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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1
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2answers
42 views

Prove $\log_n(2) + \log_2(3) + \log_3(4) + \log_4(5) + \cdots + \log_{n-1}(n) > n-1$?

This is a silly question, but I tried a little things here and I only managed to write $$\log_n(2) + \log_2(3) + \log_3(4) + \log_4(5) + \cdots + \log_{n-1}(n) > \log_n(n!)$$
2
votes
4answers
55 views

$pqrs \cdot 4 =srqp $,then what is the value of $qrs$?

This is question 26 from Australian Maths Competition 2013. $pqrs $ is a 4-digit number and has the property that $pqrs \cdot 4 = srqp$.If p=2,what's the value if the 3-digit number qrs? Here's what ...
0
votes
0answers
34 views

On oblath's theorem [on hold]

it is just my first encounter about this topic ,it is the topic that my prof gave to me in my undergrad studies.I found it interesting but there are still parts(like theorem) in this topic which make ...
0
votes
5answers
54 views

How to find $(2x+5)(2x-5)$

Given that $(x + y)•(x - y) = x^2 - y^2$ How do I do this: $(2x+y)(2x-y) = ?$ Not really sure about the tag I should use since English is not my native tongue
0
votes
2answers
46 views

Calculate $\log_{\frac{1}{2}}{(2\sqrt[3]{4})}$

I have this exercise:$$\log_{\frac{1}{2}}{(2\sqrt[3]{4})}$$ There is not equation or something else, how should I solve this type of exercise?
3
votes
4answers
92 views

Solve $2^x+4^x=2$

This is the equation, but the result is different from wolframalpha: $$2^x+4^x=2$$ $$2^x+2^{2x}=2^1$$ $$x+2x=1$$ $$x=\frac{1}{3}$$ WolframAlpha: $x=0$ Where is the error?
10
votes
5answers
202 views

Subtracting expressions with radicals

I want to subtract the expressions $20\sqrt{72a^3b^4c} - 14\sqrt{8a^3b^4c}$. I simplified this to $120ab^2\sqrt{2ac}-28ab^2\sqrt{2ac}$. My textbook says the answer is $92ab^2\sqrt{2ac}$. Why doesnt ...
2
votes
5answers
46 views

Solving $2x^4+x^3-11x^2+x+2 = 0$ [duplicate]

I am having no idea how I can solve this problem. I need help! Here's the problem $2x^4+x^3-11x^2+x+2 = 0$ I am learning Quadratic Expressions and this is what I need to solve, and I can't ...
0
votes
0answers
27 views

On Oblath's Problem [on hold]

I am trying to read the paper On Oblath's Problem, and I'm have difficulty understanding the main theorem. I can read the theorem but I don't understand it. May someone help me to make this theorem as ...
3
votes
2answers
44 views

Is $\frac {x^2 + 5x}{x} = x+5$?

We are graphing functions in class and the function $f(x) = \frac {x^2 + 5x}{x}$, came up and our teacher simplified it to $x+5$ and graphed that with a hole in the function at $x=0$. I started ...
1
vote
1answer
24 views

How do I “stretch” and “compress” a piecewise function?

I have Googled a few times and experimented on Desmos, but both attempts were to no avail, and now I come here. How is a piecewise function transformed to be "stretched" or "compressed"? What about ...
6
votes
4answers
126 views

Why does $(128)!$ equal the product of these binomial coefficients $128! = \binom{128}{64}\binom{64}{32}^2 \dots \binom21^{64}$?

I'm working through some combinatorics practice sets and found the following problem that I can't make heads or tails of. It asks to prove the following: $$128! = \binom{128}{64}\binom{64}{32}^2\...
0
votes
1answer
40 views

What fraction of the mixture by weight is pineapple?

A fruit salad is made from pineapples, pear, and peaches mixed in the ratio of $2$ to $3$ to $5$ respectively by weight. What fraction of the mixture by weight is pineapple? A) 1/5 B) 3/10 C) 2/5 D) ...
0
votes
1answer
20 views

Working out cost based on time spent - simple math [on hold]

I did a task, and my hourly rate is $£25$ , I spent a total of $36$ minutes on it, how can I work out the total amount of time spent on the task? My attempt: I can do this for simple sums such as ...
1
vote
6answers
73 views

Prove by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$

My question is: Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever $n$ is a nonnegative integer. I'm stuck at the basis step. If I ...
0
votes
1answer
37 views

“In terms of” question

A diagram shows a glass window consisting of a rectangle of hight $h$ m and width $2r$ m and semicircle of radius $r$ m. The perimeter of the window is $8$m. Express $h$ in terms of $r$. Please help ...
2
votes
4answers
70 views

value of $\displaystyle \left(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}\right)\cdot \left(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c}\right)$

If $a+b+c=0,$ Then value of $\displaystyle \left(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}\right)\cdot \left(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c}\right)$ $\bf{My\; Try::}$ Given $$\displaystyle ...
1
vote
2answers
56 views

Solve in integers the equation $\left\lfloor\frac{x^2-y^3}{x+y^2} \right\rfloor=1+x-y$

Solve in integers the equation $$\left\lfloor\frac{x^2-y^3}{x+y^2} \right\rfloor=1+x-y$$ My attempt: I used http://www.wolframalpha.com/: $x=-2; y=\{3,4,5,6,7,..\}$ or $x=-1, y=\{-10,-9,...\}$. ...
-1
votes
1answer
53 views

Find the value of x of $\frac{(3x^2-27)(8x^2)6}{4(9-3x)(x^2+3x)}=\frac{\tan (x+4)}{\log (x+\frac{1}{4})}$? [on hold]

$\frac{(3x^2-27)(8x^2)6}{4(9-3x)(x^2+3x)}=\frac{\tan (x+4)}{\log (x+\frac{1}{4})}$? How to find the value of $x$ I've been thinking for this question for quite a time. Hope can somebody solve it. ...
5
votes
2answers
60 views

Testing for symmetry about a curve/line

In High School Algebra , after studying how to plot a graph of $f(x)$ (rather called $y$) against $x$ in Cartesian coordinates, we studied several tests to determine the symmetry of the plotted graphs ...
0
votes
2answers
54 views

Restrictions on Factorial Usage

I had always understood that the factorial n! was defined as $$\prod_{k=1}^nk$$ However, this leaves several questions: Why does 0! exist?* By extension, why can't you take the factorial of a ...
1
vote
0answers
24 views

determine a specific polynomial [on hold]

I have the following expression $$\pi(x)=\pm\sqrt{-\beta x^3+\lambda x^2+\beta x+d+g(x)(1-x^2) }$$ where $\beta$, $\lambda$ and d are constants. I need to find the $g(x)$ such that the expression ...
0
votes
1answer
27 views

Central Force Problem

Given an equation $F(x)$ that represents the magnitude of some force $F$ that varies with distance from the origin, is it possible to derive the equation of motion $p(t)$ of a point particle $P$ ...
3
votes
2answers
39 views

Is it possible to convert the polar equation $\ r = k \cos (\theta n) + 2$ into cartesian form?

Is it possible to convert the polaer equation $$\ r = k \cos (\theta n) + 2$$ into cartesian form? Here, $k$ is some constant and $n$ is any positive whole number greater than $2$. The ...
1
vote
4answers
44 views

Symmetric system of $3$ equations

I need help solving the following system for all real ordered triples $(x,y,z)$ without guessing and checking: $$x+y+z=23$$ $$xy+yz+xz=144$$ $$xyz=252$$ Preferably, the solution should use methods ...
1
vote
1answer
31 views

For what integral value of $n$ is $3\pi$ the period of the function $\cos(nx)\sin(5x/n)$?

For what integral value of $n$ is $3\pi$ the period of the function $\cos(nx)\sin(5x/n)$ ? What should be the correct approach to this problem?Will taking the LCM of the periods of the two functions ...
0
votes
1answer
23 views

How do I find the period of the function $\tan{\pi/2[x]}$?

How do I find the period of the function? $$\tan{\frac{\pi}{2}[x]}$$ What are the factors that I must take care of? (Maybe its simple but i'm not getting it methodically.$2$ seems to work though) [] ...
-2
votes
1answer
40 views

$ \frac{a^3}{c} + \frac{b^3}{d} \ge 1$ prove inequality [duplicate]

Suppose that $ a,b,c,d \in \mathbb{R}$, $a,b,c,d \gt 0$ and $ c^2 +d^2=(a^2 +b^2)^3$. Prove that $$ \frac{a^3}{c} + \frac{b^3}{d} \ge 1.$$ If I rewrite the inequation like $ \frac{a^3}{c} + \frac{...
0
votes
1answer
30 views

5th degree polynomial with positive leading coefficient

I'm guessing C or D because odd degree polynomials which aren't linear extend one way to infinity and the other way to negative infinity? So what's the relevance of the a>0? As x approaches infinity ...
0
votes
0answers
25 views

Trig identities problem, figuring out the angle relation

I want to solve this question: Note after trying to solve this I had to use the tips: $\alpha +\beta +y=\pi$ However, even with this I still do not get how to go from left to right on the first ...
1
vote
2answers
102 views

prove inequation

a,b,c,d $\in \mathbb{R}$ $a,b,c,d \gt 0$ and $ c^2 +d^2=(a^2 +b^2)^3$ prove that $$ \frac{a^3}{c} + \frac{b^3}{d} \ge 1$$ If I rewrite the inequation like $ \frac{a^3}{c} + \frac{b^3}{d} \ge \frac{...
0
votes
1answer
46 views

Question about mathematical logic

Let say that you think you have 10 dollars in your pocket and you find out that you have two more dollars than what you thought you had, so you have 12 dollars. But what does it mean to say that you ...
12
votes
1answer
106 views

Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$?

Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$ ? I have noticed that if $a_n$ is the leading co-efficient of $p(x)$ then $a_n=(-1)^n ...
0
votes
0answers
51 views
+100

Applying drag to a collision prediction formula

I feel like this question might be below the minds of Math StackExchange, but I'll try anyway. (I can understand Math generally, but I'm probably not the caliber of people here.) I've been working on ...
0
votes
1answer
81 views

AMC 2012 Junior Question [on hold]

$x^2 +y^2 +z^2 = 100x+10y+z $. Find the smallest number and largest number that fit the equation.The numbers are below 1000 I am just baffled at the question.Is there a way to tackle such questions?
5
votes
1answer
82 views

Find all functions

Find all functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that $$f(xyz)=f(xy+yz+xz)(f(x)+f(y)+f(z))$$ for all non-zero reals $x, y, z$ such that $...
0
votes
2answers
74 views

High powers of complex numbers [on hold]

I have these two questions that I am trying to solve. I know that I am suppose to use De Movire's Theorem but I am getting stuck. Can you guys please help out? Thanks. Compute the following ...
2
votes
3answers
84 views

Q27 from AMC 2012(Senior)

Five consecutive integers $p,q,r,s,t$,each less than $10000$, produce a sum which is a perfect square,while the sum of $q,r,s$ is a perfect cube.What is the value of $ \sqrt{p+q+r+s+t}$ ? What I have ...
3
votes
2answers
98 views

Q26 from AMC 2012

Slim took a long road trip across Australia over a number of days($x>1$).She travelled a total of 2012 km.On the first day,she travelled a whole number of kilometers and each subsequent day she ...
-1
votes
2answers
54 views

Find the integer closest to $ a - b$ [on hold]

Let $$a = \frac{1^{2}}{1} + \frac{2^{2}}{3} + \frac{3^{2}}{5} + \ldots + \frac{1001^{2}}{2001}.$$ Let $$b = \frac{1^{2}}{3} + \frac{2^{2}}{5} + \frac{3^{2}}{7} + \ldots + \frac{1001^{2}}{2003}.$$ ...
6
votes
2answers
109 views

Show that an integer matrix with following conditions is the identity $I$

every entries of $A$ is integer every entries of $A-I$ is multiple of a prime $p$ ($p\geq3$) there exists $n\ge1$ such that $A^n=I$ show that $A=I$ I tried $A=I+p^kB$ where not every entries of $B$ ...
0
votes
1answer
30 views

Gradient at a point

I have seen that to find the gradient at a point on a curve this is the equation, e.g $G$ at a point $= \lim_{h \to 0} \frac{ f(a+h)-f(a)}{h}$ given $2x^2-5x$, find the gradient to the tangent at ...
2
votes
5answers
294 views

$33^{33}$ is the sum of $33$ consecutive odd numbers. Which one is the largest? (Q25 from AMC 2012)

The number $33^{33}$ can be expressed as the sum of $33$ consecutive odd numbers. The largest of these odd numbers is $\mathrm{A.}\ 33^{32} +32$ $\mathrm{B.}\ 33^{31} +32$ $\mathrm{C.}\...
0
votes
1answer
39 views

Variant of the geometric series

Can someone explain me how one computes $$\sum_{k=1}^n kq^k = \dfrac{nq^{n+2}-(n+1)q^{n+1}+q}{(1-q)^2}$$ and what exactly the derivative has to do with it?
0
votes
1answer
24 views

Vector questions about finding magnitudes, dot products, and angles.

I am given the following problem: Let $\Vert \overrightarrow{a}\Vert = 3$ , $\Vert \overrightarrow{b}\Vert = 2$ and $\angle \left(\overrightarrow{a},\overrightarrow{b}\right) = 60^\circ$. Find $\...
1
vote
1answer
31 views

Solving an equation on the graph

The picture shows the graph of $$ y = \frac{1}{10} (x^3 + 3x + 20). $$ I was then told to draw a straight line on the graph, then solve the equation $x^3 + 23x = 30$. I'm a little confused here ...