MathDB

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Part of 2023 Saint Petersburg Mathematical Olympiad

Problems(3)

Composition of polynomials has roots powers of 2

Source: St. Petersburg 2023 11.1

8/12/2023
Let f(x),g(x)f(x), g(x) be real polynomials of degrees 22 and 33, respectively. Could it happen that f(g(x))f(g(x)) has 66 distinct roots, which are powers of 22?
algebra
Fractional part is the sum of the rest 2022 numbers

Source: St. Petersburg 2023 10.1

8/12/2023
Do there exist 20232023 nonzero reals, not necessarily distinct, such that the fractional part of each number is equal to the sum of the rest 20222022 numbers?
algebra
2-variable inequality

Source: St. Petersburg 2023 9.1

8/12/2023
Let a,b>1a, b>1 be reals such that a+1a2≥5b−3b2a+\frac{1}{a^2} \geq 5b-\frac{3}{b^2}. Show that a>5b−4b2a>5b-\frac{4}{b^2}.
inequalities