Usually the WMC throws you in at the deep end, but occasionally it's nice to teach you a bit of maths. This week, I'd like to introduce the AM-GM inequality, which you might be familiar with as it does crop up in Olympiad-style problems.

• The arithmetic mean of n numbers a₁, a₂, ..., a_n is defined as (a₁ + a₂ + ... + a_n)/n. In other words, it's the standard way of calculating the mean of a set of numbers.
• The geometric mean is the n^th root of the product of those numbers, i.e. (a₁a₂...a_n)^1/n.
• The AM-GM inequality simply states that the GM of a set of positive numbers is always less than the AM of those numbers. (They can be equal, if all the numbers are the same.)

(a) Prove the AM-GM inequality in the case n = 2. You might like to start from the fact that (a-b)² is always positive and see where that takes you!

(b) A team of four players is trying to answer a quiz question. The players have independent probabilities of success a, b, c, d, where a + b + c + d = 1. The team wins if at least one player answers the question correctly. What is the minimum probability that the team wins?

(This is another problem from Noa, who has helped produce a variety of WMC problems this year: #426, #432, #443, #449, as well as helping to verify solutions for a few others - thank you!)