HMMT 二月 2017 · 团队赛 · 第 1 题

HMMT February 2017 — Team Round — Problem 1

[ 15 ] Let P ( x ) , Q ( x ) be nonconstant polynomials with real number coefficients. Prove that if b P ( y ) c = b Q ( y ) c for all real numbers y , then P ( x ) = Q ( x ) for all real numbers x .

解析

[ 15 ] Let P ( x ) , Q ( x ) be nonconstant polynomials with real number coefficients. Prove that if b P ( y ) c = b Q ( y ) c for all real numbers y , then P ( x ) = Q ( x ) for all real numbers x . Proposed by: Alexander Katz Answer: By the condition, we know that | P ( x ) − Q ( x ) | ≤ 1 for all x . This can only hold if P ( x ) − Q ( x ) is a constant polynomial. Now take a constant c such that P ( x ) = Q ( x ) + c. Without loss of generality, we can assume that c ≥ 0 . Assume that c > 0 . By continuity, if deg P = deg Q > 0, we can select an integer r and a real number x such that Q ( x ) + c = r . Then b P ( x ) c = b Q ( x ) + c c = r. On the 0 0 0 0 other hand, b Q ( x ) c = b r − c c < r as r was an integer. This is a contradiction. Therefore, c = 0 as 0 desired.