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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11 votes
3 answers
326 views

Is $x=2,y=13$ the unique solution?

Problem: Find all positive integers $x$ and $y$ satisfying: $$12x^4-6x^2+1=y^2.$$ If $x=1, 12x^4-6x^2+1=12-6+1=7,$ which is not a perfect square. If $x=2, 12x^4-6x^2+1=192-24+1=169=13^2$, which is a ...
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  • 1,970
9 votes
0 answers
268 views

Set $S$ of integers $\ge0$ such that $\{1,2,\cdots,n\}$ partitions into $A, B$ with $\sum_\limits{a\in A}a^k=\sum_\limits{b\in B}b^k$ for all $k\in S$

Let $s_n$ be the largest size of a set $S$ of integers $\ge0$ st there exist two subsets $A,B\subseteq \{1,2,...,n\}$ that satisfy the conditions (1) $A\cap B=\emptyset$, (2) $A\cup B=\{1,2,...,n\}$, (...
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  • 164
8 votes
0 answers
173 views

The most generic radicals-solvable quintic

It's well known that it is impossible to solve a generic quintic equation in terms of radicals involving its coefficients. However: what's the "most generic" quintic equation that is still ...
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  • 321
8 votes
0 answers
342 views

Proving an inequality with a function that uses matrix inverses

Let $A$ be a matrix with all nonnegative entries and row sums strictly less than one, let $v$ be a vector with all entries between zero and one, and let $B\equiv\left(I-A\mathrm{diag}\left(v\right)\...
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  • 59
8 votes
0 answers
747 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: ...
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  • 4,202
7 votes
1 answer
153 views

How many ways are there to choose subsets $S$ and $T$ of $A=\{1,2,3,,.....,n\}$ so that $S$ contains $T$?

How many ways are there to choose subsets $S$ and $T$ of $A=\{1,2,3,,.....,n\}$ so that $S$ contains $T$? My attempt : The number of all subsets of $A$ is $2^n$. Let's denote this subsets by $S_{\...
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7 votes
0 answers
110 views

Prove the converse of "The sum of two odd consecutive numbers is a multiple of 4"

The sum of two odd consecutive numbers is a multiple of 4. I've tried rewriting this as: If $a$ and $b$ are two consecutive odd numbers, then $a+b=4p$, where $p\in\mathbb{N}$. I'm trying to prove the ...
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7 votes
0 answers
122 views

Finding sum of the roots of $4(x-\sqrt x)^2-7x+7\sqrt x=2$

Find sum of the roots of $$4(x-\sqrt x)^2-7x+7\sqrt x=2$$ By substituting $t=x-\sqrt x$ we have $4t^2-7t-2=0$ $$4t^2-8t+t-2=0$$ $$(4t+1)(t-2)=0$$ So we get $x-\sqrt x=2$ Hence $x=4$. Or $x-\sqrt x=-\...
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  • 5,575
7 votes
1 answer
260 views

Limits of negative-power tower

Consider the following: $$n \uparrow -\Bigl((n+1) \uparrow -\bigl((n+2) \uparrow \cdots \uparrow -m \bigr)\Bigr)= n^{{{-(n+1)}^{-(n+2)}}^{\cdots^{-m}}}$$ It doesn't converge for $m \to \infty$, but ...
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  • 1,505
7 votes
0 answers
186 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 ...
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  • 9,602
7 votes
1 answer
187 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 ...
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  • 2,710
7 votes
1 answer
167 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 ...
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7 votes
3 answers
426 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 ...
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7 votes
0 answers
226 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 ...
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6 votes
0 answers
153 views

Prove that $2017$ is a divisor of $2016!(x-\frac{x^2}{2}+\frac{x^3}{3}-....-\frac{x^{2016}}{2016})$.

Let $x$ be an integer such that $2017$ is a divisor of $x^2+x+1$ . Prove that $2017$ is a divisor of $2016!(x-\frac{x^2}{2}+\frac{x^3}{3}-....-\frac{x^{2016}}{2016})$. We define "segment ...
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  • 1,130
6 votes
0 answers
158 views

Finding a Solution to a Log-Linear System of Equations, or Showing Existence of Such a Solution

I'm trying to find the solution ($x^*_1, y^*_1, x^*_2, y^*_2$) to the following system of equations: $$ gx_1=\lambda\left(\log \frac{x_2}{1-x_2}-\log \frac{x_2 + y_2}{2-x_2-y_2}\right)\\ by_1=\lambda\...
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6 votes
1 answer
56 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 ...
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6 votes
0 answers
87 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{...
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6 votes
1 answer
121 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 ...
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6 votes
0 answers
233 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},\...
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6 votes
1 answer
438 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 ...
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6 votes
0 answers
221 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 ...
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6 votes
1 answer
129 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(...
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6 votes
1 answer
519 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 ...
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6 votes
0 answers
92 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= \...
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  • 50.1k
6 votes
1 answer
310 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 ...
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  • 5,403
6 votes
0 answers
196 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=...
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  • 61
6 votes
1 answer
263 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) ...
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  • 1,881
5 votes
0 answers
63 views

To find $n$ such that the expansion of $(1+x)^n$ has three consecutive coefficients $p,q,r$ that satisfy $p:q:r = 1:7:35$.

To find $n$ such that the expansion of $(1+x)^n$ has three consecutive coefficients $p,q,r$ that satisfy $$p:q:r = 1:7:35$$ My work: Suppose the consecutive coefficients are $\binom{n}{k-1}, \binom{n}...
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  • 11.9k
5 votes
4 answers
169 views

Solve the equation $\sin x+\cos x=k \sin x \cos x$ for real $x$, where $k$ is a real constant.

As I had solved the equation when $k=1$ in Quora and MSE by two methods, I started to investigate the equation for any real constant $k$: $$ \sin x+\cos x=k \sin x \cos x, $$ I first rewrite the ...
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  • 5,156
5 votes
0 answers
79 views

How to prove the following determinant identity

Prove: $$ \begin{array}{|cccccccccc|} 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \cdots & 0 \\ x & x & x & \cdots & x & y & y & y & \cdots ...
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5 votes
1 answer
93 views

$O$ is the circumcenter of non-right $\triangle ABC$. $\frac{|AB \cdot CO|}{|AC \cdot BO|} = \frac{|AB \cdot BO|}{|AC \cdot CO|} = 3$. Find $\tan A$

Problem: $O$ is the circumcenter of $\triangle ABC$, which is not a right triangle. $$\frac{| AB \cdot CO |}{|AC \cdot BO|} = \frac{|AB \cdot BO|}{|AC \cdot CO|} = 3$$. Find $\tan A$. Here $AB \cdot ...
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  • 743
5 votes
0 answers
112 views

If $0<a, b, c, d<1$, then how do I prove the following inequality $(a+b+c+d+1)^{2} \geq 4\left(a^{2}+b^{2}+c^{2}+d^{2}\right)$?

An interesting problem, I kind of solved it, but I'm not sure that I solved it correctly. Different solutions are welcome. My attempt: Let $x=a+b+c+d$ and $y=a^{2}+b^{2}+c^{2}+d^{2} .$ We easily see ...
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  • 1,788
5 votes
2 answers
99 views

Is there a formal name for equations and inequalities containing parameters (known variables, coefficients)?

Is there a formal name for equations and inequalities containing parameters (known variables, coefficients)? Is it just "equations with parameters"? "general equations"? Here are ...
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5 votes
1 answer
47 views

Is there meaning in the minimal instances of a variable you need to write a rational expression?

This is something I've never thought about before. Given a rational function $f \in \mathbf{k}(x)$, the minimum number of $x$ you need to write down a formula for $f$ on its domain is well-defined. My ...
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  • 18.2k
5 votes
0 answers
62 views

Why is the recurrence $f_n=\frac{5f_{n-1} + 1}{25 f_{n-2}}$ cyclic?

Given: $f_1 = a$ $f_2 = b$ and $f_n = \dfrac{5f_{n-1} + 1}{25 \cdot f_{n-2}}$ You can just start doing the algebra to show $f_3 = \dfrac{5b + 1}{25a}$ $f_4 = \dfrac{5a + 5b + 1}{125ab}$ $...
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  • 123
5 votes
1 answer
78 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+ \...
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  • 61
5 votes
0 answers
103 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 ...
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5 votes
0 answers
80 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 ...
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  • 3,816
5 votes
1 answer
214 views

How to prove that $e^x +1- \sin(x) > 0$?

So I have found a solution but I don't think it's the best one. So by saying that $$ |\sin x|\le 1 (\forall x \in R) \Rightarrow -1 \le \sin x \le 1 \Rightarrow 0 \le 1- \sin x$$ and if we add $$ +e^x ...
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5 votes
0 answers
95 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 ...
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  • 1,245
5 votes
0 answers
461 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. ...
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  • 9,740
5 votes
0 answers
121 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 ...
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5 votes
0 answers
73 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 ...
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  • 9,125
5 votes
0 answers
178 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 ...
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5 votes
3 answers
113 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. ...
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5 votes
1 answer
137 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 $...
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  • 2,239
5 votes
1 answer
945 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 ...
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5 votes
0 answers
231 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: $k \ge2$ ...
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  • 251
5 votes
1 answer
68 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 $\log_{y}{x}$ ...
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