MathDB

2

Part of 2011 Saint Petersburg Mathematical Olympiad

Problems(3)

Some geometry

Source: St Petersburg Olympiad 2011, Grade 11, P2

9/15/2017
ABCABC-triangle with circumcenter OO and ∠B=30\angle B=30. BOBO intersect ACAC at KK. LL - midpoint of arc OCOC of circumcircle KOCKOC, that does not contains KK. Prove, that A,B,L,KA,B,L,K are concyclic.
geometrycircumcircle
Sum of divisors

Source: St Petersburg Olympiad 2011, Grade 10, P2

9/15/2017
nn - some natural. We write on the board all such numbers dd, that d≤1000d\leq 1000 and d∣n+kd|n+k for some 1≤k≤1000 1\leq k \leq 1000. Let S(n)S(n) -sum of all written numbers. Prove , that S(n)<106S(n)<10^6 and S(n)>106S(n)>10^6 has infinitely many solutions.
number theory
GCD and LCM

Source: St Petersburg Olympiad 2011, Grade 9, P2

9/15/2017
a,ba,b are naturals and a×GCD(a,b)+b×LCM(a,b)<2.5aba \times GCD(a,b)+b \times LCM(a,b)<2.5 ab. Prove that b∣ab|a
number theorygreatest common divisorleast common multiple