BdMO Higher Secondary Problems and Solutions 2018

Problem 1: Find solutions of the following equation that are real numbers,
x2(2 – x)2 = 1 + 2(1 – x)2

Problem 2: AB is the diameter of a circle. Tangents AD and BC are drawn so that AC and BD intersect at a point on the circle. Suppose AD = a and BC = b, and a ≠ b. What is the diameter AB?

Problem 3: Urmi rolls four fair six-sided dice. She doesn’t see the results. Her friend Ipshita tells her that the product of the numbers is 144. Ipshita also says the sum of the dice, S satisfies 14 ≤ S ≤ 18. Urmi tells Ipshita that S cannot be one of the numbers in the set {14, 15, 16, 17, 18} if the product is 144. Which number in the range {14, 15, 16, 17, 18} is an impossible value for S?

Problem 4: Sabbir is counting the minimum number of lines m, that can be drawn on the plane so that they intersect in exactly 200 distinct points. What is m?

Problem 5: Four circles are drawn with the sides of quadrilateral ABCD as diameters. The two circles passing through A meet again at A′. The two circles passing through B meet again at B′. The two circles passing through C meet again at C′. The two circles passing through D meet again at D′. Suppose, A′, B′, C′, D′ are all distinct. Is the quadrilateral A′B′C′D′ similar to ABCD? Show with proof.

Problem 6: Find all pairs of integers (m; n) satisfying the equality
3(m2 + n2) – 7(m + n) = – 4

 

Problem 7: Evaluate

Math Olympiad problems

 

Problem 8: In a tournament of n players, every pair of players play exactly one match and there is no draw. A positive integer k is called n-good if there exists a tournament of n players in which for any k number of players, there is at least one player other than the k players who beats all of them.
(i) Prove that if k is n-good then n ≥ 2k+1 – 1.
(ii) Find all n so that 2 is n-good.