MathDB

5

Part of 2000 Harvard-MIT Mathematics Tournament

Problems(7)

2000 Advanced Topics #5: 3-digit Sums of Cubes

Source:

6/21/2012
Find all 33-digit numbers which are the sums of the cubes of their digits.
geometry3D geometry
2000 Guts #5: Cube of sum of digits

Source:

10/12/2014
Find all natural numbers nn such that nn equals the cube of the sum of its digits.
geometry3D geometry
2000 Algebra #5: Price of Jacket

Source:

10/5/2014
A jacket was originally priced \textdollar 100. The price was reduced by 10%10\% three times and increased by 10%10\% four times in some order. To the nearest cent, what was the final price?
2000 HMMT RMT Geometry # 5

Source:

3/3/2024
Side AB‾=3\overline{AB} = 3. △ABF\vartriangle ABF is an equilateral triangle. Side DE‾=AB‾=AF‾=GE‾\overline{DE} =\overline{ AB} = \overline{AF} = \overline{GE}, ∠FED=60o\angle FED = 60^o, FG=1FG = 1. Calculate the area of ABCDEABCDE. https://cdn.artofproblemsolving.com/attachments/e/9/0ac1a88b4a83cdf3d562af0ce11b5ddbc5b8bc.png
geometry
2000 Oral #5: Triangle with integer coordinates

Source:

10/6/2014
Show that it is impossible to find a triangle in the plane with all integer coordinates such that the lengths of the sides are all odd.
analytic geometry
2000 HMMT RMT Team #5

Source:

3/8/2024
Given cos⁡(α+β)+sin(α−β)=0\cos (\alpha + \beta) + sin (\alpha - \beta) = 0, tan⁡β=12000\tan \beta =\frac{1}{2000}, find tan⁡α\tan \alpha.
trigonometryalgebra
2000 HMMT RMT General #5

Source:

3/8/2024
Find the interior angle between two sides of a regular octagon (degrees).
geometry