Date
|
Problem
|
Solution
|
M, 10/13
|
Find all
solutions $(x,y)$ of the equation $x^y=y^x$ for real numbers
$x,y>0$.
|
Solution |
F, 10/17
|
Evaluate
\int \frac{5}{16+9\cos^2 x}\,dx.
|
Solution |
M, 10/20
|
What is the next
term of the sequence?
1, 1, 1, 3, 1, 4, 1, 1, 3, 6, 1,
2, 3, 1, 4, 8, 1, 3, 3, 2, 4, 1, 6, ?
|
Solution |
M, 10/27
|
Take $Q$ to be a
closed unit square (i.e. each side has length 1)
together with its interior. Take five distinct points at random in $Q$.
Show that some two points are no more than $\sqrt{2}/2$ units apart.
|
Solution |
W, 10/29
|
The winning team
of the World
Series must win four games out of seven. Assuming that the
teams
are equally matched, find the probability that the Series lasts
- exactly four games,
- exactly five games,
- exactly six games, and
- exactly seven games.
|
Solution |
F, 10/31
|
A
circle of radius 1 touches the graph of $y=|2x|$ in two places.
Find the area of the region that lies between the two curves. |
Solution |
M, 11/3
|
Suppose that two
triangles
have a common angle. Show that the sums of the sines of the
angles will be larger in that triangle where the differences of the two
remaining angles is smaller. Hint:
Think of the product-to-sum formula for sines. |
Solution |
W, 11/5
|
What is the
greatest common divisor of the set of numbers
\{16^n+10n-1\mid n=1,2,3,\dots\}?
|
Solution |
F, 11/7
| The arms of an equal-arm
balance are never exactly the same length. To eliminate the error
due to this unavoidable contingency, it is proposed that objects be
weighed twice, once on each side of the balance, and the average taken.
Show,
however, that the average of the two weighings never gives the correct
weight unless the arms are exactly the same length, and determine how
the weighings ought to be handled in order to find the correct weight. | Solution |
M, 11/10
| Sketch the graph of the curve $|3x^2+y^2-12|=|x^2-y^2+4|$. | Solution |
W, 11/12
| Let $A$ and $B$ be square matrices. Prove that if $AB$ is nonsingular, then both $A$ and $B$ are nonsingular. | Solution |
F, 11/14
| Find the sum of the series $\displaystyle\sum_{n=2}^\infty \ln\left(1-\frac1{n^2}\right)$. | Solution |
M, 11/17 (last one!)
| Show that $4n^3+6n^2+4n+1$ is composite for $n=1, 2, 3, \dots$. | Solution |