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...
Missing digits of 34!

Missing digits of 34!

Understand the problem 34!=295232799cd96041408476186096435ab000000 Find (all single digits). Source of the problem BMO 2002 Topic Number Theory Difficulty Level Easy Suggested Book An Excursion in Mathematics Start with hints Hint 0Hint 1Hint 2Hint 3Hint 4 Do you...