Test to prepare for the Olympiad on the topic of operation on rational numbers

Evaluate:
−1 −[tex](-1)^{1}[/tex] − [tex](-1)^{2}[/tex] − [tex](-1)^{3}[/tex] − · · · − [tex](-1)^{99}[/tex] − [tex](-1)^{100}[/tex]

Evaluate:
2008 × 20092009 − 2009 × 20082008

From 2009 subtract half of it at first, then subtract [tex]\frac{1}{3}[/tex] of the remaining number, next subtract [tex]\frac{1}{4}[/tex] of the remaining number, and so on, until [tex]\frac{1}{2009}[/tex] of the remaining number is subtracted. What is the final remaining number?

Find the denominator of the sum
[tex]\frac{1}{5*7}[/tex] + [tex]\frac{1}{7*9}[/tex] + [tex]\frac{1}{9*11}[/tex] + [tex]\frac{1}{11*13}[/tex] + [tex]\frac{1}{13*15}[/tex]

Find the sum:
11 + 192 + 1993 + 19994 + 199995 + 1999996 + 19999997 +
199999998 + 1999999999.

Evaluate:
[tex]1^{2}[/tex] - [tex]2^{2}[/tex] + [tex]3^{2}[/tex] - [tex]4^{2}[/tex] + ... -[tex]2008^{2}[/tex] + [tex]2009^{2}[/tex]

Find the numerator of the sum:
[tex]\frac{1}{1*2*3}[/tex] + [tex]\frac{1}{2*3*4}[/tex] + ... +[tex]\frac{1}{100*101*102}[/tex]

Find the numerator of the sum:
[tex]\frac{1}{10}[/tex]+[tex]\frac{1}{40}[/tex]+[tex]\frac{1}{88}[/tex]+[tex]\frac{1}{154}[/tex]+[tex]\frac{1}{238}[/tex]

Find the difference of denominator and numerator of the sum:
[tex]\frac{1}{1+2}[/tex]+[tex]\frac{1}{1+2+3}[/tex]+...+[tex]\frac{1}{1+2+3+...+51}[/tex]

Let n be a positive integer, find the value of:
1 + [tex]\frac{1}{2}[/tex] + [tex]\frac{2}{2}[/tex] + [tex]\frac{1}{2}[/tex] + [tex]\frac{1}{3}[/tex] + [tex]\frac{2}{3}[/tex] + [tex]\frac{3}{3}[/tex] + [tex]\frac{2}{3}[/tex] + [tex]\frac{1}{3}[/tex] + ... + [tex]\frac{1}{n}[/tex] + [tex]\frac{2}{n}[/tex] + ... + [tex]\frac{n}{n}[/tex] + [tex]\frac{n-1}{n}[/tex] + ... + [tex]\frac{1}{n}[/tex]