MathDB

4

Part of 2011 AMC 10

Problems(2)

Evaluating arithmetic series

Source: AMC 10 2011 a Problem 4

6/25/2011
Let XX and YY be the following sums of arithmetic sequences: \begin{eqnarray*} X &=& 10 + 12 + 14 + \cdots + 100, \\ Y &=& 12 + 14 + 16 + \cdots + 102. \end{eqnarray*} What is the value of Y−XY - X?
<spanclass=′latex−bold′>(A)</span> 92<spanclass=′latex−bold′>(B)</span> 98<spanclass=′latex−bold′>(C)</span> 100<spanclass=′latex−bold′>(D)</span> 102<spanclass=′latex−bold′>(E)</span> 112 <span class='latex-bold'>(A)</span>\ 92\qquad<span class='latex-bold'>(B)</span>\ 98\qquad<span class='latex-bold'>(C)</span>\ 100\qquad<span class='latex-bold'>(D)</span>\ 102\qquad<span class='latex-bold'>(E)</span>\ 112
AMC
Sharing costs equally

Source: AMC 10 2011 b Problem 4

2/24/2011
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid AA dollars and Bernardo had paid BB dollars, where A<BA < B. How many dollars must LeRoy give to Bernardo so that they share the costs equally?
<spanclass=′latex−bold′>(A)</span> A+B2<spanclass=′latex−bold′>(B)</span> A−B2<spanclass=′latex−bold′>(C)</span> B−A2<spanclass=′latex−bold′>(D)</span> B−A<spanclass=′latex−bold′>(E)</span> A+B <span class='latex-bold'>(A)</span>\ \frac{A+B}{2} \qquad <span class='latex-bold'>(B)</span>\ \frac{A-B}{2} \qquad <span class='latex-bold'>(C)</span>\ \frac{B-A}{2} \qquad <span class='latex-bold'>(D)</span>\ B-A \qquad <span class='latex-bold'>(E)</span>\ A+B
AMC