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.
4,705
questions with no upvoted or accepted answers
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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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11
votes
2
answers
468
views
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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10
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0
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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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9
votes
0
answers
188
views
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 $f(m, km)$ ...
9
votes
0
answers
295
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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8
votes
0
answers
153
views
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 ...
- 18.3k
8
votes
0
answers
157
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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 ...
user1059904
8
votes
0
answers
273
views
Uniqueness of a constrained system of linear equations
I would like to determine whether there exists a solution (and if so, check uniqueness) to the following system of linear equations (with respect to $\eta = (\eta_1,...,\eta_J))$:
$$\begin{aligned} \...
- 117
8
votes
1
answer
198
views
Can a product of a number and its reverse consist of only $1$'s?
Problem:
Let $n \gt 1$. If you write the digits of $n$ in reverse, then multiply by original $n$, is it possible for the product to consist only of $1$'s?
This came from a competition I recently ...
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7
votes
0
answers
78
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Subset $B$ of $S\subseteq\mathbb{N}^+$ such that no two elements in $B$ add up to another element in $B$, and $|B|\ge \frac 13 |S|$.
Given a set $S\subseteq\mathbb{N}+$ with a finite number of elements, show that one can always choose a set $B\subseteq S$ such that for all $x, y, z\in B$ (not necessarily distinct), $x+y\neq z$ and $...
user1200609
7
votes
0
answers
220
views
Integral solutions to the equation $\left(\frac{1}{n}\right)^{-1/2}=\sqrt{a+\sqrt{15}}-\sqrt{a-\sqrt{15}}.$
Find all integral solutions to the equation
$$\left(\frac{1}{n}\right)^{-1/2}=\sqrt{a+\sqrt{15}}-\sqrt{a-\sqrt{15}}.$$
Clearly, $\displaystyle \left(\frac{1}{n}\right)^{-1/2}=\sqrt{n}$, so $n\geq ...
- 10.9k
7
votes
2
answers
540
views
How does synthetic division work?
I've read that we can divide any polynomial by a linear polynomial by synthetic division considerably faster than that by long division method.
Now, I've learnt the steps to do so but I don't quite ...
- 473
7
votes
3
answers
483
views
Prove that $\sum\limits_{cyc}\frac{a}{a^2+ab+b^2+3}\leq\frac{1}{2}$
Let $a$, $b$ and $c$ be non-negative numbers. Prove that:
$$\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}$$
I think this inequality is very interesting because ...
- 198k
7
votes
1
answer
332
views
Trigonometry or inequality problem
Today, I saw this question:
If $x,y,z \in [0,\frac\pi 2]$, $x+y+z=\frac{3\pi}{4}$ and $\sec^2(x)\sec^2(y)\sec^2(z)=8$, calculate $E=\tan x\tan y+\tan y\tan z+\tan z\tan x$
My first thought was ...
- 5,583
7
votes
0
answers
251
views
Functional inverse of $(a + b\sin\theta)^2\tan\theta$
So, I revisited to the situation involved in my previous question, with the intent of generalizing it to any two masses and charges. When I started going through the model again, beginning with the ...
- 1,505
7
votes
1
answer
178
views
Calculating $e^{ar}+e^{ar^2}+...+e^{ar^n}$
Calculate the sum,
$$e^{ar}+e^{ar^2}+...+e^{ar^n}\ \text{where} \ a,r\in \mathbb{R}$$
It's easy to calculate the sum when the powers of $e$ are in an arithmetic progression. How do we proceed when the ...
user694028
6
votes
0
answers
142
views
Find the fourth roots of the following binomial surd: $14+8\sqrt{3}$
Find the fourth roots of the following binomial surd: $X=14+8\sqrt{3}$
I attempt to find the square root first: $\sqrt{X}=\sqrt{14+8\sqrt{3}}$
$\sqrt{14+8\sqrt{3}}=\sqrt{x_1}+\sqrt{y_1}$
$(\sqrt{14+8\...
- 1,356
6
votes
0
answers
170
views
Proving $\cos(\tfrac \pi{18})$ cannot be written in the form $\frac{\sqrt a-\sqrt b}{\sqrt c -\sqrt d}$
For reasons I'm not going to get too into, I'm interested in a number of the form $p+q \cos(\tfrac \pi{18})$, where $p^2,q^2$ are rational. A question that occurred to me is whether or not such a ...
- 1,433
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votes
0
answers
97
views
On the topic of Trinomial Expansion
So I was looking at the Wikipedia of the binomial theorem when I read this:
Multi-binomial theorem
When working in more than two dimensions, it is often useful to deal with products of binomial ...
- 2,128
6
votes
0
answers
101
views
To find $n$ such that the expansion of $(1+x)^n$ has three consecutive coefficients $p,q,r$ that satisfy $p:q:r = 1:7:35$.
To find $n$ such that the expansion of $(1+x)^n$ has three consecutive coefficients $p,q,r$ that satisfy $$p:q:r = 1:7:35$$
My work:
Suppose the consecutive coefficients are $\binom{n}{k-1}, \binom{n}...
- 12.6k
6
votes
0
answers
98
views
How to prove the following determinant identity
Prove:
$$
\begin{array}{|cccccccccc|} 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \cdots & 0 \\ x & x & x & \cdots & x & y & y & y & \cdots ...
- 359
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votes
0
answers
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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 ...
- 1,180
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votes
0
answers
169
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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:
$$
gx_1=\lambda\left(\log \frac{x_2}{1-x_2}-\log \frac{x_2 + y_2}{2-x_2-y_2}\right)\\
by_1=\lambda\...
- 127
6
votes
0
answers
301
views
Find all $P(x)$ $\in$ $\mathbb{R}[x]$ such that for some $c \in \mathbb{R}$, the functional equation $ (x + 1)P(x - 1) - (x - 1)P(x) = c $ holds.
$\blacksquare~$ Problem: Find all $P(x)$ $\in$ $\mathbb{R}[x]$ such that for some $c \in \mathbb{R}$, the following functional equation holds
\begin{align*}
(x + 1)P(x - 1) - (x - 1)P(x) = c
\end{...
- 2,321
6
votes
1
answer
128
views
Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\to S$ satisfying $\sum \limits_{x\in T} x^{f(x)}\equiv a \pmod p$
This question is the third round of Iranian exam questions, which has not been answered for several years now. I think there are many people here, which may be able to solve this problem.
From AOPS
...
- 92.9k
6
votes
0
answers
243
views
Geometric/trigonometric origin of a surprising algebraic identity?
Let $f(x,y,z)=(x+y+z-1)^2-4xyz$. One can verify by inspection that for all $x,y,z$ $$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},\...
- 15.5k
6
votes
0
answers
231
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 ...
- 61
6
votes
1
answer
139
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
854
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 ...
- 3,641
6
votes
0
answers
95
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 $$\displaystyle S= \...
- 52.2k
6
votes
0
answers
167
views
Is there a name for systems of equations with min and max functions included?
In a big project I'm working on, I'm running into systems of equations that look like the following:
$$a = \min(b, c)$$
$$b = d^2 + a$$
$$c = \max(a + b, d)$$
Basically, nonlinear systems of ...
- 61
6
votes
1
answer
319
views
How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?
I would like to solve the following equation for $x\in\mathbb{R}^{n}$
$$\mathrm{diag}(x) \; A \; x = \mathbf{1}, \quad \text{with $A\in\mathbb{R}^{n\times n}$},$$
where $\mathrm{diag}(x)$ is a ...
- 123
6
votes
0
answers
116
views
Reference? filler: IRS, Rhind Papyrus, High-school algebra
I believe something like this was included as a filler in one of the MAA journals many years ago. I am searching for the exact reference (for the filler, or an earlier source).
Someone dies, and ...
- 108k
6
votes
1
answer
157
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How to solve the equation $ (x-2)^{\log_{100}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2\log_{10}(x-2)}$?
If $\displaystyle (x-2)^{\log_{10^2}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2.\log_{10}(x-2)}$, then value of $x$ is ...
My Try
Let$$\log_{10}(x-2) = y \quad \Leftrightarrow \quad (x-2)=10^y .$$
Then$$(10)^{...
- 52.2k
6
votes
0
answers
208
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:
$$\displaystyle \large x_n=...
- 61
6
votes
0
answers
233
views
How to transform series of series into series
I need to prove this equation.
$$ \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) ...
- 307
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1
answer
273
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) ...
- 1,929
6
votes
1
answer
659
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 ...
6
votes
0
answers
279
views
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 ...
- 161
6
votes
2
answers
394
views
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 ...
user1135823
5
votes
0
answers
95
views
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 ...
- 2,904
5
votes
0
answers
69
views
Why is the average magnitude of the imaginary parts of the roots of these polynomials greater than that of the real part?
Let $f_n(x)$ be a polynomial obtained by replacing $10$ with $x$ in the base $10$ expansion of $n, n \ge 10$. Eg. $f_{98}(x) = 9x+8$ and $f_{2021}(x) = 2x^3 + 2x + 1$. $f_n(x)$ has exactly $[k = \log_{...
- 18.3k
5
votes
0
answers
172
views
Weird way to factor $2x^4 - 15 x^2 -27.$
A normal person would factor $2x^4 - 15 x^2 -27$ by treating $x^2$ as the variable. I'm grading a test now and the student has come up with the extremely clever idea to add and subtract $6x^3 + 3x^2 ...
- 32.7k
5
votes
0
answers
70
views
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 ...
- 229
5
votes
0
answers
223
views
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....
- 10.7k
5
votes
1
answer
51
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 ...
- 18.8k
5
votes
0
answers
64
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
$f_3 = \dfrac{5b + 1}{25a}$
$f_4 = \dfrac{5a + 5b + 1}{125ab}$
$...
- 123
5
votes
1
answer
85
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
$$
\begin{split}
x+y &= 1+ \...
- 61
5
votes
0
answers
99
views
Are there polynomials $P, Q$ with degree no less than 2018 and with integer coefficients, such that $P(Q(x))=3Q(P(x))+1$ for all real $x$?
Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that$$P(Q(x))=3Q(P(x))+1$$for all real numbers $x$.
Attempt:
I found a ...
- 4,244
5
votes
1
answer
327
views
How to prove that $e^x +1- \sin(x) > 0$?
So I have found a solution but I don't think it's the best one.
So by saying that $$ |\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 ...
