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 RMO Suppose for some positive integers r and s, the digits of 2^r is obtained by permuting the digits of 2^s in decimal expansion. Prove that r = s RMO 2014
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
August 30, 2019August 30, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Find all integers a, b, c such that a^2 = bc + 1, b^2 = ca + 1. CRMO 2015
August 29, 2019August 29, 2019 M CUBE: Math-e-Matics by Maheshwari Olympiads, RMO Let x, y, z be real numbers, each greater than 1. Prove that (x + 1)/(y + 1) + (y + 1)/(z + 1) +(z + 1)/(x + 1) ≤ (x − 1)/( y − 1) + (y − 1)/( z − 1) + (z − 1)/( x − 1) RMO 2017
August 29, 2019August 29, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Let ABC be a triangle. Let B’ and C’ denote respectively the reflection of B and C in the internal angle bisector of ∠A. Show that the triangles ABC and AB’C’ have the same in-centre. CRMO 2015
M CUBE: MatheMatics by Maheshwari 