Michael Hartley
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If $a$ is a root of $x^2+ x + 1$, simplify $1 + a + a^2 +\dots+ a^{2017}.$
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8 votes

I haven't looked through your working, but here's a much easier method: If $a^2+a+1=0$, then $(a-1)(a^2+a+1)=0$, that is, $a^3-1=0$, ie, $a^3=1$. On the other hand, $1+a+\dots+a^{2017}=\frac{a^{2018}...

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Write all permutations in the cyclic group $C_{12}$ ​of order 12 in cycle notation.
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4 votes

$C_{12}$ is the cyclic group of order 12. So it's not $D_{12}$ (aka $D_{24}$) which has order 24. So, looking at the two elements you've identified: $e$, or "do nothing". That doesn't have ...

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The sequence $(x_n)$ : $x_{n+2}=\frac{x_{n+1}\sqrt{x_n^2+1}+x_{n}\sqrt{x_{n+1}^2+1}-x_n-x_{n+1}}{x_{n+1}x_n-(\sqrt{x_{n+1}^2+1}-1)(\sqrt{x_n^2+1}-1)}$
4 votes

By rationalising the denominator, $x_{n+2}$ simplifies to $$\frac{\sqrt{x_n^2+1}\sqrt{x_{n+1}^2+1}+x_nx_{n+1}-1}{x_n+x_{n+1}}.$$ Letting $$x_n=\frac12\left(p_n-\frac1{p_n}\right),$$ that is (say) $$...

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A Number Theoretic Game
4 votes

Let's represent the coins with a vector in $Z_2^n$, with $0$ representing heads, $1$ representing tails. Let $G$ be the group acting on $Z_2^n$ generated by a cyclic permutation of the basis vectors. ...

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If I flip two fair coins, and then tell you that one is heads, what is the probability that the other coin is also heads?
4 votes

The difference is in the first scenario, all you're told is that one of the coins is heads, but you don't know which one. This only rules out TT, so you still have three possibilities: HH, HT or TH. ...

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Showing that either $v$ is an eigenvector for $2\times 2$ matrix $A$ or else $(A − \lambda \operatorname{Id})v$ is an eigenvector for $A$.
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4 votes

Are you given that $a=d$? Because that neither implies, nor is implied by, the fact that $A$ has a repeated eigenvalue. Rather, your characteristic polynomial will be $(x-\lambda)^2$, so you know $(A-\...

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If $450^\circ<\alpha<540^\circ$ and $\cot\alpha=-\frac{7}{24},$ calculate $\cos\frac{\alpha}{2}$. Why is my solution wrong?
4 votes

It's probably easier to use $$\cos^2\alpha = \frac{1}{\sec^2\alpha} = \frac{1}{1+\tan^2\alpha} = \frac{1}{1+\frac{1}{\cot^2\alpha}}$$ to find $\cos\alpha$, this gives $\cos\alpha=\pm\frac7{25}$. The ...

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The number of real roots of an exponential equation with 4 parameters
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3 votes

If $a=b$, then write $k_1 e^{ax}-k_2e^{bx}$ as $ke^{ax}$. If $k$ and $a$ are opposite sign, or if either is 0, then $x=ke^{ax}$ has exactly one solution. Otherwise, $$x=ke^{ax}$$ if and only if $$ax=...

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Proving the sum to product formula with complex numbers starting from the left hand side
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3 votes

Obviously, you could "cheat" and use the rhs to inform your factorisation. But that's not what you want. If you write $a=e^{ix}$ and $b=e^{iy}$, then you have $$\frac{1}{2i}\left(a-\frac 1a +...

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What is the maximum number of distinct integers $A$ can have as elements?
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3 votes

The rational root theorem says that if you have a polynomial $a_n x^n + ... + a_0$ with integer coefficients, then all rational roots must be of the form $\pm\frac{p}{q}$, where $p$ divides $a_0$ and $...

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Powers of $2$ starting with $123$...Does a pattern exist?
3 votes

If a power of 2 starts with 123, then it must be between $1.23\times 10^n$ and $1.24\times 10^n$ for some $n$. So you want $k$ and $n$ for which $$1.23\times 10^n\leq 2^k < 1.24\times 10^n$$. This ...

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Relation of surface area of a sphere to its volume.
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3 votes

Not a coincidence. The same holds for circles - the circumference is the derivative of the area. Start with a sphere of radius $r$. If the skin has thickness $dr$, the volume of the skin will be ...

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$P(X^2+Y^2<1)$ when $X,Y\sim U[0,1]$
3 votes

That is a correct way to solve it. A better correct way to solve it, since you're dealing with uniform random variables on the unit square, is to ask "what is the area of the shape defined by $X^2+Y^...

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Let $V$ be the space spanned by $v_1=\cos^2x$, $v_2=\sin^2x$ and $v_3=\cos2x$.
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3 votes

Well, a basis for a vector space is a linearly independent spanning set. Do you know that $\{\cos^2(x), \sin^2(x)\}$ span $V$? Hint: elements of $V$ are of the form $a_1\cos^2(x) + a_2\sin^2(x) + ...

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Coloring a rectangle with 3 rows and 4 columns using two colors.
3 votes

Well, one way is to use Burnside's Counting Lemma. There may be easier ways, but I like Burnside's counting lemma. There are, as you noted, 924 ways to colour the grid if you ignore symmetry. The ...

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Finding Limit of an Integral: $\lim_{n\to\infty}\int_a^b f(x)\sin^3{(nx)} \:dx$
3 votes

You can't conclude $$\lim_{n\rightarrow\infty} \int_a^b g(x,n)dx$$ doesn't exist just because $$\lim_{n\rightarrow\infty} g(x,n)$$ doesn't. For example, $$\lim_{n\rightarrow\infty} \sin(nx)$$ doesn't ...

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Prove that $\sum_{i=0}^{m}{\binom{n-i}{k}}=\binom{n+1}{k+1}-\binom{n-m}{k+1}$
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2 votes

Your way is fine. Here's another way: proceed using induction. The proposition is true if $m=0$: $${\binom{n-0}{k}}=\binom{n+1}{k+1}-\binom{n-0}{k+1}$$ because $${\binom{n}{k}}+\binom{n}{k+1}=\binom{n+...

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What is the largest number of elements of $\{\sin\alpha\cos\beta,\sin\beta\cos\gamma,\sin\gamma\cos\alpha\}$ (all angles acute) that can exceed $1/2$?
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2 votes

if $\sin\alpha\cos\beta>\frac12$ then $\frac12\sin(\alpha+\beta)+\frac12\sin(\alpha-\beta)>\frac12$ However, $\frac12\sin(\alpha+\beta)\leq\frac12$, so this is only possible if $\sin(\alpha-\...

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Could the decidability of a theorem be undecidable?
2 votes

If T is undecideable within formal system F, it means (1): F does not contain a proof of T, AND F does not contain a proof of ~T. Suppose the question "T is undecideable" is undecideable. ...

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Change in Partition for Geometric Mean if Sum Increases
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2 votes

Yes. Proof: Assume $\prod x_h$ is maximised, and there exists $i,j$ such $x_i+1\leq x_j-1$ (that is, $x_i$ and $x_j$ differ by at least 2). Without loss of generality, let $i=1$ and $j=2$. Let $y_h$ ...

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Convexity of $f(x)=\frac{1}{2}x^TAx$?
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2 votes

a positive semidefinite matrix has all eigenvalues non-negative. However, $A$ has a negative eigenvalue $\lambda_{min}$.

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Prove: If $f(x)$ is a local maximum, then $f(x)$ is a global maximum.
2 votes

I would agree with your instincts. To show 1, you just need an example. The function $f(x) = x^2$ on $[-1,2]$ will do. Hint: what are the local maxima? And don't forget to prove $f$ is convex.

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Some have more than one stationary distributions?
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2 votes

"$(1,0,0),(0,1,0),(0,0,1)$ are all eigenvectors corresponding to eigenvalue 1" What you have found is not all eigenvectors, but a basis for the vector space of the eigenvectors. If you want ...

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Find the sum of $x_1+x_2+x_3$ of intercept points
2 votes

If polynomial $p(x) = ax^n + bx^{n-1} + ...$ has roots $x_i$, then $x_1+\dots+x_n=-\frac{b}{a}$, as you noted. Now, though, you don't want $p(x)=0$, you want $p(x)=mx+b$. So, to use Vieta's theorem, ...

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How to prove that $a(1-a^N)/(1-a)<N$?
2 votes

Well, $1-a^N=(1-a)(1+a+a^2+a^3+...+a^{N-1})$, and each of the terms in the second bracket is less than or equal to 1, so $1-a^N\leq(1-a)N$ Therefore, $\frac{a(1-a^N)}{1-a}\leq\frac{a(1-a)N}{1-a}\leq ...

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Calculating Probability and Standard Deviation with Central Limit Theorem
2 votes

That formula is not the right one to use. That would apply if each dice had two faces, 0 and 1, and then $p$ is the probability it shows 1. Examples of things like this: tossing a coin $n$ times ($...

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How to solve this type of permutations product question
2 votes

In this notation, numbers not mentioned are fixed. So, if p = (1,3,2), then p sends 1 to 3, 3 to 2, 2 to 1, 4 to 4, 5 to 5, 6 to 6, 7 to 7, etc.

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$a$ algebraic number, $\exists$ integer $n>0$ such that $na$ algebraic integer
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2 votes

Let $f(a)=0$ be monic, degree $n$. If the polynomial is not already a polynomial over the integers, it has a term whose coefficient is rational, with denominator $q$ when fully reduced. Suppose that's ...

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How to convert $r(t,s)=\langle t+s,t-s,t^2+s^2\rangle$ to Cartesian coordinates?
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2 votes

If $x=t+s$ and $y=t-s$, you can solve these as a system of equations for $t$ and $s$. This gives $t=\frac12(x+y)$ and $s=\frac12(x-y)$. Then you can substitute these into $z=t^2+s^2$ to get $z$ as a ...

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Induction on Fermat Numbers: $F_n = \prod_{j=0}^{n-1}F_j+2$
2 votes

Looks good to me :) NB - those are Fermat numbers, not Fermat primes.

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