MathDB

8

Part of 2022 JHMT HS

Problems(5)

2048th Roots of Unity

Source:

8/8/2024
Let ω\omega be a complex number satisfying ω2048=1\omega^{2048} = 1 and ω1024≠1\omega^{1024} \neq 1. Find the unique ordered pair of nonnegative integers (p,q)(p, q) satisfying 2p−2q=∑0≤m<n≤2047(ωm+ωn)2048. 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}.
algebracomplex numbers2022
Square and Right Triangle Tiles

Source:

8/8/2024
Find the number of ways to completely cover a 2×102 \times 10 rectangular grid of unit squares with 2×12 \times 1 rectangles RR and 2\sqrt{2} - 2\sqrt{2} - 22 triangles TT such that the following all hold:
[*] a placement of RR must have all of its sides parallel to the grid lines, [*] a placement of TT must have its longest side parallel to a grid line, [*] the tiles are non-overlapping, and [*] no tile extends outside the boundary of the grid.
(The figure below shows an example of such a tiling; consider tilings that differ by reflections to be distinct.) [asy] unitsize(1cm); fill((0,0)--(10,0)--(10,2)--(0,2)--cycle, grey); draw((0,0)--(10,0)--(10,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((1,2)--(3,0)); draw((1,0)--(3,2)); draw((3,2)--(5,0)); draw((3,0)--(5,2)); draw((2,1)--(4,1)); draw((5,0)--(5,2)); draw((7,0)--(7,2)); draw((5,1)--(7,1)); draw((8,0)--(8,2)); draw((8,0)--(10,2)); draw((8,2)--(10,0)); [/asy]
combinatorics2022
Intersection of a Circumcircle and an Incircle

Source:

8/8/2024
In equilateral △ABC\triangle ABC, point DD lies on BC‾\overline{BC} such that the radius of the circumcircle Γ1\Gamma_1 of △ACD\triangle ACD is 77 and the radius of the incircle Γ2\Gamma_2 of △ABD\triangle{ABD} is 22. Suppose that Γ1\Gamma_1 and Γ2\Gamma_2 intersect at points XX and YY. Find XYXY.
geometrycircumcircle2022
Points Yielding Minimum Perimeter of 25

Source:

8/9/2024
Let P=(−4,0)P = (-4, 0) and Q=(4,0)Q = (4, 0) be two points on the xx-axis of the Cartesian coordinate plane, and let XX and YY be points on the xx-axis and yy-axis, respectively, such that over all ZZ on line XY↔\overleftrightarrow{XY}, the perimeter of △ZPQ\triangle ZPQ has a minimum value of 2525. What is the smallest possible value of XY2XY^2?
perimeteroptimizationcalculus2022
Number of Ant Paths

Source:

8/8/2024
An ant is walking on a sidewalk and discovers 1212 sidewalk panels with leaves inscribed in them, as shown below. Find the number of ways in which the ant can traverse from point AA to point BB if it can only move
[*] up, down, or right (along the border of a sidewalk panel), or [*] up-right (along one of two margin halves of a leaf)
and cannot visit any border or margin half more than once (an example path is highlighted in red). [asy] unitsize(1cm); int r = 4; int c = 5;
for (int i = 0; i < r; ++i) { for (int j = 0; j < c; ++j) { pair A = (j,i); } } for (int i = 0; i < r; ++i) { for (int j = 0; j < c; ++j) { if (j != c-1) { draw((j,i)--(j+1,i)); } if (i != r-1) { draw((j,i)--(j,i+1)); } } }
for (int i = 1; i < r+1; ++i) { for (int j = 0; j < c-2; ++j) { fill(arc((i,j),1,90,180)--cycle,deepgreen); fill(arc((i-1,j+1),1,270,360)--cycle,deepgreen); draw((i-1,j)--(i,j+1), heavygreen+linewidth(0.5)); draw((i-2/3,j+1/3)--(i-2/3,j+1/3+0.1), heavygreen); draw((i-1/3,j+2/3)--(i-1/3,j+2/3+0.1), heavygreen); draw((i-2/3,j+1/3)--(i-2/3+0.1,j+1/3), heavygreen); draw((i-1/3,j+2/3)--(i-1/3+0.1,j+2/3), heavygreen); draw(arc((i,j),1,90,180)); draw(arc((i-1,j+1),1,270,360)); } } draw((0,3)--(0,1), red+linewidth(1.5)); draw((0,3)--(0,1), red+linewidth(1.5)); draw(arc((1,1),1,90,180), red+linewidth(1.5)); draw((1,2)--(1,1)--(2,1), red+linewidth(1.5)); draw(arc((2,2),1,270,360), red+linewidth(1.5)); draw(arc((4,2),1,90,180), red+linewidth(1.5)); draw((4,3)--(4,0), red+linewidth(1.5)); dot((0,3)); dot((4,0)); label("AA", (0,3), NW); label("BB", (4,0), SE); [/asy]
combinatorics2022