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4

You just have to find the area of the curvilinear quadrilateral having its vertices in the four red points.

2

All prime divisors of $\frac{(pr)^{p}-1}{pr-1}$ for any positive integer $r$ are of the form $pk+1$. (and so all positive divisors are of the form $pk+1$, but it's not needed). Proof: Define $\text{ord}_n(a)$ to be the least positive integer $m$ such that $a^m\equiv 1\pmod{n}$. First a lemma: if $x^k\equiv 1\pmod{n}$, then $\text{ord}_n(x)\mid k$. To ...

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Solution: Let $S$ be the set of numbers not divisible by 23. I claim that $|S| \leq 22$; indeed in each residue class mod 23, there can be at most one number in $S$, else we have:$lcm(23a+r,23b+r) \geq 529ab/gcd(b-a) \geq 529$. The number of numbers which are divisible by 23 and at most 400 is at most 18. Thus the total number is at most 40.

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Like what Jesko said: There are ${9 \choose 3}$ ways to pick the 3 spots for the 3 white marbles and ${6 \choose 6}$ ways to pick the 6 remaining spots for the black marbles = ${9 \choose 3}$ * ${6 \choose 6}$ = 84 * 1 = 84 ways

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