MathDB

P2

Part of Russian TST 2022

Problems(2)

Interesting FE

Source: Russian TST 2022, Day 3 P2

3/21/2023
Determine all functions f:R→Rf:\mathbb{R}\to\mathbb{R} satisfying f(xy+f(x))+f(y)=xf(y)+f(x+y),f(xy+f(x))+f(y)=xf(y)+f(x+y),for all real numbers x,yx,y.
functional equationalgebraTST
Quadrilateral and incircles

Source: Russian TST 2022, Day 7 P2

3/21/2023
The quadrilateral ABCDABCD is inscribed in the circle Γ\Gamma. Let IBI_B and IDI_D be the centers of the circles ωB\omega_B and ωD\omega_D inscribed in the triangles ABCABC and ADCADC, respectively. A common external tangent to ωB\omega_B and ωD\omega_D intersects Γ\Gamma at KK and LL{}. Prove that IB,ID,KI_B,I_D,K and LL{} lie on the same circle.
geometryConcyclicincircle