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Systems of Equations (Hard)

Late AMC 10/12 through early AIME — dense computation.

Easy Medium Hard AIME / USAJMO / AMO

Systems of Equations: core idea

Nonlinear.

Example problem: hint and answer

$x+y=11,xy=24$ . Find $x^2+y^2$ .

Hint: Eliminate variables, or use symmetric identities when they are given. Answer: 73.

Practice (9 problems — type your answers)

1. Chickens and rabbits total $35$ heads and $94$ legs. Find rabbits.
2. $2x+3y=19,x-y=2$ . Find $x+y$ .
3. Two positive numbers have sum $17$ and difference $5$ . Find product.
4. $x+y+z=12,x-y=1,y+z=7$ . Find $x$ .
5. Nineteen $3$ - or $5$ -dollar tickets cost $79$ . How many are $5$ -dollar?
6. $x^2+y^2=25,x+y=7$ . Find $xy$ .
7. $a+b=8,a^2+b^2=34$ . Find $|a-b|$ .
8. $x-y=4,x^2-y^2=40$ . Find $x+y$ .
9. $3x+2y=16,5x-2y=8$ . Find $xy$ .

Systems of Equations Practice Problems — Hard