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Malaysia IMONST R2 2020 p2

geometry problems from Malaysia IMO National Selection Test , Round 2, with aops links in the names



2020

Prove that if a and b are legs, c is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula r = \frac12 (a + b - c).

Prove that for any integer n\ge 6 we can divide an equilateral triangle completely into n smaller equilateral triangles.

Given a trapezium with two parallel sides of lengths m and n, where m, n are integers, prove that it is possible to divide the trapezium into several congruent triangles.

Given are four circles \Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4. Circles \Gamma_1 and \Gamma_2 are externally tangent at point A. Circles \Gamma_2 and \Gamma_3 are externally tangent at point B. Circles \Gamma_3 and \Gamma_4 are externally tangent at point C. Circles \Gamma_4 and \Gamma_1 are externally tangent at point D. Prove that ABCD is cyclic.

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