Connor Harris
  • Member for 8 years, 3 months
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Why does this pattern in powers happen?
18 votes

This is a well-known result analogous to the formula $$ \frac{d^k}{dx^k} x^k = k!$$for differentiation of polynomials. I'll try to state it clearly: Definition: Let the discrete derivative of a ...

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What is the period of Langton's ant on a torus?
12 votes

Not an answer, but a quick table with the number of turns until the recurrence of the initial position on rectangular grids small enough to be testable by computer. The ant is initially facing up (...

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Does the infinite nested logarithm $\ln(2\ln(3\ln(4\ln(5\ln(6...)))))$ converge?
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12 votes

For all integers $n \geq 2$, note that $$n(n+1) < e^n. \tag{*}$$ We may show this by expanding $e^n > 1 + n + \frac{n^2}{2} + \frac{n^3}{6}$, which implies: $$e^n- n(n+1) > 1 - \frac{n^2}{2} +...

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Show that $\mathbb{R}^{n}$ ($n>2$) minus a countable number of lines is still path-connected
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8 votes

Reduce the problem to showing that $\mathbb{R}^2$ minus a countable set of points is path-connected, using one of these methods. Let $S^2 \subset \mathbb{R}^n$ be a 2-sphere that contains $p$ and $q$ ...

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Arrange numbers 1-12 around a circle so that any three consecutive numbers have a sum $\leq 20$?
7 votes

The arrangement is indeed impossible. As the answer to the question you linked mentions, the sums of each segment of three adjacent numbers all themselves have to add up to $3 (1 + 2 + \cdots + 12) = ...

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Is a positive, monotone and sub-additive function concave?
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7 votes

Subadditivity is implied by a requirement that $g(x) := \frac{f(x)}{x}$ be monotonically decreasing. If $g(x) \geq g(x+y)$ and $g(y) \geq g(x+y)$, then $$f(x) + f(y) = x g(x) + y g(y) \geq x g(x+y) + ...

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Find all integers $x$ such that $2^p + 3^p = x^2$ where $p$ is prime
7 votes

There's no solution. Consider reduction modulo 3.

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If n is a positive integer that is four digits long and is relatively prime to 100!, why must n be prime?
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6 votes

Hint: suppose some number $n$ is not prime, but $n$ has no prime factors less than 100. That is, $n$ is the product of at least two prime numbers greater than 100. What's the smallest that $n$ could ...

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Prove that there exists no differentiable real function $g(x)$ such that $g(g(x))=-x^3+x+1$.
6 votes

Recall that a fixed point of a function $f$ is a solution to $f(x) = x$. The only fixed point of $g \circ g$ is $1$, so the only fixed point of $g$, if any, is also $1$, as any fixed point of $g$ is ...

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How to prove that: $\int_{-1}^{1} \frac{1}{x^2-1}dx$ diverges?
6 votes

Here's a less orthodox answer. Consider just the interval $[0, 1)$, and divide it into intervals $[0, 1/2), [1/2, 2/3), \ldots, [1 - \frac{1}{n}, 1 - \frac{1}{n+1}), \ldots$, each of width $\frac{1}{n(...

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Solve for $a,b,c,d \in \Bbb R$, given that $a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac 25 =0$
5 votes

Multiply by $2$ and rearrange to \begin{align*}(a-b)^2 + (b-c)^2 + (c-d)^2 + (d-a)^2 + 2ad - 2d + \frac{4}{5} = 0. \tag{$\star$}\end{align*} For fixed $a$ and $d$, the minimum value of $(a-b)^2 + (b-c)...

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Prove that $10|n+3n^3+7n^7+9n^9$
5 votes

Denote $f(n) = n + 3n^3 + 7n^7 + 9n^9$. The terms in $f(n)$ are either all odd or all even, so $f(n)$ is even. By Fermat's little theorem, $9n^9 \equiv 4n \bmod 5$ and $7n^7 \equiv 2n^3 \bmod 5$. ...

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Find the function that is implicitly defined in the relation
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5 votes

The main branches of $\sqrt{\cdot}$ and $\cos^{-1}$ have nonnegative ranges, so their sum is zero only if they are both zero themselves. This forces $y = x$.

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Any deeper insights for why $\int \frac{1}{x}dx = \ln|x|+C$?
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5 votes

If we replace the indefinite integral with a definite one so as to keep one point (say, $(1, 0)$) fixed as we vary the exponent, then there's no jump at all. For all $s \neq 0$, define $$f_s(x) := \...

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Does $c^{a} + c^{b} = a^{c} + b^{c}$ when $a \ne b$?
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5 votes

The only triples of positive integers $(a, b, c)$ for which $a \neq b$ and $c^a + c^b = a^c + b^c$, up to interchange of $a$ and $b$, are $(1, 3, 2)$, $(2, 4, 2)$, and $(2, 4, 4)$. Allowing $a = b$ ...

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check whether $\sum_{n=1}^{\infty} \frac{1}{\sqrt{(2n-1)(2n+1)}}$ converges
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5 votes

The series diverges, because $$\sum_{n=1}^\infty \frac{1}{\sqrt{4n^2 - 1}} > \sum_{n=1}^\infty \frac{1}{2n}.$$

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Computing $ \int_0^1 \frac{1 + 3x +5x^3}{\sqrt{x}}\ dx$
5 votes

Hint: the formula $$\int x^c\, dx = \frac{x^{c+1}}{c+1} + C$$ applies to all real numbers $c \neq -1$, not just integers.

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Find all possible values of $f(2018)$
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4 votes

$f$ can only be the 2-adic valuation on $\mathbb{Q}$. That is, $f(0) = 0$ (a simple deduction from $f(0 \times 2) = f(0) f (2)$), and if $q = \pm 2^k \frac{a}{b}$ with $a$ and $b$ both odd, then $f(q) ...

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Proof that no set is equinumerous to its power set
4 votes

It's completely possible that $x \in g(x)$ for every $x \in A$. But if this is the case, then the set $B$ is just the empty set $\emptyset$, which is a perfectly good element of $\mathcal{P}A$ that ...

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Find all function $f(n)$ satisfying $f(n)^2 = n f(f(n))$
4 votes

If $f(n) = k_n n$, then $(k_n n)^2 = n f(k_n n)$, so $f(k_n n) = k_n^2 n = k_n (k_n n)$. Inductively, therefore, $f(k_n^r n) = k_n^{r+1} n$. And $k_n$ has to be an integer: if it weren't, then there ...

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Proving there is a bijection from set of real continuous function on $[0,1]$ denoted by $C[0,1]$ to the set of reals $\mathbb{R}$
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4 votes

A brief sketch of an argument: Continuous functions are determined by their values on any dense subset of their domain, such as the rationals. The rationals are countable, so pick some enumeration $...

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sum of primes squared equals a prime squared
4 votes

As user113102 hints, if $n$ is odd, then $n^2 \equiv 1 \bmod 8$. Thus, in the equation $$p_3^2 + p_4^2 + p_5^2 + p_6^2 + p_7^2 = p_8^2 - 8$$ where all $p_i$ are odd primes, the LHS is congruent to $5$ ...

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Closed form for $K(n)=[0;\overline{1,2,3,...,n}]$
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4 votes

Following up on Daniel Schepler's comment. Let $$P_n(x) = \frac{1}{1 + \frac{1}{2 + \ddots \frac{1}{n+x}}}.$$ This is basically the RHS of the recurrence equation for $K(n)$. Then: \begin{align*} P_1(...

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Range of balls needed in lottery for 0 and 1 match to be equally likely with 5 balls drawn
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4 votes

Here's what I think you're asking, generalized slightly. There are $n$ balls numbered $1, \ldots, n$. Lottery players choose $k$ distinct numbers with $k < n$, then $k$ balls are drawn. Tickets are ...

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Showing $[\mathbb{Q}(2^{1/3},2^{1/2}):\mathbb{Q}(2^{1/2})]=3$
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4 votes

Method 1: Recall that $[L:K][K:F] = [L:F]$, note that $\mathbb{Q}(2^{1/3}, 2^{1/2}) = \mathbb{Q}(2^{1/6})$, and show that $X^6 - 2$ is irreducible over $\mathbb{Q}$ using Eisenstein or similar. ...

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Which is bigger, $ \log_{1000} 1001$ or $\log_{999} 1000 $?
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4 votes

Note that $$\frac{d}{dx} \frac{\log(x+1)}{\log x} = \frac{\frac{\log x}{x+1} - \frac{\log (x+1)}{x}}{\log^2 x} < 0,$$ so $\log_{999} 1000$ is larger.

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Is my answer logically equivalent to the other?
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4 votes

They're not the same. In your solution, for example, $x$ could be a teacher; in that case, for all $y$, $S(x) \wedge F(y)$ is false, so the implication $(S(x) \wedge F(y)) \Rightarrow \neg A(x, y)$ is ...

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Graph Combinatorics - Upper and lower bound on the number of group isomorphism classes
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4 votes

The number of graphs on $n$ distinct vertices is $2^{(n^2 - n)/2}$, giving the upper bound. To show the lower bound, consider a set $\mathcal{S}$ of non-isomorphic graphs on $n$ vertices. Construct ...

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1998 USAMO problem #4
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4 votes

If two adjacent squares on an edge are different colors, they can only be made the same color (this is what they mean by "fixed") if one square is in the rectangle being inverted and one of them isn't....

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Determinant of a matrix with integer power entries
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4 votes

For point (2), if you can find quantities $a, b, c, d$, not all zero, such that $$ a m^2 + b(m+1)^2 + c(m+2)^2 + d(m+3)^2 = 0$$ for any integer $1 \leq m \leq n$, then this gives you a nontrivial ...

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