Interesting Problems I've Run Across.

This is a collection of interesting problems I've seen. It is organized in roughly chronological order of exposure to the problem. I have provided references when I have been aware of them. Most of these are problems that I solved at one point in time, and I ought to be able to reproduce a solution to any one of them.

1. Is there a "good" formula for *sin(10 ^{o})*? (That is,
one involving only square roots, cube roots, etc. of real numbers).

2. Let *S = { [n √ 2 ] : n ≥ 1 }* and let
*T = { [n (2 + √ 2) ] : n ≥ 1 }*. Prove that
every positive integer is in exactly one of *S* or *T*.
(Here *[x]* denotes the largest integer less than or equal to *x*).

3. A calculator is broken so that the only keys that work
are *sin*, *cos*, *tan*, *sin ^{-1}*,

4. Prove that if *a*, *b*, and *c* are all odd
integers, then the roots of *ax ^{2} + bx + c = 0* are
irrational. (Problem 57, USSR Olympiad problem book).

5. Find the smallest positive integer that can be written as the sum of two triangular numbers in 48 different ways. (Problem 665 from the November 1999 issue of the College Mathematics Journal. My solution was published in the November 2000 issue).

6. An alien appears to two mathematicians, Sam and Polly. The alien
says, "I'm thinking of two numbers *X* and *Y* with *3
≤ X ≤ Y ≤ 97*. I will tell their sum to Sam and their
product to Polly." The alien does this and then disappears. The
following conversation occurs.

Sam: You don't know what *X*
and *Y* are.

Polly: That's true, but now I do.

Sam: And
now I do too.

Find *X* and *Y*. (This was an extra
credit problem given in my discrete math class).

7. Suppose that a *P* is a convex polyhedron with the following
two properties: (i) exactly three edges meet at every vertex, (ii)
every face is a pentagon or a hexagon. Prove that the number of
pentagons equals 12.

8. Suppose that *F* is a *C ^{1}* vector field
on

9. Consider a paper punch that can be centered at any point of the plane that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point? (Problem A4 from the 1990 Putnam exam).

10. Let *S* be the set of positive integers that, when written in
base 10, does not contain the digit 9. Show that the sum of *1/n*
over all *n ∈ S* converges and is
less than 80. (Problem 157, USSR Olympiad Problem Book).

11. Let *A* be the sum of the digits of the
number *4444 ^{4444}*, and

12. Suppose that *a* and *b* are integers with *ab ≠
-1*. Prove that if
*1 + ab* divides *a ^{2} + b^{2}*
then

13. Let *V _{n}* be the volume of the

14. Give an elementary proof that there are infinitely many primes congruent to 9 mod 10.

15. Let *a _{1} = 1*,

Compute

16. Suppose that *S* is a set of an odd number of real
numbers. Suppose that for all *x ∈ S*,
*S - {x}* can be divided into two sets of equal size with
equal sum. Prove that all elements of *S* are equal. (Homework
problem from Conjecture and Proof course. Also, Problem 11002 from the
American Mathematical Monthly, March 2003).

17. Is there a bounded subset *H* of *R ^{2}* that is
isometric to a proper subset of itself? (Homework problem from
Conjecture and Proof).

18. If *0 < ε < 1*, prove that there
is an open dense subset of *[0,1]* with
measure *ε*. (Problem 2.7 from Rudin's
"Real and Complex Analysis").

19. For *n ≥ 1*,
let *a _{n}* be the number of solutions
to

20. Prove that for all *n ≥ 1*,
*1 + 5 ^{n} + 5^{2n} + 5^{3n} +
5^{4n}* is composite. (Problem 10947 from the May 2002
issue of the American Mathematical Monthly).

21. If *G* is a group of order 36, prove that *G* has a
normal Sylow 2-subgroup or a normal Sylow 3-subgroup. (Problem 6.2.18
from Dummit and Foote's "Abstract Algebra").

22. Classify finite groups *G* with the property that *|G|*
is the smallest positive integer *n* for which *G* is
isomorphic to a subgroup of *S _{n}*.

23. Prove that if *G* is a finite simple group
and *|G| ≤ 1000*, then *|G| =
60, 168, 360, 504* or *660*. (Suggested as "a good project for
a student of group theory" by Marty Isaacs in "Abstract Algebra, A
Graduate Course", pg. 38).

24. The pair *(17/21,37/21)* is a solution of *x^3 + y^3 =
6* with *x* and *y* positive rational numbers. Find a
second solution with *x, y* both positive rationals, or prove
that none exists. (Homework problem from elliptic curves. Also, part
of the London Sunday Telegraph's New Year's Quiz for 1995).

25. Prove that if *p* is a prime congruent to 1 mod 4 then there
are integers *x* and *y* so that
*x ^{2} - py^{2} = -4*.

26. Prove that if *p* is a prime that is congruent to 1 mod 4, then
the class number of
*Q(√ -p)* is even.

27. Suppose that *f(z)
= ∑ _{n} a_{n}
q^{n}, q = e^{2 π i z}* is a nonzero cusp form on a
congruence subgroup with

28. Suppose *p* and *q* are distinct primes congruent to 1
mod 4 with the property that *p* is not a square
mod *q*. Prove that there are integers *x* and *y* so
that *x ^{2} - pqy^{2} = -4*.

30. Find an infinitely differentiable function *f* so
that *f(0) = 1, f(1) = 0* and
*f ^{(n)}(0) = f ^{(n)}(1) = 0* for all

31. Let *a _{n}* be a sequence of complex numbers and
suppose that the series

32. For a positive integer *n*, denote by *σ(n)* the
sum of the positive divisors of *n*. Compute
*∑ _{n=1}^{∞} σ(n) e^{-2 π
n}*.

33. Show that if *p* is prime, then *x ^{p} - x - 1*
is irreducible over

34. Let *R* be a commutative ring with unity. Prove that there
is an injective homomorphism from *R* into a direct product of fields
if and only if *R* contains no nonzero nilpotent elements.

35. Show that every positive rational number can be written as the sum
of some subset of *{ 1, 1⁄2, 1⁄3, ...}*.

36. Find the smallest positive integer *n* so that
*1 + 1⁄2 + 1⁄3 + 1⁄4 + ... + 1⁄n > 1000*.

37. Find a right triangle with rational side lengths whose area is 997.

38. Find all integers *x* so that *(x + (x+72) ^{1/2})^{1/3}* is a whole number.

39. Prove that there are infinitely many prime numbers *p* so that
*p ^{p}+1* can be written as a sum of two integer squares.

40. Prove that for every integer *n* there exist integers *x*,
*y*, and *z* so that *n = x ^{2} + y^{2} + z^{3}*.

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