for all 
.
.
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.
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.
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.
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 
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
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 
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.
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.
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.
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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