MathDB

10

Part of 2019 AIME Problems

Problems(2)

2019 Roots

Source: 2019 AIME I 10

3/14/2019
For distinct complex numbers z1,z2,…,z673z_1,z_2,\dots,z_{673}, the polynomial (x−z1)3(x−z2)3⋯(x−z673)3 (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 can be expressed as x2019+20x2018+19x2017+g(x)x^{2019} + 20x^{2018} + 19x^{2017}+g(x), where g(x)g(x) is a polynomial with complex coefficients and with degree at most 20162016. The value of ∣∑1≤j<k≤673zjzk∣ \left| \sum_{1 \le j <k \le 673} z_jz_k \right| can be expressed in the form mn\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
AMCAIMEAIME I2019 AMC2019 AIME Icomplex numbers
0, 1, 2, 3, 0, ...

Source: 2019 AIME ii #10

3/22/2019
There is a unique angle θ\theta between 0∘0^{\circ} and 90∘90^{\circ} such that for nonnegative integers nn, the value of tan⁡(2nθ)\tan{\left(2^{n}\theta\right)} is positive when nn is a multiple of 33, and negative otherwise. The degree measure of θ\theta is pq\tfrac{p}{q}, where pp and qq are relatively prime integers. Find p+qp+q.
AMCAIMEAIME II2019 AIME II2019 AMC