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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.

12
votes
0answers
316 views

Can you obtain $\pi$ using elements of $\mathbb{N}$, and finite number of basic arithmetic operations + exponentiation?

Is it possible to obtain $\pi$ from finite amount of operations $\{+,-,\cdot,\div,\wedge\}$ on $\mathbb{N}$ (or $\mathbb{Q}$, the answer will still be the same), note that set of all real numbers ...
11
votes
0answers
903 views

Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and calculus....
10
votes
0answers
140 views

Sum equals the product

A while ago, just playing with some numbers I noticed that $1+2+3=1\cdot2\cdot3$, so I started thinking about the non-zero integer solutions of the equation $$\prod_{i=1}^na_i=\sum_{i=1}^na_i$$ For ...
9
votes
0answers
299 views

Solution by radicals of $(1+x)^n=x^m$

Given the equation $$(1+x)^n=x^m$$ where $m$ and $n$ are two different natural numbers, I was trying to find as many solution as possible expressing them without transcendent functions. WLOG we can ...
8
votes
0answers
135 views

Does this “almost all integers in order” sequence have a closed form?

Can you help me define a formula for the following sequence (first $130$ terms) : ...
6
votes
0answers
93 views

Eliminate $\alpha,\beta,\gamma$ from the system of equations

Eliminate $\alpha,\beta,\gamma$ from the following system of equations. $$a\cos(\alpha)+b\cos(\beta)+c\cos(\gamma)=0$$ $$a\sin(\alpha)+b\sin(\beta)+c\sin(\gamma)=0$$ $$a\sec(\alpha)+b\sec(\beta)+c\...
6
votes
0answers
117 views

Solving for solutions to $\left(a+b\right)^2=a.b$

I ran across this stackoverflow question regarding finding solutions to $$\left(a+b\right)^2=a.b$$ for $a,b\in\Bbb{N}$ where $a.b$ denotes concatenation of $a$ and $b$. Since concatenation isn't ...
6
votes
0answers
231 views

British Maths Olympiad (BMO) 2004 Round 1 Question 4 alternative solutions?

The question states: Alice and Barbara play a game with a pack of $2n$ cards. On each of which is written a positive integer. The pack is laid out in a row, with the numbers facing upwards. Alice ...
6
votes
0answers
167 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},\...
6
votes
0answers
189 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 ...
6
votes
0answers
144 views

How many triples $(x,y,n)$ are there such that $x^n - y^n = 2^{100}$

How many triples $(x,y) \in \mathbb{N^+}^2$ and $n \gt 1$ are there such that $x^n - y^n = 2^{100}$ I dont know how to start. Any hint will be helpful. I know the identity $x^n-y^n = (x-y)(...
6
votes
0answers
76 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= \...
6
votes
0answers
174 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=...
5
votes
0answers
31 views

Does there exist a function $f_{\Box,\Box}(\Box)$ making the formula $a + (b \oplus c) = (f_{b,c}(a)+b) \oplus (f_{c,b}(a)+c)$ true?

Let $a$ and $b$ denote the resistances of two resistors. If they're put in series, the total resistance is $a+b$. If they're put in parallel, the total resistance is $$a \oplus b := \frac{1}{\frac{1}{...
5
votes
0answers
118 views

If $f(x)$ is a polynomial function with integer coefficients find min value of $f(12)$

$f(x)$ is a polynomial function with integer coefficients satisfying $f(1)=5$ and $f(2)=7$. What is the smallest possible positive value of $f(12)$? I have no idea on how to begin with this question. ...
5
votes
0answers
100 views

Thinking like in ancient Greece

The next result is attributed to Archimedes: The equation $x^3-ax^2+(4/9)a^2b$ has a positive root if and only if $a>3b$, where $a$ and $b$ are positive real numbers. This problem appeared in the ...
5
votes
0answers
150 views

Show $\lim_{n\to \infty}{|A_{n}|+|A_{n-2}| \above 1.5pt |A_{n-1}|}=2$

Question: Let $A(n)$ be a finite square $n \times n$ matrix with entries $a_{ij}=1$ if $i+j$ is a prime; otherwise equals to $0$. I write $|A(n)|$ to count the number of $1$'s in $A(n)$. Is it ...
5
votes
0answers
212 views

Is it possible to simplify a nested radical in the form $\sqrt[3]{\sqrt[3]{A}-B}$ into $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$

I'm wondering if there is a way to simplify any nested radical in the form $\sqrt[3]{\sqrt[3]{A}-B}$ into $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$. Examples such as $\sqrt[3]{7\sqrt[3]{20}-1}=\sqrt[3]{\...
5
votes
0answers
118 views

Probability of another 3 integers with same sum and product as the first 3 integers

Let us suppose $3$ integers are selected at random from a large range, say $$-1000\leq x\leq y\leq z\leq 1000$$ Now, we define the sum and product: $$\begin{align*}s&=x+y+z \\p&=xyz\end{align*...
5
votes
0answers
99 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 ...
4
votes
0answers
59 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: ...
4
votes
0answers
51 views

Interesting applications of density to prove difficult theorems

If we wanted to prove certain statements for every element of a set $S$, a possible approach is to prove the statement for a certain dense subset $S'\subset S$ (with respect to a certain metric), then ...
4
votes
0answers
80 views

What is rule for when solving algebraic equations?

I'm a high school student trying to get critical intuition when learning algebraic equation solving. For $x$ any complex number and $c$ constant, simple polynomial such as $x^n -c=0$ are easily ...
4
votes
0answers
122 views

Prove that if $a,b,c \in \mathbb{R^+}\text{ and } abc=8\text{ then } {ab+4\over a+2}+{bc+4\over b+2}+{ca+4\over c+2}\ge6$

Question: Prove that if $a,b,c \in \mathbb{R^+}\text{ and } abc=8\text{ then } {ab+4\over a+2}+{bc+4\over b+2}+{ca+4\over c+2}\ge6$ My Approach: Now we have: $${ab+4\over a+2}={2\times (ab+4)\...
4
votes
0answers
71 views

Given $n$ integers, is it always possible to choose $m$ from them so that their sum is a multiple of $m$?

The original question: given $6666$ integers, (positive, negative or $0$) is it always possible to choose $2018$ from them so that the chosen numbers add to a multiple of $2018$? (positive multiple, ...
4
votes
0answers
111 views

How to find self intersection of this polar curve: $r = 1 + 2\cos(\theta) - 4\cos^2 (\theta)$

I am trying to find the values of $\theta$ where the following polar curve intersect itself:$$r = 1 + 2\cos(\theta) - 4\cos^2 (\theta).$$ I am studying this function on $[0, 2\pi] $ since it is ...
4
votes
0answers
88 views

Real root of $x(x+t)^2-4=0$ $\ge$ greatest real root of $4x^6-6x^4+4t^3x^3+t^6-3t^4=0$

I have a problem that I can not solve although I know that the statement is true. For all $t\ge0$ the real root of $x(x+t)^2-4=0$ is greater than or equal to the largest real root of $4x^6-6x^4+4t^...
4
votes
0answers
68 views

Does $(\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0$ hold for $n \leq 8$?

Let positive reals $\{a_i\}$, where not all $a_i$ are equal. Does $$ f(\{a_i\}) = (\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0 $$ hold for $n \le 8$? It is ...
4
votes
0answers
119 views

On Pandigital numbers

The other day I ran across a very nice video from Numberphile in YouTube, which proves that the following formula is approximate to $e$ by $18457734525360901453873570$ digits, which is pretty amazing. ...
4
votes
0answers
118 views

If $2f(\frac{x}{2})+3f(\frac{2-x}{3})=g(x)$ then find $25ad-bc$

$2f(\frac{x}{2})+3f(\frac{2-x}{3})=g(x)$, $0 \leq x < 3$, $f''(x)>0$; if $g(x)$ is strictly increasing in $(a,b)$ and $g(x)$ is strictly decreasing $(c,d)$ then $25ad-bc$ is ? My Attempt: ...
4
votes
0answers
81 views

Find the $least$ number $N$ such that $N=a^{a+2b} = b^{b+2a}, a \neq b$.

When I graphed the relation $a^{a+2b}=b^{b+2a}$ , it gives a graph similar to $y=x$. However, the question explicitly states that $a \neq b$. So does that mean that no such $N$ exists ? What happens ...
4
votes
0answers
77 views

Equality of Floors of some Partial Sums

Let $S_n=\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}$ denote the $n$th partial sum in the series expansion for $e=\sum\frac{1}{k!}$. I want to prove that $\lfloor n\cdot(S_n+1/n!)\rfloor=\lfloor n\...
4
votes
0answers
85 views

Is this a valid way for performing polynomial division?

While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials. Suppose you have a quartic ...
4
votes
0answers
84 views

If $f(n)= \binom{n}{0}a^{n-1}-..+(-1)^{n-1}\binom{n}{n-1}a^{0}$ ,Then $f(2007)+f(2008) $

If $\displaystyle a= \frac{1}{3^{223}}+1$ and $\displaystyle f(n)= \binom{n}{0}a^{n-1}-\binom{n}{1}a^{n-2}+...........+(-1)^{n-1}\binom{n}{n-1}a^{0}$ Then value of $f(2007)+f(2008) = $ $\bf{My\; ...
4
votes
0answers
166 views

Finding the minimum value of a radical expression

If $a$, $b$ and $c$ are positive real numbers, find the minimum value of $\sqrt { \frac { a }{ b+c } } +\sqrt [ 3 ]{ \frac { b }{ c+a } } +\sqrt [ 4 ]{ \frac { c }{ a+b } } $. I am not able to ...
4
votes
0answers
295 views

Motivation of Vieta's transformation

The depressed cubic equation $y^3 +py + q = 0$ can be solved with Vieta's transformation (or Vieta's substitution) $y = z - \frac{p}{3 \cdot z}.$ This reduces the cubic equation to a quadratic ...
4
votes
0answers
150 views

How to solve this equation in $ \mathbb{C} $?

From a small simple calculation , we get the following formulas: $ \begin{cases} e^x = \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n}}{(3n)!} \Big) + \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n+...
4
votes
0answers
244 views

Writing sum of square roots with symmetric polynomials

I want to write the function $$ F_N=\sum_{i=1}^N\sqrt{x_i} $$ in terms of the $N$ elementary symmetric polynomials of the $N$ positive variables $x_1,\dots,x_N$. The $N=1$ case is trivial, as we ...
4
votes
0answers
94 views

Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
4
votes
0answers
121 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 ...
4
votes
0answers
58 views

Existence of Solutions of Two Cubic Equations in a Particular Region

If I have two cubic equations in two variables, $ax^3 + bx^2 y + cxy^2+\dots=0$ and another one with different coefficients, and I know that $(x,y)=(0,0)$ or $(1,1)$ are solutions, are there any nice, ...
4
votes
0answers
57 views

Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.

To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following: (somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
4
votes
0answers
147 views

Polynomial bound

Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that $$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$ Suppose that $P(x)> 0$ for all ...
4
votes
0answers
97 views

Digits of two irrational numbers, given their power with fixed number of digits

I have $a, b \in \mathbb{R} \setminus \mathbb{Q}$, I want to know the result of $a^b$, but I don't know exact $a, b$ because I write them in numeric form. My question is how many digits of $a, b$ have ...
4
votes
0answers
424 views

Mandelbrot set's border in parametric form

I've post this question just because I'm curious, Mandelbrot set is defined as: $ z_{n+1} = z^2_n + c $, if $n \rightarrow \infty $ and it doesn't diverge we get the border. This border is unlimited ...
4
votes
0answers
192 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) ...
4
votes
0answers
176 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 ...
4
votes
0answers
246 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 ...
3
votes
0answers
50 views

Problem about linear combinations of real numbers over $\Bbb Q$

I have the following problem that might be silly but I am not able to find a solution. Let $a_1,\dots,a_n$ be real numbers such that $a_1,\dots,a_n,\pi$ are linearly independent over $\Bbb Q$. Let $...
3
votes
0answers
64 views

Maximum possible number of extrema of the function?

Consider a function : $$ f(x)= P(x)e^{-(x^4+2x^2)} $$in the domain $x \in (-\infty,\infty)$, $P(x)$ is any polynomial of degree $k$. What is the maximum possible number of extrema of the function. ...