MathDB

9

Part of 2019 CMIMC

Problems(4)

2019 A/NT9: 29 Divides a GCD

Source:

1/27/2019
Let a0=29a_0=29, b0=1b_0=1 and an+1=an+an−1⋅bn2019,bn+1=bnbn−1a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1} for n≥1n\geq 1. Determine the smallest positive integer kk for which 2929 divides gcd⁡(ak,bk−1)\gcd(a_k, b_k-1) whenever a1,b1a_1,b_1 are positive integers and 2929 does not divide b1b_1.
2019complex numbersnumber theorygreatest common divisor
2019 C/CS9: Mellon Transportation Company

Source:

1/27/2019
There are 15 cities, and there is a train line between each pair operated by either the Carnegie Rail Corporation or the Mellon Transportation Company. A tourist wants to visit exactly three cities by travelling in a loop, all by travelling on one line. What is the minimum number of such 3-city loops?
2019combinatorics
2019 G9: Locus of Circumcenters of Right Triangles

Source:

1/27/2019
Let ABCDABCD be a square of side length 11, and let P1,P2P_1, P_2 and P3P_3 be points on the perimeter such that ∠P1P2P3=90∘\angle P_1P_2P_3 = 90^\circ and P1,P2,P3P_1, P_2, P_3 lie on different sides of the square. As these points vary, the locus of the circumcenter of △P1P2P3\triangle P_1P_2P_3 is a region R\mathcal{R}; what is the area of R\mathcal{R}?
geometrycircumcircle
2019 T9: Multiplicative Bijection on the Naturals

Source:

1/27/2019
Let f:N→Nf:\mathbb{N}\to \mathbb{N} be a bijection satisfying f(ab)=f(a)f(b)f(ab)=f(a)f(b) for all a,b∈Na,b\in \mathbb{N}. Determine the minimum possible value of f(n)/nf(n)/n, taken over all possible ff and all n≤2019n\leq 2019.
2019team