MathDB

1

Part of 2017 Saint Petersburg Mathematical Olympiad

Problems(3)

St. Petersburg MO 2017 Grade 9 P1

Source: St. Petersburg MO 2017 Grade 9 P1

5/3/2018
Sasha’s computer can do the following two operations: If you load the card with number aa, it will return that card back and also prints another card with number a+1a+1, and if you consecutively load the cards with numbers aa and bb, it will return them back and also prints cards with all the roots of the quadratic trinomial x2+ax+bx^2+ax+b (possibly one, two, or none cards.) Initially, Sasha had only one card with number ss. Is it true that, for any s>0s> 0, Sasha can get a card with number s\sqrt{s}?
algebra
St. Petersburg MO 2017 Grade 10 P1

Source: St. Petersburg MO 2017 Grade 10 P1

5/3/2018
It’s allowed to replace any of three coefficients of quadratic trinomial by its discriminant. Is it true that from any quadratic trinomial that does not have real roots, we can perform such operation several times to get a quadratic trinomial that have real roots?
algebra
St. petersburg 2017

Source:

5/3/2018
A1,A2,...,Am are subsets of X and we have |Ai|=mk (m,k natural numbers) prove that we can separate X into k sets such that every set has at least one member of each Ai.
combinatorics