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
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
August 29, 2019 M CUBE: Math-e-Matics by Maheshwari RMO Show that the equation a^3 + (a+ 1)^3 + (a+ 2)^3 + (a+ 3)^3 + (a+ 4)^3 + (a+ 5)^3 + (a+ 6)^3 = b^4 + (b+ 1)^4 has no solutions in integers a, b. RMO 2017
August 29, 2019 M CUBE: Math-e-Matics by Maheshwari RMO Let Ω be a circle with a chord AB which is not a diameter. Let Γ1 be a circle on one side of AB such that it is tangent to AB at C and internally tangent to Ω at D. Likewise, let Γ2 be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to Ω at F. Suppose the line DC intersects Ω at X 6= D and the line F E intersects Ω at Y 6= F. Prove that XY is a diameter of Ω RMO 2017
August 29, 2019 M CUBE: Math-e-Matics by Maheshwari RMO Consider n 2 unit squares in the xy-plane centred at point (i, j) with integer coordinates, 1 ≤ i ≤ n, 1 ≤ j ≤ n. It is required to colour each unit square in such a way that whenever 1 ≤ i < j ≤ n and 1 ≤ k < l ≤ n, the three squares with centres at (i, k),(j, k),(j, l) have distinct colours. What is the least possible number of colours needed? RMO 2017
M CUBE: MatheMatics by Maheshwari 