Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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11
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3answers
392 views

$f(f(x))$ has no fixed points if $f(x)$ has no fixed points

Assume that $f(x)=x$ has no real roots where $$f(x) = ax^2+bx+c$$ Prove that $f(f(x))=x$ has no real roots as well. What I've done is, calculating $f(f(x))$: ...
11
votes
2answers
246 views

A polynomial determined by two values

From a St. Petersburg school olympiad, 11th grade. Prove or disprove: a non constant polynomial P with non-negative integer coefficients is uniquely determined by its values P(2) and P(P(2)).
11
votes
2answers
696 views

My Daughter's 4th grade math question got me thinking

Given a number of 3in squares and 2in squares, how many of each are needed to get a total area of 35 in^2? Through quick trial and error (the method they wanted I believe) you find that you need 3 ...
11
votes
5answers
3k views

How can I write an equation that matches any sequence?

One thing I have been wondering about lately is how to write an equation that describes a pattern of numbers. What I mean is: x 0 1 2 y 1 5 9 If ...
11
votes
3answers
182 views

How do we solve $a \le b^{r}-r$ for $r$?

Given two values $a$ and $b$, how should one go about solving the following inequality for $r$: $$a \le b^r -r .$$ Applying $\log_b$ on both sides of the inequality doesn't help me much since that ...
11
votes
1answer
499 views

Interesting negative decimal number notation

I was studying logarithms, and had to solve the problem: If $\log 8 = 0.90$, find $\log 0.125$. I found out the answer to be $-0.90$. That was easy. But my text book has given the answer as: ...
11
votes
2answers
359 views

Number of real positive roots of a polynomial?

Consider the polynomial $$f(x)=x((1+x^n)^n+a^n)-a(1+x^n)^n,$$ where $n\geq 2$ is a positive integer and $a$ is a positive real number. I'm interesting in deducing the number of positive real roots ...
11
votes
5answers
1k views

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same?
11
votes
4answers
616 views

Proof that $n^3 + 3n^2 + 2n$ is a multiple of $3$.

I'm struggling with this problem: For any natural number $n$, prove that $n^3 + 3n^2 + 2n$ is a multiple of $3$. That $n^3 + 3n^2 + 2n$ is a multiple of $3$ means that: $n^3 + 3n^2 + 2n = 3 ...
11
votes
4answers
474 views

Prove maximum value of $(z-xy)(x-yz)(y-zx)$ is $\frac{1}{64}$ given $x,y,z \in (0,1)$

Prove maximum value of $(z-xy)(x-yz)(y-zx)$ is $\frac{1}{64}$ given $x,y,z \in (0,1)$ I can make it $\frac{1}{64}$ by setting $x,y,z = \frac{1}{2}$, but I have no idea how to show that's the maximum. ...
11
votes
3answers
330 views

Collatz-ish Olympiad Problem

The following is an Olympiad Competition question, so I expect it to have a pretty solution: For a positive integer $d$, define the sequence: \begin{align} a_0 &= 1\\ a_n &= ...
11
votes
4answers
720 views

Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$.

Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$ I have no idea how to deal with this crazy question. One idea came into my mine is factorization, but I can't go on... Can anyone help me please? ...
11
votes
1answer
285 views

Finding $x^4 + y^4 + z^4$ using geometric series

This is a problem from the 2001 Stanford Math Tournament Algebra section. $$$$Given that $$x+y+z=3$$ $$x^2 + y^2 + z^2 = 5$$$$x^3+y^3+z^3=7$$Find $x^4+y^4+z^4$. $$$$My friend claimed that he was able ...
11
votes
2answers
415 views

Given $n! = c$, how to find $n$?

I'm dealing with a time-complexity problem in which I know the running time of an algorithm: $$t = 1000 \mathrm{ms} .$$ I also know that the algorithm is upper bounded by $O(n!)$. I want to know ...
11
votes
3answers
182 views

Prove that x is rational

Let $x$ be a real number with the properties that $x^3+x$ and $x^5+x$ are rational. Prove that $x$ is rational. Denote $a=x^3+x$; $b=x^5+x$. We can multiply and add them together until we get the ...
11
votes
2answers
134 views

Prove $\sqrt[n]{m}\leq\sqrt[3]{3}\lor\sqrt[m]{n}\leq\sqrt[3]{3}$ for $n,m\in\mathbb{N}>1$.

Prove that for any integers $m$ and $n$ greater than $1$, at least one of the numbers $\sqrt[n]{m}$ or $\sqrt[m]{n}$ is not greater than $\sqrt[3]{3}$. My attempt goes something along the lines ...
11
votes
1answer
985 views

Is solving an equation the same as finding the roots?

For example, would solving for $x$ in $x^2=8x+7$ be the same as finding the roots of the equation? Also, would finding the roots of this be the same as finding the zeros?
11
votes
2answers
254 views

Prove this inequality $a^{\frac{a}{b}}b^{\frac{b}{c}}c\geq1$

Please help me to prove this inequality. Assume $a,b,c>0$ and $abc\geq1$ then $a^{\frac{a}{b}}b^{\frac{b}{c}}c\geq1$. Thanks.
11
votes
1answer
425 views

Find the sum $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + …+ \frac{1}{\sqrt{99}+\sqrt{100}}$

I would like to check I have this correct Find the sum $$\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + ...+ \frac{1}{\sqrt{99}+\sqrt{100}}$$ Hint: rationalise the denominators ...
11
votes
1answer
251 views

If the product of $x$ positive integers is $n!$ What is the smallest possible value their sum can have?

If the product of $x$ positive integers is $n!$ What is the smallest possible value their sum can have? I was wondering what could be the most efficient strategy to solve this problem for ...
10
votes
11answers
2k views

Solving $5^n > 4,000,000$ without a calculator

If $n$ is an integer and $5^n > 4,000,000.$ What is the least possible value of $n$? (answer: $10$) How could I find the value of $n$ without using a calculator ?
10
votes
6answers
960 views

Proving that $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$

How am I suppose to prove that: $$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$$ Do I use the way like how we count $1+2+ \cdots+100$ to estimate? So $1/5050 \lt ...
10
votes
4answers
2k views

Can you prove this identity?

Show that $$(x-y)^3+(y-z)^3+(z-x)^3 = 3(x-y)(y-z)(z-x)$$ This can be shown through expansion but there is a more elegant solution I cannot discover anything I would consider elegant. Can anyone ...
10
votes
7answers
1k views

Solve trigonometric equation: $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$

Dealing with a physics Problem I get the following equation to solve for $\alpha$ $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$ Putting this in Mathematica gives the result: $a==2 ...
10
votes
4answers
2k views

What is the term for a factorial type operation, but with summation instead of products?

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name ...
10
votes
5answers
672 views

Solve equations $\sqrt{t +9} - \sqrt{t} = 1$

Solve equation: $\sqrt{t +9} - \sqrt{t} = 1$ I moved - √t to the left side of the equation $\sqrt{t +9} = 1 -\sqrt{t}$ I squared both sides $(\sqrt{t+9})^2 = (1)^2 (\sqrt{t})^2$ Then I got $t + 9 ...
10
votes
7answers
501 views

Can someone show me why this factorization is true?

$$x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \dots + xy^{n-2} + y^{n-1})$$ Can someone perhaps even use long division to show me how this factorization works? I honestly don't see anyway to "memorize ...
10
votes
5answers
394 views

Why is $\sqrt{4} = 2$ and Not $\pm 2$? [duplicate]

I've always been told that if $\ x^2 = 4,$ $ =>x = \pm2$ But recently, Prof. mentioned that if $x = \sqrt{4}$, Then $x = +2(only)$ I am very skeptical about this because they both mean the ...
10
votes
5answers
300 views

What is the coefficient of the $x^3$ term in the expansion of $(x^2+x-5)^7$ (See details)?

I fail to see a simple way to answer this. As such, this is my long winded approach: Using the multinomial theorem, $$(x_1 + x_2 + \cdots + x_m)^n = \sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, ...
10
votes
4answers
507 views

The value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$?

How to find value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$ ? I've calculated it by MATLAB for some finite terms and I've got : $0.3001 - 0.4201i$, but I don't know how to ...
10
votes
5answers
495 views

imaginary numbers - how can I understand them - especially as they occur in 'roots' of polynomials?

In another question here, about roots of equations being imaginary, the accepted answer said something interesting about "imaginary" (as a technical word in math) not meaning "not real". I ...
10
votes
5answers
244 views

function identity: does $\frac{x^2-4}{x-2} = x+2$

Can you tell me how to resolve the (apparent) paradox that the function: $$f(x) = \frac{x^2-4}{x-2}$$ is identical to the function: $$g(x) = x+2$$ because by factoring the numerator: $$f(x) = ...
10
votes
4answers
494 views

Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction

How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
10
votes
2answers
189 views

Proof that $\frac{1}{a_1} +\frac{1}{a_2} +…+\frac{1}{a_{20}}$ is an integer

Assume that, for $n\ge1$,$$a_n=\sqrt{1+\left(1+\frac{1}{n}\right)^2 } +\sqrt{1+\left(1-\frac{1}{n}\right)^2 } $$ How to prove that $$\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_{20}}$$ is an ...
10
votes
4answers
341 views

If $2^x=3^y=6^{-z}$ then prove that:$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$

If $$2^x=3^y=6^{-z}$$ and $x,y,z \neq 0 $ then prove that:$$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$ I have tried starting with taking logartithms, but that gives just some more equations. Any ...
10
votes
5answers
396 views

Can this function be rewritten to improve numerical stability?

I'm writing a program that needs to evaluate the function $$f(x) = \frac{1 - e^{-ux}}{u}$$ often with small values of $u$ (i.e. $u \ll x$). In the limit $u \to 0$ we have $f(x) = x$ using L'Hôpital's ...
10
votes
3answers
3k views

Why should you never divide both sides by a variable when solving an equation?

I'm currently working through an algebra book, and during the chapter about rational expressions and inequalities, the author has a side note in which he states: Never divide both sides of the ...
10
votes
2answers
166 views

Given a cubic and quadratic share a root, prove $(ac-b^{2})(bd-c^{2})\geq 0$

Here is an interesting problem. Perhaps someone would be so kind as to give me a shove in the right direction?. If $ax^{3}+3bx^{2}+3cx+d$ and $ax^{2}+2bx+c$ share a common root, then prove that ...
10
votes
5answers
334 views

Elementary proof that $\pi < \sqrt{5} + 1$

I wanted to show that $$ \frac{\pi}{4\phi} < \frac{1}{2} $$ Where $\phi$ is the golden ratio. I have confirmed the results numerically, and by simple algebra the inequality simplifies down to $$ ...
10
votes
6answers
238 views

Finding the roots of $x^3 - 93x - 308$ extremely quickly?

I was at a quiz bowl competition and one of the questions was to find the roots of this polynomial. In one or two seconds after reading the question, somebody on the other team buzzed and got the ...
10
votes
3answers
204 views

What's $(-1)^{2/3}\; $?

I know that $\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{2} \right )^{1/3}=1$ But Matlab computes this as $- 0.5 + 0.8660254038i$ a complex number.Why?
10
votes
5answers
4k views

Good book for high school algebra

I'm gonna take a Calculus course next year, my professor suggest me to review high school algebra. I want to know, which book is good for refresh knowledge on high school algebra?
10
votes
2answers
661 views

Relearning from the basics to Calculus and beyond.

Assume someone has very limited knowledge of math. (low level high school, 5-6 years ago) How would they learn from the basics of algebra, geometry and trigonometry to a solid foundation for calculus ...
10
votes
2answers
202 views

Composition of two functions is constant

Suppose $f$ and $g$ are such that $f(g(x)) = 1$. Does this imply that $g(f(x))$ is constant?
10
votes
2answers
130 views

calculate $x^{206}+x^{200}+x^{90}+x^{84}+x^{18}+x^{12}+x^{6}+1$ given $(x+x^{-1})^2 = 3$

If $\left(x+\dfrac 1 x\right)^2=3$ then the value of $$x^{206}+x^{200}+x^{90}+x^{84}+x^{18}+x^{12}+x^{6}+1.$$ I'm trying to solve it like this $$x^2+\dfrac {1}{x^2}=1\text{ and }; x^6+\dfrac ...
10
votes
1answer
2k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
10
votes
2answers
188 views

Basic Mathematics. Trouble with proof, powers and odd numbers.

Greets, In the exercises, at the end of chapter 1.4, Basic Mathematics, Serge Lang 6) Prove: If $n$ is odd, then $\quad (-1)^n = -1$ How? The working I did $$\begin{align}( -1)^n &= ( -1 ...
10
votes
5answers
369 views

How to calculate $2^{\sqrt{2}}$ by hand efficiently?

I've been trying to calculate $2^{\sqrt{2}}$ by hand efficiently, but whatever I've tried to do so far fails at some point because I need to use many decimals of $\sqrt{2}$ or $\log(2)$ to get a ...
10
votes
1answer
539 views

Prove that minimum of $\lambda \sin \theta + (1 - \lambda) \cos \theta \le -\dfrac{1}{\sqrt 2}$

I need a little nudge to the finish for the last bit of this problem. Express $\lambda \sin \theta + (1 - \lambda) \cos \theta$ in the form $R \sin (\theta + \phi)$, where $R(R>0)$ and $\tan ...
10
votes
4answers
163 views

Coefficients of a polynomial also are the roots of the polynomial?

How many real solutions $(r_1, r_2, \cdots, r_n)$ are there such that $(r_1, r_2, \cdots, r_n)$ are the roots of the polynomials $x^{n} + r_1 x^{n-1} + r_2 x^{n-2} + \cdots + r_n$ For $n = 2, 3, 4$ I ...