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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18
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0answers
264 views

Two interview questions

I recently came across two interview questions for admission in B.Math at an university. I gave the two questions a try and want to know if my solutions are correct or not. Q1: Given that ...
9
votes
0answers
347 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
412 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 ...
7
votes
0answers
78 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
214 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 ...
5
votes
0answers
82 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 ...
5
votes
0answers
81 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 ...
5
votes
0answers
145 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
80 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
125 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
73 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
105 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
321 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
94 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, ...
4
votes
0answers
66 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 ...
4
votes
0answers
33 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
70 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) = $ ...
4
votes
0answers
89 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 ...
4
votes
0answers
111 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} ...
4
votes
0answers
192 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
66 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
469 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. ...
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
240 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
171 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
185 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
13 views

Maximum value of the smallest number of operations to obtain configuration from original configuration

Let $n$ be a positive integer. There are $n(n+1)/2$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each ...
3
votes
0answers
56 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 ...
3
votes
0answers
30 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
28 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,\ ...
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
62 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
71 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
51 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
110 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
85 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
46 views

Triangles, sine and cosine problem

Hi everyone I tried solving this countless times but I always get the wrong answer! what I did first is 600/tan(46) - 600/tan(40) and that sounded reasonable to find the answer! but I keep getting it ...
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
73 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
115 views

Prove that: $ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}b_{i}b_{j} \right)$

Let $a_{1}, \cdots, a_{n}, b_{1}, \cdots, b_{n}$ be positive real numbers. Prove that: $$ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq ...
3
votes
0answers
210 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
102 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
204 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 ...