MathDB

3

Part of 2023 JBMO TST - Turkey

Problems(2)

One variable function in Turkey JBMO TST

Source: Turkey JBMO TST 2023 Day 1 P3

4/30/2023
Find all f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} such that f(x+f(x))=f(−x)f(x+f(x))=f(-x) and for all x≤yx \leq y it satisfies f(x) ≤ f(y)f(x) \leq f(y)
functionalgebra
Paralell lines and tangent to (ABC)

Source: Turkey JBMO TST 2023 Day 2 P3

4/30/2023
Let ABCABC is triangle and D∈ABD \in AB,E∈ACE \in AC such that DE//BCDE//BC. Let (ABC)(ABC) meets with (BDE)(BDE) and (CDE)(CDE) at the second time K,LK,L respectively. BKBK and CLCL intersect at TT. Prove that TATA is tangent to the (ABC)(ABC)
geometrytangent