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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Explaining the fence-post formula [duplicate]

From the book: Set Theory for Begginners I call the formula “n-m+1” the fence post formula . If you construct a 3-foot fence by placing a fence-post every foot ,then the fence will consist of 4 fence ...
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3 votes
3 answers
94 views

Prove that $\frac{a}{a^2+\lambda}+\frac{b}{b^2+\lambda}+\frac{c}{c^2+\lambda} \leq \frac{3}{\lambda +1}$

If $a+b+c=3$ ,$ a,b,c>0$ and $\lambda \geq 1$, prove that : $$\frac{a}{a^2+\lambda}+\frac{b}{b^2+\lambda}+\frac{c}{c^2+\lambda} \leq \frac{3}{\lambda +1}.$$ my attempt: using CBC twice and ...
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1 vote
3 answers
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Why do you have to check all the 'sections' of an inequality when solving for x?

For instance, the inequality $x^2+3x+2>0$ factors into $(x+2)(x+1)>0$ If this were just an equation (i.e ...=0) you would know the solutions are x={-2, -1}. But, because it's an inequality, you ...
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-4 votes
0 answers
18 views

If the ratio of principle and the monthly simple interest is 192:1. What is the yearly rate of simple interest? [closed]

Please help me If the ratio of principle and the monthly simple interest is 192:1. How can I find the yearly rate of simple interest?
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3 votes
1 answer
34 views

Computing the log of a sum of exponentials

in a Coursera course by UW I've come across this piece of code computing the log of a sum of exponentials. ...
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0 votes
1 answer
37 views

Let $a_n$ be the number of $X$-strings of length $n$ that do not contain $344$ as a sub-string. Find a recurrence relation for $a_n$.

Let $X = \{1,2,3,4\}$ and $a_n$ be the number of $X$-strings of length $n$ that do not contain $344$ as a sub-string. Find a recurrence relation for $a_n$. We can construct a string of length $n$ in ...
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0 answers
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Can we draw a closed path made up of 9 line segments , each of which intersects exactly one of the other segments?

The solution given in Fomin's book is as follows. If such a closed path were possible, then all the line segments could be partitioned into pairs of intersecting segments. But then the number of ...
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3 votes
1 answer
101 views

$365(x)^{364}-365(x)^{365}+x^{365}=0.9$

I can't seem to solve this, I tried using multiple software but it says it doesn't support this kind of equation: $$365(x)^{364}-365(x)^{365}+x^{365}=0.9$$ Context: Hey! I was having fun with tricked ...
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0 votes
2 answers
106 views

Proving $ \sqrt{n} = a + \frac{-n + a^2}{a + \sqrt{n}}$

I'm a high school student, I do math advanced and extention, I was told to prove the equation: $$ \sqrt{n} = a + \frac{-n + a^2}{a + \sqrt{n}} $$ My teacher originally gave me the equation as: $$\sqrt{...
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2 answers
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An algebraic question from math olympiad .

There is a 1 gm weight on each side of a balance. If Harry Potter casts a spell with his magic wand on any of the weights, then the mass of that particular weight doubles, but the other weight remains ...
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0 answers
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Inner rectangle with prespecified axis

I have a bunch of points that forms multiple clusters and I am looking to find an inner rectangle for each of the clusters. The sides of the inner rectangle must be parallel to some predefined sides. ...
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  • 399
1 vote
2 answers
74 views

Is there a name for this technique involving breaking a term into multiple terms?

I recently saw a solution to the quadratic equation $x^2-5x-6=0$ that involved re-writing the middle term, $-5x$, into two terms, $x-6x$, so that the expression could be factored and $x$ solved for, ...
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  • 353
1 vote
0 answers
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Why does average of the sum of numbers to n not equal the average of all numbers to n-1 and n [duplicate]

I am very sorry since this is the first time I am here and I unsure how to type any math. Anyway. The sum of all numbers to any n is (n^2+n)/2, and I divided that by n to get the average at that point....
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1 vote
1 answer
61 views

prove that $\arctan\frac{\cos x-\sin x}{\cos x+\sin x}=\frac{\pi}{4}-x$, where $0<x<\pi$

I tried to solve this $$\begin{align} \arctan\frac{\cos x-\sin x}{\cos x+\sin x}&=\arctan\frac{1-\tan x}{1+\tan x}\\&=\arctan\frac{\tan\frac{\pi}{4}-\tan x}{1+\tan\frac{\pi}{4}\tan x}\\&=\...
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1 vote
1 answer
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Is there a closed form of $x_2x_3\cdots x_n + x_1x_3\cdots x_n + \dots + x_1x_2\cdots x_{n-1}$?

Given real numbers $x_1,\dots,x_n \in \mathbb{R}$, does there exist a closed form for the expression $$A_n := x_2x_3\cdots x_n + x_1x_3\cdots x_n + \dots + x_1x_2\cdots x_{n-1} = \sum_{i=1}^n \prod_{j=...
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  • 2,063
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2 answers
81 views

Show that $[(n+1)(2n+1)]^n > 6^n n!$ without using induction

Show that for all n>1 we have that $[(n+1)(2n+1)]^n > 6^n n!$ . n belongs to integers . What i considered was as $(n+1)(2n+1)$ function is increasing . So we have $(n+1)(2n+1)>6 \forall n&...
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7 votes
1 answer
220 views

Puzzling function

Let $f$ be a function whose domain is the set of positive integers, and for positive integers $a$, $b$ and $n$, if $a + b = 2^{n}$, then $f(a) + f(b) = n^2$. What is $f(2021)$? I started by testing ...
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1 vote
0 answers
38 views

Simplifying atan(x)-atan(y)

From this question post and this youtube video "proof", I am convince that for $x>0$ and $y>0$, the following holds true: $$ \tan^{-1}(x)-\tan^{-1}(y)=\tan^{-1}\left(\frac{x-y}{1+xy}\...
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-1 votes
1 answer
29 views

match between a function graph and it's derivative graph [closed]

I need help with this question I only manage to match 3 functions: 1-B 2-C 3-D
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3 votes
2 answers
98 views

Is $0$ the root of the equation $\frac{x^2}{ x}= 0$?

I want to understand if I understand the concept of equation correctly. For this I want to know if $0$ is the root of the equation $\frac{x^2}{x} = 0$. If we simplify the equation then we can get ...
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1 vote
1 answer
52 views

Eliminating x = 0 from an equation when x can be equal to all real numbers

If we have the equation: $$2+0x=a+0x$$ The solution to this equation for $x$ is all real numbers. If I then multiply each side by $x$ I get: $$2x=ax$$ The solution to this equation for $x$ is still ...
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  • 13
0 votes
2 answers
107 views

Why is the Arccosine of $30$ degrees undefined?

Recently, while working on Trigonometry, a problem came up in which I was asked to evaluate the value of $\cos(\arccos(30^\circ))$, and I stated that the value of this function was $30$ degrees (...
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-4 votes
0 answers
45 views

How to solve these equations to find the value of $n$, $n\log n = 10^6$ ,$2^n = 10^6$ and $n! = 10^6$? [closed]

Recently, I am learning algorithms and I found these type of equations formed while calculating running time of algorithms. Please help me how to solve these equations to find the value of $n$? $$n\...
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0 votes
1 answer
67 views

How to solve equations involving reciprocal square roots of quadratics? [closed]

$$\frac{2x-8}{\:\sqrt{2x^2-16x+34}}+\frac{2x-3}{\sqrt{2x^2-6x+5}}=0$$ Is it possible to solve this equations? If yes then how?
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1 vote
1 answer
73 views

Prove if sum of complex numbers = 0 and their magnitudes = 1, then sum of their squares = 0

This problem is from Chapter 6 (Basics of Complex Numbers) of the AoPS Precalculus textbook. Show that if complex numbers $ w_1+w_2+w_3 = 0$ and $|w_1|=|w_2|=|w_3| = 1$, then $w_1^2 + w_2^2 + w_3^2 = ...
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4 votes
3 answers
103 views

Find the integer solution: $a+b+c=3d$, $\: a^{2} + b^{2} + c^{2}= 4d^{2}-2d+1$

Find the integer solutions: $$a+b+c=3d$$ $$a^{2} + b^{2} + c^{2}= 4d^{2}-2d+1$$ Attempt: Notice that $a=b=c=d=1$ is a solution. Other facts: Notice that $a^{2} + b^{2} + c^{2} > 0$, so $4d^{2}-...
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0 votes
0 answers
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A Question Regarding the Differences of Terminologies and Theorems Related to Polynomial Division

This will be a long post and there will be a TL;DR at the end. I've recently been re-reading topics on polynomial division to brush up my knowledge on them but sometimes I get a little confused and ...
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-2 votes
1 answer
82 views

How does $\dfrac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}}$ become $1$? [closed]

$$\dfrac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}}$$ The answer to this question is "1" but I have no idea how !! Please show the steps to solve the problem.
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0 votes
0 answers
22 views

Deriving from Doppler's formula

I am studying Doppler sonography. I do understand doppler's formula: $$ f=f_{0}\frac{v}{v-u}, $$ where $f_{0}$ is emitted frequency, f is received frequency, v is speed of the ultrasound puls and u is ...
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5 votes
2 answers
75 views

Find the maximum of $\frac{abc}{(4a+1)(9a+b)(4b+c)(9c+1)}$,where$a,b,c>0$

$a,b,c>0$, find the maximum of : $$\frac{abc}{(4a+1)(9a+b)(4b+c)(9c+1)}$$ I try to find the minimum of $\frac{(4a+1)(9a+b)(4b+c)(9c+1)}{abc}=\frac{4a+1}{\sqrt{a}}\cdot\frac{9a+b}{\sqrt{ab}}\cdot\...
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  • 139
5 votes
5 answers
284 views

prove or disprove that:$a^2+b^2+c^2\leq \frac{27}{4}$

I tried to solve the below problem, I spend more than 5h just for prove it but without any result , this is the best attempt I did ,just I need to show that $a^2+b^2+c^2\leq \frac{27}{4}$ if $a,b,c&...
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4 votes
2 answers
80 views

Is it possible to maximize $\frac{3t^2}{t^3+4}$ (where $t>0$) without taking derivative?

To find the maximum of $f(t)=\dfrac{3t^2}{t^3+4}$ (for $t>0$) we can simply equate the derivative with zero, $$f'(t)=0\Rightarrow 6t(t^3+4)-3t^2(3t^2)=0\Rightarrow -3t^4+24t=0\Rightarrow t=2$$ And $...
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  • 1,007
1 vote
1 answer
84 views

How to extract variable values from interrelated algebraic equations using programming?

I need help with solving a set of equations that are interrelated. The equations are: $$ R(\omega_{s})+j X(\omega_{s})=R_{\mathrm{s}}+\frac{\left(R_{0}+\frac{1}{j \omega_{s} C_{0}}\right)\left(R_{m}+\...
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1 vote
1 answer
38 views

Train Traveling from Aytown to Beetown Accident (AHSME 1955)

I was working on the question: A train traveling from Aytown to Beetown meets with an accident after 1 hour. The train is stopped for 30 minutes, after which it proceeds at four-fifths of its usual ...
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-3 votes
1 answer
41 views

fence-post formula [closed]

Would you explain this fence-post formula : n-m+1 exactly ,why is +1?
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3 votes
1 answer
124 views

Why am I getting extraneous solutions solving $\frac{1+\sqrt{1-x}}{x-\sqrt{1-x^{2}}}=2 x$?

I came across a beautiful problem: Solve $$\frac{1+\sqrt{1-x}}{x-\sqrt{1-x^{2}}}=2 x$$ My approach: Obviously $x=0$ is not a solution. Now we have: $$1+\sqrt{1-x}=2x^2-2x\sqrt{1-x^2} \tag1$$ ...
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-1 votes
0 answers
34 views

Extracting the leading non zero element from a list

Given a circular list of 10 numbers. Each element $e_i$ in the list ($0 \leq e_i \leq1)$. There are always either 1 or 2 non zero elements in the list. Case 1: In case we have 1 non zero element, it ...
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0 votes
0 answers
26 views

Queuing models - Solving problem with different service time 1/μ

I'm struggling with a problem because the service time is different depending on the number of tickets purchased. hereinafter the case: "The owner wants to reduce the waiting and serving time for ...
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0 votes
0 answers
31 views

What is the proof for the $(\sqrt[n]{a})^n$ equaling $a$ or $|a|$ if $n$ is odd or even?

If I use $64$ and $-64$ as the radicands, and have $2$ or $3$ as the indices, I know that it all works out arithmetically (except for $\sqrt[2]{-64}$ which has no real solutions) but I'm not sure how ...
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  • 115
2 votes
2 answers
55 views

Finding the set of parameters for which the inequality holds for all $x,y$

I encountered the following question about inequalities which I am curious how to solve. The simplest case is to consider the inequality $$|x|+|y|+|x+y|+ax+by\geq 0$$ where $x,y,a,b\in\mathbb{R}$. The ...
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0 votes
2 answers
94 views

Cities $A$ and $B$ are $999$km apart. Road signs every km show distances to each city. How many signs use only two digits (without trial and error)?

Recently, I came across a question that seemed as if it can only be solved using trial error. Here it is: There are two cities, $A$ and $B$, their distance is $999$km. Along a road that connects $A$ ...
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1 vote
0 answers
56 views

finding a linear map over natural numbers

Consider a map $f$ from $\mathbb{N}^N$ to $\mathbb{Z}$ as follows: $$ f(n_1, n_2, \cdots, n_N) = n_1 m_1 + n_2 m_2 + \cdots + n_N m_N. $$ Where $n_i$'s are natural numbers with $0\leq n_i \leq M$. The ...
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-2 votes
0 answers
46 views

Sketching $X\cup Y$, $X\cap Y$, $X-Y$, $Y-X$, where $X=\{(x,y):x^2+y^2\leq1\}$ and $Y=\{(x,y):-1\leq y\leq 0\}$

Image contains the question as well as its answer. Expecting someone to verify the sketch.
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-1 votes
2 answers
68 views

How can you solve algebraic equations (eg, $10x+y=3xy$, knowing that $x$ and $y$ are integers between $1$ and $9$) without trial and error?

Often, I will come across questions which involve equations that need to be solved by trial and error. For example, the equation $$10x+y=3xy$$ In this case, the question provides us that $x\neq y$ and ...
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0 votes
2 answers
73 views

Range of rational function in different ways.

I was looking for range of a rational function, I used three ways to get the answer and all of them are giving different results . e.g. f(x)=1/(1-x²) then, (1) ...
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2 votes
0 answers
102 views

Will there always be prime numbers of the form $X_i =i^2+i+1$?

The sequence is: $$1,3,7,13,21,31,...$$ $$ OR $$ $$(1),(1+2),(1+2+4),(1+2+4+6), (1+2+4+6+8), (1+2+4+6+8+...),...$$ $$OR$$ $$X_i =i^2+i+1$$ $$OR$$ $$X_i = 1+2T_i$$ ($T$ for triangular numbers) $$\begin{...
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  • 1,399
6 votes
2 answers
69 views

Explaining symmetry in sequences defined by $\frac{a_{n+1}}{n+1}=\frac{a_n}{n}-\frac12$. (Eg, for $a_1=5$, we get $5,9,12,14,15,15,14,12,9,5$)

I've noticed recently the following fact: consider ratios $\frac{a_n}{n}$ with $n = 1,\dots,10$. If we require that $$ \frac{a_{n+1}}{n+1} = \frac{a_n}{n} - \frac12 $$ that is, each successive ratio ...
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  • 34.1k
-1 votes
2 answers
25 views

Questin on algebraic manipulation of exponential

i don't understand how we went from the 2nd line to the 3rd line in order to have $\cos x$ and $\sin x$. This is used in order to prove that $\cos z$ is analytic, i know there is easier ways to ...
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10 votes
2 answers
171 views

Given a sequence with $a_1=1$ and $a_{n+1}=a_n-\frac13a_n^2$. Is there an easier way to get an upper bound of $1/a_{100}$?

Suppose that a sequence $\{a_n\}_{n=1}^\infty$ satisfies $a_1=1$ and $$a_{n+1}=a_n-\frac13a_n^2,\qquad n=1,2,3,\cdots.$$ Which of the following is true? (A) $2<100a_{100}<\frac52$ (B) $\frac52&...
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  • 6,608
1 vote
0 answers
23 views

Solve Inequalities to find boundary for one variable

X<0 and Y<0 : as we are having sorted array and we are picking two pairs. arr: -7, -3 ,3, 4, 7 so I can have number in this way ...
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