# BdMO Secondary Problems and Solutions 2019

Problem 1: At some math Olympiad, 72 medals were handed out. Afterwards, it was found that Sumon had lost the receipt! He only remembers that the total price of the medals was a 5-digit number, and the three middle digits were all 9. If the price of all the medals were the same integer, what was the amount spent for each medal?

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

Problem 3: 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 4: In this figure ABCD is a trapezium where AB||CD. P, Q, R, S are the midpoint of AB, BD, DC, CA respectively. AC and BD intersect at point O. Area of △AOB = 2019 and area of △COD = 2020.What is the area of quadrilateral PQRS?

Problem 5: 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 6: Consider a 9 × 9 array of white squares. Find the largest n is in N with the property: No matter how one chooses n out of 81 white squares and color in black, there always remains a 1 × 4 array of white squares (either vertical or horizontal).

Problem 7: M and N are two positive integers where M is not equal to N. LCM of (M and N) = M2 – N2 + MN. Show that MN is a perfect cubic number.

Problem 8: Prove that for all positive integers n we can find a permutation of {1, 2, 3, … …, n} such thatthe average of two numbers doesn’t appear in-between them. For example, {1, 2, 3, 4} works, but{1, 4, 3, 2} doesn’t because 2 is between 1 and 3.

Problem 9: 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 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.