# BdMO Higher Secondary Problems and Solutions 2019

Problem 1: Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.

Problem 2: Prove that, if a, b, c are positive real numbers,

Problem 3: Let α and ꙍ be 2 circles such that ꙍ goes through the center of α. ꙍ intersects α at A and B. Let P be any point on the circumference ꙍ. The lines PA and PB intersects α again at E and F respectively. Prove that AB = EF.

Problem 4: A is a positive real number. Find the set of possible values of the infinite x0n+ x1n+ x2n + … … where x0, x1, x2 are all positive real numbers so that the infinite series x0 + x1 + x2 + has sum A.

Problem 5: Prove that for all positive integers n we can find a permutation of {1, 2, … …, n} such that the average of two numbers doesn’t appear in-between them. For example, {1, 2, 3, 4} works, but {1, 2, 3, 4} doesn’t because 2 is between 1 and 3.

Problem 6: When a function f(x) is differentiated n times, the function we get is denoted by f(n)(x). If f(x) = ex/x, Find the value of

Problem 6: Given three concentric circles ꙍ1, ꙍ2, ꙍ3 with radius r1, r2, r3 such that r1 + r3 ≥ 2r2, construct a line that intersects ꙍ1, ꙍ2, ꙍ3 at A, B, C respectively such that AB = BC.

Problem 8: The set of natural numbers N are partitioned into a finite number of subsets. Prove that
there exists a subset S so that for any natural number n, there are infinitely many multiples of n in
S.

Problem 9: Let ABCD be a convex quadrilateral. The internal angle bisectors of ∠BAC and ∠BDC
meets at P ∠APB = ∠CPD. Prove that, AB + BD = AC + CD.

Problem 10: In chess, a normal knight goes two steps forward and one step to the side, in some orientation. Thanic thought that he should spice the game up a bit, so he introduced a new kind of piece called a warrior. A warrior can either go three steps forward and one step to the side, or two steps forward and two steps to the side in some orientation.

In a 2020 × 2020 chessboard, prove that the maximum number of warriors so that none of them attack each other is less than or equal to 2/5 of the number of cells.