A fraction involving primes

A fraction involving primes

Understand the problem Given that the number  is an integer where  and  are prime positive numbers, determine . Source of the problem Danube 2014 Topic Number Theory Difficulty Level Easy Suggested Book An Excursion in Mathematics Start with hints Hint 0Hint 1Hint...
A taste of affine transformations

A taste of affine transformations

Understand the problem Let  be the area of the triangle . A non-regular convex polygon  is called guayaco if exists a point  in its interior such thatShow that, for every integer , a guayaco polygon of  sides exists. Source of the problem Cono sur olympiad 2017 Topic...
Solving a congruence

Solving a congruence

Understand the problem Prove that the number of ordered triples   in the set of residues of such that , where  and  is prime is . Source of the problem Brazilian Olympiad Revenge 2010 Topic Number Theory Difficulty Level Medium Suggested Book Elementary Number Theory...
Inequality involving sides of a triangle

Inequality involving sides of a triangle

Understand the problem Let  be the lengths of sides of a (possibly degenerate) triangle. Prove the inequality Let  be the lengths of sides of a triangle. Prove the inequality Source of the problem Caucasus Mathematical Olympiad Topic Inequalities Difficulty Level Easy...
Vectors of prime length

Vectors of prime length

Understand the problem Given a prime number  and let  be  distinct vectors of length  with integer coordinates in an  Cartesian coordinate system. Suppose that for any , there exists an integer  such that all three coordinates of  is divisible by . Prove that . Source...