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 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
August 29, 2019August 29, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Find all fractions which can be written simultaneously in the forms (7k − 5)/ (5k − 3) and (6l − 1)/ (4l − 3) , for some integers k, l. CRMO 2015
August 29, 2019August 29, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Find all real numbers a such that 3 < a < 4 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 29, 2019August 29, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Let ABC be a right triangle with ∠B = 90◦ . Let E and F be respectively the mid-points of AB and AC. Suppose the incentre I of triangle ABC lies on the circumcircle of triangle AEF. Find the ratio BC/AB. CRMO 2015
August 29, 2019August 29, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO Suppose 28 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 29, 2019August 29, 2019 M CUBE: Math-e-Matics by Maheshwari CRMO In a cyclic quadrilateral ABCD, let the diagonals AC and BD intersect at X. Let the circumcircles of triangles AXD and BXC intersect again at Y . If X is the incentre of triangle ABY , show that ∠CAD = 90◦ . CRMO 2015
M CUBE: MatheMatics by Maheshwari 