2

Part of 2019 Middle European Mathematical Olympiad

Problems(2)

Polynomial inequality

Source: 2019 MEMO Problem T-2

\alpha

be a real number. Determine all polynomials

P

with real coefficients such that

P(2x+\alpha)\leq (x^{20}+x^{19})P(x)

holds for all real numbers

x

.
Proposed by Walther Janous, Austria

algebrapolynomialmemoMEMO 2019roots

Bohemian vertices of a convex polygon

Source: 2019 MEMO Problem I-2

n\geq 3

be an integer. We say that a vertex

A_i (1\leq i\leq n)

of a convex polygon

A_1A_2 \dots A_n

is Bohemian if its reflection with respect to the midpoint of

A_{i-1}A_{i+1}

A_0=A_n

A_1=A_{n+1}

) lies inside or on the boundary of the polygon

A_1A_2\dots A_n

. Determine the smallest possible number of Bohemian vertices a convex

n

-gon can have (depending on

n

).
Proposed by Dominik Burek, Poland

combinatorial geometrycombinatoricsMEMO 2019memo