MathDB

2

Part of 2023 Stanford Mathematics Tournament

Problems(6)

SMT 2023 Algebra #2

Source:

5/3/2023
Compute the number of values of xx in the interval [−11π,−2π][-11\pi,-2\pi] that satisfy 5cos⁡(x)+45sin⁡(x)+3=0\tfrac{5\cos(x)+4}{5\sin(x)+3}=0.
SMT 2023 Algebra Tiebreaker #2

Source:

5/3/2023
f(x)f(x) is a nonconstant polynomial. Given that f(f(x))+f(x)=f(x)2f(f(x)) + f(x) = f(x)^2, compute f(3)f(3).
SMT 2023 Discrete #2

Source:

5/3/2023
A 3×33\times3 grid is to be painted with three colors (red, green, and blue) such that
[*] no two squares that share an edge are the same color and [*] no two corner squares on the same edge of the grid have the same color.
As an example, the upper-left and bottom-left squares cannot both be red, as that would violate condition (ii). In how many ways can this be done? (Rotations and reflections are considered distinct colorings.)
SMT 2023 Discrete Tiebreaker #2

Source:

5/3/2023
Every cell in a 5×55\times5 grid of paper is to be painted either red or white with equal probability. An edge of the paper is said to have a "tree" if the set of cells depicted in the diagram below are all painted red when the paper is rotated so that the edge lies at the bottom. Given that at least one edge of the paper has a tree, what is the expected number of edges that have a tree? https://cdn.artofproblemsolving.com/attachments/1/2/f81d8da53d7bc6819fc1dfe4acb9567d545856.png
SMT 2023 Geometry #2

Source:

8/9/2023
Triangle △ABC\vartriangle ABC has side lengths AB=39AB = 39, BC=16BC = 16, and CA=25CA = 25. What is the volume of the solid formed by rotating △ABC\vartriangle ABC about line BCBC?
geometry3D geometry
SMT 2023 Geometry Tiebreaker #2

Source:

8/9/2023
Triangle △ABC\vartriangle ABC has side lengths AB=3AB = 3, AC=2AC = 2 and angle ∠CBA=30o\angle CBA = 30^o. Let the possible lengths of BCBC be ℓ1\ell_1 and ℓ2\ell_2, where ℓ2>ℓ1\ell_2 > \ell_1. Compute ℓ2ℓ1\frac{\ell_2}{\ell_1} .
geometry