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Feb
2
comment Limit in number theory
This would get down to $c = 1/2 + \epsilon$ as well.
Jan
29
comment Expected value of sum of cards
This seems a bit tricky to get an exact answer to but simulation would be easy. Do you need an exact answer or just an approximate one?
Jan
28
comment Pair of consecutive squarefree numbers with $n$ distinct prime factors
This is related to oeis.org/A052215 (which unfortunately doesn't give any results beyond what's figured out in the answers to this question).
Jan
27
comment How many numbers are there of 2n digits that the sum of the digits in the first half equals the sum of the digits in the second half
I believe you mean the Central Limit Theorem, not the Law of Large Numbers, but this is the first thing that came to mind for me as well.
Jan
27
comment 1995 USAMO Problems/Problem 2
You seem to be referring to a solution that's out there somewhere - can you link to it?
Jan
25
comment What inversion sets of permutations of $123\ldots n$ are possible?
A comment (although I'm not sure how to get from this to an answer): every permutation of $123 \cdots n$ has a different inversion set.
Jan
22
comment Prove that $\frac{2^{122}+1}{5}$ is a composite number
$1709 = 14*122 + 1$ is also a factor, as is $3456749$. (I am not inspired - I found these by trial division.)
Jan
19
comment Will you eventually run out of fair coins which either triplicate or disappear when flipped?
You're describing something an awful lot like a Galton-Watson process: en.wikipedia.org/wiki/Galton–Watson_process
Jan
13
comment Why does this sequence converge to $\pi$?
It appears that $s_k \approx \cos ((k/n) (\pi/2))$. I'd guess that the recurrence here is an approximate numerical solution to some differential equation.
Jan
13
comment What is the average number of connected components in a Secret Santa graph?
Something seems off here - the average number of cycles in a permutation of $[n]$ is $H_n$. I think there's an extra factor of $1/e$ in your answer and the average number of cycles in a derangement of $[n]$ is close to $H_n-1$.
Jan
13
comment Choose a composition from the previous composition
Hint: can you use multiple kinds of bars?
Jan
12
comment Evaluation of $\displaystyle \int_{1}^{3}\left[\sqrt{1+(x-1)^3}+(x^2-1)^{\frac{1}{3}}\right]dx$
I agree with Lucian. In a sense this is a question about the geometry of curves in the plane - the integral, which turns out to be 6, can actually be written as the area of a single rectangle.
Jan
11
comment Least value of $C$ for which $CT_n+1$ is always a triangular number.
Hint: an integer $m$ is triangular if and only if $8m+1$ is square.
Jan
11
comment Probability that a quadratic equation with random coefficients has real roots
We can generate a random point $(u, v, w)$ on the sphere by taking $X, Y, Z$ independent random standard normals and then taking $(u, v, w) = (X, Y, Z)/\sqrt{X^2+Y^2+Z^2}$. I simulated $10^6$ points on the sphere and found that $351503$ of them had $u^2 + 2v^2 \le 2w^2$, so the probability is in the neighborhood of $0.352$. This hopefully helps confirm an analytic solution if someone attempts it.
Dec
14
comment How can I create 14 unique team rosters from 24 students in groups of 2?
That's an interesting question, designing scheduling algorithms that are robust to students dropping. I wish I knew something about this!
Nov
23
comment Number theory primes and congruences: $p^{32} -1$ is divisible by $16320$
A relevant link to the OEIS: oeis.org/A079612. In fact $p^{32}-1$ is divisible by 16320 whenever $p$ has no factors in common with $16320$.
Nov
20
comment Verify the polynomial of degree $≤ 4$ is exact: $\int_0^1f(x)dx ≈ {1\over90}[7f(0) + 32f({1\over 4}) + 12f({1\over 2}) + 32f({3\over 4}) + 7f(1)]$.
As a warmup, you might try to prove that $\int_0^1 f(x) \: dx = {1 \over 6} [f(0) + 4f(1/2) + f(1)]$ when $f$ is a polynomial of degree 3 or less. (This is Simpson's rule for integration.)
Nov
5
comment Show that T(n)=4×T(n−1)−T(n−2)
It would help to be more specific about the problem. What is $T$?
Oct
23
comment What is the optimal strategy in the “Factor Game”?
The fact that there's a grid doesn't matter, does it? The game plays out exactly the same on, say, a, 10-by-6 grid and a 15-by-4 grid, which both have 60 squares.
Oct
14
comment A set of n integers is a complete residue modulo n if no two elements are congruent mod n.
This is usually called a "complete residue system", and often it's defined to be a set of $n$ integers with no two congruent mod $n$. If you've been asked to prove this, you probably have been provided some other definition of "complete residue system". What definition do you have?