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.

learn more… | top users | synonyms (2)

24
votes
0answers
428 views

Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by ...
11
votes
0answers
132 views

Can we find this infinite root in term of elementary function?

Let $f(x)=\left(x+f(x+1)\right)^\frac{1}{x}$. What is the value of $f(2)$ ? More precisely, how to find the value of $$\sqrt{2+\sqrt[3]{3+\sqrt[4]{4+\cdots}}}~?$$ Thank you.
9
votes
0answers
337 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 ...
8
votes
0answers
279 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 ...
8
votes
0answers
74 views

Minimising an expression - involving polynomial

I found this one on a forum but it has been unanswered from long there. I am curious to know if there is a solution to this problem. Here it is: Let n be a positive integer. Determine the smallest ...
8
votes
0answers
197 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 ...
8
votes
0answers
2k 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 ...
7
votes
0answers
81 views

How are inequalities from IMO built?

I notice that there are lots of apparently difficult inequalities in IMO. Are there some techniques to manipulate well-known inequalities in order to built a difficult exercise? What are the main ...
7
votes
0answers
213 views

Proving a geometric inequality without Lagrange multipliers

Let $e=(1,1,\ldots,1)$ 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 $x=(x_1,x_2,\ldots,x_n) \in \mathbb R^n$ such ...
6
votes
0answers
369 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
94 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 ...
5
votes
0answers
105 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
143 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
76 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
251 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
105 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
4
votes
0answers
101 views

$15a+6b+4c+8d=0$ implies $ax^3+bx^2+cx+d$ has a positive root

Let $a,b,c,d$ be real numbers such that $15a+6b+4c+8d=0$. Show that $f(x)=ax^3+bx^2+cx+d$ has a positive root. I want to try to use the intermediate value theorem, showing that $f(s)<0$ and ...
4
votes
0answers
176 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
76 views

How to find $f$ and $g$ if $f\circ g$ and $g\circ f$ are given?

The question is: Let $f:\mathbb R\rightarrow \mathbb R$ and $g:\mathbb R\rightarrow \mathbb R$ be two functions such that $(f\circ g)(x)=4x^2+4x+1$ and $(g\circ f)(X)=x^2+2x+2$. Find $f(x)$ and ...
4
votes
0answers
74 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
48 views

Order of operations involving mod (%)

I was taking a programming test last night that had a math equation that simplified to 11 % 2 * 3, no () or likewise. When I compute it, being taught modulous occurs at the same level of ...
4
votes
0answers
36 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
101 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 ...
4
votes
0answers
50 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
49 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
86 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
67 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
157 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
161 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
74 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
68 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
34 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 ...
3
votes
0answers
99 views

Generating functions to solve number of integer solution problem

If I have $x_1 + x_2 + x_3 =10$ with $1\leq x_1 \leq 5, \; 2 \leq x_2 \leq 6, \;3 \leq x_3 \leq 9$ I know that I compute $(t^1+\dots + t^5)(t^2 +\dots + t^6)(t^3+\dots +t^9)$ and look at the ...
3
votes
0answers
52 views

Determining whether points are collinear

$(1,1)(3,9)(6,21)$ The way I figured that this should be solved is by finding the slope of: $(1,1)(3,9)$ Then, $(3,9)(6,21)$ Finally $(1,1)(6,21)$ Which are 4, 4,and 4 respectively. So I ...
3
votes
0answers
93 views

Simplifying $\sqrt[3]{a\pm\sqrt{b}}$

Let $$x=\sqrt{a\pm\sqrt{b}}$$ We know that $$x=\sqrt{\frac{a+\sqrt{a^2-b}}{2}}\pm\sqrt{\frac{a-\sqrt{a^2-b}}{2}}$$ But, what about cubic root? Let $$y=\sqrt[3]{a\pm\sqrt{b}}$$ Is there any formula to ...
3
votes
0answers
79 views

How prove: $a=x$ and $b=x^x$ for $x^{a+b}=a^b b$?

Let $x, a, b$ natural numbers such that $x^{a+b}=a^b b$. How prove: $a=x$ and $b=x^x$?
3
votes
0answers
52 views

Websites for math tests/quizzes

Next semester I'm taking calculus at college and I was looking for websites that have quizzes/test for things like trigonometry, trig formulas, pre-calculus, calculus readiness, etc. so I can get ...
3
votes
0answers
49 views

Clock based Question

It is between 2 and 3'o clock; but a person looking at the clock and mistaking the hour hand for the minute hand fancies that the time of the day is 57 minutes earlier than reality. What is the true ...
3
votes
0answers
124 views

The n-th k-gonal number

I was doing some school work and got bored so I started messing with k-gonal numbers. I started with the triangular numbers, square numbers and looked for patterns. I noticed something. Let ...
3
votes
0answers
51 views

Solving a system of 3 variables

How to solve or what is the algorithm to solve a system of equations like this: $$\eqalign{ (x +\phantom{3} z)^2 + (y +\phantom{3} w)^2 &= 52\cr (x + 3z)^2 + (y + 3w)^2 &= 296\cr (x ...
3
votes
0answers
34 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
72 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
55 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
316 views

How to calculate this triple summation?

I need to calculate the following summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m-j}\choose{k-j}}}{k\choose j}r^{k-j+i}$$ I do not know if it is a well-known summation or not. ...
3
votes
0answers
69 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
80 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
56 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
37 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
341 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
82 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 ...