MathDB

8

Part of 2000 Harvard-MIT Mathematics Tournament

Problems(7)

2000 Guts #8: Ages

Source:

10/12/2014
Johny's father tells him: "I am twice as old as you will be seven years from the time I was thrice as old as you were". What is Johny's age?
2000 Advanced Topics #8: Non-Isomorphic Graphs

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6/21/2012
How many non-isomorphic graphs with 99 vertices, with each vertex connected to exactly 66 other vertices, are there? (Two graphs are isomorphic if one can relabel the vertices of one graph to make all edges be exactly the same.)
2000 Algebra #8: Ages of Daughter

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10/5/2014
A man has three daughters. The product of their ages is 168168, and he remembers that the sum of their ages is the number of trees in his yard. He counts the trees but cannot determine any of their ages. What are all possible ages of his oldest daughter?
2000 Oral #9: Three dimensional vectors

Source:

10/6/2014
Let v1⃗,v2⃗,v3⃗,v4⃗\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4} and v5⃗\vec{v_5} be vectors in three dimensions. Show that for some i,ji,j in 1,2,3,4,51,2,3,4,5, vi⃗⋅vj⃗≥0\vec{v_i}\cdot \vec{v_j}\ge 0.
vectortrigonometrygeometry3D geometryspheregeometric transformationrotation
2000 HMMT RMT Geometry # 8

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3/3/2024
A sphere is inscribed inside a pyramid with a square as a base whose height is 152\frac{\sqrt{15}}{2} times the length of one edge of the base. A cube is inscribed inside the sphere. What is the ratio of the volume of the pyramid to the volume of the cube?
geometry
2000 HMMT RMT Team #8

Source:

3/8/2024
A woman has $1.58\$1.58 in pennies, nickels, dimes, quarters, half-dollars and silver dollars. If she has a different number of coins of each denomination, how many coins does she have?
number theory
2000 HMMT RMT General #8

Source:

3/8/2024
Bobo the clown was juggling his spherical cows again when he realized that when he drops a cow is related to how many cows he started off juggling. If he juggles 11, he drops it after 6464 seconds. When juggling 22, he drops one after 5555 seconds, and the other 5555 seconds later. In fact, he was able to create the following table:
https://cdn.artofproblemsolving.com/attachments/1/0/554a9bace83b4b3595c6012dfdb42409465829.png
He can only juggle up to 2222 cows. To juggle the cows the longest, what number of cows should he start off juggling? How long (in minutes) can he juggle for?
algebra