RMO – M CUBE: MatheMatics by Maheshwari

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

Let ABCD be a quadrilateral in which AB is parallel to CD and perpendicular to AD; AB=3CD; and the area of the quadrilateral is 4. If a circle can be drawn touching all sides of the quadrilateral, then find its radius.

RMO 2006 Solution: Let P, Q, R, S be the points of contact of in-circle with the sides AB, BC, CA, DA respectively. Since AD is perpendicular to AB and AB || DC we see that, AP=AS=SD=DR=r, radius of the in-circle. Let BP=BQ=y and CQ=CR=x . Using AB=3CD, we get (r+y)=3(r+x)

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 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, 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

August 29, 2019 M CUBE: Math-e-Matics by Maheshwari RMO

Let P(x) = x 2 + 1 2 x + b and Q(x) = x 2 + cx + d be two polynomials with real coefficients such that P(x)Q(x) = Q(P(x)) for all real x. Find all the real roots of P (Q(x))= 0

RMO 2017

August 29, 2019August 29, 2019 M CUBE: Math-e-Matics by Maheshwari RMO

Let AOB be a given angle less than 180◦ and let P be an interior point of the angular region determined by ∠AOB. Show, with proof, how to construct, using only ruler and compasses, a line segment CD passing through P such that C lies on the ray OA and D lies on the ray OB, and CP : P D = 1 : 2.

RMO 2017

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

How many 6- digit numbers are there such that: (a) The digits of each number are all from the set {1,2,3,4,5} (b) Any digit that appears in the number appears at least twice? [Example 225252 is an admissible number while 222133 is not]

RMO 2007

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

Let a,b,c be three natural numbers such that a<b<c and gcd(c-a,c-b)=1. Suppose there exists an integer d such that a+d,b+d,c+d form the sides of a right-angled triangle. Prove that there exists integer l, m such that c+d=l^2+m^2

RMO 2007 SOLUTION: OR

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