You may also like

problem icon

Weekly Challenge 43: A Close Match

Can you massage the parameters of these curves to make them match as closely as possible?

problem icon

Weekly Challenge 44: Prime Counter

A weekly challenge concerning prime numbers.

problem icon

Weekly Challenge 28: the Right Volume

Can you rotate a curve to make a volume of 1?

Inner Equality

Stage: 5 Short Challenge Level: Challenge Level:1
Well done to Amrit, Adithya, Daven and Sergio who all sent in solutions to this problem. Here are the first four inequalities:

$$ 10 < a+ b- c - d < 20 $$
$$ 0 < a- c < 10 $$
$$ -10 < a - c + d - b < 10 $$
$$ 0 < abcd < 625 $$
Aditha's solution explains how to get each of them, you can read the pdf

Amrit explained how to work out the fifth inequality:

For the last inequality, we need to prove the AM-GM inequality
A number squared is always greater than 0 unless the number is 0 or in this case if a=c
Plugging in a=c into the last inequality, we have
Looking at the AM-GM inequality, we want a and c to be as far apart as
possible. So |a| has to be 5 and |c| has to be 0 or vice versa.
Applying this, we have $\frac{|a|+|c|}{2}-\sqrt|ac|<\frac{5}{2}$