MathDB

2

Part of 2000 Harvard-MIT Mathematics Tournament

Problems(7)

2000 Advanced Topics #2: Simplification of Complex Numbers

Source:

6/21/2012
Simplify (−1+i32)6+(−1−i32)6\left(\dfrac{-1+i\sqrt{3}}{2}\right)^6+\left(\dfrac{-1-i\sqrt{3}}{2}\right)^6 to the form a+bia+bi.
2000 Guts #2: Infinite geometric series

Source:

10/12/2014
If X=1+x+x2+x3+⋯X=1+x+x^2+x^3+\cdots and Y=1+y+y2+y3+⋯Y=1+y+y^2+y^3+\cdots, what is 1+xy+x2y2+x3y3+⋯1+xy+x^2y^2+x^3y^3+\cdots in terms of XX and YY only?
geometric series
2000 Algebra #2: Factorization

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10/5/2014
Evaluate 20003−1999⋅20002−19992⋅2000+199932000^3-1999\cdot 2000^2-1999^2\cdot 2000+1999^3
2000 HMMT RMT Geometry # 2

Source:

3/3/2024
In a triangle the sum of squares of the sides is 9696. What is the maximum possible value of the sum of the medians?
geometry
2000 Oral #2: Solutions to 10th Degree equation

Source:

10/6/2014
How many positive solutions are there to x10+7x9+14x8+1729x7−1379x6=0x^{10}+7x^9+14x^8+1729x^7-1379x^6=0? How many positive integer solutions?
2000 HMMT RMT Team #2

Source:

3/8/2024
The price of a gold ring in a certain universe is proportional to the square of its purity and the cube of its diameter. The purity is inversely proportional to the square of the depth of the gold mine and directly proportional to the square of the price, while the diameter is determined so that it is proportional to the cube root of the price and also directly proportional to the depth of the mine. How does the price vary solely in terms of the depth of the gold mine?
algebra
2000 HMMT RMT General #2

Source:

3/8/2024
The temperatures foFf^o F and coCc^o C are equal when f=95c+32f = \frac95 c + 32. What temperature is the same in both oF^o F and oC^o C?
algebra