Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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9
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0answers
50 views

What to keep in mind when attempting proof of basic properties of divisibility/what techniques are useful/what's the intuition for showing them?

So I am currently trying to prove some basic divsiibility relations, as follows. If $a \mid b$ and $a \mid c$, then $a \mid (b + c)$. If $a \mid b$ and $s \in \mathbb{Z}$, then $a \mid sb$. ...
9
votes
0answers
366 views

How can I construct a solution for this system of many inequalities?

Let there be types $\omega\in\{0,1\}^n$ drawn according to some probability distribution. Suppose that these types are relayed through some imperfect message service. Specifically, any type $\omega$'s ...
9
votes
0answers
433 views

Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and calculus....
8
votes
0answers
47 views

Intuition behind the proof of the validity of the Euclidean algorithm

As the question title suggests, could anybody explain to me their intuition behind the proof of the validity of the Euclidean algorithm?
8
votes
0answers
98 views

Use a direct proof to show that if $x\gt 1$ then $x^5 +x+1\gt 2$.

Use a direct proof to show that if $x\gt 1$ then $x^5 +x+1\gt 2$. It's obvious that if $x\gt 1$ then $x+1\gt 2$ , also $x^5\gt0 \;as \;x\gt1\gt0$, thus $x^5+x+1\gt 2$. Is it fine ? I can't ...
7
votes
0answers
87 views

What is the criterion for a matrix containing vectors and their permutations being invertible?

Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix $...
7
votes
0answers
222 views

Trigonometry or inequality problem

Today, I saw this question: If $x,y,z \in [0,\frac\pi 2]$, $x+y+z=\frac{3\pi}{4}$ and $\sec^2(x)\sec^2(y)\sec^2(z)=8$, calculate $E=\tan x\tan y+\tan y\tan z+\tan z\tan x$ My first thought was ...
6
votes
0answers
109 views

Proving that $u_n $ is arithmetic sequence

Let $u_n$ be a sequence defined on natural numbers (the first term is $u_0$) and the terms are natural numbers ($u_n\in \mathbb{N}$ ) We defined the following sequences: $$\displaystyle \large x_n=...
5
votes
0answers
72 views

How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and ...
5
votes
0answers
95 views

Probability that the roots of a quadratic equation are real

Roots of the quadratic equation $x^2+5x+3=0$ are $4\sin^2\alpha+a$ and $4\cos^2\alpha+a$. Another quadratic equation is $x^2+px+q=0$ where $p,q\in\mathbb{N}$ and $p,q\in[1,10]$. Find the probability ...
5
votes
0answers
82 views

Probability of another 3 integers with same sum and product as the first 3 integers

Let us suppose $3$ integers are selected at random from a large range, say $$-1000\leq x\leq y\leq z\leq 1000$$ Now, we define the sum and product: $$\begin{align*}s&=x+y+z \\p&=xyz\end{align*...
5
votes
0answers
163 views

Solving a question by using special products (Students debate to Teacher)

So today,we got back our exam papers,and we found a question marked wrongly and teacher said that it is wrong.We all students do NOT believe this.So here is what happened. Before reading the next ...
5
votes
0answers
91 views

Prove $\log_ab+\log_bc+\log_ca\geq1+\log_{ab}bc+\log_{bc}ab$

Prove inequality $$\log_ab+\log_bc+\log_ca\geq1+\log_{ab}bc+\log_{bc}ab$$ for $a>1,b>1,c>1.$ Inequality is interesting because of asymmetry and inhomogeneity and I think the solution might ...
5
votes
0answers
128 views

How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?

I would like to solve the following equation for $x\in\mathbb{R}^{n}$ $$\mathrm{diag}(x) \; A \; x = \mathbf{1}, \quad \text{with $A\in\mathbb{R}^{n\times n}$},$$ where $\mathrm{diag}(x)$ is a ...
5
votes
0answers
74 views

Reference? filler: IRS, Rhind Papyrus, High-school algebra

I believe something like this was included as a filler in one of the MAA journals many years ago. I am searching for the exact reference (for the filler, or an earlier source). Someone dies, and ...
5
votes
0answers
331 views

Bloom of Thymaridas

I'm interested in learning more about the Bloom of Thymaridas, a description of which can be found here. Obviously the mathematics behind the identity is not particularly deep from a modern ...
4
votes
0answers
89 views

Find the range of $a$ for $a^2 - bc - 8a + 7 = 0$ and $b^2 + c^2 + bc - 6 a + b = 0$.

Let $a, b, c$ be real numbers such that \begin{cases} a^2 - bc - 8a + 7 = 0\\ b^2 + c^2 + bc - 6 a + b = 0\\ \end{cases} Find the range of $a$. By adding the two equations I have $$(a - 7)^2 + (b + ...
4
votes
0answers
100 views

Given $a+b+..=a^7+b^7+..=0$ show that $a(a+b)..=0$

Question: Suppose $a,b,c,d$ are real numbers such that $a+b+c+d=a^7+b^7+c^7+d^7=0$ Show that $a(a+b)(a+c)(a+d)=0$ My attempt: Using $a+b+c+d=0$, I get $a(a+b)(a+c)(a+d)= 0 \implies a=0, \text{...
4
votes
0answers
36 views

Is this a valid way for performing polynomial division?

While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials. Suppose you have a quartic ...
4
votes
0answers
71 views

If $f(n)= \binom{n}{0}a^{n-1}-..+(-1)^{n-1}\binom{n}{n-1}a^{0}$ ,Then $f(2007)+f(2008) $

If $\displaystyle a= \frac{1}{3^{223}}+1$ and $\displaystyle f(n)= \binom{n}{0}a^{n-1}-\binom{n}{1}a^{n-2}+...........+(-1)^{n-1}\binom{n}{n-1}a^{0}$ Then value of $f(2007)+f(2008) = $ $\bf{My\;...
4
votes
0answers
94 views

Finding the minimum value of a radical expression

If $a$, $b$ and $c$ are positive real numbers, find the minimum value of $\sqrt { \frac { a }{ b+c } } +\sqrt [ 3 ]{ \frac { b }{ c+a } } +\sqrt [ 4 ]{ \frac { c }{ a+b } } $. I am not able to ...
4
votes
0answers
48 views

sum of the Series $\sum^{n}_{r=0}(-1)^r\binom{n}{r}\left[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+…\bf{m\; terms}\right]$

The sum of the Series $\displaystyle \sum^{n}_{r=0}(-1)^r\binom{n}{r}\left[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+........\bf{m\; terms}\right]$ $\bf{My\; Try::}$Let $$\displaystyle S= ...
4
votes
0answers
118 views

How to solve this equation in $ \mathbb{C} $?

From a small simple calculation , we get the following formulas: $ \begin{cases} e^x = \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n}}{(3n)!} \Big) + \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n+...
4
votes
0answers
193 views

Writing sum of square roots with symmetric polynomials

I want to write the function $$ F_N=\sum_{i=1}^N\sqrt{x_i} $$ in terms of the $N$ elementary symmetric polynomials of the $N$ positive variables $x_1,\dots,x_N$. The $N=1$ case is trivial, as we ...
4
votes
0answers
83 views

Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
4
votes
0answers
69 views

Is there a name for systems of equations with min and max functions included?

In a big project I'm working on, I'm running into systems of equations that look like the following: $$a = \min(b, c)$$ $$b = d^2 + a$$ $$c = \max(a + b, d)$$ Basically, nonlinear systems of ...
4
votes
0answers
55 views

Existence of Solutions of Two Cubic Equations in a Particular Region

If I have two cubic equations in two variables, $ax^3 + bx^2 y + cxy^2+\dots=0$ and another one with different coefficients, and I know that $(x,y)=(0,0)$ or $(1,1)$ are solutions, are there any nice, ...
4
votes
0answers
99 views

Polynomial bound

Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that $$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$ Suppose that $P(x)> 0$ for all ...
4
votes
0answers
87 views

Digits of two irrational numbers, given their power with fixed number of digits

I have $a, b \in \mathbb{R} \setminus \mathbb{Q}$, I want to know the result of $a^b$, but I don't know exact $a, b$ because I write them in numeric form. My question is how many digits of $a, b$ have ...
4
votes
0answers
246 views

Mandelbrot set's border in parametric form

I've post this question just because I'm curious, Mandelbrot set is defined as: $ z_{n+1} = z^2_n + c $, if $n \rightarrow \infty $ and it doesn't diverge we get the border. This border is unlimited ...
4
votes
0answers
172 views

How to transform series of series into series

I need to prove this equation. $$ \sum_{k=0}^{i-2} \left( e \space α(k+1)\space\frac{(-1)^{i+k+2}}{(i-k-2)!} \right) = \sum_{k=0}^{i-2} \frac{(i-k)^k}{k!} \space e^{i-k} (-1)^k\space where,\space α(i) ...
4
votes
0answers
187 views

How many natural numbers $x, y$ are possible if $(x - y)^2 = \frac{4xy}{(x + y - 1)}$.

How many natural numbers $x$, $y$ are possible if $(x - y)^2 = \frac{4xy}{x + y - 1}$. Does this system has infinite solutions which can be generalized for some integer $k \geq 2?$ $(x - y)^2(x + y) ...
3
votes
0answers
64 views

Is it possible to simplify a nested radical in the form $\sqrt[3]{\sqrt[3]{A}-B}$ into $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$

I'm wondering if there is a way to simplify any nested radical in the form $\sqrt[3]{\sqrt[3]{A}-B}$ into $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$. Examples such as $\sqrt[3]{7\sqrt[3]{20}-1}=\sqrt[3]{\...
3
votes
0answers
61 views

Find the polynomial $p(x)$

A polynomial $p(x)$ gives a remainder of $1$ when divided by $x^{100}$ and a remainder of $2$ when divided by $(x-2)^3$. Evaluate $p(x)$. By the Remainder Theorem, $p(x)$ can be written as $$p(x)...
3
votes
0answers
34 views

Selling oranges when people queue up in a line

We have $a$ oranges to give to $b$ people. Each person has a value $f(n)$ for receiving $n$ oranges, where $f$ is a nondecreasing, nonnegative function that is the same for everyone. Let $X$ be the ...
3
votes
0answers
29 views

In search of a College Leve Pre Algebra Applications Textbook

I am looking for a textbook at the college level that mainly focuses on applying algebra to situations. I want students to know how to set up the equations, not just solve them.
3
votes
0answers
18 views

Confusion regarding direct and indirect propotion.

If $x\ \propto \ y$ and $y\ \propto \ \dfrac{1}{z}$, then which of the following is correct $\color{green}{a.)\ x\ \propto \ z} \\ b.)\ x\ \propto \ \dfrac{1}{z}$ I did $x=k_{1}y,\ y=\dfrac{k_{2}}{...
3
votes
0answers
63 views

Prove that $S = 8S_1-4S_2$ in the geometrical figure

Let squares be constructed on the sides $BC,CA,AB$ of triangle $ABC$, all to the outside of the triangle, and let $A_1,B_1,C_1$ be their centers. Starting from the triangle $A_1B_1C_1$ one ...
3
votes
0answers
132 views

Approach to this integral

According to a standard literature $$\frac{1}{\sqrt{2\pi}q}\int^{-d}_{−∞} \sum ^∞_{k=1} \frac{(A^2/2q^2)^k}{2^k(k!^2)} e^\frac{−r^2}{2q^2}He_{2k}(\frac{r}{q})dr=\frac{1}{\sqrt{2\pi}} \sum ^∞_{k=1} \...
3
votes
0answers
67 views

What is $\textit{the}$ discriminant of a degree $n$ polynomial?

In my high school algebra class the teacher (who is me) says that the discriminant of a quadratic polynomial $ax^2 + bx + c$ is $b^2 - 4ac$. I have read in the Wikipedia article that the discriminant ...
3
votes
0answers
76 views

Motivation of Vieta's transformation

The depressed cubic equation $y^3 +py + q = 0$ can be solved with Vieta's transformation (or Vieta's substitution) $y = z - \frac{p}{3 \cdot z}.$ This reduces the cubic equation to a quadratic ...
3
votes
0answers
52 views

What can be said about a function with rotational symmetry of order other than 2?

It is well known that an odd function is a function whose graph has rotational symmetry of order $2$ (about the origin). Suppose the graph of $f:U \to \Bbb{R}$ has rotational symmetry of some higher ...
3
votes
0answers
112 views

Finding exact roots

I know of the rational root theorem to find all rational zeros and Newtons method of approximating zeros, but what if all the solutions are irrational/imaginary and you need exact answers for the ...
3
votes
0answers
59 views

What is the maximum amount of solutions to $f(x+1)f(x)= ax^2+bx+c$.

$f(x)$ is going to be in the form $mx+h$ thus, $(mx+m+h)(mx+h) = ax^2+bx+c$. With basic algebra $m= \pm \sqrt{a}$. Also $(m+h)(h)=c$. I would guess that because $(m+h)h=c$ has two solutions max if $m$ ...
3
votes
0answers
88 views

Formula for nth number of the following sequence:

I have two number sequences but have failed to find a formula for the nth term and, also, the formula for the sum of the sequence. First sequence: ...
3
votes
0answers
65 views

How does one solve $y^y-x^x=x$ for $x$ as a function of $y$?

In order to find the answer to this question I started thinking that as a first step to obtain the first and second column, one would have to solve the equation: $$y^y-x^x=x$$ for $x$ as a function ...
3
votes
0answers
74 views

Number of integral solutions to a polynomial

Given a polynomial of $n$th order, represented by $$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}+a_{n-1}x+a_{n}=0$$ Is it possible to find the number of integral solutions/roots to ...
3
votes
0answers
220 views

Solving an exponential equation without the quadratic formula

High school math student here. In my homework I was asked to solve $16^x +4^{x+1} - 3= 0$ and I used substitution to get $x=\log_4{(-2+\sqrt7)}$. However, this was in the chapter on ...
3
votes
0answers
105 views

If $A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is,

If $\displaystyle A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots\cdots\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is, where $\lfloor A\rfloor = A-\{A\}.$ $\...
3
votes
0answers
188 views

Closed formula for the numbers of the form $\sqrt{1+\sqrt{4+\sqrt{9}}}$

how can i find the formula for the nth term of this series? SQ = square root $\sqrt{1} = 1$ $\sqrt{1 +\sqrt{4}} = \sqrt{3}$ $\sqrt{1 +\sqrt{4+\sqrt{9}}} \approx 1.909385061$ $\sqrt{1 +\sqrt{4+\...