8

Part of 2020 Tuymaada Olympiad

Problems(2)

Partitioning strips with segments

Source: 2020 Tuymaada Junior P8

In a horizontal strip

1 \times n

n

unit squares the vertices of all squares are marked. The strip is partitioned into parts by segments connecting marked points and not lying on the sides of the strip. The segments can not have common inner points; the upper end of each segment must be either above the lower end or further to the right. Prove that the number of all partitions is divisible by

2^n

. (The partition where no segments are drawn, is counted too.)
(E. Robeva, M. Sun)

Polynomial equation involving x^{n + 1} and (x+1)^{n + 1}

Source: Tuymaada 2020 Senior, P8

The degrees of polynomials

P

Q

with real coefficients do not exceed

n

. These polynomials satisfy the identity

P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1.

Determine all possible values of

Q \left( - \frac{1}{2} \right)

algebrapolynomial