MathDB

A7

Part of 2020 Princeton University Math Competition

Problems(4)

2020 PUMaC Algebra A7

Source:

1/1/2022
Suppose that pp is the unique monic polynomial of minimal degree such that its coefficients are rational numbers and one of its roots is sin⁡2π7+cos⁡4π7\sin \frac{2\pi}{7} + \cos \frac{4\pi}{7}. If p(1)=abp(1) = \frac{a}{b}, where a,ba, b are relatively prime integers, find ∣a+b∣|a + b|.
algebra
2020 PUMaC Combinatorics A7

Source:

1/1/2022
Let ff be defined as below for integers n≥0n \ge 0 and a0,a1,...a_0, a_1, ... such that ∑i≥0ai\sum_{i\ge 0}a_i is finite: f(n;a0,a1,...)={a2020, n=0∑i≥0aif(n−1;a0,...,ai−1,ai−1,ai+1+1,ai+2,...)/∑i≥0ain>0f(n; a_0, a_1, ...) = \begin{cases} a_{2020}, & \text{ $n = 0$} \\ \sum_{i\ge 0} a_i f(n-1;a_0,...,a_{i-1},a_i-1,a_{i+1}+1,a_{i+2},...)/ \sum_{i\ge 0}a_i & \text{$n > 0$} \end{cases}. Find the nearest integer to f(20202;2020,0,0,...)f(2020^2; 2020, 0, 0, ...).
combinatorics
2020 PUMaC Geometry A7

Source:

12/31/2021
Let ABCABC be a triangle with sides AB=34AB = 34, BC=15BC = 15, AC=35AC = 35 and let Γ\Gamma be the circle of smallest possible radius passing through AA tangent to BCBC. Let the second intersections of Γ\Gamma and sides ABAB, ACAC be the points X,YX, Y . Let the ray XYXY intersect the circumcircle of the triangle ABCABC at ZZ. If AZ=pqAZ =\frac{p}{q} for relatively prime integers pp and qq, find p+qp + q.
geometry
2020 PUMaC NT A7

Source:

1/1/2022
Let ϕ(x,u)\phi (x, u) be the smallest positive integer nn so that 2u2^u divides xn+95x^n + 95 if it exists, or 00 if no such positive integer exists. Determine∑i=0255ϕ(i,8) \sum_{i=0}^{255} \phi(i, 8).
number theory