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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0
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1answer
35 views

Difference between non - negative and positive integral solution :

Difference between non - negative and positive integral solution : (a) Number of non negative integral solution of equation $x+2y+3z+4w =n$ = Coefficient of $x^n$ in ...
0
votes
1answer
31 views

What is an upper bound on $e^{-W_{-1}(c_1)}$?

What is an upper bound on $e^{-W_{-1}(c_1)}$ and $e^{-W(c_1)}$, where $W$ is the Lambert W function?
4
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1answer
118 views

Growth of ratio based on sum of squared binomial identity

It is a well-known identity that $$\binom{n}{0}^2+\binom{n}{1}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}.$$ By symmetry of the binomial coefficients, this means the ratio ...
2
votes
5answers
53 views

Why is $\frac{1}{n}x(1-x)\leq \frac{1}{4n}$ for $x\in[0,1]$?

Why is $\frac{1}{n}x(1-x)\leq \frac{1}{4n}$ for $x\in[0,1]$? I don't see why this is true, just from looking at the left hand side of the inequality.
1
vote
1answer
36 views

compute the number of ordered triples.

Compute the number of order triples of positive integers $(a,b,c)$ such that $$a\le b\le c \qquad \text{ and } \qquad abc =2014^3$$ I have no clue on how to solve this type of question, please teach ...
2
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1answer
37 views

high school senior A division contest question [duplicate]

The product of my age 7 years ago and my age 7 years from now is a positive perfect square. Compute my age now. I set up the equation, $(x+7)(x-7)=a^2$, so $x^2 -49= a^2$. So I know that x has to be ...
1
vote
4answers
39 views

Simplfiying $x(5xy+2x-1)=y(5xy+2y-1)$

I want to simplify: $x(5xy+2x-1)=y(5xy+2y-1)$ to $(x-y)(5xy+something-1)=0$ but I can't figure out what to do with the $2x$ and $2y$ on both sides.
1
vote
0answers
66 views

Rationality of sum of roots (or rational function thereof) of a system of algebraic equations.

I am looking for a reference/hints of proof towards statements of the kind; Given an irreducible system of $n$ polynomial equations over $\mathbb Q$ in $n$ variables $$P_i(x_1,...,x_n)=0,\quad ...
2
votes
1answer
32 views

How to find the range of the function $\frac{e^x log_{e} x 5^{x^2+2} (x^2-7x+10)}{2x^2-11x+12}$

How to find the range of the function $$\frac{e^x log_{e} x 5^{x^2+2} (x^2-7x+10)}{2x^2-11x+12}$$ We can see the domain of the function is $(\frac{3}{2}, 4) \cup (4, \infty)$ as the denominator is ...
0
votes
1answer
29 views

natural way of thinking this logarithm inequality

I know that this inequality hold for $x\geq0$ and $n\in\mathbb{N}$ $$\ln(x+n)<nx+n$$ in fact, when $x=0$, for all $n$, $\ln(n)<n$ if $x>0$, $$\frac{d}{dx}[\ln(x+n)]=\frac{1}{x+n}\qquad ...
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votes
2answers
40 views

Problem on multiplication formulae.

Given $a^3 + b^{3}+ c^{3}= (a+b+c)^{3} $. Prove that for any natural number $n$, $$a^{2n+1}+b^{2n+1}+c^{2n+1}=(a+b+c)^{2n+1}$$ I first tried mathematical induction but did not proceed anywhere. Can ...
3
votes
4answers
97 views

Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$

Prove the identity: $$n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$$ I tried using the binomial coefficients identity $2^n = \sum_{k=1}^n {n \choose k}$ but got stuck along the way.
1
vote
1answer
33 views

Solve to find $y(x)$ of the $\frac{1}{\sum_{n=0}^{\infty }y^n}-\sum_{n=0}^{\infty }x^n=0$

Solve the equation to find the $y$ as a function to respect $x$ without $n$ $$\frac{1}{\sum_{n=0}^{\infty }y^n}-\sum_{n=0}^{\infty }x^n=0$$
1
vote
0answers
41 views

Aristarchus and the Moon's distance

I'm trying to plug the number into the equation below, but I'm getting 67 earth radii instead of 60 radii. What am I getting wrong? http://www.phy6.org/stargaze/Shipprc2.htm Let $R$ be the radius ...
3
votes
1answer
54 views

How to invert a simple exponential growth formula

I think this is simple but my math skills are limited. I have a basic exponential growth formula: $$y=x \cdot (1-p)^n$$ and I have $y$ and $x$ and $n$ values and I need value of $p$. Then when I solve ...
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3answers
62 views

Finding all prime numbers $p$ such that $p^a + p^b$ is a perfect square

Find all prime numbers $p$ and positive integers $a$ and $b$ such that $p^a + p^b$ a perfect square. How can I find this. I have no idea about this problem.
0
votes
1answer
212 views

Why Do Imaginary Numbers Exist [duplicate]

I understand how to solve problems dealing with imaginary numbers, but I don't understand the reason why they exist and what they really do. Could somebody please explain to me what the point of them ...
0
votes
2answers
18 views

How do you define a function to get the output you need for a particular parameter?

Suppose we have a function $y(x)$ such that $y (\frac{-e^{-2\lambda} + e^{-\lambda}}{1-e^{-2\lambda}}) = \lambda$ How can I determine $y(x)$? Are there steps that outline how to solve such a ...
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2answers
72 views

high school math, senior A division contest

Larry selects a 20-digit number while David selects a 14-digit number. When larry divides his number by David's number, the quotient is an integer with n digits. Compute all possible value of n. This ...
0
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1answer
39 views

Parallel line with a known point

Line m goes through a point D(2, -4). Line m is parallel to line l: $5x+3y=-17$. Describe line m with an equation of type $ax+by=c$. The solution should be $c=5*2+3*-4=-2$ so $\text{m: }5x+3y=-2$ ...
0
votes
2answers
71 views

Simple formula on $X_n^{(k)} = \sum_{1 \le i_1 < … < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$ (to show $X_n^{(k)}$ is martingale)

Let $$X_n^{(k)} = \sum_{1 \le i_1 < ... < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$$ If I take $k=2$ and $S_n = Y_1 + \dots + Y_n$ I have of course: $$X_n^{(2)} = \frac{1}{2} (S_n^2 - ...
2
votes
1answer
24 views

equation transformation for y=√x

I need help with this problem not sure if it's correct I have y=√x It says shifts up 6 Reflects on the y axis Then shifted right 4 units I ended up getting y=-√x-4+6 If this not correct please ...
2
votes
1answer
73 views

Converting rational equations into polynomial equations.

A question on my practice exam asks to solve the equation: $$ \frac{1}{x^4} - \frac{1}{x^2} = 12 $$ In the answers section it says to first convert the equation to the equivalent equation:$$ 12x^4 + ...
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votes
2answers
121 views

Sum of 1/1 + 1/2 + 1/3 … +1/n

any hint how to resolve $$f(n) = \frac 11 + \frac 12 +\frac 13 + \dots + \frac 1n$$ What I'm trying to do is to find connection between $$f(n),\,f(n+1)$$ different of $$f(n+1) = f(n)+\frac ...
3
votes
1answer
55 views

Finding the equation of a cubic polynomial $f(x)$, then solving $f(x) = 120$.

Determine algebraically the value(s) of $x$ where the cubic function that has zeroes at $2$, $3$, and $-5$, and passes through the point $(4,36)$ has a value of $120$. It is clear that I need to ...
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2answers
33 views

Roots of a quadratic equation

Can a quadratic equation have irrational roots? By extension, can any equation have irrational roots? If not, why? If it can, how would you visualize it? (geometrically). I want to add that I am a ...
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3answers
101 views

Translating a Word Problem into an Algebraic Equation

Find two consecutive odd integers such that three times the smaller one exceeds two times the larger one by $7$.
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5answers
163 views

Number of real roots of the equation $(6-x)^4+(8-x)^4 = 16$

Find number of real roots of the equation $(6-x)^4+(8-x)^4 = 16$ $\bf{My\; Try::}$ Let $f(x) = (6-x)^4+(8-x)^4\;,$ and we have to find real values of $x$ for which $f(x) = 16$. Now we will form ...
1
vote
1answer
55 views

How do I find the horizontal asymptote of $f(x)=\frac{\sin (x) }{x}$?

I can instantly see that there will be a vertical asymptote at $x=0$, however I am finding it quite a challenge to find a horizontal asymptote. I've drawn the graph and it seems as if the amplitude of ...
0
votes
2answers
53 views

What is the property that allow the transformation $\frac{16a^3}{8ac}=\frac{16}8\cdot\frac{a^3}a\cdot\frac1c$?

In a monomial division like this: $$\frac{16a^3}{8ac}=\frac{16}8\cdot\frac{a^3}a\cdot\frac1c$$ Why I can do this $\dfrac1c$? Where this 1 come from?
1
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0answers
89 views

Help writting financial distribution formula

I need help writing a function to calculate the financial contribution of a product subscription into a given month. Not so straight forward however, since it has to consider months with fixed length ...
0
votes
1answer
72 views

Books Authored by P.S. Modenov

Does anyone have any digital copies of the English translated versions of the books written by Soviet mathematician, Peter S. Modenov?
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4answers
92 views

positive fractions, denominator 4, difference equals quotient

(4,2) are the only positive integers whose difference is equal to their quotient. Find the sum of two positive fractions, in their lowest terms, whose denominators are 4 that also share this same ...
3
votes
3answers
56 views

How to find the value of this function

If $f:\mathbb{R} \rightarrow\mathbb{R}$ is a function which satisfies $$f(x)+f(y)=\frac{f(x-y)}{2}\cdot \frac{f(x+y)}{2}$$ for all $x\in\mathbb{R}$ and $f(1)=3$, what is $f(6)$?
5
votes
1answer
658 views

Fermat's Last Theorem: Contemplation.

So my friend and I are studying elliptic curves and Fermat's Last Theorem appeared several times in the subject matter. So the proof of Fermat's Last Theorem was settled by Andrew Wiles with the work ...
0
votes
1answer
28 views

For what value of a does the curve have minimum and maximum points?

For what value of a does the curve $$f(x)=5x^3 + ax^2 + 10x $$ have minimum and maximum points? Multiple Choices are: A) |a|>15, B) |a|>$\sqrt{150}, C) |a|>1500, D)|a|>\sqrt{30}, E) ...
0
votes
1answer
47 views

How can I tell if the coefficient is negative?

Simple question that I can't find an answer to. I am trying to figure out what my book means by 'if the coefficient is negative'. They don't explain this at all, so that is why I am asking. Here is ...
3
votes
2answers
118 views

Limits with trigonometric functions without using L'Hospital Rule.

I want to find the limits $$\lim_{x\to \pi/2} \frac{\cos x}{x-\pi/2} $$ and $$\lim_{x\to\pi/4} \frac{\cot x - 1}{x-\pi/4} $$ and $$\lim_{h\to0} \frac{\sin^2(\pi/4+h)-\frac{1}{2}}{h}$$ without ...
0
votes
2answers
46 views

Simplifying an inequality: $4x(x-2) \lt 2(2x-1)(x-3)$

I have: $$4x(x-2) \lt 2(2x-1)(x-3)$$ For the last part, do I multiply both things in $()$ by two then solve them like I normally would? If I solve them and then multiply will it work the same? Is that ...
1
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3answers
120 views

Inequality problem involving QM-AM-GM-HM or Cauchy Schwarz inequality

Question: Prove that if $x$, $y$, $z$ are positive real numbers then the following inequality holds: ...
1
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6answers
96 views

What is the limit for the radical $\sqrt{x^2+x+1}-x $?

I'm trying to find oblique asmyptotes for the function $\sqrt{x^2+x+1}$ and I manage to caclculate that the coefficient for the asymptote when x approaches infinity is 1. But when i try to find the ...
14
votes
4answers
171 views

Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$

Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$ This problem is copied from Math Olympiad Treasures ...
1
vote
1answer
28 views

Maximum of k real numbers

Let $a_1, a_2, \cdots, a_k$ are real numbers. Prove that $$\sum_{1 \leq m \leq k} a_{m} - \sum_{1 \leq m < n \leq k } \min (a_{m},a_{n}) +\sum_{1 \leq m < n <p \leq k } \min ...
1
vote
3answers
113 views

Is there a way to expand the cube of a trinomial easily? [closed]

Is there a way to expand $(x+y+z)^3$ easily/fast? It takes some time for me to expand this type of quantities, I was just thinking if there's like a shortcut. Motivation: whenever I do ...
3
votes
2answers
148 views

Why is it important to extend the trigonometric functions to all angles?

The title saids it all. I understand the process to evaluate a trig function for any angle but I don't get why it is actually important...
0
votes
4answers
60 views

Quadratic equation $3x^2 + x - 2 = 0$

I have $3x^2 + x - 2 = 0$ and the answers are supposed to be $-1$ and $2/3$. It was in the quadratic formula chapter so I tried to use that but since the middle x is only 1 for a coefficient, it ends ...
3
votes
2answers
196 views

Fast way to come up with solutions to $x(x-1)(x-2)(x-3)=1$?

I can solve this equation $x(x-1)(x-2)(x-3)=1$ using the usual method but I am looking for a fast analytical method to solve this. Any hints ?
0
votes
1answer
44 views

Prove $\cos\left(\frac{\pi}{2}-x\right) = \sin(x)$ from the unit circle definition

I want to prove that $$\sin\left(\frac{\pi}{2}-x\right) = \cos x \tag{1}$$ I can't use triangles and I need to prove it for all numbers. Observation: By the $\cos$ identity: $$\cos(a-b) = \cos ...
1
vote
1answer
43 views

Angular speed and RPM of truck wheels

A truck with $48$ in. diameter wheels is traveling at $50$ mph a) Find the angular speed of the wheels in rad/min b) How many revolutions per minute does the wheel make? So for (a), I know that one ...
0
votes
1answer
18 views

$\frac{L}{a_{\overline{n}\rceil i}}(n-a_{\overline{n}\rceil i})$ vs. $Li \frac{n-1}{2}$ which is larger?

I am having trouble deciding which of the expression is larger. The following is the original problem and I may not have the expression entirely correct, but I am pretty confident. A loan of $L$ ...