b, c) \) of integers. Define a function \(f : X \mapsto X \) by \( f(a, b, c) = (a + b + c, ab + bc + ca, abc) \). Find all triples \((a, b, c) \) in \(X \) such that \( f(f(a, b, c)) = (a, b, c) \).
  4. Some 46 squares are randomly chosen from a 9 × 9 chess board and are coloured red. Show that there exists a 2 × 2 block of 4 squares of which at least three are coloured red.
  5. In a cyclic quadrilateral \( ABCD, AB = a, BC = b, CD = c \), \( ∠ABC = 120^0 \) , and \( ∠ABD = 30^0 \). Prove that
    (i) \( c \ge a + b \);
    (ii) \( |\sqrt{c + a} −\sqrt{c + b}| = \sqrt{c − a − b} \).
  6. (a) Prove that if \(n \) is a positive integer such that \(n \ge 4011^2 \), 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 \(n ge M \), there exists an integer \(l \), such that \( n < l^2 < (1 +\frac{1}{2005})n \).

===INMO 2007===

  1. In a triangle ”ABC” right-angled at ”C”, the median through ”B” bisects the angle between ”BA” and the bisector of ”∠B”. Prove that \( \frac{5}{2} < \frac{AB}{BC} < 3 \).
  2. . 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, of 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.
  3. Let ”m” and ”n” be positive integers such that the equation \( x^2−mx+n = 0 \) 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.)
  4. Let \( \sigma = (a_1, a_2, a_3, . . . , a_n) \) be a permutation of \( (1, 2, 3, . . . , n) \). A pair \( (a_i, a_j ) \) is said to correspond to an inversion of \( \sigma \), if \( i < j \) but \( ai > aj \) . (Example: In the permutation \((2, 4, 5, 3, 1) \), there are 6 inversions corresponding to the pairs \( (2, 1), (4, 3), (4, 1), (5, 3), (5, 1), (3, 1) \)
    How many permutations of \( (1, 2, 3, . . . , n), (n \ge 3) \), have exactly two inversions.
  5. 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 the perpendicular from P on AC. If \( \frac{AP}{PD}=\frac{BP}{PE}= λ,\frac{BD}{AD}= m and z = m^2(1 + λ) \),
    prove that \( z^2− (λ^3− λ^2− 2)z + 1 = 0 \). Hence show that \( λ ≥ 2 \) and \( λ = 2 \) if and only if ABC is equilateral.
  6. If x, y, z are positive real numbers, prove that \( (x+y+z)^2(yz+zx+xy)^2≤ 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2) \).
  7. Let \( f :Z \mapsto Z \)$be a function satisfying \(f(0) \neq 0 \) , \(f(1)=0 \) and
    1. \( f(xy) + f(x)f(y) = f(x) + f(y) \),
    2. \( (f(x-y) – f(0))f(x)f(y) = 0 \) for all x ,  y in Z  simultaneously.
    1. Find the set of all possible values of the function f.
    2. If \( f(10) \neq 0 \) and \( f(2) = 0 \), find the set of all integers n such that \( f(n) \neq 0 \) .

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