Difference between revisions of "2020 USOJMO Problems/Problem 6"
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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.) | ||
Revision as of 12:54, 7 July 2020
Problem
Let 
be an integer. Let 
be a nonconstant 
-variable polynomial with real coefficients. Assume that whenever 
are real numbers, at least two of which are equal, we have 
. Prove that cannot be written as the sum of fewer than 
monomials. (A monomial is a polynomial of the form 
, where 
is a nonzero real number and 
, 
, 
, 
are nonnegative integers.)
