MathDB

24

Part of 2017 AMC 10

Problems(2)

Two Polynomials

Source: 2017 AMC10A #24, 2017 AMC12A #23

2/8/2017
For certain real numbers aa, bb, and cc, the polynomial g(x)=x3+ax2+x+10g(x) = x^3 + ax^2 + x + 10 has three distinct roots, and each root of g(x)g(x) is also a root of the polynomial f(x)=x4+x3+bx2+100x+c.f(x) = x^4 + x^3 + bx^2 + 100x + c. What is f(1)f(1)?
<spanclass=′latex−bold′>(A)</span> −9009<spanclass=′latex−bold′>(B)</span> −8008<spanclass=′latex−bold′>(C)</span> −7007<spanclass=′latex−bold′>(D)</span> −6006<spanclass=′latex−bold′>(E)</span> −5005<span class='latex-bold'>(A)</span>\ -9009 \qquad<span class='latex-bold'>(B)</span>\ -8008 \qquad<span class='latex-bold'>(C)</span>\ -7007 \qquad<span class='latex-bold'>(D)</span>\ -6006 \qquad<span class='latex-bold'>(E)</span>\ -5005
AMCAMC 10AMC 12algebrapolynomial2017 AMC 10A2017 AMC 12A
Hyperbola on AMC 10 ?!

Source: 2017 AMC 10B #24

2/16/2017
The vertices of an equilateral triangle lie on the hyperbola xy=1,xy=1, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
<spanclass=′latex−bold′>(A)</span> 48<spanclass=′latex−bold′>(B)</span> 60<spanclass=′latex−bold′>(C)</span> 108<spanclass=′latex−bold′>(D)</span> 120<spanclass=′latex−bold′>(E)</span> 169<span class='latex-bold'>(A)</span> \text{ 48} \qquad <span class='latex-bold'>(B)</span> \text{ 60} \qquad <span class='latex-bold'>(C)</span> \text{ 108} \qquad <span class='latex-bold'>(D)</span> \text{ 120} \qquad <span class='latex-bold'>(E)</span> \text{ 169}
conicshyperbolaAMC 102017 AMC 10B