# 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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### 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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### 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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### 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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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 $B\equiv\left(I-A\mathrm{diag}\left(v\right)\... 8 votes 0 answers 747 views ###$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: ... 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_{\... 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 ... 122 views

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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: $k \ge2$ ...
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 $\log_{y}{x}$ ...