2014 UNCO Math Contest II Problems/Problem 2

Problem

Define the Cheshire Cat function $\fbox{:)}$ by

$\begin{align*} \fbox{:)}(x) &= -x \quad \text {if } x \text{ is even and} \\ \fbox{:)}(x) &= x \quad \text{ if }x \text{ is odd} \end{align*}$

Find the sum $\fbox{:)}(1) + \fbox{:)}(2) + \fbox{:)}(3) + \fbox{:)}(4) + . . .+ \fbox{:)}(289)$

Solution

This is asking for the sum of 1-2+3-4...+289. We know that the sum of the first N even numbers is N^2+N. Because there are 144 even numbers less than 289, we take 144^2+144 and get 20880. Because we're subtracting these, we have -20880. We are also adding the first 145 odd numbers, and the sum of the first N odd numbers is N^2. This is 145^2, so we get 21025. Adding these gives us 145, which is the answer.

2014 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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2014 UNCO Math Contest II Problems/Problem 2

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