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Progetto Olimpiadi della Matematica - GARA di FEBBRAIO

Algebra Difficulty 4.4 AIME Find the answer Italy

Problem:
Consider the polynomial $p(x)=\left(1+x^{3^{1}}\right)\left(1+x^{3^{2}}\right)\left(1+x^{3^{3}}\right)\left(1+x^{3^{4}}\right)\left(1+x^{3^{5}}\right)\left(1+x^{39}\right)$ , and suppose we expand the product, thus obtaining an expression of the form $a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{402} x^{402}$ , where for example $a_{0}=a_{402}=1$ . How many of the coefficients $a_{0}, \ldots, a_{402}$ are different from zero?

Pick one

Solution

Solution:
The answer is $(\mathbf{C})$ . First, observe that the nonzero coefficients are at most $2^{6}=64$ , depending on whether in each factor we take the term 1 or the term in $x$ . However, we can observe that $x^{3} \cdot x^{9} \cdot x^{27}=x^{39}$ , so there are terms in the expansion of the product that contribute to only one term in the resulting polynomial. How many are these repeated terms (which we will then have to remove from the 64 indicated above)? There are 4, namely $x^{39}$ , $x^{39} \cdot x^{3^{4}}$ , $x^{39} \cdot x^{3^{5}}$ and $x^{39} \cdot x^{3^{4}} \cdot x^{3^{5}}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.