APMO Problems and Solutions 2018

Problem 1: Let H be the orthocenter of the triangle ABC. Let M and N be the midpoints of the sides AB and AC, respectively. Assume that H lies inside the quadrilateral BMNC and that the circumcircles of triangles BMH and CNH are tangent to each other. The line through H parallel to BC intersects the circumcircles of the triangles BMH and CNH in the points K and L, respectively. Let F be the intersection point of MK and NL and let J be the incenter of triangle MHN. Prove that F J = FA.

Problem 2: Let f(x) and g(x) be given by

Math Olympiad problems

 

Prove that

|f(x) – g(x)| > 2

for any non-integer real number x satisfying 0 < x < 2018.

Problem 3: A collection of n squares on the plane is called tri-connected if the following criteria are satisfied:
(i) All the squares are congruent.
(ii) If two squares have a point P in common, then P is a vertex of each of the
squares.
(iii) Each square touches exactly three other squares.

How many positive integers n is there with 2018 n 3018, such that there exists a collection of n squares that is tri-connected?

Problem 4: Let ABC be an equilateral triangle. From the vertex A we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle α, it leaves with a directed angle 1800– α. After n bounces, the ray returns to A without ever landing on any of the other two vertices. Find all possible values of n.

Problem 5: Find all polynomials P(x) with integer coefficients such that for all real numbers s and t, if P(s) and P(t) are both integers, then P(st) is also an integer.

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