MathDB

7

Part of 1999 Harvard-MIT Mathematics Tournament

Problems(6)

1999 Advanced Topics #7: Find a Quadratic

Source:

6/21/2012
Find an ordered pair (a,b)(a,b) of real numbers for which x2+ax+bx^2+ax+b has a non-real root whose cube is 343343.
quadraticsgeometry3D geometry
Time to demolish a building [HMMT Problem]

Source:

11/10/2011
Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in 33, Anne and Carl in 55. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?
HMMTalgebrasystem of equations
1999 Calculus #7: Right Triangle Inside Semicircle

Source:

6/21/2012
If a right triangle is drawn in a semicircle of radius 1/21/2 with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area?
calculusgeometryderivative
1999 HMMT Team #7

Source:

3/8/2024
Let 11−x−x2−x3=∑i=0∞anxn\frac{1}{1-x-x^2-x^3} =\sum^{\infty}_{i=0} a_nx^n, for what positive integers nn does an−1=n2a_{n-1} = n^2?
algebra
1999 HMMT Geometry # 7

Source:

3/3/2024
A dart is thrown at a square dartboard of side length 22 so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance 11 of a corner?
geometry
1999 Oral #7: Powers on Factorials

Source:

10/20/2012
Evaluate ∑n=1∞n5n!.\sum_{n=1}^\infty \dfrac{n^5}{n!}.
factorial