2012 AMC 12B Problem 5

Question

Two integers have a sum of

26.

When two more integers are added to the first two integers the sum is

41.

Finally when two more integers are added to the sum of the previous four integers the sum is

57.

What is the minimum number of even integers among the

6

integers?

1

2

3

4

5

Answer: A

Concepts:parity

Difficulty rating: 1200

Small hint:

Look at the sum of each new pair:

26,

41-26,

and

57-41

Big hint:

A pair has an odd sum only when exactly one of its two integers is even

Solution:

The three successive pairs have sums

26,

41-26=15,

and

57-41=16.

A pair sums to an even number when its two integers share parity, and to an odd number when exactly one is even. Only the middle pair sums to an odd number, so it must contain at least one even integer.

The other two pairs can be all odd, so as few as

1

even integer is possible, for example

1,25,1,14,1,15.

Thus, the correct answer is A.