MathDB

2.4

Part of 2022 Saudi Arabia BMO + EGMO TST

Problems(2)

2f(x)f(x + y)- f(x^2) = x}/2 (f(2x) + 4f(f(y)))

Source: 2022 Saudi Arabia November Camp Test 2.4 BMO + EGMO TST

5/13/2024
Find all functions f:R→Rf : R \to R such that 2f(x)f(x+y)−f(x2)=x2(f(2x)+4f(f(y)))2f(x)f(x + y) -f(x^2) =\frac{x}{2}(f(2x) + 4f(f(y))) for all x,y∈Rx, y \in R.
algebrafunctional equationfunctional
f(x + 2y + f(x + y)) = f(2x) + f(3y),

Source: 2022 Saudi Arabia January Camp Test 2.4 BMO + EGMO TST

5/13/2024
Consider the function f:R+→R+f : R^+ \to R^+ and satisfying f(x+2y+f(x+y))=f(2x)+f(3y),  ∀  x,y>0.f(x + 2y + f(x + y)) = f(2x) + f(3y), \,\, \forall \,\, x, y > 0.
1. Find all functions f(x)f(x) that satisfy the given condition.
2. Suppose that f(4sin⁡4x)f(4cos⁡4x)≥f2(1)f(4\sin^4x)f(4\cos^4x) \ge f^2(1) for all x∈(0π2)x \in \left(0\frac{\pi}{2}\right) . Find the minimum value of f(2022)f(2022).
algebrafunctionalfunctional equation