Important pairs of polynomials

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Important pairs of polynomials

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Source: All Russian 2025 11.3

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egxa

#1 h Apr 18, 2025, 9:42 AM • 1 Y

Y by Mysteriouxxx

A pair of polynomials $F(x, y)$ and $G(x, y)$ with integer coefficients is called $\emph{important}$ if from the divisibility of both differences $F(a, b) - F(c, d)$ and $G(a, b) - G(c, d)$ by $100$ , it follows that both $a - c$ and $b - d$ are divisible by 100. Does there exist such an important pair of polynomials $P(x, y)$ , $Q(x, y)$ , such that the pair $P(x, y) - xy$ and $Q(x, y) + xy$ is also important?

This post has been edited 2 times. Last edited by egxa, Apr 18, 2025, 10:11 AM

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#2 h Apr 18, 2025, 2:25 PM

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Good problem

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#3 h Apr 18, 2025, 4:24 PM • 1 Y

Y by sami1618

The answer Is no.
Suppose the pair of polynomials F and G is important. Consider all pairs of remainders when the numbers F(a,b) and G(a,b) are divided by 100, where a and b range over all possible pairs of integers from 0 to 99. According to the problem's condition, all such remainder pairs are distinct. Since there are 100² total number pairs, each remainder pair modulo 100 appears exactly once. This means all 4 possible parity combinations for the numbers F(a,b) and G(a,b) must be achieved.

Since the parity of a polynomial's value with integer coefficients at point (a,b) depends only on the parity of numbers a and b, we conclude that the values (F(0,0);G(0,0)), (F(1,0);G(1,0)), (F(0,1);G(0,1)), and (F(1,1);G(1,1)) must cover all four possible parity combinations.

However, note that for both the polynomial pair F=P, G=Q and the pair F(x,y)=P(x,y)-xy, G(x,y)=Q(x,y)+xy, the first three parity combinations are identical, and only the fourth combination differs. Therefore, both such polynomial pairs cannot simultaneously be important.

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