MathDB

6

Part of 2004 Harvard-MIT Mathematics Tournament

Problems(6)

2004 Algebra #6

Source:

12/26/2011
Find all real solutions to x4+(2−x)4=34x^4+(2-x)^4=34.
2004 Calculus #6

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11/29/2011
For x>0x>0, let f(x)=xxf(x)=x^x. Find all values of xx for which f(x)=f′(x)f(x)=f'(x).
calculuslogarithms
2004 Combinatorics #6

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12/30/2011
A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Bill and Karl must serve together or not at all, and Alice and Jane refuse to serve with each other?
2004 General, part 1 #6

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3/8/2024
Find the ordered quadruple of digits (A,B,C,D)(A,B,C,D) with A>B>C>DA > B > C > D, such that \begin{tabular}{ccccc} & A & B & C & D \\ - & D & C & B & A \\ \hline = & B & D & A & C \\ \end{tabular}
number theory
2004 HMMT Geometry # 6

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3/3/2024
In trapezoid ABCDABCD shown, ADAD is parallel to BCBC, and AB=6AB = 6, BC=7BC = 7, CD=8CD = 8; AD=17AD = 17. If sides ABAB and CDCD are extended to meet at EE, find the resulting angle at EE (in degrees).
https://cdn.artofproblemsolving.com/attachments/c/c/987dde1d841a7b976b01f597511e5c1e828e5a.png
geometry
2004 General, part 2 #6

Source:

3/8/2024
aa and bb are positive integers. When written in binary, aa has 20042004 11's, and bb has 20052005 11's (not necessarily consecutive). What is the smallest number of 11's a+ba + b could possibly have?
number theory