BdMO 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: The vertices of a regular nonagon (9-sided polygon) are labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.

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.

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