MathDB

3

Part of 2024 Kazakhstan National Olympiad

Problems(2)

Dumpty point problem

Source: Kazakhstan National Olympiad 2024 (9 grade), P3

3/21/2024
An acute triangle ABCABC (AB≠ACAB\ne AC) is inscribed in the circle ω\omega with center at OO. The point MM is the midpoint of the side BCBC. The tangent line to ω\omega at point AA intersects the line BCBC at point DD. A circle with center at point MM with radius MAMA intersects the extensions of sides ABAB and ACAC at points KK and LL, respectively. Let XX be such a point that BX∥KMBX\parallel KM and CX∥LMCX\parallel LM. Prove that the points XX, DD, OO are collinear.
geometry
Wrapped, hard functional equation.

Source: Kazakhstan National Olympiad 2024 (10-11 grade), P3

3/19/2024
Find all functions f:R+→R+f: \mathbb R^+ \to \mathbb R^+ such that f(x+f(xy)x)=f(xy)f(y+1y) f \left( x+\frac{f(xy)}{x} \right) = f(xy) f \left( y + \frac 1y \right) holds for all x,y∈R+.x,y\in\mathbb R^+.
functionalgebrafunctional equation