# Math Problems

These trivia questions and puzzles were taken from mowsey's original #math FAQ.

## Trivia and Trivialities

• Q: What is 0^0? A: It is not universally defined, see this article (ftp://rtfm.mit.edu/pub/faqs/sci-math-faq/specialnumbers/0to0) for details.
• Q: Why does 0.999=1? ne1?? ????? A: Try this article (ftp://rtfm.mit.edu/pub/faqs/sci-math-faq/specialnumbers/0.999eq1)

## Puzzles

### Number theory

• Determine the number of odd numbers in the nth row of Pascal's triangle.
• Determine which rows of Pascal's triangle contain an arithematic progression a,a+b,a+b+b.
• Show that the Fibonacci numbers have the property, f(gcd(m,n))=gcd(f(m),f(n)). f(1)=f(2)=1,f(n+2)=f(n+1)+f(n). by Polytope
• If C(n) is a sequence of non-zero integers such that gcd(C(m),C(n)) = C(gcd(m,n)), then the generalized binomial coefficients B(n,k) = C(n) C(n-1) ... C(n-k+1) / C(k) C(k-1) ... C(1) ] are all integers. by Polytope, from Concrete Mathematics, Knuth, Graham, Patashnik.
• Show that the nth Fibonacci number is prime only if n=4 or n is prime. Which Fibonacci numbers are prime? Are there infinitely many such primes? by Polytope (possibly open) Note: The 9311th Fibonacci number is prime.
• prove that (ap choose bp) = (a choose b) mod p. Here a,b are integers and p is a prime.
• prove that (ap choose bp) = (a choose b) mod p^2.
• when is (ap choose bp) = (a choose b) mod p^3?
by Galois
• let m_1, ...., m_n be relatively prime in pairs (that is m_i and m_j are relatively prime for any i different from j), and let x_1, ..., x_n be any integers, show that there is an x such that x = x_1 (mod m_1), x = x_2 (mod m_2), x = x_3 (mod m_3), ..., x= x_n (mod m_n) by CRT
• When is (sum k=1 to n) 1/k an integer? by Poinky
• When is 1+2+...+n an even square? by Poinky
• When can a positive integer be written as the sum of consequetive positive integers (obviously requiring more than one number in the sum)?
• Show that the denominators of the continued fraction convergent to (a+sqrt(a^2+4))/2 satisfy (q_n,q_m)=q_(n,m) for integers a. by Poinky
• Problem: Prove that for 0< a < 10 , q(n) = aq(n-1)+q(n-2), q(0)=0, q(1)=1, then: sum(k=0 to +inf) q(k)/10^(k+1) = 1/(100-10a-1) (a=1 is the Fibonacci-number-case) by Poinky
• If p is prime and greater than 5, then p*p divided by 120 has a remainder of either 1 or 49.
• Show that the set of ratios p/q of prime numbers 0 < p < q is dense in the interval from 0 to 1.
• If a and b are positive integers, (1+ab) | (a^2+b^2) implies (a^2+b^2)/(1+ab) is a square. by Niven and Zuckerman and brouwer

### Enumerative combinatorics

• A blind bartender and an antagonist play a game. The antagonist sets four cups on a tray, each one either right-side up, or upside down. At each turn the bartender announces that he will touch 2 of the cups, and specifies them by their position on the tray, for instance "the one near me on my left, and the one far from me on my right." He can tell if a cup is up or down, and can decide to flip either or both or neither of those cups over. There is however one catch. The antagonist can rotate the tray after the bartender announces his choice, but before the bartender can touch the cups. The bartender will still be able to touch two cups, but it will be the two cups that are now in the positions he called out. If, after announcing that he is done flipping over the cups, the four cups are either all up or all down, the bartender wins, otherwise a new round begins by the bartender making a new announcement of which two cups to touch. Can the blind bartender ever win? by Dr. Ehrenborg
• Mafiaboy has a row of 2n boxes, some of which may contain marbles. He wants to know if there is a string of n consecutive boxes which all contain marbles. Show that he only needs to open n+1 boxes to determine if this is the case. by Polytope
• You have been kidnapped and blindfolded by ruthless criminals with sadistic zeal for puzzles. They threaten to kill you unless you agree to play a game. They tell you there are N coins on the table, and K of them are heads up. You may specify that A distinct coins be flipped over, and that exactly B flipped and C unflipped coins be put into one pile, and the rest in another. If there are the same number of heads in each pile you win, but otherwise...

### Algebra

• Decompose n->n+3 into the product of two cycles
• Show Q(sqrt(2+sqrt(2))) is galois of degree 4
• Show Aut(\R/K) = 0 for any field K between Q and R inclusive.
• Show that Aut(K(x)/K) = PGL(2,K). Show that Aut(D(x)/D) = PGL(2,Frac(D)).
• Let R be an integral domain. Show that E is an injective R module iff E is an injective S^-1 R module. If one assume already that E is a S^-1 R module, but no longer requires R to have no zero divisors, show that E is injective over R iff it is injective over S^-1 R. What can one say if E is not already an S^-1 R module?
• Show that the set of holomorphic functions on a compact set is a euclidean domain, but that the set of holomorphic functions on an open set is not even UFD. Addition and mutliplication are pointwise.
• Suppose a field contains a primitive nth root of unity, where n is odd. Show it also contains a primitive 2nth root of unity. by brouwer
• Let F be a field. Prove that x^4 + 1 factors in F[x] iff at least one of 2, -1, or -2 has a square root in F. by Polytope
• If every element of a group has order dividing 2, show the group has at least two distinct automorphisms as long as it has at least two distinct elements.
• If R is a commutative ring with 1, and rxy=r, then are there u and v such that uv=1 and rx=ru? by drini

### Analysis

• Let f be nonconstant and analytic on {z in C : |z| <= 1} and let |k|=1 be such that |f(k)| = max {|f(z)| : |z|=1}. Show f'(k) isn't 0. by brouwer
• If f is entire and f(x)=f(x+a)=f(x+b) for a,b in C a/b not in R, then f is constant.
• If f is defined and continuous on the closed disk and analytic on the interior, show that if f is constant on any connected set having more than 1 point, then f is identically constant.
• Define two polynomials: f(x) = x^2+ax+b and g(x) = x^2+px+q. Determine a,b,p and q so that f(x)^g(x) = 1 has the solutions x = 1,2,3,4,5 by Poinky
• Find a continuous function g on R so that g(x)+g(x+1)+g(x+2)=0. by kakutani
• Express x^2 as the sum of 3 periodic functions. by kakutani

### Topology

• Show that the irrationals have a complete metric that induces the same topology as the subspace topology from R. Show that if X is completely metrisable, then so is X\C for any countable C. by Evandar
• Prove or disprove, in R^n if A and B are seperated (Cl(A) intersect B = Cl(B) intersect A = empty set) then there are disjoint open sets containing A and B. by koro
(This solution is from ^loner^ of undernet #math via koro): Let X be a metric space and let A,B be separated subsets of X. Let T=cl(A) intersec cl(B). Note that since cl(A) inter B = A inter cl(B) = void, A is a subset of cl(A)-T and B is a subset of cl(B)-T so that the closure of A in X-T contains A and the closure of B in X-T contains B, and note that those closures are disjoint since [cl(A)-T] inter [cl(B)-T] = void. Now cl(A)-T and cl(B)-T are closed in X-T which is a metric space (and hence is normal), so that there are disjoint opens U and V in X-T such that A\subset cl(A)-T \subset U and B\subset cl(B)-T\subset V. But X-T is open, (since T is closed) so that U and V are also open in X. Q.E.D.
• Suppose every subset of X is either open or closed. If X is haussdorf then X can have at most one accumulation point. by Koro
• A continuous function on a countable compact Hausdorff space must have a periodic point.

### Geometry

• For every n > 5, it is possible to cut a square into n smaller squares. For every n > 47, it is possible to cut a cube into n smaller cubes. by Polytope
• How to divide an ice-cream-packet into exactly k (arbitrary integer) parts with the aid of a knife. by Poinky
• Cut a 12x12x12 cube into 4 pieces, and reassemble the pieces into an 8x8x27 block. by IBM
• Let x_1,...,x_n be points in R^2, and let f:R^2-->R be a function such that for any rigid motion T in the plane, it holds \sum_{i=1}^n{f(Tx_n)} = 0. Prove that f = 0. by koro
• For which n do there exist n+1 points in Z^n such that every pair is some certain distance d apart? by Koro

### Statistics

• Let's have random variable x and sample x1,x2,...,xn. E(x)=y and var(x)=s^2. Now we have estimate y = a1*x1+a2*x2+...+an*xn, where ai>0 and a1+a2+...+an = 1. Show that var(y) is smallest, when y is mean value of x1,x2,...,xn. In other words, ai = 1/n by UC

### TeX

• Q: How do I include graphics? What is EPS?