MathDB

4

Part of 2022 Kosovo National Mathematical Olympiad

Problems(4)

pq-p-q+3 perfect square

Source: Kosovo Mathematical Olympiad 2022, Grade 9, Problem 4

3/6/2022
Find all prime numbers pp and qq such that pq−p−q+3pq-p-q+3 is a perfect square.
number theoryprime numbers
(a^3+b^3+c^3)/3=>(abc+2021)/2022

Source: Kosovo Mathematical Olympiad 2022, Grade 10, Problem 4

3/6/2022
Let a,ba,b and cc be positive real numbers such that a+b+c+3abc≥(ab)2+(bc)2+(ca)2+3a+b+c+3abc\geq (ab)^2+(bc)^2+(ca)^2+3. Show that the following inequality hold, a3+b3+c33≥abc+20212022.\frac{a^3+b^3+c^3}{3}\geq\frac{abc+2021}{2022}.
inequalitiesalgebra
Geometric Inequality Kosovo Math Olympiad 2022 Grade 11

Source: Kosovo Mathematical Olympiad, Grade 11, Problem 4

3/5/2022
Assume that in the △ABC\triangle ABC there exists a point DD on BCBC and a line ll passing through AA such that ll is tangent to (ADC)(ADC) and ll bisects BD.BD. Prove that a2≥b+c.a\sqrt{2}\geq b+c.
geometric inequalityinequalities
k!+3^m=3^n

Source: Kosovo Mathematical Olympiad 2022, Grade 12, Problem 4

3/6/2022
Find all positive integers k,mk,m and nn such that k!+3m=3nk!+3^m=3^n
number theory