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10

Part of 2009 AIME Problems

Problems(2)

The AIME committee

Source: 2009 AIME I #10

3/18/2009
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from 1 1 to 15 15 in clockwise order. Committee rules state that a Martian must occupy chair 1 1 and an Earthling must occupy chair 15 15. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is N⋅(5!)3 N\cdot (5!)^3. Find N N.
AMCAIMEfunctionparameterizationlinear algebramatrix2009 AIME I
Lighthouse Distances

Source: AIME 2009II Problem 10

4/2/2009
Four lighthouses are located at points A A, B B, C C, and D D. The lighthouse at A A is 5 5 kilometers from the lighthouse at B B, the lighthouse at B B is 12 12 kilometers from the lighthouse at C C, and the lighthouse at A A is 13 13 kilometers from the lighthouse at C C. To an observer at A A, the angle determined by the lights at B B and D D and the angle determined by the lights at C C and D D are equal. To an observer at C C, the angle determined by the lights at A A and B B and the angle determined by the lights at D D and B B are equal. The number of kilometers from A A to D D is given by prq \displaystyle\frac{p\sqrt{r}}{q}, where p p, q q, and r r are relatively prime positive integers, and r r is not divisible by the square of any prime. Find p\plus{}q\plus{}r,
analytic geometrytrigonometrynumber theoryrelatively primegeometryangle bisectorPythagorean Theorem