MathDB

A10

Part of 2008 Princeton University Math Competition

Problems(3)

2008 PUMaC Geometry A10/B10

Source:

11/9/2012
A cuboctahedron is the convex hull of (smallest convex set containing) the 1212 points (±1,±1,0),(±1,0,±1),(0,±1,±1)(\pm 1, \pm 1, 0), (\pm 1, 0, \pm 1), (0, \pm 1, \pm 1). Find the cosine of the solid angle of one of the triangular faces, as viewed from the origin. (Take a figure and consider the set of points on the unit sphere centered on the origin such that the ray from the origin through the point intersects the fi gure. The area of that set is the solid angle of the fi gure as viewed from the origin.)
geometrytrigonometry3D geometryspherecombinatorial geometry
2008 PUMaC Algebra A10

Source:

9/24/2019
Find the sum of all integer values of nn such that the equation x(yz)2+y(zx)2+z(xy)2=n\frac{x}{(yz)^2} + \frac{y}{(zx)^2} + \frac{z}{(xy)^2} = n has a solution in positive integers.
algebra
2008 PUMaC Combinatorics A10

Source:

10/4/2019
In his youth, Professor John Horton Conway lived on a farm with 20092009 cows. Conway wishes to move the cows from the negative xx axis to the positive xx axis. The cows are initially lined up in order 1,2,...,20091, 2, . . . , 2009 on the negative xx axis. Conway can give two possible commands to the cows. One is the PUSH command, upon which the first cow from the negative xx axis moves to the lowest position on the positive yy axis. The other is the POP command, upon which the cow in the lowest position on the yy axis moves to the positive xx axis. For example, if Conway says PUSH POP 20092009 times, then the resulting permutation of cows is the same, 1,2,...,20091, 2, . . . , 2009. If Conway says PUSH 20092009 times followed by POP 20092009 times, the resulting permutation of cows is 2009,...,2,12009, . . . , 2, 1. How many output permutations are possible after Conway finishes moving all the cows from the negative xx axis to the positive xx axis?
combinatorics