MathDB

6

Part of 2018 Saint Petersburg Mathematical Olympiad

Problems(3)

Old and famous

Source: St Petersburg Olympiad 2018, Grade 11, P6

6/21/2018
α,β\alpha,\beta are positive irrational numbers and [α[βx]]=[β[αx]][\alpha[\beta x]]=[\beta[\alpha x]] for every positive xx. Prove that α=β\alpha=\beta
irrational numbernumber theory
inequalities

Source: St. Petersburg Mathematical Olympiad 2018 P5

4/26/2018
Let a,b,c,d>0a,b,c,d>0 . Prove that a4+b4+c4+d4≥4abcd+4(a−b)2abcda^4+b^4+c^4+d^4 \geq 4abcd+4(a-b)^2 \sqrt{abcd}
algebrainequalitiesSaint Petersburg
Division problem

Source: St Petersburg Olympiad 2018, Grade 9, P6

7/13/2018
a,ba,b are odd numbers. Prove, that exists natural kk that bk−a2b^k-a^2 or ak−b2a^k-b^2 is divided by 220182^{2018}.
number theory