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Saudi Arabian Mathematical Competitions

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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