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Difference between revisions of "2025 USAMO Problems/Problem 2"

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Let <math>n</math> and <math>k</math> be positive integers with <math>k<n</math>. Let <math>P(x)</math> be a polynomial of degree <math>n</math> with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers <math>a_0, a_1, \dots, a_k</math> such that the polynomial <math>a_kx^k+\cdots+a_1x+a_0</math> divides <math>P(x)</math>, the product <math>a_0a_1\cdots a_k</math> is zero. Prove that <math>P(x)</math> has a nonreal root.
 
Let <math>n</math> and <math>k</math> be positive integers with <math>k<n</math>. Let <math>P(x)</math> be a polynomial of degree <math>n</math> with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers <math>a_0, a_1, \dots, a_k</math> such that the polynomial <math>a_kx^k+\cdots+a_1x+a_0</math> divides <math>P(x)</math>, the product <math>a_0a_1\cdots a_k</math> is zero. Prove that <math>P(x)</math> has a nonreal root.
  
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== Solution ==
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== Problem ==
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Assume for contradiction that all roots of <math>P(x)</math> are real. Let the distinct non-zero real roots be <math>r_1, r_2, \ldots, r_n</math>.
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Let <math>n</math> and <math>k</math> be positive integers with <math>k<n</math>. Let <math>P(x)</math> be a polynomial of degree <math>n</math> with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers <math>a_0, a_1, \dots, a_k</math> such that the polynomial <math>a_kx^k+\cdots+a_1x+a_0</math> divides <math>P(x)</math>, the product <math>a_0a_1\cdots a_k</math> is zero. Prove that <math>P(x)</math> has a nonreal root.
  
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Case <math>k=2</math>
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== Solution ==
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For any pair of roots <math>r_i, r_j</math>, consider:
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We proceed by contradiction. Assume that all roots of <math>P(x)</math> are real. Let the distinct roots be <math>r_1, r_2, \ldots, r_n</math>, all nonzero since the constant term is nonzero.
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<cmath> Q(x) = (x-r_i)(x-r_j) = x^2 - (r_i+r_j)x + r_ir_j </cmath>
 
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The product of coefficients is:
 
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<cmath> -r_ir_j(r_i+r_j) = 0 </cmath>
 
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Since <math>r_i,r_j \neq 0</math>, we must have <math>r_i + r_j = 0</math> for all pairs.
 
  
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But for three roots <math>r_1, r_2, r_3</math>, this gives:
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Consider any subset of <math>k</math> roots <math>\{r_{i_1}, r_{i_2}, \ldots, r_{i_k}\}</math> and form the polynomial:
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<cmath> r_1 + r_2 = 0 </cmath>
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<cmath> Q(x) = \prod_{j=1}^k (x - r_{i_j}) = x^k + a_{k-1}x^{k-1} + \cdots + a_0 </cmath>
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<cmath> r_1 + r_3 = 0 </cmath>
 
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<cmath> r_2 + r_3 = 0 </cmath>
 
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which implies <math>r_1 = r_2 = r_3 = 0</math>, contradicting the nonzero constant term.
 
  
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\subsection*{General <math>k</math>:}
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By Vieta's formulas:
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For any <math>k</math> roots <math>r_{i_1}, \ldots, r_{i_k}</math>, the polynomial:
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\begin{itemize}
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<cmath> Q(x) = \prod_{m=1}^k (x-r_{i_m}) </cmath>
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\item <math>a_0 = (-1)^k \prod_{j=1}^k r_{i_j} \neq 0</math>
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must have some coefficient (other than constant term) equal to zero. For <math>k=3</math>, this requires:
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\item <math>a_{k-1} = -\sum_{j=1}^k r_{i_j}</math>
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<cmath> r_i + r_j + r_m = 0 </cmath>
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\item <math>a_{k-2} = \sum_{1\leq m<n\leq k} r_{i_m}r_{i_n}</math>
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for all triples, which is impossible for distinct non-zero reals when <math>n \geq 4</math>.
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\end{itemize}
  
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Thus, <math>P(x)</math> must have at least one nonreal root. \hfill (by Jonathan Wang)
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The given condition requires that <math>a_0a_1\cdots a_k = 0</math>. Since <math>a_0 \neq 0</math>, at least one other coefficient must be zero.
 +
 
 +
Case <math>k=2</math>:
 +
For any pair of roots <math>(r_i, r_j)</math>, we have:
 +
<cmath> Q(x) = x^2 - (r_i+r_j)x + r_ir_j </cmath>
 +
The condition implies <math>-r_ir_j(r_i+r_j) = 0</math>, so <math>r_i + r_j = 0</math> for all pairs. But with <math>n \geq 3</math>, considering three roots <math>r_1, r_2, r_3</math> gives:
 +
<cmath> r_1 + r_2 = 0 \quad \text{and} \quad r_1 + r_3 = 0 \implies r_2 = r_3 </cmath>
 +
contradicting distinct roots.
 +
In General <math>k</math>:
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For any <math>k</math> roots, some symmetric sum must be zero. For <math>k=3</math>, this would require:
 +
<cmath> r_i + r_j + r_m = 0 \quad \text{for all triples} </cmath>
 +
which leads to contradictions when <math>n \geq 4</math> as it would force roots to be equal.
 +
 
 +
Thus, our initial assumption is false, and <math>P(x)</math> must have at least one nonreal root.（By Jonathan)
 +
 
 +
== See Also ==
 +
{{USAMO newbox|year=2025|num-b=1|num-a=3}}
 +
{{WAA Notice}}
  
 
== See Also ==
 
== See Also ==
 
{{USAMO newbox|year=2025|num-b=1|num-a=3}}
 
{{USAMO newbox|year=2025|num-b=1|num-a=3}}
 
{{WAA Notice}}
 
{{WAA Notice}}

Revision as of 23:47, 27 March 2025

Problem

Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0, a_1, \dots, a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.

Problem

Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0, a_1, \dots, a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.

Solution

We proceed by contradiction. Assume that all roots of $P(x)$ are real. Let the distinct roots be $r_1, r_2, \ldots, r_n$, all nonzero since the constant term is nonzero.

Consider any subset of $k$ roots $\{r_{i_1}, r_{i_2}, \ldots, r_{i_k}\}$ and form the polynomial: \[Q(x) = \prod_{j=1}^k (x - r_{i_j}) = x^k + a_{k-1}x^{k-1} + \cdots + a_0\]

By Vieta's formulas: \begin{itemize} \item $a_0 = (-1)^k \prod_{j=1}^k r_{i_j} \neq 0$ \item $a_{k-1} = -\sum_{j=1}^k r_{i_j}$ \item $a_{k-2} = \sum_{1\leq m<n\leq k} r_{i_m}r_{i_n}$ \end{itemize}

The given condition requires that $a_0a_1\cdots a_k = 0$. Since $a_0 \neq 0$, at least one other coefficient must be zero.

Case $k=2$: For any pair of roots $(r_i, r_j)$, we have: \[Q(x) = x^2 - (r_i+r_j)x + r_ir_j\] The condition implies $-r_ir_j(r_i+r_j) = 0$, so $r_i + r_j = 0$ for all pairs. But with $n \geq 3$, considering three roots $r_1, r_2, r_3$ gives: \[r_1 + r_2 = 0 \quad \text{and} \quad r_1 + r_3 = 0 \implies r_2 = r_3\] contradicting distinct roots. In General $k$: For any $k$ roots, some symmetric sum must be zero. For $k=3$, this would require: \[r_i + r_j + r_m = 0 \quad \text{for all triples}\] which leads to contradictions when $n \geq 4$ as it would force roots to be equal.

Thus, our initial assumption is false, and $P(x)$ must have at least one nonreal root.（By Jonathan)

See Also

2025 USAMO (Problems • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

Template:WAA Notice

See Also

2025 USAMO (Problems • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

Template:WAA Notice

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