MathDB

7

Part of 2000 Harvard-MIT Mathematics Tournament

Problems(7)

2000 Guts #7: Expected value of monomial

Source:

10/12/2014
Suppose you are given a fair coin and a sheet of paper with the polynomial xmx^m written on it. Now for each toss of the coin, if heads show up, you must erase the polynomial xrx^r (where rr is going to change with time - initially it is mm) written on the paper and replace it with xr−1x^{r-1}. If tails show up, replace it with xr+1x^{r+1}. What is the expected value of the polynomial I get after mm such tosses? (Note: this is a different concept from the most probable value)
algebrapolynomialprobabilityexpected value
2000 Advanced Topics #7: Equations in Positive Integers

Source:

6/21/2012
Assume that a,b,c,da,b,c,d are positive integers, and ac=bd=34\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{3}{4}, a2+c2−b2+d2=15\sqrt{a^2+c^2}-\sqrt{b^2+d^2}=15. Find ac+bd−ad−bcac+bd-ad-bc.
2000 Algebra #7: Multiplicatively Perfect Numbers

Source:

10/5/2014
A number nn is called multiplicatively perfect if the product of all the positive divisors of nn is n2n^2. Determine the number of positive multiplicatively perfect numbers less than 100100.
geometry3D geometry
2000 HMMT RMT Geometry # 7

Source:

3/3/2024
Let ABCABC be a triangle inscribed in the ellipse x24+y29=1\frac{x^2}{4} +\frac{y^2}{9}= 1. If its centroid is the origin (0,0)(0,0), find its area.
conicsellipsegeometry
2000 Oral #7: Polyhedrons of volume 1

Source:

10/6/2014
A regular tetrahedron of volume 11 is filled with water of total volume 716\frac{7}{16}. Is it possible that the center of the tetrahedron lies on the surface of the water? How about in a cube of volume 11?
geometry3D geometrytetrahedron
2000 HMMT RMT Team #7

Source:

3/8/2024
87128712 is an integral multiple of its reversal, 21782178, as 8712=4∗21788712=4 * 2178. Find another 44-digit number which is a non-trivial integral multiple of its reversal.
number theory
2000 HMMT RMT General #7

Source:

3/8/2024
Find [19992000][ \sqrt{19992000}] where [a][a] is the greatest integer less than or equal to xx.
algebra