Combinations (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Combinations: core idea

Bijection proofs.

**(Proof track.)** Give a complete justification. How many ways select three vertices of a decagon that form no side?

Hint: Give a bijection, a cases argument, or a complement count. Answer: 50.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. How many integer solutions to $a+b+c=15$ have $a,b,c\ge2$ ?
2. **(Proof track.)** Give a complete justification. Choose six from ten if at least one of two named people is chosen. How many?
3. **(Proof track.)** Give a complete justification. How many subsets of $\{1,\dots,9\}$ have odd sum?
4. **(Proof track.)** Give a complete justification. How many paths from $(0,0)$ to $(6,6)$ stay weakly below $y=x$ ?
5. **(Proof track.)** Give a complete justification. How many nonnegative solutions to $x+y+z+w=8$ ?
6. **(Proof track.)** Give a complete justification. How many ways distribute eight identical balls among three boxes with none empty?
7. **(Proof track.)** Give a complete justification. How many four-person committees from eight have exactly two of three captains?
8. **(Proof track.)** Give a complete justification. How many diagonals of a $20$ -gon are not diameters?
9. **(Proof track.)** Give a complete justification. How many ways choose two disjoint three-person teams from eight people?