Vietnam National Olympiad 2014

1 Let be two positive sequences defined by and 

for all .

Prove that they are converges and find their limits. 

2 Given the polynomial

   

where is a positive integer. Prove that can't be written as a product of non-constant polynomials with integer coefficients.

3 Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let be the number of pairs of adjacent red vertices and be the number of pairs of adjacent blue vertices.

a) Find all possible values of pair

b) Determine the number of pairwise non-similar colorings of the polygon satisfying 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon. 

4 Let be an acute triangle, be the circumcircle, and Let be the midpoint of arc (not containing ). lies on such that intersects at the second point and intersects at intersects at

a) Prove that

b) lies on such that is parallel to intersects at The circumcircle of triangle intersects at the second point Prove that passes through the midpoint of segment

Day 2 - 04 January 2014

1 Given a circle and two fixed points on and an arbitrary point on such that the triangle is acute. lies on ray lies on ray such that and Let be the intersection of and intersects at

a) Prove that are collinear.

b) is the midpoint of Let be the intersection of and is the line passing through and perpendicular to is the intersection of and intersects at Prove that passes through a fixed point. 

2 Find the maximum of

where are positive real numbers. 

3 Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.

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