INMO – M CUBE: MatheMatics by Maheshwari

September 20, 2019September 20, 2019 M CUBE: Math-e-Matics by Maheshwari INMO

Call a natural number n faithful, if there exist natural numbers a < b < c such that a divides b, b divides 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.

INMO 2011 Solution

September 20, 2019 M CUBE: Math-e-Matics by Maheshwari INMO

Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.

INMO 2011 Solution 1: Solution 2: Here is a pure geometric solution. Consider the triangle BDF, CED and AFE with BD, CE and AF as bases. The sides DF, ED and FE make equal angles θ with the bases of respective triangles. If B ≥ C ≥ A, then it is easy to see that … Continue reading Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.

August 22, 2019 M CUBE: Math-e-Matics by Maheshwari INMO, INMO

Do there exist three distant positive real numbers a,b,c such that the numbers a,b,c, b+c-a, a+b-c and a+b+c form a 7-term arithmetic progression in some order?

INMO 2002

August 22, 2019August 22, 2019 M CUBE: Math-e-Matics by Maheshwari INMO, INMO

Do there exists 100 lines in the plane, no three of them concurrent, such that they intersect exactly in 2002 points?

INMO 2002

August 22, 2019September 20, 2019 M CUBE: Math-e-Matics by Maheshwari INMO, INMO

Determine the least positive value taken by the expression a^3+b^3+c^3-3abc as a,b,c vary over all positive integers. Find also all triples (a,b,c) for which least value is attained

INMO 2002

August 21, 2019 M CUBE: Math-e-Matics by Maheshwari INMO, INMO

Let R denote the set of all real numbers. Find all functions f : R → R satisfying the condition f(x + y) = f(x)f(y)f(xy) for all x, y in R

INMO 2001

August 21, 2019August 21, 2019 M CUBE: Math-e-Matics by Maheshwari INMO, INMO

All possible 6 digit numbers, in each of which the digits occur in non-increasing order (from left to right, eg: 877550) are written as a sequence in increasing order. Find the 2005-th number in this sequence.

INMO 2005

August 18, 2019 M CUBE: Math-e-Matics by Maheshwari INMO, Olympiads

Let a, b, c be three real numbers such that 1 ≥ a ≥ b ≥ c ≥ 0. Prove that if λ is a root of the cubic equation x^3 + ax^2 + bx + c = 0 (real or complex),then |λ| ≤ 1.

August 18, 2019August 18, 2019 M CUBE: Math-e-Matics by Maheshwari INMO, Olympiads

In a convex quadrilateral PQRS, PQ = RS, (√ 3+1)QR = SP and ∠RSP −∠SPQ = 30 . Prove that ∠PQR − ∠QRS = 90

August 17, 2019August 21, 2019 M CUBE: Math-e-Matics by Maheshwari INMO, Olympiads

The in-circle 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 P Q bisects the sides AB and AC of triangle ABC

INMO 2000

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