MathDB

3

Part of 2014 CentroAmerican

Problems(2)

Maximum value of sum of fractions

Source: Centroamerican Olympiad 2014, problem 3

6/10/2014
Let aa, bb, cc and dd be real numbers such that no two of them are equal,
ab+bc+cd+da=4\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4 and ac=bdac=bd. Find the maximum possible value of
ac+bd+ca+db.\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.
inequalitiesinequalities proposed
Funny numbers

Source: Centroamerican Olympiad 2014, problem 6

6/11/2014
A positive integer nn is funny if for all positive divisors dd of nn, d+2d+2 is a prime number. Find all funny numbers with the largest possible number of divisors.
modular arithmeticnumber theory proposednumber theory