MathDB

9

Part of 2010 AIME Problems

Problems(2)

Nonlinear System

Source: AIME 2010I Problem 9

3/17/2010
Let (a,b,c) (a,b,c) be the real solution of the system of equations x^3 \minus{} xyz \equal{} 2, y^3 \minus{} xyz \equal{} 6, z^3 \minus{} xyz \equal{} 20. The greatest possible value of a^3 \plus{} b^3 \plus{} c^3 can be written in the form mn \frac{m}{n}, where m m and n n are relatively prime positive integers. Find m \plus{} n.
3D geometryquadraticsinequalitiesAMCAIMEfunction
Hexagon within the hexagon

Source: 2010 AIMEII Problem 9

4/1/2010
Let ABCDEF ABCDEF be a regular hexagon. Let G G, H H, I I, J J, K K, and L L be the midpoints of sides AB AB, BC BC, CD CD, DE DE, EF EF, and AF AF, respectively. The segments AH AH, BI BI, CJ CJ, DK DK, EL EL, and FG FG bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of ABCDEF ABCDEF be expressed as a fraction mn \frac {m}{n} where m m and n n are relatively prime positive integers. Find m \plus{} n.
ratiogeometryanalytic geometrytrigonometryrotationsymmetrynumber theory