MathDB

1

Part of 2023 Austrian MO National Competition

Problems(2)

symmetric R to R FE

Source: Austria MO Final round 2023 P1

5/27/2023
Given is a nonzero real number α\alpha. Find all functions f:R→Rf: \mathbb{R} \to \mathbb{R} such that f(f(x+y))=f(x+y)+f(x)f(y)+αxyf(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy for all x,y∈Rx, y \in \mathbb{R}.
algebra
4-variable inequality with square root

Source: 2023 Austrian Federal Competition For Advanced Students, Part 1 p1

5/4/2023
Let a,b,c,da, b, c, d be positive reals strictly smaller than 11, such that a+b+c+d=2a+b+c+d=2. Prove that (1−a)(1−b)(1−c)(1−d)≤ac+bd2.\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}.
algebrainequalities proposed