MathDB

10

Part of 2022 JHMT HS

Problems(5)

GCD Double Infinite Sum

Source:

8/8/2024
Compute the exact value of ∑a=1∞∑b=1∞gcd⁡(a,b)(a+b)3. \sum_{a=1}^{\infty}\sum_{b=1}^{\infty} \frac{\gcd(a, b)}{(a + b)^3}. If necessary, you may express your answer in terms of the Riemann zeta function, Z(s)=∑n=1∞1nsZ(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, for integers s≥2s \geq 2.
number theorygreatest common divisor2022
Rectangle Pixelations

Source:

8/8/2024
Let RR be the rectangle in the coordinate plane with corners (0,0)(0, 0), (20,0)(20, 0), (20,22)(20, 22), and (0,22)(0, 22), and partition RR into a 20×2220\times 22 grid of unit squares. For a given line in the coordinate plane, let its pixelation be the set of grid squares in RR that contain part of the line in their interior. If PP is a point chosen uniformly at random in RR, then compute the expected number of sets of grid squares that are pixelations of some line through PP.
rectangleexpected value2022
Circle Through Incenter and Orthocenter

Source:

8/8/2024
In △JMT\triangle JMT, JM=410JM=410, JT=49JT=49, and ∠MJT>90∘\angle{MJT}>90^\circ. Let II and HH be the incenter and orthocenter of △JMT\triangle JMT, respectively. The circumcircle of △JIH\triangle JIH intersects JT↔\overleftrightarrow{JT} at a point P≠JP\neq J, and IH=HPIH=HP. Find MTMT.
geometryincentercircumcircle2022
Maximum Value of Infinite Sum

Source:

8/9/2024
The maximum value of 2∑n=1∞sin⁡(nθ)44n 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} over all real numbers θ\theta can be expressed as a common fraction pq\tfrac{p}{q}. Compute p+qp + q.
calculustrigonometry2022
Lattice Point Connections

Source:

8/8/2024
Let Λ\Lambda denote the set of points (x,y)(x,y) in 2D space with integer coordinates such that 0≤x≤40\leq x\leq 4 and 0≤y≤20\leq y\leq 2. That is, Λ={(x,y)∈Z2:0≤x≤4, 0≤y≤2}. \Lambda=\{ (x,y) \in \mathbb{Z}^2: 0\leq x\leq 4, \ 0\leq y\leq 2 \}. Find the number of ways to connect points of Λ\Lambda with segments of length 2\sqrt{2} or 5\sqrt{5} such that the interior of any unit square with vertices in Λ\Lambda contains part of exactly one segment; an example is shown below (connections that differ by reflections are distinct). [asy] unitsize(1cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((4,2)); draw((0,0)--(1,1)); draw((0,2)--(2,1)); draw((1,1)--(2,0)); draw((2,0)--(3,2)); draw((3,1)--(4,2)); draw((3,0)--(4,1)); [/asy]
combinatorics2022