Vieta's Formulas (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).

Easy Medium Hard AIME / USAJMO / AMO

Vieta's Formulas: core idea

Newton identities.

Example problem: hint and answer

**(Proof track.)** Prove Newton–Girard formulae relating power sums to elementary symmetric polynomials. For the cubic $x^3-3x+1=0$ , enter $p_2=r^2+s^2+t^2$ .

Hint: $p_2=e_1^2-2e_2$ . Answer: 6.

Practice (9 problems — type your answers)

No proof submission required — answers only.

1. **(Proof track.)** Prove that if $r,s$ are roots of $x^2-px+q=0$ , then $r^n+s^n$ satisfies a linear recurrence. Enter $r^5+s^5$ when $p=3$ , $q=1$ .
2. **(Proof track.)** Prove a factorization of $(x-1)(x-2)(x-3)(x-4)+1$ into quadratics and enter the product of the constant terms of the two monic quadratics.
3. **(Proof track.)** Using Vieta, prove that the roots of $x^3+ax^2+bx+c=0$ form a geometric progression iff $a^3c=b^3$ (under nonzero conditions). For $b=-12$ , $c=-8$ , enter the corresponding $a$ .
4. **(Proof track.)** Prove that $r^3+s^3+t^3-3rst=(r+s+t)(\cdots)$ . For roots of $x^3-6x^2+11x-6=0$ , enter $r^3+s^3+t^3$ .
5. **(Proof track.)** Show how to express $\sum 1/r$ via Vieta for a monic cubic. For $x^3-6x^2+11x-6$ , enter $\sum 1/r$ .
6. **(Proof track.)** Prove the discriminant criterion for a cubic to have a repeated root, then for $x^3-3x+2$ enter the number of distinct real roots.
7. **(Proof track.)** Using symmetric polynomials, prove that if $r+s=rs=k$ , then $r^2+s^2=k^2-2k$ . Enter $r^2+s^2$ for $k=4$ .
8. **(Proof track.)** Prove that all roots of $x^4-5x^2+4=0$ are real and enter the sum of their squares.
9. **(Proof track.)** Establish the relation between resultants and common roots; as a check, enter the resultant of $x^2-1$ and $x-1$ (as polynomials in $x$ ).