August 30, 2019August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO A polynomial is called a Fermat polynomial if it can be written as the sum of the squares of two polynomials with integer coefficients. Suppose that f(x) is a Fermat polynomial such that f(0) = 1000. Prove that f(x) + 2x is not a Fermat polynomial CRMO 2013
August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Find all 4-tuples (a, b, c, d) of natural numbers with a ≤ b ≤ c and a! + b! + c! = 3^d CRMO 2013
August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Find the number of eight-digit numbers the sum of whose digits is 4. CRMO 2013
August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Suppose that m and n are integers such that both the quadratic equations x^2 + mx − n = 0 and x^2 − mx + n = 0 have integer roots. Prove that n is divisible by 6. CRMO 2013
August 30, 2019August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO, Olympiads Let ABC be a triangle with ∠A = 90◦ and AB = AC. Let D and E be points on the segment BC such that BD : DE : EC = 3 : 5 : 4. Prove that ∠DAE = 45◦ . CRMO 2013
August 30, 2019August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Find all primes p and q such that p divides q^2 − 4 and q divides p^2 − 1. CRMO 2013
August 30, 2019August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Let ABC be an acute angled triangle. The circle Γ with BC as diameter intersects AB and AC again at P and Q, respectively. Determine ∠BAC given that the ortho-center of triangle AP Q lies on Γ CRMO 2013
August 30, 2019August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Find all real numbers a such that 4 < a < 5 and a(a−3{a}) is an integer. (Here {a} denotes the fractional part of a. For example {1.5} = 0.5; {−3.4} = 0.6.) CRMO 2015
August 30, 2019August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Two circles Γ and Σ in the plane intersect at two distinct points A and B, and the centre of Σ lies on Γ. Let points C and D be on Γ and Σ, respectively, such that C, B and D are collinear. Let point E on Σ be such that DE is parallel to AC. Show that AE = A CRMO 2015
August 30, 2019August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite? CRMO 2015
M CUBE: MatheMatics by Maheshwari 