May 6th Review Problems

Solve for $x$: $$ 10^{35} x \equiv 4 \mod{7} $$
Find the smallest non-negative $x$ such that: $$ x \equiv 13 \mod{25} \\ x \equiv 7 \mod{9} $$
To the previous problem add the condition: $x \equiv 1 \mod{4}$
Show that any list of 20 integers must have two whose difference is divisible by 13.
Seventy cars sit on a parking lot. Thirty have stereo systems, 30 have A/C, 40 have sun roofs. Thirty of these cars have at least 2 of these 3 options. Ten cars have all three. How many have at least one option? How many have only one option?
What is the coefficient of $x^{25}$ in the binomial expansion of $ (2x - \frac{3}{x^2})^{58}$?
In how many ways can two white rooks, two black bishops, eight black pawns, and eight white pawns be placed on a prescribed 20 squares of a chessboard?
Prove that $ \binom{2n}{2} = 2\binom{n}{2} + n^2 $ using logic not formula.