Systems of Equations (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).

Systems of Equations: core idea

Contest systems.

**(Proof track.)** Give a complete justification. Positive $x,y$ satisfy $x+y=10,x^3+y^3=370$ . Find $xy$ .

Hint: Look for a symmetric identity before solving more variables than necessary. Answer: 21.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. Solve $x+y+z=6,xy+yz+zx=11,xyz=6$ ; find $x^2+y^2+z^2$ .
2. **(Proof track.)** Give a complete justification. A two-digit number plus its reversal is $121$ ; its digits differ by $3$ . Find larger digit.
3. **(Proof track.)** Give a complete justification. Find $x+y$ if $x-y=2$ and $x^2+y^2=34$ .
4. **(Proof track.)** Give a complete justification. Two lines have intercept sums $12$ and meet at $(3,4)$ . Find their product of slopes.
5. **(Proof track.)** Give a complete justification. If $a+b=7,ab=10$ , find $a^3+b^3$ .
6. **(Proof track.)** Give a complete justification. A $5\%$ and $20\%$ solution mix to $12\%$ in 10 L. Find liters of $20\%$ solution.
7. **(Proof track.)** Give a complete justification. Find the larger positive $x$ satisfying $x+y=9,1/x+1/y=1/2$ .
8. **(Proof track.)** Give a complete justification. If $x+y=4$ and $x^2-y^2=12$ , find $x^2+y^2$ .
9. **(Proof track.)** Give a complete justification. Three pens and two notebooks cost $19$ ; two pens and five notebooks cost $23$ . Find pen price.