2024 SSMO Speed Round Problems/Problem 3

Problem

The polynomial $x^3 - 15x^2 + 4x + 4$ has distinct real roots $r$ , $s$ , and $t$ . Find the value of $\left|(r^2 + s^2 + t^2)(rst)\right|.$

Solution

By vietta's, we have that $r^2+s^2+t^2 = (r+s+t)^2 - 2(rs+st+rt) = 15^2 - 2(4) = 217$ and $rst = -4$ . Thus, $|(r^2+s^2+t^2)(rst)| = |217\cdot -4| = \boxed{868}$ .

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2024 SSMO Speed Round Problems/Problem 3

Problem

Solution