MathDB

1

Part of 1999 Harvard-MIT Mathematics Tournament

Problems(6)

1999 Advanced Topics #1: Illegible Receipts

Source:

6/21/2012
One of the receipts for a math tournament showed that 7272 identical trophies were purchased for $\$-99.999.9-, where the first and last digits were illegible. How much did each trophy cost?
1999 Algebra #1: Number of Solutions to Function

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6/21/2012
If a@b=a3−b3a−ba@b=\dfrac{a^3-b^3}{a-b}, for how many real values of aa does a@1=0a@1=0?
functionquadraticsalgebra
1999 Calculus #1: Twice Differentiable Function

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6/21/2012
Find all twice differentiable functions f(x)f(x) such that f′′(x)=0f^{\prime \prime}(x)=0, f(0)=19f(0)=19, and f(1)=99f(1)=99.
calculusfunctionintegration
1999 HMMT Geometry # 1

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3/3/2024
Two 10×2410 \times 24 rectangles are inscribed in a circle as shown. Find the shaded area. https://cdn.artofproblemsolving.com/attachments/1/7/c97fb0e6f45a52fa751777da6ebc519839e379.png
geometry
1999 HMMT Team #1

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3/8/2024
A combination lock has a 33 number combination, with each number an integer between 00 and 3939 inclusive. Call the numbers n1n_1, n2n_2, and n3n_3. If you know that n1n_1 and n3n_3 leave the same remainder when divided by 44, and n2n_2 and n1+2n_1 + 2 leave the same remainder when divided by 44, how many possible combinations are there?
number theory
1999 Oral #1: Infinite Bisections

Source:

10/20/2012
Start with an angle of 60∘60^\circ and bisect it, then bisect the lower 30∘30^\circ angle, then the upper 15∘15^\circ angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original 60∘60^\circ angle into two angles. Find the measure (degrees) of the smaller angle.
limit