MathDB

4

Part of 2010 Kosovo National Mathematical Olympiad

Problems(4)

Kosovo MO 2010 Grade 10, Problem 4

Source: Kosovo MO 2010 Grade 10, Problem 4

6/7/2021
Let a,b,ca,b,c be non negative integers. Suppose that cc is even and a5+4b5=c5a^5+4b^5=c^5. Prove that b=0b=0.
number theory
Kosovo MO 2010 Grade 9, Problem 4

Source:

6/7/2021
Prove that 3\sqrt 3 is irrational.
number theory
Kosovo MO 2010 Grade 11, Problem 4

Source: Kosovo MO 2010 Grade 11, Problem 4

6/7/2021
Prove that 53\sqrt[3]{5} is irrational.
algebra
Kosovo MO 2010 Grade 12, Problem 4

Source: Kosovo MO 2010 Grade 12, Problem 4

6/7/2021
Let (p1,p2,...,pn)(p_1,p_2,..., p_n) be a random permutation of the set {1,2,...,n)\{1,2,...,n). If nn is odd, prove that the product (p1−1)⋅(p2−2)⋅...⋅(pn−n)(p_1-1)\cdot (p_2-2)\cdot ...\cdot (p_n-n) is an even number. @below fixed.
number theory