MathDB

11.5

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(3)

f(x+y)>= f(x)+ f(y) if f(x)/x increasing (I Soros Olympiad 1994-95 R1 11.5)

Source:

8/1/2021
Function f(x)f(x). which is defined on the set of non-negative real numbers, acquires real values. It is known that f(0)≤0f(0)\le 0 and the function f(x)/xf(x)/x is increasing for x>0x>0. Prove that for arbitrary x≥0x\ge 0 and y≥0y\ge 0, holds the inequality f(x+y)≥f(x)+f(y)f(x+y)\ge f(x)+ f(y) .
functioninequalitiesalgebra
f(x) + f(2x^2 - 1) = 2x + a (I Soros Olympiad 1994-99 Round 2 11.5)

Source:

5/26/2024
Is there a function f(x)f(x) defined for all xx and such that for some aa and all xx holds the equality f(x)+f(2x2−1)=2x+a?f(x) + f(2x^2 - 1) = 2x + a?
algebrafunctionalfunctional equation
m \ne k_1^n+ k_2^n+... k_n^n, m not equal to sum of perfect powers

Source: I Soros Olympiad 1994-95 Ukraine R2 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/6/2024
Prove that for any natural n>1n>1 there are infinitely many natural numbers mm such that for any nonnegative integers k1k_1,k2k_2, ......,kmk_m, m≠k1n+k2n+...knn,m \ne k_1^n+ k_2^n+... k_n^n,
number theoryPerfect Powers