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Part of 2023/2024 Tournament of Towns

Problems(4)

Source: 45th International Tournament of Towns, Senior A-Level P1, Fall 2023

For every polynomial of degree 45 with coefficients

1,2,3, \ldots, 46

(in some order) Tom has listed all its distinct real roots. Then he increased each number in the list by 1 . What is now greater: the amount of positive numbers or the amount of negative numbers? Alexey Glebov

Polymonialalgebra

chess knight in $8 \times 8$ board

Source: 45th International Tournament of Towns, Junior A-Level P1, Fall 2023

1. Every square of a

8 \times 8

board is filled with a positive integer, such that the following condition holds: if a chess knight can move from some square to another then the ratio of numbers from these two squares is a prime number. Is it possible that some square is filled with 5 , and another one with 6 ?
Egor Bakaev

Baron Munchhausen's polynomial

Source: 45th International Tournament of Towns, Senior O-Level P1, Fall 2023

1. Baron Munchhausen was told that some polynomial

P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}

P(x)+P(-x)

has exactly 45 distinct real roots. Baron doesn't know the value of

n

. Nevertheless he claims that he can determine one of the coefficients

a_{n}, \ldots, a_{1}, a_{0}

(indicating its position and value). Isn't Baron mistaken? Boris Frenkin

algebrapolynomial

Source: 45th International Tournament of Towns, Junior O-Level P1, Fall 2023

1. A strip for playing "hopscotch" consists of ten squares numbered consecutively

1,2, \ldots, 10

. Clarissa and Marissa start from the center of the first square, jump 9 times to the centers of the other squares so that they visit each square once, and end up at the tenth square. (Jumps forward and backward are allowed.) Each jump of Clarissa was for the same distance as the corresponding jump of Marissa. Does this mean that they both visited the squares in the same order? Alexey Tolpygo