MathDB

8

Part of 2004 Harvard-MIT Mathematics Tournament

Problems(6)

2004 Algebra #8

Source:

12/26/2011
Let xx be a real number such that x3+4x=8x^3+4x=8. Determine the value of x7+64x2x^7+64x^2.
2004 Calculus #8

Source:

11/29/2011
If xx and yy are real numbers with (x+y)4=x−y(x+y)^4=x-y, what is the maximum possible value of yy?
calculusfunctiongeometrygeometric transformationrotationanalytic geometrygraphing lines
2004 Combinatorics #8

Source:

12/31/2011
Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?
probability
2004 HMMT Geometry # 8

Source:

3/3/2024
A triangle has side lengths 1818, 2424, and 3030. Find the area of the triangle whose vertices are the incenter, circumcenter, and centroid of the original triangle.
geometry
2004 General, part 1 #8

Source:

3/8/2024
You have a 10×1010\times 10 grid of squares. You write a number in each square as follows: you write 11, 22, 33 ,... ... ,1010 from left to right across the top row, then 1111, 1212, ......, 2020 across the second row, and so on, ending with a 100100 in the bottom right square. You then write a second number in each square, writing 11, 22, 33 ,... ... ,1010 in the first column (from top to bottom), then 1111, 1212, ......, 2020 in the second column, and so forth. When this process is finished, how many squares will have the property that their two numbers sum to 101101?
combinatorics
2004 General, part 2 #8

Source:

3/8/2024
A freight train leaves the town of Jenkinsville at 1:001:00 PM traveling due east at constant speed. Jim, a hobo, sneaks onto the train and falls asleep. At the same time, Julie leaves Jenkinsville on her bicycle, traveling along a straight road in a northeasterly direction (but not due northeast) at 1010 miles per hour. At 1:121:12 PM, Jim rolls over in his sleep and falls from the train onto the side of the tracks. He wakes up and immediately begins walking at 3:53:5 miles per hour directly towards the road on which Julie is riding. Jim reaches the road at 2:122:12 PM, just as Julie is riding by. What is the speed of the train in miles per hour?
algebra