MathDB

1

Part of 2004 Harvard-MIT Mathematics Tournament

Problems(6)

2004 Algebra #1

Source:

12/26/2011
How many ordered pairs of integers (a,b)(a,b) satisfy all of the following inequalities?
\begin{eqnarray*} a^2 + b^2 &<& 16 \\ a^2 + b^2 &<& 8a \\ a^2 + b^2 &<& 8b \end{eqnarray*}
inequalitiesBad Latex
2004 Calculus #1

Source:

11/29/2011
Let f(x)=sin⁡(sin⁡(x))f(x)=\sin(\sin(x)). Evaluate lim⁡h→0f(x+h)−f(h)x \lim_{h \to 0} \dfrac {f(x+h)-f(h)}{x} at x=πx=\pi.
calculustrigonometrylimit
2004 Combinatorics #1

Source:

12/30/2011
There are 1000 rooms in a row along a long corridor. Initially the first room contains 1000 people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?
2004 HMMT Geometry # 1

Source:

3/3/2024
In trapezoid ABCDABCD, ADAD is parallel to BCBC. ∠A=∠D=45o\angle A = \angle D = 45^o, while ∠B=∠C=135o\angle B = \angle C = 135^o. If AB=6AB = 6 and the area of ABCDABCD is 3030, find BCBC. https://cdn.artofproblemsolving.com/attachments/0/8/d667522259c773435bc53f5988831aceaef7b7.png
geometry
2004 General, part 1 #1

Source:

3/8/2024
There are 10001000 rooms in a row along a long corridor. Initially the first room contains 10001000 people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?
combinatorics
2004 General, part 2 #1

Source:

3/8/2024
Find the largest number nn such that (2004!)!(2004!)! is divisible by ((n!)!)!((n!)!)!.
number theory