MathDB

12

Part of 2013 AIME Problems

Problems(2)

Triangle and Hexagon

Source: 2013 AIME I Problem 12

3/15/2013
Let △PQR\triangle PQR be a triangle with ∠P=75∘\angle P = 75^\circ and ∠Q=60∘\angle Q = 60^\circ. A regular hexagon ABCDEFABCDEF with side length 1 is drawn inside △PQR\triangle PQR so that side AB‾\overline{AB} lies on PQ‾\overline{PQ}, side CD‾\overline{CD} lies on QR‾\overline{QR}, and one of the remaining vertices lies on RP‾\overline{RP}. There are positive integers aa, bb, cc, and dd such that the area of △PQR\triangle PQR can be expressed in the form a+bcd\tfrac{a+b\sqrt c}d, where aa and dd are relatively prime and cc is not divisible by the square of any prime. Find a+b+c+da+b+c+d.
geometrytrigonometryAMCAIMEnumber theoryrelatively primearea of a triangle
Integer cubics with all roots of magnitude 20 and 13

Source: AIME II 2013, Problem 12

4/4/2013
Let SS be the set of all polynomials of the form z3+az2+bz+cz^3+az^2+bz+c, where aa, bb, and cc are integers. Find the number of polynomials in SS such that each of its roots zz satisfies either ∣z∣=20\left\lvert z \right\rvert = 20 or ∣z∣=13\left\lvert z \right\rvert = 13.
algebrapolynomialtrigonometryAMCAIMEVietacomplex numbers