MathDB

21

Part of 2019 AMC 12/AHSME

Problems(2)

Complex Bashing

Source: 2019 AMC 12A #21

2/8/2019
Let z=1+i2.z=\frac{1+i}{\sqrt{2}}. What is (z12+z22+z32+⋯+z122)⋅(1z12+1z22+1z32+⋯+1z122)?(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?
<spanclass=′latex−bold′>(A)</span>18<spanclass=′latex−bold′>(B)</span>72−362<spanclass=′latex−bold′>(C)</span>36<spanclass=′latex−bold′>(D)</span>72<spanclass=′latex−bold′>(E)</span>72+362<span class='latex-bold'>(A) </span> 18 \qquad <span class='latex-bold'>(B) </span> 72-36\sqrt2 \qquad <span class='latex-bold'>(C) </span> 36 \qquad <span class='latex-bold'>(D) </span> 72 \qquad <span class='latex-bold'>(E) </span> 72+36\sqrt2
2019 AMC 12A2019 AMCAMC 12AMCcomplex numbers
Roots equals Coefficients

Source: 2019 AMC 12B #21

2/14/2019
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is ax2+bx+c,a≠0,ax^2+bx+c,a\neq 0, and the roots are rr and s,s, then the requirement is that {a,b,c}={r,s}\{a,b,c\}=\{r,s\}.)
<spanclass=′latex−bold′>(A)</span>3<spanclass=′latex−bold′>(B)</span>4<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>6<spanclass=′latex−bold′>(E)</span>infinitely many<span class='latex-bold'>(A) </span> 3 \qquad<span class='latex-bold'>(B) </span> 4 \qquad<span class='latex-bold'>(C) </span> 5 \qquad<span class='latex-bold'>(D) </span> 6 \qquad<span class='latex-bold'>(E) </span> \text{infinitely many}
2019 AMC 12BAMCAMC 12AMC 12 Bquadraticsalgebrapolynomial