MathDB

Problem 4

Part of 2024 Ukraine National Mathematical Olympiad

Problems(4)

Quadratic functions

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 8.4

3/19/2024
The board contains 2020 non-constant linear functions, not necessarily distinct. For each pair (f,g)(f, g) of these functions (190190 pairs in total), Victor writes on the board a quadratic function f(x)⋅g(x)−2f(x)\cdot g(x) - 2, and Solomiya writes on the board a quadratic function f(x)g(x)−1f(x)g(x)-1. Victor calculated that exactly VV of his quadratic functions have a root, and Solomiya calculated that exactly SS of her quadratic functions have a root. Find the largest possible value of S−VS-V.
Remarks. A linear function y=kx+by = kx+b is called non-constant if k≠0k\neq 0.
Proposed by Oleksiy Masalitin
algebraquadratics
Fantastic tangency

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 9.4

3/19/2024
Points E,FE, F are selected on sides AC,ABAC, AB respectively of triangle ABCABC with AC=ABAC=AB so that AE=BFAE = BF. Point DD is chosen so that D,AD, A are in the same halfplane with respect to line EFEF, and △DFE∼△ABC\triangle DFE \sim \triangle ABC. Lines EF,BCEF, BC intersect at point KK. Prove that the line DKDK is tangent to the circumscribed circle of △ABC\triangle ABC.
Proposed by Fedir Yudin
geometrytangency
Functional equation from best Ukrainian author

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 10.4

3/19/2024
Find all functions f:R→Rf:\mathbb{R} \to \mathbb{R}, such that for any x,y∈Rx, y \in \mathbb{R} holds the following:
f(x)f(yf(x))+yf(xy)=xf(xy)+y2f(x)f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)
Proposed by Mykhailo Shtandenko
functionalgebrafunctional equation
Sorry...

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 11.4

3/19/2024
Point XX is chosen inside a convex ABCDABCD so that ∠XBC=∠XAD,∠XCB=∠XDA\angle XBC = \angle XAD, \angle XCB = \angle XDA. Rays AB,DCAB, DC intersect at point OO, circumcircles of triangles BCO,ADOBCO, ADO intersect at point TT. Prove that line TXTX and the line through OO perpendicular to BCBC intersect on the circumcircle of △AOD\triangle AOD.
Proposed by Anton Trygub
geometrycircumcircle