06/08/2020 (2011: National Sprint Round, Problem 25)

Q: Each number in the set {5, 9, 10, 13, 14, 18, 20, 21, 25, 29}was formed by adding two of the numbers in the set {a, b, c, d, e} where a < b < c < d < e. What is the value of c?

A: 7

We know that the first number (5) is the sum of the two least numbers, a and b. We also know that the greatest number is (29) is the sum of the two greatest numbers, d and e. That means the sum of a, b, d, and e is 5 + 29 = 34.

We find the sum of the first set  of numbers to be 5 + 9 + 10 + 13 + 14 + 18 + 20 + 21 + 25 + 29 = 164.

We also know that the sum of all the numbers is the sum of every possible combination of 2 separate values of a, b, c, d, and e, meaning there are 4 "a"s, 4 "b"s, 4 "c"s, 4 "d"s, and 4 "e"s. That means the sum of the numbers in the first set divided by 4 is a + b + c + d + e. We divide 164 by 4 and get 41. That means a + b + c + d + e = 41.

a + b + c + d + e - (a + b + d + e) = c, so c = 41 - 34 = 7.