MathDB

4

Part of 2018 Saint Petersburg Mathematical Olympiad

Problems(3)

Equation without roots

Source: St Petersburg Olympiad 2018, Grade 11, P4

6/21/2018
(b+c)x2+(a+c)x+(a+b)=0(b+c)x^2+(a+c)x+(a+b)=0 has not real roots. Prove that 4ac−b2≤3a(a+b+c)4ac-b^2 \leq 3a(a+b+c)
algebrainequalities
Polynomial with integer roots

Source: St Petersburg Olympiad 2018, , Grade 10, P4

6/21/2018
f(x)f(x) is polynomial with integer coefficients, with module not exceeded 5∗1065*10^6. f(x)=nxf(x)=nx has integer root for n=1,2,...,20n=1,2,...,20. Prove that f(0)=0f(0)=0
algebrapolynomialabstract algebra
Coloring beads

Source: St Petersburg Olympiad 2018, Grade 9, P4

7/13/2018
On the round necklace there are n>3n> 3 beads, each painted in red or blue. If a bead has adjacent beads painted the same color, it can be repainted (from red to blue or from blue to red). For what nn for any initial coloring of beads it is possible to make a necklace in which all beads are painted equally?
combinatorics