MathDB

3

Part of 2004 Harvard-MIT Mathematics Tournament

Problems(6)

2004 Algebra #3

Source:

12/26/2011
Compute ⌊200532003⋅2004−200332004⋅2005⌋ \left\lfloor \dfrac {2005^3}{2003 \cdot 2004} - \dfrac {2003^3}{2004 \cdot 2005} \right\rfloor
floor function
2004 Calculus #3

Source:

11/29/2011
Find lim⁡x→∞(x3+x23−x3−x23). \lim_{x \to \infty} \left( \sqrt[3]{x^3 + x^2}-\sqrt[3]{x^3-x^2} \right).
calculuslimit
2004 HMMT Geometry # 3

Source:

3/3/2024
A swimming pool is in the shape of a circle with diameter 6060 ft. The depth varies linearly along the east-west direction from 33 ft at the shallow end in the east to 1515 ft at the diving end in the west (this is so that divers look impressive against the sunset) but does not vary at all along the north-south direction. What is the volume of the pool, in ft3^3?
geometry
2004 Combinatorics #3

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12/30/2011
A class of 1010 students took a math test. Each problem was solved by exactly 77 of the students. If the first nine students each solved 44 problems, how many problems did the tenth student solve?
2004 General, part 1 #3

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3/8/2024
Suppose ff is a function that assigns to each real number xx a value f(x)f(x), and suppose the equation f(x1+x2+x3+x4+x5)=f(x1)+f(x2)+f(x3)+f(x4)+f(x5)−8f(x_1 + x_2 + x_3 + x_4 + x_5) = f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5) - 8 holds for all real numbers x1,x2,x3,x4,x5x_1, x_2,x_3, x_4, x_5. What is f(0)f(0)?
algebra
2004 General, part 2 #3

Source:

3/8/2024
How many ordered pairs of integers (a,b)(a, b) satisfy all of the following inequalities? a2+b2<16a^2 + b^2 < 16 a2+b2<8aa^2 + b^2 < 8a a2+b2<8ba^2 + b^2 < 8b
number theoryalgebra