MathDB

20

Part of 2012 AMC 12/AHSME

Problems(2)

Product of Polynomials with Powers of Two as Exponents

Source: 2012 AMC 12A Problem #20

2/8/2012
Consider the polynomial P(x)=∏k=010(x2k+2k)=(x+1)(x2+2)(x4+4)⋯(x1024+1024).P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024). The coefficient of x2012x^{2012} is equal to 2a2^a. What is aa?
<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 24 <span class='latex-bold'>(A)</span>\ 5\qquad<span class='latex-bold'>(B)</span>\ 6\qquad<span class='latex-bold'>(C)</span>\ 7\qquad<span class='latex-bold'>(D)</span>\ 10\qquad<span class='latex-bold'>(E)</span>\ 24
algebrapolynomialAMC
Trapezoid and possible areas

Source: 2012 AMC12B #20

2/23/2012
A trapezoid has side lengths 3,5,7,3, 5, 7, and 1111. The sum of all the possible areas of the trapezoid can be written in the form of r1n1+r2n2+r3r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3, where r1,r2,r_1, r_2, and r3r_3 are rational numbers and n1n_1 and n2n_2 are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to r1+r2+r3+n1+n2?r_1 + r_2 + r_3 + n_1 + n_2?
<spanclass=′latex−bold′>(A)</span> 57<spanclass=′latex−bold′>(B)</span> 59<spanclass=′latex−bold′>(C)</span> 61<spanclass=′latex−bold′>(D)</span> 63<spanclass=′latex−bold′>(E)</span> 65 <span class='latex-bold'>(A)</span>\ 57\qquad<span class='latex-bold'>(B)</span>\ 59\qquad<span class='latex-bold'>(C)</span>\ 61\qquad<span class='latex-bold'>(D)</span>\ 63\qquad<span class='latex-bold'>(E)</span>\ 65
geometrytrapezoidparallelogramarea of a triangleHeron's formulaAMC