MathDB

3

Part of 2018 Moscow Mathematical Olympiad

Problems(3)

Polynomial equation

Source: Moscow Olympiad 2018, Grade 11, P3

4/7/2018
Are there such natural nn, that exist polynomial of degree nn and with nn different real roots, and a) P(x)P(x+1)=P(x2)P(x)P(x+1)=P(x^2) b) P(x)P(x+1)=P(x2+1)P(x)P(x+1)=P(x^2+1)
algebrapolynomial
Some geometry

Source: Moscow Olympiad 2018, Grade 10, P3

7/13/2018
OO is circumcircle and AHAH is the altitude of △ABC\triangle ABC. PP is the point on line OCOC such that AP⊥OCAP \perp OC. Prove, that midpoint of ABAB lies on the line HPHP.
geometrycircumcircle
Number of roots

Source: Moscow Olympiad 2018, Grade 9, P3

7/13/2018
a1,a2,...,aka_1,a_2,...,a_k are positive integers and 1a1+1a2+...+1ak>1\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}>1. Prove that equation [na1]+[na2]+...+[nak]=n[\frac{n}{a_1}]+[\frac{n}{a_2}]+...+[\frac{n}{a_k}]=n has no more than a1∗a2∗...∗aka_1*a_2*...*a_k postivie integer solutions in nn.
number theory