1

Part of 2015 Saint Petersburg Mathematical Olympiad

Problems(3)

Source: St Petersburg Olympiad 2015, Grade 10, P1

There is child camp with some rooms. Call room as

4-

4

children live here. Not less then half of all rooms are

4-

rooms , other rooms are

3-

rooms. Not less than

2/3

3-

rooms. Prove that not less than

35\%

of all children are boys.

Source: saint petersburg 2015,grade 11,P1

x,y

are real numbers such that

x^2+y^2=1 , 20x^3-15x=3

Find the value of

|20y^3-15y|

Quadratic with f(f(sqrt2))=0

Source: St. Petersburg Math Olympiad, 2015, round II, grade 9, P1

Is there a quadratic trinomial

f(x)

with integer coefficients such that

f(f(\sqrt{2}))=0

quadraticsalgebrapolynomial