Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

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34
votes
2answers
5k views

Showing that $\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$?

$$\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$$ In the second nested radical, the repeating pattern is $(-,-,+)$. I approached this ...
17
votes
0answers
492 views

Can you obtain $\pi$ using elements of $\mathbb{N}$, and finite number of basic arithmetic operations + exponentiation?

Is it possible to obtain $\pi$ from finite amount of operations $\{+,-,\cdot,\div,\wedge\}$ on $\mathbb{N}$ (or $\mathbb{Q}$, the answer will still be the same), note that set of all real numbers ...
15
votes
0answers
350 views

If $a^2+a b+b^2=40$ and $a^2-\sqrt{a b}+b=5$, then find $a^2+\sqrt{a b}+b$

I was given this problem to solve with elementary methods (High School level). Knowing that $$\begin{align} a^2+a b+b^2 &=40 \\ a^2-\sqrt{a b}+b &=\phantom{0}5 \end{align}$$ find $$a^2+\sqrt{...
8
votes
1answer
101 views

Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\to S$ satisfying $\sum \limits_{x\in T} x^{f(x)}\equiv a \pmod p$

This question is the third round of Iranian exam questions, which has not been answered for several years now. I think there are many people here, which may be able to solve this problem. From AOPS ...
7
votes
0answers
58 views

Find all $P(x)$ $\in$ $\mathbb{R}[x]$ such that for some $c \in \mathbb{R}$, the functional equation $ (x + 1)P(x - 1) - (x - 1)P(x) = c $ holds.

$\blacksquare~$ Problem: Find all $P(x)$ $\in$ $\mathbb{R}[x]$ such that for some $c \in \mathbb{R}$, the following functional equation holds \begin{align*} (x + 1)P(x - 1) - (x - 1)P(x) = c \end{...
7
votes
1answer
81 views

Spivak Calculus Chapter 1 Problem 5 (ii)

Prove : If $a < b$ then $-b < -a$ My proof : $a + (-b) < b + (-b) $ $a - b < 0$ $a - b + (-a) < 0 + (-a)$ $a + (-a) -b < -a$ $-b < -a$ Is my proof correct?
7
votes
1answer
121 views

Use of the fact that every function is sum of an odd and an even function.

It is well know that every real variable function $f$ can be written as a sum of an odd and an even function, namely $h$ and $g$ where: $$h(x) = {f(x)-f(-x)\over 2}\;\;\;\;\;\;\;\;\;\;\;\;g(x) = {f(x)+...
7
votes
0answers
146 views

Integral solutions to the equation $\left(\frac{1}{n}\right)^{-1/2}=\sqrt{a+\sqrt{15}}-\sqrt{a-\sqrt{15}}.$

Find all integral solutions to the equation $$\left(\frac{1}{n}\right)^{-1/2}=\sqrt{a+\sqrt{15}}-\sqrt{a-\sqrt{15}}.$$ Clearly, $\displaystyle \left(\frac{1}{n}\right)^{-1/2}=\sqrt{n}$, so $n\geq ...
7
votes
1answer
165 views

Can a product of a number and its reverse consist of only $1$'s?

Problem: Let $n \gt 1$. If you write the digits of $n$ in reverse, then multiply by original $n$, is it possible for the product to consist only of $1$'s? This came from a competition I recently ...
7
votes
0answers
208 views

Geometric/trigonometric origin of a surprising algebraic identity?

Let $f(x,y,z)=(x+y+z-1)^2-4xyz$. One can verify by inspection that for all $x,y,z$ $$f(x^2,y^2,z^2)=16 f\left(\frac{1+x}{2},\frac{1+y}{2},\frac{1+z}{2}\right)\cdot f\left(\frac{1-x}{2},\frac{1-y}{2},\...
7
votes
1answer
120 views

How to work out the trajectories of the cannons in Mario 64

So recently I've played one of my childhood games again, namely Super Mario 64, and as anyone who has played it as well knows, you will find cannons at specific locations that allow Mario to send ...
7
votes
1answer
241 views

Roots of $x+\frac 1x=2e^{i\phi}$

Let $\alpha$ and $\beta$ be the roots of $$x+\frac{1}{x}=2e^{i\phi}, \ 0<\phi<\pi$$ (a) Show that $\alpha+i$ and $\beta+i$ have the same argument and $|\alpha-i|=|\beta-i|$. (b) Find the locus ...
7
votes
0answers
169 views

How many triples $(x,y,n)$ are there such that $x^n - y^n = 2^{100}$

How many triples $(x,y) \in \mathbb{N^+}^2$ and $n \gt 1$ are there such that $x^n - y^n = 2^{100}$ I dont know how to start. Any hint will be helpful. I know the identity $x^n-y^n = (x-y)(...
7
votes
0answers
197 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 ...
7
votes
4answers
9k views

What is the difference between “Polynomial” and “Multinomial” in two or more variables?

What is the difference between "Polynomial" and "Multinomial" in two or more variables? Since, by definition: Multinomial: An algebraic expression having two or more (unlike) ...
6
votes
0answers
36 views

Unifying abstraction of duality between $A - B$ and $A + B$

I'm wondering whether there's an abstraction that unifies the special cases of dual or complementary equations of the form $A - B$ and $A + B$ that I've seen in math. Here are some examples: 1: Even ...
6
votes
0answers
133 views

$\frac{(x-y)^3}{x+y}\neq g(f(x)-f(y))$?

$$h(x,y)=\frac{(x-y)^3}{x+y}$$ Prove that there does not exist 1D real functions $f,g$ such that $h(x,y)=g(f(x)-f(y))$. The problem seems really really easy because it is obvious that $x+y\neq f(x)-...
6
votes
0answers
334 views

$a$ and $b$ are solutions of $ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $, $a+b=?$

$a$ and $b$ are solutions of $$ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $$ What is $a+b=?$ $$ $$ Are there better approaches than the one below? Solution: ...
6
votes
1answer
206 views

Prove that if $a,b,c \in \mathbb{R^+}\text{ and } abc=8\text{ then } {ab+4\over a+2}+{bc+4\over b+2}+{ca+4\over c+2}\ge6$

Question: Prove that if $a,b,c \in \mathbb{R^+}\text{ and } abc=8\text{ then } {ab+4\over a+2}+{bc+4\over b+2}+{ca+4\over c+2}\ge6$ My Approach: Now we have: $${ab+4\over a+2}={2\times (ab+4)\...
6
votes
1answer
295 views

How does synthetic division work?

I've read that we can divide any polynomial by a linear polynomial by synthetic division considerably faster than that by long division method. Now, I've learnt the steps to do so but I don't quite ...
6
votes
0answers
205 views

Product of squared sines: $\prod_{k=1}^{n-1}\prod_{j=1}^{n-1}\left[ \sin^2\left(\frac{k\pi}{2 n}\right) +\sin^2\left(\frac{j\pi}{2 n}\right)\right]$

I have a double product $$a(n)=2^{2 (n-1)^2} \prod_{k=1}^{n-1} \prod_{j=1}^{n-1} \left[ \sin^2 \left(\frac{k\pi}{2 n}\right) +\sin^2 \left(\frac{j\pi}{2 n}\right) \right]$$ which always gives ...
6
votes
0answers
97 views

Equality of Floors of some Partial Sums

Let $S_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}$ denote the $(n+1)^{st}$ partial sum in the series expansion for $e=\sum_{k\ge 0}\frac{1}{k!}$. I want to prove that $\lfloor n\cdot(...
6
votes
1answer
320 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 ...
6
votes
0answers
436 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 ...
6
votes
0answers
87 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= \...
6
votes
1answer
291 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
184 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=...
6
votes
1answer
253 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) ...
5
votes
1answer
73 views

The parametric ratio $\frac{x}{y}$ with known $x+y$ and $x\cdot y$

$x$ and $y$ are in fact $\lambda_1$ and $\lambda_2$, the bigger and smaller eigenvalues of a parametric matrix $A'A$, and $t$ is a very small constant. I have that $$ \begin{split} x+y &= 1+ \...
5
votes
0answers
63 views

Field of study dedicated to algebra 'tricks' such as change of variable, obscure identity substitution, etc.?

I'm an undergraduate math major and sometimes I find proofs that seem to use algebraic 'tricks' to reach their conclusions. The 'trick' that I see most often is a change of variable or the use of an ...
5
votes
0answers
135 views

Evaluate the sum $\binom{2n}{n}+\binom{3n}{n}+\binom{4n}{n}+\cdots+\binom{kn}{n}$

Evaluate the sum $$\binom{2n}{n}+\binom{3n}{n}+\binom{4n}{n} + \cdots +\binom{kn}{n}$$ My Attempt: Given sum = coefficient of $x^n$ in the expansion $$\{(1+x)^{n}+(1+x)^{2n}+(1+x)^{3n}+\cdots+(1+x)^{...
5
votes
0answers
44 views

Are there polynomials $P, Q$ with degree no less than 2018 and with integer coefficients, such that $P(Q(x))=3Q(P(x))+1$ for all real $x$?

Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that$$P(Q(x))=3Q(P(x))+1$$for all real numbers $x$. Attempt: I found a ...
5
votes
0answers
74 views

How do I see if $a^3+b^3=c^3+d^3$ has any solutions where $ 1 \le a,b,c,d \in \mathbb{Z} \le 1000$ and $a \ne b \ne c \ne d$?

How do I see if $a^3+b^3=c^3+d^3$ has any solutions where $ 1 \le a,b,c,d \in \mathbb{Z} \le 1000$ and $a \ne b \ne c \ne d$ ? I know I can write a program to brute force this and find out, but is ...
5
votes
1answer
151 views

A pre-calculus problem about a quadratic function

This is from a test (I'm not a high school student) given to rising high school juniors. The problem was designed to take less than 20 minutes, preferably 5-15. Judging from its source, this is not a ...
5
votes
0answers
199 views

If $f(x)$ is a polynomial function with integer coefficients find min value of $f(12)$

$f(x)$ is a polynomial function with integer coefficients satisfying $f(1)=5$ and $f(2)=7$. What is the smallest possible positive value of $f(12)$? I have no idea on how to begin with this question. ...
5
votes
2answers
202 views

How do I solve this algebra problem

The question goes, solve in real number. $x^5 - 5 x^3y - 5x^2 + 5xy^2 + 5y = 16 \tag{1}$ $ y^5 + 5xy^3 + 5y^2 + 5x^2y + 5x = -57 \tag{2}$ I tried simplifying the first equation to, $$ x^5 + 5\...
5
votes
0answers
109 views

Thinking like in ancient Greece

The next result is attributed to Archimedes: The equation $x^3-ax^2+(4/9)a^2b$ has a positive root if and only if $a>3b$, where $a$ and $b$ are positive real numbers. This problem appeared in the ...
5
votes
0answers
60 views

In proving that $\sqrt{a}$ is always irrational, $\forall a\in\left\{\Bbb R^+ : 1< a\neq b^2\right\}$… a different way.

I was trying to prove the following statement: $$\sqrt{a}\text{ is always irrational, }\forall a\in\left\{\mathbb{R}^+ : 1<a\neq b^2\right\}.\tag{$b\in\mathbb{Z}$}$$ I know there is at least one ...
5
votes
0answers
168 views

Show $\lim_{n\to \infty}{|A_{n}|+|A_{n-2}| \above 1.5pt |A_{n-1}|}=2$

Question: Let $A(n)$ be a finite square $n \times n$ matrix with entries $a_{ij}=1$ if $i+j$ is a prime; otherwise equals to $0$. I write $|A(n)|$ to count the number of $1$'s in $A(n)$. Is it ...
5
votes
3answers
106 views

Intersection of two x powers

Many months ago in class I came up with the problem: $$x^{(x+1)} = (x+1)^x$$ Using the solve function on my calculator I have found that the answer is around 2.29... This is backed up by the graph. ...
5
votes
3answers
355 views

Prove that $\sum\limits_{cyc}\frac{a}{a^2+ab+b^2+3}\leq\frac{1}{2}$

Let $a$, $b$ and $c$ be non-negative numbers. Prove that: $$\frac{a}{a^2+ab+b^2+3}+\frac{b}{b^2+bc+c^2+3}+\frac{c}{c^2+ca+a^2+3}\leq\frac{1}{2}$$ I think this inequality is very interesting because ...
5
votes
1answer
113 views

Why do this algorithm for finding an equation whose roots are cubes of the roots of the given equation works?

Let a polynomial, $p(x)$, of degree $n$ is given. Our aim is to find another polynomial, $q(x)$, whose roots are the cubes of the roots of $p(x)$. Our algorithm go like this: Step 1 Replace $x$ by $...
5
votes
1answer
321 views

When y is a function of itself

When playing around with equations, I've twice found myself in the dilemma where my dependent variable is dependent on itself. In the first instance of this occurring, I spent hours trying things but ...
5
votes
0answers
201 views

Strange Algebra

I am working on a mathematical induction worksheet and my professor gave us the key and I have run across something that makes zero sense to me so please explain if you can. Additional info: $k \ge2$ ...
5
votes
0answers
247 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]{\...
5
votes
1answer
65 views

two symmetric functions, when they have only one solution

My Question: For what $y$ is the equation $\log_{y}{x}=y^x$, does there exist only one solution. Some thoughts of mine: What I noticed was that for almost any $a$, both functions $\log_{y}{x}$ ...
5
votes
1answer
222 views

Finding the minimum value of a radical expression

If $a, b, c\ge 0$ with $(a+b)(b+c)(c+a) \ne 0$, find the minimum of $\sqrt { \frac { a }{ b+c } } +\sqrt [ 3 ]{ \frac { b }{ c+a } } +\sqrt [ 4 ]{ \frac { c }{ a+b } }$. The minimum is $\frac{3}{\...
5
votes
1answer
104 views

showing that an inequality holds

I am trying to figure out how to show that for $n\geq 3$, $$(2^n-1)^{\frac{n}{2(n-1)}}\geq (2^{n-1}-1)^{\frac{n-1}{2(n-2)}}+1.$$ I've tried basic algebra and induction, but the inductive hypothesis ...
5
votes
0answers
156 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+...
5
votes
1answer
213 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 ...

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