MathDB

3

Part of 2015 IFYM, Sozopol

Problems(4)

Number theory Problem 01

Source:

12/10/2017
Solve in natural numbers: x3−y3=xy+61x^3-y^3=xy+61
number theory
inequality 1

Source:

4/3/2011
Let a,b,c>0 a,b,c>0 prove that:a3(a+b)3+b3(b+c)3+c3(c+a)3≥38 \frac{a^{3}}{(a+b)^{3}}+\frac{b^{3}}{(b+c)^{3}}+\frac{c^{3}}{(c+a)^{3}}\geq \frac{3}{8} Good luck! :D
inequalitiesinequalities proposed
Function R+ in R+

Source:

5/31/2013
Find all functions f:R+⟶R+f:\mathbb R^{+} \longrightarrow \mathbb R^{+} so that
f(xy+f(xy))=xy+xf(y)f(xy + f(x^y)) = x^y + xf(y) for all positive reals x,yx,y.
functionalgebra unsolvedalgebra
Problem 3 of Finals

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

1/11/2020
A cube 10x10x10 is constructed from 1000 white unit cubes. Polly and Velly play the following game: Velly chooses a certain amount of parallelepipeds 1x1x10, no two of which have a common vertex or an edge, and repaints them in black. Polly can choose an arbitrary number of unit cubes and ask Velly for their color. What’s the least amount of unit cubes she has to choose so that she can determine the color of each unit cube?
game strategycombinatorics