MathDB

4

Part of 2000 Harvard-MIT Mathematics Tournament

Problems(7)

2000 Advanced Topics #4: Sum Divisible by 3

Source:

6/21/2012
Five positive integers from 11 to 1515 are chosen without replacement. What is the probability that their sum is divisible by 33?
probability
2000 Guts #4: Coordinates of triangle verticies

Source:

10/12/2014
Let ABCABC be a triangle and HH be its orthocenter. If it is given that BB is (0,0)(0,0), CC is (1,2)(1,2) and HH is (5,0)(5,0), find AA.
analytic geometrygeometrycircumcircle
2000 Algebra #4: Multiplications using Powers of x

Source:

10/5/2014
What is the fewest number of multiplications required to reach x2000x^{2000} from xx, using only previously generated powers of xx? For example x→x2→x4→x8→x16→x32→x64→x128→x256→x512→x1024→x1536→x1792→x1920→x1984→x2000x\rightarrow x^{2}\rightarrow x^{4}\rightarrow x^{8}\rightarrow x^{16}\rightarrow x^{32}\rightarrow x^{64}\rightarrow x^{128}\rightarrow x^{256}\rightarrow x^{512}\rightarrow x^{1024}\rightarrow x^{1536}\rightarrow x^{1792}\rightarrow x^{1920}\rightarrow x^{1984}\rightarrow x^{2000} uses 1515 multiplications.
2000 HMMT RMT Geometry # 4

Source:

3/3/2024
Find the total area of the non-triangle regions in the figure below (the shaded area). https://cdn.artofproblemsolving.com/attachments/1/3/cf85eb41aacc125bcd3e42d5f8c512b1e9f353.png
geometry
2000 Oral #4: Numbers on a chessboard

Source:

10/6/2014
On an nn by nn chessboard, numbers are written on each square so that the number in a square is the average of the numbers on the adjacent squares. Show that all the numbers are the same.
2000 HMMT RMT Team #4

Source:

3/8/2024
All arrangements of letters VNNWHTAAIEVNNWHTAAIE are listed in lexicographic (dictionary) order. If AAEHINNTVWAAEHINNTVW is the first entry, what entry number is VANNAWHITEVANNAWHITE?
combinatorics
2000 HMMT RMT General #4

Source:

3/8/2024
Tickets for the football game are $10\$10 for students and $15\$15 for non-students. If 30003000 fans attend and pay $36250\$36250, how many students went?
algebra