MathDB

4

Part of 1990 Greece National Olympiad

Problems(3)

f(x+y)=f(x^2)+f(y^2)

Source: 1990 Greece MO Grade XI p4

9/6/2024
Find all functions f:R+→Rf: \mathbb{R}^+\to\mathbb{R} such that f(x+y)=f(x2)+f(y2)f(x+y)=f(x^2)+f(y^2) for any x,y∈R+x,y \in\mathbb{R}^+
functionalgebrafunctional
last two digits of 6^{1989} - 1990 Greece MO Grade X p4

Source:

9/6/2024
Since this is the 66th Greek Math Olympiad and the year is 19891989, can you find the last two digits of 619896^{1989}?
number theorylast digitsDigits
{x} +{1/x} =1 for rationals

Source: 1990 Greece MO Grade XII p4

9/6/2024
Froa nay real xx, we denote [x][x], the integer part of xx and with {x}\{x\} the fractional part of xx, such that x=[x]+{x}x=[x]+\{x\}. a) Find at least one real xx such that{x}+{1x}=1\{x\}+\left\{\frac{1}{x}\right\}=1 b) Find all rationals xx such that {x}+{1x}=1\{x\}+\left\{\frac{1}{x}\right\}=1
algebrafloor functionInteger Partfractional partinteger and fractional part