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Difference between revisions of "2020 USOJMO Problems/Problem 6"
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==Problem==
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Let <math>n \geq 2</math> be an integer. Let <math>P(x_1, x_2, \ldots, x_n)</math> be a nonconstant <math>n</math>-variable polynomial with real coefficients. Assume that whenever <math>r_1, r_2, \ldots , r_n</math> are real numbers, at least two of which are equal, we have <math>P(r_1, r_2, \ldots , r_n) = 0</math>. Prove that <math>P(x_1, x_2, \ldots, x_n)</math> cannot be written as the sum of fewer than <math>n!</math> monomials. (A monomial is a polynomial of the form <math>cx^{d_1}_1 x^{d_2}_2\ldots x^{d_n}_n</math>, where <math>c</math> is a nonzero real number and <math>d_1</math>, <math>d_2</math>, <math>\ldots</math>, <math>d_n</math> are nonnegative integers.)
 
Let <math>n \geq 2</math> be an integer. Let <math>P(x_1, x_2, \ldots, x_n)</math> be a nonconstant <math>n</math>-variable polynomial with real coefficients. Assume that whenever <math>r_1, r_2, \ldots , r_n</math> are real numbers, at least two of which are equal, we have <math>P(r_1, r_2, \ldots , r_n) = 0</math>. Prove that <math>P(x_1, x_2, \ldots, x_n)</math> cannot be written as the sum of fewer than <math>n!</math> monomials. (A monomial is a polynomial of the form <math>cx^{d_1}_1 x^{d_2}_2\ldots x^{d_n}_n</math>, where <math>c</math> is a nonzero real number and <math>d_1</math>, <math>d_2</math>, <math>\ldots</math>, <math>d_n</math> are nonnegative integers.)
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==See Also==
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{{USAJMO newbox|year=2020|num-b=5|after=Last Problem}}
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{{MAA Notice}}

Latest revision as of 18:16, 6 October 2023

Problem

Let $n \geq 2$ be an integer. Let $P(x_1, x_2, \ldots, x_n)$ be a nonconstant $n$-variable polynomial with real coefficients. Assume that whenever $r_1, r_2, \ldots , r_n$ are real numbers, at least two of which are equal, we have $P(r_1, r_2, \ldots , r_n) = 0$. Prove that $P(x_1, x_2, \ldots, x_n)$ cannot be written as the sum of fewer than $n!$ monomials. (A monomial is a polynomial of the form $cx^{d_1}_1 x^{d_2}_2\ldots x^{d_n}_n$, where $c$ is a nonzero real number and $d_1$, $d_2$, $\ldots$, $d_n$ are nonnegative integers.)

See Also

2020 USAJMO (Problems • Resources)
Preceded by
Problem 5 		Followed by
Last Problem
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

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