MathDB

4

Part of 2009 Kosovo National Mathematical Olympiad

Problems(3)

Kosovo MO 2009 Grade 10, Problem 4

Source: Kosovo MO 2009 Grade 10, Problem 4

6/7/2021
Prove that if in the product of four consequtive natural numbers we add 11, we get a perfect square.
number theory
Kosovo MO 2009 Grade 11, Problem 4

Source: Kosovo MO 2009 Grade 11, Problem 4

6/7/2021
Prove that n11−nn^{11}-n is divisible by 1111.
number theory
Kosovo MO 2009 Grade 12, Problem 4

Source: Kosovo MO 2009 Grade 12, Problem 4

6/7/2021
(a)(a) Let a1,a2,a3a_1,a_2,a_3 be three real numbers. Prove that (a1−a2)(a1−a3)+(a2−a1)(a2−a3)+(a3−a1)(a2−a2)≥0(a_1-a_2)(a_1-a_3)+(a_2-a_1)(a_2-a_3)+(a_3-a_1)(a_2-a_2)\geq 0. (b)(b) Prove that the inequality above doesn't hold if we use four number instead of three.
algebra