# 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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### 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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### How to solve $\frac{dy}{dx}-1=\frac{x^2}{y}$

How to solve $$\frac{dy}{dx}-1=\frac{x^2}{y}$$ I tried letting $y^2=u$ which gives $$y\frac{dy}{dx}=\frac{1}{2}\frac{du}{dx}$$ So the equation is now $$\frac{du}{dx}-2\sqrt{u}=2x^2$$ Any help?
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### $a$ 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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### Determine all ordered pairs $(m, k)$ for which $f(m, km)$ is a perfect square

Question Let $m$ and $k$ be positive integers. Let $f(m, km)$ be the number of rectangles (including squares) on a $m$ by $km$ checkerboard. Determine all ordered pairs $(m, k)$ for which $f(m, km)$ ...
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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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### Does the inequality $c_a \le xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a \le C_a$ hold?

Update: Posted in MO since it is unanswered in MSE Let $0 \le x,y \le 1$ and $a$ be a real. Consider the function $$f(x,y,a) = xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a \tag 1$$ For a fixed $a$, the ...
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### Finding a Digit of One Root of $x\sqrt{8}+\frac{1}{x\sqrt{8}}-\sqrt{8}$ Knowing The Digit of The Other Root

Consider $f(x)=x\sqrt{8}+\frac{1}{x\sqrt{8}}-\sqrt{8}$. The function has 2 real roots, say, $x_1$ and $x_2$. If the $1994$th digit in the decimal expansion of $x_1$ is $6$, what is the $1994$th digit ...
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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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### 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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### 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 ...
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### Is it possible to express the product of two real numbers in terms of their difference?

Some motivation: I have a (simplified) function $f(x) = \frac{k}x$, where $k$ is some real constant and $x$ is a real number. The result of the function, however, is always truncated to an integer for ...
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### Parametrization of integer solutions of the equation $a^2+b^2=c^2+d^2=2x^2$

I need the general form of integer solutions to this equation $$a^2+b^2=c^2+d^2=2x^2$$ Here is my partial solution:- The parametrization of the integer solutions of the equation $$p^2+q^2=2y^2$$ is ...
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### Verify proof: $a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$

A short disclaimer: I do know this question has been asked multiple times here and several answers (including combinatorics) have been given already. However, among all these posts, I did not find ...
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### Algebraic Method

Is there a way to solve the following equation algebraically? $(x+3/5)^{5x-3}=16$ So far, I have figured out a solution to be 7/5. This was solved by comparing the exponential form of 16 but I am not ...
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### Number Theory Question from Swedish Maths Olympiad: Find all integers $n\geq 8$ such that $n^{\frac{1}{n-7}}$ is also an integer.

I encountered this video about number theory question from Swedish Maths Olympiad: Find all integers $n\geq 8$ such that $n^{\frac{1}{n-7}}$ is also an integer. The video shows step by step solution....
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 ...