MathDB

4

Part of 2016 Azerbaijan National Mathematical Olympiad

Problems(2)

Eventually integers will be observed

Source: Azerbaijan NMO 2016. Senior P4

7/27/2023
Let A=1⋅3⋅5⋅...⋅(2n−1)2⋅4⋅6⋅...⋅(2n)A = \frac{1 \cdot 3 \cdot 5\cdot ... \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot (2n)} Prove that in the infinite sequence A,2A,4A,8A,...,2kA,….A, 2A, 4A, 8A, ..., 2^k A, …. only integers will be observed, eventually.
number theorycombinatorics
Function with 2016 variables

Source: Azerbaijan NMO 2016. Senior P5

7/27/2023
Let R\mathbb R be the set of real numbers. Determine all functions f:R→Rf:\mathbb R\to\mathbb R that satisfy the equation ∑i=12015f(xi+xi+1)+f(∑i=12016xi)≤∑i=12016f(2xi)\sum_{i=1}^{2015} f(x_i + x_{i+1}) + f\left( \sum_{i=1}^{2016} x_i \right) \le \sum_{i=1}^{2016} f(2x_i) for all real numbers x1,x2,...,x2016.x_1, x_2, ... , x_{2016}.
functionalgebra