PRMO 2018 Question 16

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16. What is the value of

$$\sum_{\substack{1 \le i \le j \le 10 \\ i + j = \text{odd}}} (i + j) - \sum_{\substack{1 \le i \le j \le 10 \\ i + j = \text{even}}} (i + j)?$$

[PRE-RMO – 2018]

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To compute the sum when i +j is odd, let's figure out how many times do we need to add something to 10:

10 will be added to 1,3,5,7,9
9 will be added to 2,4,6,8 (don't redo 10 since it's already done above)
8 => 1,3,5,7 (Just note the numbers smaller than this since the bigger ones are already added above).
7 => 2,4,6
6 => 1,3,5
5 => 2,4
4 => 1,3
3 => 2
2 => 1

How many times does each number occur?
Each number occurs 5 times.
Because each odd number will be added to each of 5 even numbers.
And each even number will be added to each of 5 odd numbers.

Now to compute the sum when i + j is even:
10 will be added to 2,4,6,8
9 => 1,3,5,7
8 => 2,4,6
7 => 1,3,5
6 => 2,4
5 => 1,3
4 => 2
3 => 1
2 => 1

Now each number occurs 4 times since each even number can be only added to the remaining 4 even numbers. Same for odd numbers.

So in the first sum, each number is added 5 times and in the second one 4 times.
So net sum  = 1 + 2 ... 10 = 55