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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16
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226 views

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes. I've been thinking about this question, but ...
7
votes
0answers
161 views

Proving a geometric inequality without Lagrange multipliers

Let $e=(1,1,...,1)$ be the $n$-dimensional vector consisting only of ones. Let $r=\sqrt{\dfrac{n}{n-1}}$ and $\alpha \in (0,1)$ fixed. Given a vector $x=(x_1,x_2,...,x_n) \in \mathbb R^n$ such that ...
7
votes
0answers
1k views

Using distance formula to find slope, any reason to use the concluding equation?

So, today I was observing a class that I will be a TA for this semester and the professor started to talk about the distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Well, my mind wandered a little ...
6
votes
0answers
140 views

Summation - relatively simple?

I have a question which might be too simple for this site but I really tried many ideas without coming to a solution. This is assignment from elementary school in which I am trying to help and the ...
6
votes
0answers
107 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 ...
6
votes
0answers
318 views

Series sum formula

Is there any general formula to sum following series: $$S = 1^1 + 2^2 + 3^3 + \dotsb+(n - 1)^{n - 1} + n^n, n \in N$$ I mean for $S = f(n)$, is there a formula to compute $f(n)$? Regards, vishal.
5
votes
0answers
84 views

How to solve $A \; x = 1/x$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?

I would like to solve the following equation for $x\in\mathbb{R}^{n}$ $$A \; x = 1./x, \quad \text{with $A\in\mathbb{R}^{n\times n}$},$$ where $1./x$ denotes the ``element-wise inverse of the vector ...
5
votes
0answers
88 views

System of 3 equations

I am doing thermal calculation in electronics and when trying to device a general formula for equivalent system resistance to air flow of a part of real system, I ended with this system of three ...
5
votes
0answers
189 views

The same bit of trivial algebra in two different places?

The Villarceau circles are things whose existence is surprising. To find radii of Villarceau circles, I stupidly went through a bit of trigonometry and got a much simpler result than I expected, and ...
5
votes
0answers
73 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 ...
5
votes
0answers
209 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
64 views

Transitioning to Higher Level Mathematics

I am just finishing grade 12 pre-calculus at my school and have strong interest in math. The problem is, it seems some important elements of higher level math are not in my schools curriculum that are ...
4
votes
0answers
25 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
33 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 ...
4
votes
0answers
44 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
77 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
58 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
126 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
153 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
24 views

find all natural $n$ such that equation is a sixth power

We have $24n^3+6n+7$ and we need to find all natural $n$ for which this equation is a sixth power of natural number. My try: let $k \in N$ $24n^3+6n+7=k^6$ $24n^3+6n+6=k^6-1$ ...
3
votes
0answers
62 views

How to determine the length of $x$ in this sketch?

Consider the following construction: Assume that you are given the lengths of the sides $a,b,c,d,e$ as well as the size of the angle $\phi$. The task is to determine the length of $x$ in terms of the ...
3
votes
0answers
49 views

Prove that there is a real number $r>0$ such that…

Prove that there is a real number $r>0$ such that: There is no point in $\mathbb{R}^3$ with 3 rational coordinates, whose distance from $(0,0,0)$ equals $r$. In other words, if we build a sphere ...
3
votes
0answers
70 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 ...
3
votes
0answers
63 views

Factoring question from March $2013$ AMATYC exam

For how many pairs of positive integers $(n, \space m)$ with $n, \space m < 100$ are both of the polynomials $x^2 + mx + n$ and $x^2 + mx - n$ factorable over the integers? I have found four ...
3
votes
0answers
72 views

Solving $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$

If $m$ and $n$ are distinct positive integers then does the equation $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$ $\space$has any solution , for $x,y,z$ , in positive integers with $x,y,z$ all not equal ?
3
votes
0answers
54 views

fractional and integer system of equations

Solution for real ${a\;,b\;,c}$ in $a[a]+c\{c\}-b\{b\}=0.16$ $b[b]+a\{a\}-c\{c\} = 0.25$ $c[c]+b\{b\}-a\{a\} = 0.49$ Where $[x] =$ Integer part of $x$ and $\{x\} =$ fractional part of $x$ My ...
3
votes
0answers
34 views

Buchberger`s algorithm - Performance?

I want to solve an equation system with 3 complex variables, 3 complex equation and maximum degree 3. Is it reasonable to do this with Buchberger`s algorithm or should I better do it with an ...
3
votes
0answers
170 views

Complex slope of line $a\bar{z}+\bar{a}z+b = 0$

How can we prove......... [1] The Complex slope of the line $a\bar{z}+\bar{a}z+b = 0$ is $\displaystyle \omega = -\frac{a}{\bar{a}}$ [2] Complex slope of line joining the points $z_{1}$ and $z_{2}$ ...
3
votes
0answers
74 views

Triangular numbers and the harmonic mean

This morning somebody called my attention to the fact that $4T_n$ is the harmonic mean of $T_{2n}$ and $T_{2n+1}$, where $T_n=n(n+1)/2$ is the $n$th triangular number. I verified this algebraically ...
3
votes
0answers
167 views

Recognizing subadditivity

Let $f: (0,\infty) \to \mathbb{R}$ be some continuous function. We say that $f$ is subadditive if the bound \begin{align} f(x+y) \leq f(x) + f(y) \tag{1} \end{align} holds. I was attempting to ...
3
votes
0answers
57 views

Determining algebraically a point of intersection.

A student I was tutoring posed the question: "I know how to solve $$e^{-x} = \ln x$$ graphically, however how do you solve this algebraically?" I have been fiddling around with it for a while and I ...
3
votes
0answers
41 views

Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.

To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following: (somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
3
votes
0answers
40 views

Forward differencing (and back)

I read that "forward differencing" is a method for "evaluating polynomials at uniformly spaced intervals". That is, if I have parabola like $y = x^2$ and increase $x$ by 1, I can use it. My question ...
3
votes
0answers
62 views

Existence of roots of a polynomial equation when coefficients have varying weights

I have two $n-$degree polynomials $f_{1}(p)$ and $f_{2}(p)$, where the domain of $p\in[0,1]$. I know that $\exists$ $0 < p_{1} < 1$ such that: $f_{1}(p_{1}) = f_{2}(p_{1})$. Let ...
3
votes
0answers
137 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 ...
3
votes
0answers
114 views

How to solve this equation with another way?

I have the equation $$\left(\dfrac{x+1}{x-2}\right)^2 + \left(\dfrac{x+1}{x-3}\right) = 12\left(\dfrac{x-2}{x-3}\right)^2$$ I want to solve this equation in the set of all real numbers. First way. ...
3
votes
0answers
90 views

Functional inverse of $(a + b\sin\theta)^2\tan\theta$

So, I revisited to the situation involved in my previous question, with the intent of generalizing it to any two masses and charges. When I started going through the model again, beginning with the ...
3
votes
0answers
65 views

Uses for the generalised f-mean, functions with larger/smaller f-means

What are some uses of the generalized f-mean outside of the geometric mean and the power means? Also, is there a known way to compare two functions and find out which will yield a larger f-mean (ex: ...
3
votes
0answers
188 views

Maximum size of a rotated-then-cropped rectangle

With regard to topic/question New size of a rotated-then-cropped rectangle: The answer by Isaac, the maximum area is $b^2\csc\alpha\sec\alpha$ when $x=0.5b\csc\alpha = 0.5b/\sin\alpha$ seems to ...
3
votes
0answers
133 views

Algorithm/Formula to compute adding and/or removing compound and/or non-compound percentages from a value?

I will first start with a scenario, I have to apply some adjustments to a particular value. These adjustments are either compound or non-compounded and they can either be added or subtracted to the ...
2
votes
0answers
35 views

Understanding underlying algebra for calculus convergence problem

I'm working on series convergence/divergence problems in my Calc 2 class, and (as has happened often), I'm hung up on some underlying algebra. The first step in the solution manual for a problem I'm ...
2
votes
0answers
45 views

Don't know when to add negative numbers

I'm definitely not a math person and only did general mathematics in high school, and unfortunately, not paying as much attention to that as I should have. Well, I'm doing Discrete Mathematics in my ...
2
votes
0answers
49 views

What would be a good book to self study basic algebra?

I am asking about basic algebra so that I can tie it into learning about number theory and set theory. before I tackle Geometry, and College Algebra/Analytical Geometry. I will top it off with ...
2
votes
0answers
59 views

Bretschneider-Brahmagupta-Heron Proof

Derive Bretschneider's formula, Brahmagupta's formula and Heron's formula in one memorable elegant proof. I ask this question merely to see the creativity of the MSE community when it comes to ...
2
votes
0answers
25 views

Proportions in Venn Diagrams

P, L and G are 3 different games and a player can play any 1, 2 or 3 of them. I have complete data for P and L , but only have 86% of the trials for G have been completed. I want to figure out what ...
2
votes
0answers
40 views

How to solve the quadratic equation problem with a strict proof

Consider $f(x)=x^2-(a+b)x+ab$ with $n\le a\le b\le n+1$ where $n$ is a positive integer. Find the range of $\min\{f(n),f(n+1)\}$ Sorry I just made a mistake,now is fixed. This problem is obvious in ...
2
votes
0answers
47 views

Is there a word for a number that can be expressed as an exponential with the same base and exponent?

Some examples: \begin{align*} 1 &= 1^1 \\ 4 &= 2^2 \\ 27 &= 3^3 \\ 256 &= 4^4 \\ 3125 &= 5^5 \\ \end{align*} and so on. Is there a name for these types of numbers? It seems like ...
2
votes
0answers
54 views

Proof that $p(z)^2=a^2$ always has a nonreal solution.

Let $p(z)$ be a nonconstant integer polynomial of degree $n$ such that $p(0)=0$ and let $a$ be a nonzero real number. It seems that $$p(z)^2=a^2$$ Always has a nonreal solution (in $z$) if ...
2
votes
0answers
66 views

What is the line of thinking to get $[(n^2+3n+1)^2-5n(n+1)^2]$ from $(n^4+n^3+n^2+n+1)$?

There is this one little tiny step along the working that I don't quite understand, but I think it is better if I write the whole problem and solution for clarity. Problem: Factorise $5^{1995}-1$ ...
2
votes
0answers
31 views

Roots of a polynomial. Rotating the unknown quantity

Let $r\in\mathbb{R}$, $p\in\mathbb{C}\left[ x\right] $ and $q\left( x\right) =p\left( ix\right) $. The following reasoning is false: $p\left( r\right) =0$ iff $p\left( i\left( -ir\right) ...