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.
45,204
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Consider a quadratic equation $az^2+bz+c=0$, where $a, b, c$ are complex number. Then Find condition for one purely imaginary root.
Consider a quadratic equation $az^2+bz+c=0$, where $a, b, c$ are complex number.
Then condition that above equation has one purely imaginary root
(A)$(a\bar b+\bar ab)(b\bar c+ \bar b c )+(c \bar a -\...
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is tan(π/2 - x)=cot(x) because tan in the 2nd area is "negative" and also cot(-x) equals to -cot(x)?
trying to figure out if that negative/positive calculation is true.
and so at the end there will be two negative and the cot(x) will be positive.
is that so?
thank you.
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15
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A transcendental function?
Let $(u_n)_n=(p_n/q_n)_n$ $\left(\text{with $\mathrm{GCD}(p_n,q_n)=1$ and }\ln|u_n|=o(2^n)\right)$ a sequence of rational numbers that is not ultimately zero.
I want to prove that the function $f(z)=\...
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2
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25
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Using quadratic equations to solve ratio questions
There are two parts to the question but I have answered the first part.
The exchange rate in 1964 was $x$ dollars to £1. An American tourist remembered that in 1952 he needed $1.5$ dollars more for ...
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2
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Compare $~a,b,c~$ with $~1,2,3~$ - Revisited. The virtue of meta-cheating
This is a self-answer question that revisits a question that the original poster deleted, themself.
The original question is
Let $0 \le a \le b \le c$ such that $a+b+c \geq 6,ab+bc+ca=11,abc=6$.
...
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44
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I can't figure out how the average can be proven here to be greater than the variable shown
I have a solution to a math problem and I can't figure out the algebra for the life of me. How is
$$\frac{i+j}{2} > k$$ ?
My intution tells me that since the average of $\frac{3}{2}$ and $\frac{2}{...
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1
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Let $T$ be the following set of ordered triplets,$T=\{(a,b,c):a,b,c\in N\}$. Find the number of elements in $T$ such that $L.C.M(a,b,c)=72$.
Let $T$ be the following set of ordered triplets,$T=\{(a,b,c):a,b,c\in N\}$. Find the number of elements in $T$ such that $L.C.M(a,b,c)=72$.
My Attempt
Let $a=2^{x_1}3^{y_1}$,$b=2^{x_2}3^{y_2}$ and $c=...
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1
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Can I infer the hypotenuse given only $a > b$?
Today, I came up with a problem. The problem is this:
Let $a$, $b$ and $c$ be the sides of a right-angled triangle, such that $a > b$. The nature of $c$ is unknown. Can I infer the hypotenuse given ...
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4
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Bounds on the maximum real root of a polynomial with coefficients $-1,0,1$
Suppose I have a polynomial that is given a form
$$
f(x)=x^n - a_{n-1}x^{n-1} - \ldots - a_1x - 1
$$
where each $a_k$ can be either $0,1$.
I've tried a bunch of examples and found that the maximum ...
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2
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How do we derive the expression?
I am studying a book which gives the following expressions:
$\alpha = \frac{\beta_1 d}{2} + \frac{\beta_2 d}{2}$ and $\gamma = \frac{\beta_1 d}{2n} - \frac{\beta_2 d}{2n}$.
Then it needs to solve for $...
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How to develop the algebraic expression? [closed]
Find the development of the algebraic expression: $\frac{(2+3)(2^2+3^2)...(2^{2048}+3^{2048})+2^{4096}}{3^{2048}}$
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Does the Distributive Property of Multiplication Over Addition apply to Absolutely Convergent Infinite Sums
Does the Distributive Property of Multiplication Over Addition apply to Absolutely Convergent Infinite Sums
For example, is it true that $\sum\limits_{n=1}^\infty (k\cdot a_n) = k\left(\sum\limits_{n=...
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2
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Understanding a proof that, if $|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$, then the minimum value of $|a-b|$, over distinct reals $a$ and $b$, is $2$.
I saw this epic question in Advanced Problems in Mathematics by Vikas Gupta:
If $$|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$$ then find the minimum value of $|a-b|$, where $a$ and $b$ are distinct real numbers....
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1
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Help recreating algebraic reasoning on the Euclidean metric.
I am trying to recreate and formalize with Lean4 this hint from "Euclidean plane and its relatives" by Petrunin:
$$
\sqrt{(x_{1} + x_{2}) ^ {2} + (y_{1} + y_{2}) ^ {2}} \leq \sqrt{x_{1} ^ {2}...
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Inverse of a Function With a Specific Domain [closed]
What is the Inverse of a Function $x^2 - 9$, For $x \geqslant -9$?
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1
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explanation of this cosine solution
I was studying Physics and I stumbled upon this equation: $$U_0 \cos{x}(\cos{x} - 1) = \alpha U_0$$ So you should get the value of x to do some other stuff.
The equation doesn't seem to complex, ...
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3
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Two Quadratics, the roots of one are the coefficients of the other and vice versa.... can they exist?
I spent hours on this... feel pretty pathetic. I read this on an old AMATYC exam:
Let $\quad c, \quad$ and $\quad d\quad$ be the roots of $\quad x^2+ax+b$
and let $\quad a, \quad$ and $\quad b\quad$ ...
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1
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2
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Finding the quotient and remainder when dividing $f = 2018X^{2019} - 2019X^{2018} + 1$ to $(X-1)^2$ [duplicate]
Let $f = 2018X^{2019} - 2019X^{2018} + 1, f \in \mathbb{R}[X]$. Find the quotient and the remainder when dividing $f$ to $(X-1)^2$.
The answer to this problem is that $f = (X-1)^2(2018X^{2017} + ...
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0
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25
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Hudde's cubic proof
I've been following the proof of Hudde's description of Cardano's method of cubic roots shown here https://proofwiki.org/wiki/Cardano's_Formula. Does anyone know where this proof comes from? I can't ...
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5
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Solve $x^2+4y^2+80 = 15x+30y \\ xy=6 $
Solve for $x$ and $y$:
$$x^2+4y^2+80 = 15x+30y \\ xy=6 $$
I have no idea how to solve this. I tried setting $x=\frac{6}{y}$ and plugging in, but all I get is this:
$(\frac{6}{y})^2+4y^2+80 = 15(\frac{...
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2
answers
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The velocity of light "c" a function of "e" ? Examine the following equations
The following 3 equations show that the velocity of light 'c' can be shown to have a connection to the mathematical constant 'e'
Where c is the velocity of light in meters per second
c =299,779,724 m/...
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2
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49
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Addition or subtraction of speeds?
Can we also take $(x+15)$ and $(x-15)$ in the following problem?
Problem:
The speed of a boat in still water is $15$ km/h. It goes $30$ km upstream and return downstream to the original point in $4$ ...
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2
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When to use $\implies$ or $\iff$ [closed]
Whenever writing a solution for a equation when you go on to the next step what sign to use $ \implies $ or$ \iff$ like
$\textbf{Scenario 1}$
$x^2=36 \implies x= \pm 6$
$\textbf{Scenario 2}$
$x^2 =36 \...
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2
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2
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119
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Solve simultaneous quadratic equations $ x^2-2xy=21$ and $xy+y^2 = 18$
Solve:
$$
\left\{\begin{array}{l}
x^2-2xy=21 \\
xy+y^2=18
\end{array}
\right.
$$
$\rightarrow$ $\left\{\begin{array}{l}x(x-2y)=21 \cdot \cdot \cdot(1) \\ y(x+y) = 18 \cdot \cdot \cdot(2)\end{...
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0
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Find the Highest common factor of two polynomial [closed]
Find the HCF of:
$$x^3 + x^3y, x^3 + y^3 $$
Answer is $$ x + y $$ but I don't know how. Can someone explain this?
4
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3
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Prove that the following inequality is true for all $m \in (0, 3)$
Prove that the following inequality is true for all $m \in (0, 3)$
$$\sqrt {{m^2} + 1} + \sqrt {{{\left( {3 - m} \right)}^2} + 1} \le \sqrt {\frac{{2\left( {4{m^2} - 12m - 15} \right)}}{{(m - 3)m}}} ...
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Underoot equation in a fraction [closed]
${\sqrt7\over \sqrt{16+6(\sqrt7)}-\sqrt{16-6(\sqrt7)}}$
[1]: https://i.stack.imgur.com/eO7fy.jpg
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0
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How to combine sums? [closed]
TLDR (Question about the image) - Why is this correct? Where did "$-r$" go?
Context -
I had a proof (Cyclic Convolution Theorem) as part of my Signals & Systems course.
As part of the ...
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0
answers
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Sum of Infinite recursion sequence [closed]
Let
$a_1 = 2 $ ,
$a_{n+1} = a_n^2 - a_n +1 $
(n is a Natural number).
Find $$\sum_{i=0}^\infty \frac{1}{a_n}$$
2
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1
answer
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An unclear transition with logarithms
I watched a lecture, and it included the following segment.
There are two functions: $f(n)$ and $g(n)$.
Let's say we want to find any asymptotic relationship between those functions.
We do a ...
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1
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Setting up and solving an equation of torque given some data
A heavy hatch on a ship is made of a uniform plate of steel that measures 1.2 m X 1.2 m and has a mass of 400 kg. The hatch is hinged along one side;
it is horizontal when closed and opens upward. A ...
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Prove the Inequality: $\vert x_1\vert ^\alpha+\vert x_2\vert ^\alpha\geq\Vert X\Vert _2 ^\alpha$, for $\alpha\in (1,2)$
Let $X\in\mathbb{R}^2$, where $X=[x_1,x_2]^T$. If $\alpha\in (1,2)$, then
$$\vert x_1\vert ^\alpha+\vert x_2\vert ^\alpha\geq\Vert X\Vert ^\alpha,$$
where $\vert\cdot\vert$ is the absolute value, and $...
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1
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When substitution works and when it fails (differentiation)
I would like to solve a problem, where I differentiate a function by a fraction, something like this:
$$\frac{\partial (x_1^\delta / x_2^\gamma)}{\partial (x_1/x_2)}$$
I was told that such a problem ...
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Request for Literature Recommendations on Isotonic Mappings
I am looking for literature recommendations on classical and generalized isotonic mappings of sets. Specifically, I am interested in any scientific and technical literature on this topic that is ...
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Arithmetic Progression/Sequence | Does it include terms prior to the initial term?
I’m trying to find the right term to describe points along a linear equation f(x) = mx + b, but where the domain is restricted to the set of integers. EDIT: Also the slope and y-intercept are integers....
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0
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Solve the equation $\sqrt[2021]{\frac{2x+2021}{x+4042}}+\sqrt[2021]{\frac{x+4042}{2x+2021}}=\cos\frac{(2021+x)\pi}{47}+\cos\frac{(2021-x)\pi}{43}$
Solve the equation $$\sqrt[2021]{\dfrac{2x+2021}{x+4042}}+\sqrt[2021]{\dfrac{x+4042}{2x+2021}}=\cos\dfrac{(2021+x)\pi}{47}+\cos\dfrac{(2021-x)\pi}{43}$$
I'd say my observations of the problem are ...
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How to simplify this equation of acceleration
How does:
$$I\frac{a}{R}= -R(m_1g+m_1a)+R(m_2g-m_2a)$$
becomes:
$$a = \left(\frac{m_2-m_1}{m_1+m_2+\frac{I}{R^2}} \right)g$$
for context this is the equation for the acceleration of the atwood machine ...
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Is the trace of a constant simply that constant?
I can't seem to find an answer to this online, basically what I am asking is
$$\mathrm{Tr} ((4)_{1\times 1})=4?$$ or in general is $\mathrm{Tr} ((k)_{1\times 1})=k?$ Where $k$ is a constant.
I think ...
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2
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Solve for $x$ and $y$ :
$$bx^3 = 10a^2bx + 3a^3y$$
$$ay^3 = 10ab^2y + 3b^3x$$
I tried putting $bx = p$ and $ay = q$ resulting into system of equations:
$$p^3 = 10a^2b^2p + 3a^2b^2q$$
$$q^3 = 10a^2b^2q + 3a^2b^2p$$
...
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Need help figuring out a couple of steps behind equation logic
The problem describes the logic behind limits in math. f1
Based on an area under a curve. The overall idea is pretty clear. We split the OX line into several parts with the following coordinates f2
...
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1
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108
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If $x$ and $y$ are numbers such that $x + y, x^2 + y^2, x^3 + y^3$ and $x^4 + y^4$ are all integers. Show that $x^n + y^n$ is an integer.
If $x$ and $y$ are numbers such that $x + y, x^2 + y^2, x^3 + y^3$ and $x^4 + y^4$ are all integers. Show that $x^n + y^n$ is an integer for all positive integers $n$.
Using induction we have the ...
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2
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Is this statement correct : $\frac{dv}{dt}\times dx=\frac{dx}{dt}\times dv$
So I just came across a physics derivation where the treat the $dv$ and $dx$ operators like fractions while I have always heard it's a mistake. But so far what I came up with:
$$\begin{align*}
\frac{...
0
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2
answers
81
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In how many ways can $2^5\times 3^7$ be factored into three setwise coprime integers?
Let $T$ be the following set of ordered triples, $T=\{(a,b,c):a,b,c\in \mathbb{N}\}$.
Find the number of elements in $T$ such that $abc=2^5\times 3^7$ and $\gcd(a,b,c)=1$
My Attempt
If $a=2^x;b=3^y;c=...
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2
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1
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The Weight Power Mean, how to understand it?
Around this time, I was using the OTIS Excerpt of Evan Chen to prepare (or at least learn competitive math) for the math Olympiad. At the start of the Algebra section, I was introduced to the AM-GM ...
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0
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44
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maximum with factorials
Let $k>2$. Can we determine $\displaystyle \max_{m\in\mathbb N}\frac1{k10^{m!}}-\frac2{10^{(m+1)!}}$ ?
Or at least a non trivial lower bound of this maximum?
That is in relation with thread
Thanks ...
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0
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24
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Simplifying a subindex equation
Consider the following equation
$$
\frac{e^{\sum_i \alpha_i x_i}}{\sum_i x_i}=\sum_k\frac{e^{\sum_i \alpha_i y_{i, k}}}{\sum_i y_{i, k}}
$$
Is it possible to write $x_i$ as a function of the terms $y_{...
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1
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System of parametric equations $E=F=0$ with unknowns $x,y$ [closed]
Solve for $x$ and $y$ where $a$, $b$ are positive reals:
$$\frac{x - a}{a ^ 2} + \frac{y - b}{b ^ 2} = \frac1{x - b} - \frac1{y - a} - \frac1{a - b} = 0$$
I've tried to simplify and solve from many ...
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0
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36
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Determining the asymptotic relationship between two logarithmic functions.
This question is around computer science, but too mathmatichal to ask anywhere else. The log is base 2.
These are the two given functions:
$g(n) = 2^{\lg^2n}$
$f(n) = \sum_{i=1}^{n^2}\lg\left(i\right)$...
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2
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How are the domain and range of a quadratic function determined?
How are the domain and range of a quadratic function determined?For example, what are the domain and range of $x^2-4x+10$
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1
answer
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Invariant problem starting with the sequence $1,2, ... ,100$
Initially, we are given the sequence $1,2, ... ,100$. Every minute, we erase any two numbers $u$ and $v$ and replace them with the value $uv + u + v$. Clearly, we will be left with just one number ...
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