[ugrads] Fun Math Problems??

Aleksey Zelenberg azelenberg at iit.edu
Wed Aug 29 22:24:37 CDT 2018

*Dear math students*

*This semester I will be organizing an informal weekly session to discuss
really interesting (and often quite difficult) math problems. If you're
interested in participating in competitions, or if you'd simply enjoy a
relaxing atmosphere to think about and discuss math with fellow students,
reply to this email or stop by my office (REC110) to say hi!*

*Here's a few to get you started:*

1. Evaluate [image: \int_1^2 \frac{\ln x}{2-2x-x^2}dx].

2. Suppose that [image: m, n, r] are positive integers such that
[image: 1+m+n\sqrt{3}=(2+\sqrt{3})^{2n-1}.]

Show that [image: m] must be a perfect square.

3. Players [image: 1, 2, 3, \ldots, n] are seated around a table and each
has a single penny. Player 1 passes a penny to Player 2, who then passes
two pennies to Player 3. Player 3 then passes one penny to Player 4, who
passes two pennies to Player 5, and so on, players alternately passing one
penny or two to the next player who still has some pennies. A player who
runs out of pennies drops out of the game and leaves the table. Find an
infinite set of numbers [image: n] for which some player ends up with
all [image:
n] pennies.

4. Determine, with proof, the largest number that is the product of
positive integers whose sum is 2016.

5. A calculator is broken so that the only keys that still work are the [image:
\sin], [image: \cos], [image: \tan], [image: \sin^{-1}], [image: \cos^{-1}],
[image: \tan^{-1}] buttons.  The display initially shows [image: 0].  Given
any positive rational number [image: q], show that pressing some finite
sequence of buttons will yield [image: q].  Assume that the calculator does
real number calculations with infinite precision. All functions are in
terms of radians.

6. For any sequence of real numbers [image: A = (a_1,a_2,\ldots)],
define [image:
\Delta A] to be the sequence [image: (a_2-a_1,a_3-a_2,\ldots)] whose [image:
n]th term is [image: a_{n+1}-a_n].  Suppose that all of the terms of the
sequence [image: \Delta(\Delta A)] are 1, and that [image: a_{19}=a_{94}=0].
Find [image: a_1].

Aleksey Zelenberg, PhD, Adjunct Professor
Department of Applied Mathematics
Illinois Institute of Technology
John T. Rettaliata Engineering Center, Room 110
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