Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME, harder Prove it Saudi Arabia

Do there exist two polynomials PP and QQ with integer coefficient such that
i) both PP and QQ have a coefficient with absolute value bigger than 20212021,
ii) all coefficients of P⋅QP \cdot Q by absolute value are at most 11.

Solution

Note that the polynomial
(1−x2)(1−x4)(1−x8)…(1−x2n) (1 - x^2)(1 - x^4)(1 - x^8) \dots (1 - x^{2n})
has all coefficients equal 00, +1+1 or −1-1. Also note, that
(1−x2)(1−x4)(1−x8)…(1−x2n)=∏i=0n−1(1−x2i)⋅∏i=0n−1(1+x2i)=(1−x)n⋅∏i=0n−1(1+xi)n−i \begin{aligned} & (1 - x^2)(1 - x^4)(1 - x^8) \dots (1 - x^{2n}) \\ &= \prod_{i=0}^{n-1} (1 - x^{2i}) \cdot \prod_{i=0}^{n-1} (1 + x^{2i}) \\ &= (1 - x)^n \cdot \prod_{i=0}^{n-1} (1 + x^i)^{n-i} \end{aligned}
Due to Newton binomial formula (1±x)n(1 \pm x)^n have coefficient equal nn, also all coefficients of
(1+x)n(1+x2)n−1…(1+x2n−2)2(1+x2n−1) (1 + x)^n (1 + x^2)^{n-1} \dots (1 + x^{2n-2})^2 (1 + x^{2n-1})
are positive and bigger than coefficients of (1+x)n(1 + x)^n. □\square

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