5

Part of 2022 Tuymaada Olympiad

Problems(2)

Source: Tuymaada 2022 Junior P-5

24 \times 8

table contains some permutation of the numbers

1, 2, \cdots , 8.

In each column the numbers are multiplied. What is the minimum possible sum of all the products? (C. Wu)

permutationscombinarorics

Source: Tuymaada 2022 Senior P-5

Prove that a quadratic trinomial

x^2 + ax + b (a, b \in R)

cannot attain at ten consecutive integral points values equal to powers of

2

with non-negative integral exponent. (F. Petrov )

trinomialquadratic trinomialalgebra