MathDB

3

Part of 2022 Saudi Arabia IMO TST

Problems(2)

f(ab + bc + ca) =f(a)f(b) +f(b)f(c)+f(c)f(a) in Q^+

Source: 2022 Saudi Arabia IMO TST 1.3

11/1/2022
Find all non-constant functions f:Q+→Q+f : Q^+ \to Q^+ satisfying the equation f(ab+bc+ca)=f(a)f(b)+f(b)f(c)+f(c)f(a)f(ab + bc + ca) =f(a)f(b) +f(b)f(c)+f(c)f(a) for all a,b,c∈Q+a, b,c \in Q^+ .
functional equationfunctionalalgebra
concyclic wanted, common external tangents of 2 circles meet on circle

Source: 2022 Saudi Arabia IMO TST 4.3

11/1/2022
Let A,B,C,DA,B,C,D be points on the line dd in that order and AB=CDAB = CD. Denote (P)(P) as some circle that passes through A,BA, B with its tangent lines at A,BA, B are a,ba,b. Denote (Q)(Q) as some circle that passes through C,DC, D with its tangent lines at C,DC, D are c,dc,d. Suppose that aa cuts c,dc, d at K,LK, L respectively and bb cuts c,dc, d at M,NM, N respectively. Prove that four points K,L,M,NK, L, M,N belong to a same circle (ω)(\omega) and the common external tangent lines of circles (P)(P), (Q)(Q) meet on (ω)(\omega).
geometryconcurrencyConcyclic