JavaMan
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Surprisingly elementary and direct proofs
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The Szemeredi-Trotter incidence theorem gives an upper bound on the total number of incidences between a finite set of points and a finite set of lines in the plane. An incidence is a pair $(\ell, x)$...

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Proof using trigonometry that circle circumference is $2 \pi R$
11 votes

The question as it stands is not well posed as you are asking the wrong question. In order to prove that the circumference formula $C = 2\pi r$ holds for all circle of radius $r$, we first have to ...

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Legendre symbol proof
1 votes

All congruences are done modulo $p$. Assume that $x = \pm a^{n+1}$. Then, \begin{align*} x^2 \equiv a^{2n + 2} = a^{\frac{4n+4}{2}} = a^{\frac{p+1}{2}}. \end{align*} We need to show that $a^{\...

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Solving systems of equations using matrices
0 votes

There is an algorithm for this process called Gauss-Jordan elimination. Applying Gauss-Jordan to the example given: 1) The first step is to get a leading one in the first row of your matrix. The ...

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How to prove that $1/n!$ is less than $1/n^2$?
0 votes

Try the ratio test: $$ a_n = \frac{1}{n!} \quad \implies \quad \left| \frac{a_{n+1}}{a_n} \right| = \frac{n!}{(n+1)!} = \frac{1}{n+1} $$ so that $$ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \...

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The following graph has 45 vertices.
2 votes

Hint: The Handshaking Theorem gives the precise relation between the number of edges in a graph and the sum of the degree of the vertices of a graph. If you know the degree of each vertex, then, ...

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Finding $\lim_{x \to 0}\frac{\tan x-x}{x^3}$
2 votes

You could also note that $$ \lim_{x \to 0} \frac{2 \tan x \sec^2 x}{6x} = \lim_{x \to 0}\frac{2 \sec^2 x}{6} $$ since $$ \lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} \frac{\sin x}{x} \cdot \frac{...

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65-card deck consisting of 13 ranks and 5 suits
3 votes

A few to get you started: The number of hands in a Super 4 of a kind would be calculated as: $$ {5 \choose 4} \cdot {12 \choose 1} \cdot {13 \choose 1} = 780 $$ There are ${5 \choose 4}$ ways to get ...

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Determine number of non-negative integer solutions for both equalities
2 votes

This doesn't answer your question directly, but I can't help mentioning that Generating functions are helpful with these kinds of problems. For the first problem, note that $x_i$ can be either $0, 1, ...

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Combinatorics/Probability Distribution Example Question
3 votes

Generating functions are helpful here. You need to find the number of solutions to the equation $$ x_1 + x_2 + x_3 + x_4 + x_5 + 2x_6 \leq 5 $$ where all variables are nonnegative $(x_i \geq 0)$. ...

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prove that : $ \sum_{n=0}^\infty |x_n|^2 = +\infty \Rightarrow \sum_{n=0}^\infty |x_n| = +\infty $
1 votes

Hint: If $\sum a_n < \infty$, then $\displaystyle \lim_{n \to \infty} a_n = 0$. In particular for large $n$, we have $|a_n| \leq 1$. Then $$ |a_n|^2 = |a_n| |a_n| \leq \dots $$

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Does this have a name: If an odd prime $p$ does not divide $a$, then $p$ divides $a^n + 1$ or $a^n - 1$
Accepted answer
4 votes

This is well known. Let $p = 2n + 1$ be an odd prime. Then, it is known that if $\gcd(a,p) = 1$, then $a^{p-1} = a^{2n} \equiv 1 \pmod{p}$, by Fermat's Little Theorem. Hence, $(a^n)^2 \equiv 1 \...

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Vector (Linear Combination)
0 votes

First, let me say that you need to be more careful with your work. Writing $EQ(1) + EQ(2) = 2a + 6b -10c = 4$ is a little sloppy. Rather, you should write what $EQ(1) + EQ(2) \implies 2a + 6b - 10c ...

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Finding the integers modulo $n$ such that $x+y=2$, $2x-3y=3$
2 votes

Note that $$x + y = 2 \iff 2x - 3(2-x) = 3 \iff 5x = 9$$ This equation has a solution when...

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Is there a character to depict a large number?
Accepted answer
4 votes

It is common in number theory to write $c \gg 1$ to mean that $c$ is a sufficiently large (but finite) constant. The $\gg$ notation is Vinogradov's Notation.

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Does this sequence converge to $0$?
1 votes

No. A sequence converges to zero if and only if we can eventually make all the terms which are large enough as close to zero as we want. In symbols, $a_n \to a$ if for all $\epsilon > 0$, there ...

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Proof that $n^2 < 2^n$
10 votes

If you want to use induction, I assume you have checked the base case $n = 5$. To do the inductive step, assume that the statement holds for some $k$: $k^2 < 2^k$, and then under this assumption, ...

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How to solve $ 13x \equiv 1 ~ (\text{mod} ~ 17) $?
2 votes

A system approach is to find integers $s$ and $t$ (via the Euclidean algorithm) such that $13s + 17t = 1$ (note that we can do this as $\gcd(13,17) = 1$). Then, $$1 = 13s + 17t \equiv 13s \pmod{17}$$...

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Trigonometry equation
0 votes

Use the fact that $AB = 0 \implies A = 0 \text{ or } B = 0$. You have already found when $\csc(x) = 0$, so $\frac{\pi}{4}$ and $\frac{3\pi}{4}$ are both solutions. Now, when does $\cos(x) = 0$? ...

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How can we prove the Inequality : $ \frac {n!}{ 2^{n-1}((\frac {n-1}{2})!)^2} \leq \sqrt{n}$
0 votes

For completeness, here is the proof by induction: Base Case: The case $ n = 1$ is easy to check. Inductive Step: Assume that $$\frac{k!}{2^{k-1} \left( \frac{k-1}{2} \right)!^2} \leq \sqrt{k}.$$ ...

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whats the absolute time interval?
Accepted answer
1 votes

Your work is fine. You then need to find the $t$-values that satisfy both conditions: $-4 \leq t \leq 2$ and $t \in (-\infty, -2) \cup (0, \infty)$. Alternatively, you could consider two cases: $t + ...

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how to find inverse of a matrix in $\Bbb Z_5$
7 votes

Perhaps the easiest way here (though not the easiest way in general) is to write $$A^{-1} = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right)$$ ...

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Growth of $ n^{\ln n}$ versus polynomial, exponential, and logarithmic forms
Accepted answer
3 votes

For polynomial and logarithmic comparisons, these should be quite clear by looking at the limit itself. With polynomial, you have $\frac{n^k}{n^{\ln n}}$. Since $k$ is fixed and $\ln n \to \infty$ ...

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Radical questions algebra
0 votes

To get to the heart of the matter, since these are square roots, you are looking for the largest factor of the radicand that is a perfect square. For example, with $\sqrt{32ab^2}$ you are looking for ...

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discretization of continuous function and its maximum
Accepted answer
0 votes

If $f(x^*)$ is the maximum of the function over $[0,1]$, then $f(x^*)\geq f(x)$ for all $x\in[0,1]$. In particular, $f(x^∗)≥f(x_i)$ for $i=1,…,n$.

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Rolle and Mean Value Theorem
1 votes

The way that mathematics is actually done and the way that mathematics books are written are completely different. No mathematician would ever just sit down and guess a formula to be proven later. ...

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Notes for Beginner Fourier Analysis?
1 votes

Tom Wolff published a collection of notes titled "Lectures on Harmonic Analysis" which he made freely available. Since his passing, the notes have been kept on Izabella Laba's website. The notes are ...

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Two series questions
Accepted answer
2 votes

Note that if $\alpha \in (-1,1)$, then $1 + \alpha^k \leq 2$ so that $$ \frac{1}{1 + \alpha^k} \geq \frac{1}{2}. $$ For the second one, since $\frac{1}{3} \leq \frac{1}{2}$, then: $$ \frac{1}{2} + \...

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Probability of 3 heads in 4 coin flips
2 votes

Hint: What is the probability that you will get exactly $0$ heads? What is the probability that you will get exactly one head? If it helps, there are $2^4$ possibilities for the sequence of four ...

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Washer method to find the volume of an object.
Accepted answer
2 votes

It is helpful to first draw a picture: Notice that $x = x^2$ implies that $x^2 - x = 0$ so that $x = 0$ or $x = 1$. So we are rotating the area around the line $y = 2$. We think of the volume we ...

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