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HMMT 十一月 2021 · THM 赛 · 第 5 题

HMMT November 2021 — THM Round — Problem 5

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Let n be the answer to this problem. The polynomial x + ax + bx + c has real coefficients and exactly k real roots. Find the sum of the possible values of k .
解析
  1. Let n be the answer to this problem. The polynomial x + ax + bx + c has real coefficients and exactly k real roots. Find the sum of the possible values of k . Proposed by: Sean Li Answer: 10 n 2 Solution: Note that the roots to the above polynomial must satisfy x = − ax − bx − c . Therefore, it n suffices to consider how many times a parabola can intersect the graph x . For n ≤ 2, a parabola can n intersect x 0, 1, or 2 times, so the sum of the possible values of k is 3. Therefore, we know we must n have n > 2. If n is odd, then a parabola can intersect x 0, 1, 2, or 3 times, so the sum of the possible values of k is 6, which isn’t odd. Thus, we must have n is even and n > 2. In this case a parabola can n intersect x 0, 1, 2, 3, or 4 times, so the sum of the possible values of k in this case is 10. Thus, we must have n = 10. n 2 Here is a more rigorous justification of the above reasoning for n > 2: consider f ( x ) = x + ax + bx + c . ′′′ n − 3 Note that f ( x ) = n ( n − 1)( n − 2) x , which has at most one real root. However, it is known that ′ if a differentiable function f ( x ) has k real roots, f ( x ) must have at least k − 1 real roots, with at least one in between any pair of real roots of f ( x ) (Proof sketch: apply Rolle’s theorem many times.). ′′′ Therefore, if f ( x ) has at least five real roots, applying the above result three times yields that f ( x ) has at least two real roots, a contradiction. Moreover, if f ( x ) has four real roots and n is odd, then since nonreal roots come in pairs we know that at least one of these real roots c has multiplicity at ′ least 2. Therefore, f ( x ) has three real roots in between the roots of f ( x ), plus one real root at c . Thus ′ ′′′ f ( x ) has at least four real roots, implying that f ( x ) has at least two real roots, again a contradiction.