MathDB

6

Part of 2010 Saint Petersburg Mathematical Olympiad

Problems(3)

Inequality

Source: St Petersburg Olympiad 2010, Grade 11, P6

9/14/2017
For positive numbers is true that ab+ac+bc=a+b+cab+ac+bc=a+b+c Prove a+b+c+1≥4abca+b+c+1 \geq 4abc
algebraInequality
Extract primes

Source: St Petersburg Olympiad 2010, Grade 10, P6

9/14/2017
Natural number NN is given. Let pNp_N - biggest prime, that ≤N \leq N. On every move we replace NN by N−pNN-p_N. We repeat this until we get 00 or 11. Prove that exists such number NN, that we need exactly 10001000 turns to make 00
number theory
Inequality

Source: St Petersburg Olympiad 2010, Grade 9, P6

9/14/2017
For positive is true 3abc≥a+b+c\frac{3}{abc} \geq a+b+c Prove 1a+1b+1c≥a+b+c\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq a+b+c
algebrainequalities