MathDB

7

Part of 2004 Harvard-MIT Mathematics Tournament

Problems(6)

2004 Calculus #7

Source:

11/29/2011
Find the area of the region in the xyxy-plane satisfying x6−x2+y2≤0x^6-x^2+y^2 \le 0.
calculusgeometry
2004 Algebra #7

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12/26/2011
If xx, yy, kk are positive reals such that 3=k2(x2y2+y2x2)+k(xy+yx),3=k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+k\left(\dfrac{x}{y}+\dfrac{y}{x}\right), find the maximum possible value of kk.
function
2004 Combinatorics #7

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12/31/2011
We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?
geometry3D geometrytetrahedrongraph theory
2004 HMMT Geometry # 7

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3/3/2024
Yet another trapezoid ABCDABCD has ADAD parallel to BCBC. ACAC and BDBD intersect at PP. If [ADP]=[BCP]=1/2[ADP]=[BCP] = 1/2, find [ADP]/[ABCD][ADP]/[ABCD]. (Here the notation [P1...Pn][P_1 ...P_n] denotes the area of the polygon P1...PnP_1 ...P_n.)
geometry
2004 General, part 1 #7

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3/8/2024
Let ACEACE be a triangle with a point BB on segment ACAC and a point DD on segment CECE such that BDBD is parallel to AEAE. A point YY is chosen on segment AEAE, and segment CYCY is drawn. Let XX be the intersection of CYCY and BDBD. If CX=5CX = 5, XY=3XY = 3, what is the ratio of the area of trapezoid ABDEABDE to the area of triangle BCDBCD? https://cdn.artofproblemsolving.com/attachments/9/2/d6c723e54dbc4c88c12aa6a6ee91ae9e3ea581.png
geometry
2004 General, part 2 #7

Source:

3/8/2024
Farmer John is grazing his cows at the origin. There is a river that runs east to west 5050 feet north of the origin. The barn is 100100 feet to the south and 8080 feet to the east of the origin. Farmer John leads his cows to the river to take a swim, then the cows leave the river from the same place they entered and Farmer John leads them to the barn. He does this using the shortest path possible, and the total distance he travels is dd feet. Find the value of dd.
algebrageometry