MathDB

3

Part of 2012 Kosovo National Mathematical Olympiad

Problems(4)

Kosovo MO 2012 Grade 9, Problem 3

Source: Kosovo MO 2012 Grade 9, Problem 3

6/7/2021
The integers a1,a2,...,a2012a_1,a_2,...,a_{2012} are given. Exactly 2929 of them are divisible by 33. Prove that the sum a12+a22+...+a20122a_1^2+a_2^2+...+a_{2012}^2 is divisible by 33.
number theory
Kosovo MO 2012 Grade 10, Problem 3

Source: Kosovo MO 2012 Grade 10, Problem 3

6/7/2021
Let n≢2(mod3)n\not\equiv 2\pmod{3}. Is ⌊n+2n3⌋+7,∀n∈N\sqrt{\lfloor n+\tfrac {2n}{3}\rfloor+7},\forall n \in \mathbb {N}, a natural number?
number theory
Kosovo MO 2012 Grade 11, Problem 3

Source: Kosovo MO 2012 Grade 11, Problem 3

6/7/2021
Prove that for any integer n≥2n\geq 2 it holds that (2nn)>4n2n\dbinom {2n}{n}>\frac {4^n}{2n}.
algebra
Kosovo MO 2012 Grade 12, Problem 3

Source: Kosovo MO 2012 Grade 12, Problem 3

6/7/2021
Solve the recurrence R0=1,Rn=nRn−1+2n⋅n!R_0=1, R_n=nR_{n-1}+2^n\cdot n!.
algebra