MathDB

4

Part of 2013 Saint Petersburg Mathematical Olympiad

Problems(3)

St.Peterburg, P4 Grade 11, 2013

Source:

4/17/2014
Find all pairs (p,q)(p,q) of prime numbers, such that 2p−12p-1, 2q−12q-1, 2pq−12pq-1 are perfect square.
F. Petrov, A. Smirnov
number theory proposednumber theory
Roots on the board

Source: St Petersburg Olympiad 2013, Grade 10, P4

10/13/2017
There are 100100 numbers from (0,1)(0,1) on the board. On every move we replace two numbers a,ba,b with roots of x2−ax+b=0x^2-ax+b=0(if it has two roots). Prove that process is not endless.
algebra
Cents in glasses

Source: St Petersburg Olympiad 2013, Grade 9, P4

10/13/2017
There are 100100 glasses, with 101,102,...,200101,102,...,200 cents.Two players play next game. In every move they can take some cents from one glass, but after move should be different number of cents in every glass. Who will win with right strategy?
combinatorics