MathDB

1.6

Part of 2022 CMIMC

Problems(3)

2022 Alg/NT Div 1 P6

Source:

2/28/2022
Find the probability such that when a polynomial in Z2027[x]\mathbb Z_{2027}[x] having degree at most 20262026 is chosen uniformly at random, x2027−x∣Pk(x)−x  ⟺  2021∣kx^{2027}-x | P^{k}(x) - x \iff 2021 | k (note that 20272027 is prime). Here Pk(x)P^k(x) denotes PP composed with itself kk times.
Proposed by Grant Yu
algebranumber theory
2022 Geo Div 1 P6

Source:

2/28/2022
Let Γ1\Gamma_1 and Γ2\Gamma_2 be two circles with radii r1r_1 and r2,r_2, respectively, where r1>r2.r_1>r_2. Suppose Γ1\Gamma_1 and Γ2\Gamma_2 intersect at two distinct points AA and B.B. A point CC is selected on ray AB→,\overrightarrow{AB}, past B,B, and the tangents to Γ1\Gamma_1 and Γ2\Gamma_2 from CC are marked as points PP and Q,Q, respectively. Suppose that Γ2\Gamma_2 passes through the center of Γ1\Gamma_1 and that points P,B,QP, B, Q are collinear in that order, with PB=3PB=3 and QB=2.QB=2. What is the length of AB?AB?
Proposed by Kyle Lee
geometry
2022 Combo Div 1 P6

Source:

2/28/2022
Barry has a standard die containing the numbers 1-6 on its faces.
He rolls the die continuously, keeping track of the sum of the numbers he has rolled so far, starting from 0. Let EnE_n be the expected number of time he needs to until his recorded sum is at least nn.
It turns out that there exist positive reals a,ba, b such that lim⁡n→∞En−(an+b)=0\lim_{n \rightarrow \infty} E_n - (an + b) = 0
Find (a,b)(a,b).
Proposed by Dilhan Salgado
combinatorics