MathDB

B4

Part of 2011 Princeton University Math Competition

Problems(3)

2011 PUMaC Algebra B4

Source:

9/24/2019
Let ff be an invertible function defined on the complex numbers such that z2=f(z+f(iz+f(−z+f(−iz+f(z+…)))))z^2 = f(z + f(iz + f(-z + f(-iz + f(z + \ldots))))) for all complex numbers zz. Suppose z0≠0z_0 \neq 0 satisfies f(z0)=z0f(z_0) = z_0. Find 1/z01/z_0. (Note: an invertible function is one that has an inverse).
algebra
2011 PUMaC Combinatorics B4

Source:

9/24/2019
A function f:{1,2,…,n}→{1,…,m}f:\{1,2, \ldots, n\} \to \{1, \ldots, m\} is multiplication-preserving if f(i)f(j)=f(ij)f(i)f(j) = f(ij) for all 1≤i≤j≤ij≤n1 \le i \le j \le ij \le n, and injective if f(i)=f(j)f(i) = f(j) only when i=ji = j. For n=9,m=88n = 9, m = 88, the number of injective, multiplication-preserving functions is NN. Find the sum of the prime factors of NN, including multiplicity. (For example, if N=12N = 12, the answer would be 2+2+3=72 + 2 + 3 = 7.)
combinatorics
2011 PUMaC Geometry B4

Source:

9/24/2019
Let ω\omega be a circle of radius 66 with center OO. Let ABAB be a chord of ω\omega having length 55. For any real constant cc, consider the locus L(c)\mathcal{L}(c) of all points PP such that PA2−PB2=cPA^2 - PB^2 = c. Find the largest value of cc for which the intersection of L(c)\mathcal{L}(c) and ω\omega consists of just one point.
geometry