MathDB

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Part of 2023 All-Russian Olympiad

Problems(3)

Two quadratics

Source: All-Russian MO 2023 Final stage 9.1

4/23/2023
Given are two monic quadratics f(x),g(x)f(x), g(x) such that f,g,f+gf, g, f+g have two distinct real roots. Suppose that the difference of the roots of ff is equal to the difference of the roots of gg. Prove that the difference of the roots of f+gf+g is not bigger than the above common difference.
quadraticsalgebra
Rotate triangle about circumcenter by 120 degrees

Source: All-Russian MO 2023 10.1

4/24/2023
Sidelines of an acute-angled triangle TT are colored in red, green, and blue. These lines were rotated about the circumcenter of TT clockwise by 120∘120^\circ (we assume that the line has the same color after rotation). Prove that three points of pairs of lines of the same color are the vertices of a triangle which is congruent to TT.
rotationgeometrycircumcircle
Trigonometry makes rational number

Source: All-Russian MO 2023 Final stage 11.1

4/23/2023
If x∈Rx\in\mathbb{R} satisfy sinsin x+tanx+tan x∈Qx\in\mathbb{Q}, coscos x+cotx+cot x∈Qx\in\mathbb{Q} Prove that sinsin 2x2x is a root of an integral coefficient quadratic function
trigonometryquadratics