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1. Alternating sum of consecutive integers ●○○○○ 10 ptsWhat is the value of the expression $100 - 99 + 98 - 97 + 96 - 95 + \cdots + 4 - 3 + 2 - 1$?
2. Marbles for distinct counts ●○○○○ 10 ptsLisa has 10 friends and 34 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least one marble and no two friends receive the same number of marbles?
3. Alternating sum of integers up to 2007 ●○○○○ 10 ptsFind the value of $1 - 2 + 3 - 4 + \cdots + 2005 - 2006 + 2007$.
4. Sum of fractions with shared denominators ●●○○○ 20 ptsFind the value of $\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{20}\right)+\left(\frac{2}{3}+\cdots+\frac{2}{20}\right)+\cdots+\left(\frac{18}{19}+\frac{18}{20}\right)+\frac{19}{20}$.
5. Alternating block sum of squares ●●○○○ 20 ptsCompute $100^{2}+99^{2}-98^{2}-97^{2}+96^{2}+95^{2}-94^{2}-93^{2}+\ldots+4^{2}+3^{2}-2^{2}-1^{2}$.
6. Candle height at half burn time ●●○○○ 20 ptsA large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. (The candle burns down each individual centimeter at a fixed rate.) Suppose it takes $T$ seconds for the candle to burn down completely. Compute the candle's height in centimeters $\tfrac{T}{2}$ seconds after it is lit.