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, cos^-1, and tan^-1. The display initially shows 0. Given any positive rational number q, show that pressing some finite sequence of buttons will yield q. Assume the calculator does real number calculations with infinite precision. All functions are in terms of radians. (USAMO, 1995).

4. Prove that if a, b, and c are all odd integers, then the roots of ax² + 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 C¹ vector field on R³. If ∇ · F = 0, prove that there is some C¹ vector field G so that F = ∇ × G. (Problem 7.6.17 from Susan Jane Colley's "Vector Calculus").

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 4444⁴⁴⁴⁴, and B the sum of the digits of A. Compute the sum of the digits of B. (IMO, 1975).

12. Suppose that a and b are integers with ab ≠ -1. Prove that if 1 + ab divides a² + b² then (a² + b²)/(1+ab) is a perfect square, or is -5. (Extension of a problem from the IMO, 1988).

13. Let V_n be the volume of the n-dimensional unit sphere. Evaluate V₁ + V₂ + V₃ + · · ·.

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

15. Let a₁ = 1, a₂ = 1/2 + 1/3, and continue the sequence by constructing a_n+1 by replacing each fraction 1/d in the expression for a_n by (1/(d+1)) + (1/(d² + d + 1)).
Compute lim_{n → ∞} a_n. (Problem 10901 from the American Mathematical Monthly, November 2001).

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 R² 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 x² ≡ x (mod n). Evaluate ∑_n a_n n^-2.

20. Prove that for all n ≥ 1, 1 + 5ⁿ + 5²ⁿ + 5³ⁿ + 5⁴ⁿ 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 x² - py² = -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 qⁿ, q = e^{2 π i z} is a nonzero cusp form on a congruence subgroup with a(n) real. Prove that there are infinitely many n so that a(n) < 0.

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 x² - pqy² = -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 n ≥ 1.

31. Let a_n be a sequence of complex numbers and suppose that the series ∑_n a_n n^-s converges for all complex numbers s with Re(s) > α (assume also that α > 0). Prove that for all ε > 0 there is a constant C_ε so that | ∑_{n ≤ x} a_n| < C_ε n^{α + ε}. (Problem XIV.4.15(b) from Gamelin's "Complex Analysis").

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 Q.

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 = x² + y² + z³.