MIOTP Weekly Challenge (Week -2) IMO INMO USAMO PUTNAM Olympiad Training

                                       MIOTP(WEEK-2)

                                       MATHFREAKS

1.Let ABC be a triangle with its centroid G. Let D and E be points on segments AB and AC, respectively, such that, AB/AD   +   AC/AE=3. Prove that the points D,G and E are collinear.

2. Find all three digit natural numbers of the form (abc)10 such that (abc)10, (bca)10 and (cab)10 are in geometric progression. {Here (abc)10 is representation in base 10.}   (RMO INDIA 2015)

3.Find all pairs (a, b) of positive integers, such that (ab)²-4(a+b) is the square of an integer.

4.For any positive integer n define

E(n) = n(n + 1)(2n + 1)(3n + 1)· · ·(10n + 1).

Find the greatest common divisor of E(1), E(2), E(3), . . . , E(2009).

5.Prove that there does not exist any positive integer n < 2310 such that n(2310 – n) is a multiple of 2310.(RMO INDIA-2014)

6.For any natural number n, let S(n) denote the sum of the digits of n. Find the number of all 3-digit numbers n such that

                                     S(S(n))=2.

7.The sum of the solutions of the equation

 | √x – 2 | + √x ( x – 4) + 2 = 0, (√x > 0) is equal to?

8. If cos (a + b) =3/5, sin (a – b) = 5/13 and 0 < a,b< π/4, then tan (2a) is equal to?

9)Given a fraction in its lowest term lying between 0 and 1 having numerator x and denominator y, how many such fractions are possible such that xy=27! 

                                        Level 2

1) Find the sum of

2) A spider has one shoe and one sock for each of its 8 legs, In how many different ways the spider can put on its socks and shoe assuming that the spider wears sock before shoe?[AMC12 2001]

3)How many words of length n can be formed from the letters A,B,C, such that the number of A’s is even?

                                                

       4)   Let ABCD be a convex quadrilateral such that the diagonals AC and BD are

perpendicular, and let P be their intersection. Prove that the reflections of P with

respect to AB, BC, CD, and DA are concyclic (i.e., lie on a circle).