# Tagged Questions

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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### Beautiful problem of a set of a,b,c.

A set of a,b,c was changed to this set: $a^4-2b^2, b^4-2c^2, c^4-2a^2$. It happened that these two sets are identical. Find a,b,c, if a+b+c=-3. $a^2(a^2-2)+b^2(b^2-2)+c^2(c^2-2)=a+b+c=-3$ I guess, ...
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### Newtonian potential for ellipsoid

Is there an explicit expression of the Newtonian potential for ellipsoid? As the expression for ball is clear by its symmetry. Definition of Newtonian potential of ellipsoid $\Omega$ at x is defined ...
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### Proof of a summation of $k^4$

I am trying to prove $$\sum_{k=1}^n k^4$$ I am supposed to use the method where $$(n+1)^5 = \sum_{k=1}^n(k+1)^5 - \sum_{k=1}^nk^5$$ So I have done that and and after reindexing and a little algebra, ...
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### Is $xyz=0$ a joint variation

Is $xyz=0$ a joint variation I know that a joint variation is $\dfrac{x}{yz} = k$ I just want to know if $k$ is allowed to be zero
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### What is the inverse of $f(x)=x^{x^x}$?

I'm curious to find the inverse of $f(x)=x^{x^x}$ As an added extra, I'm already familiar with the Lambert Product Log function.
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### Finding prime solutions to $100q+80 = p^3 + q^2$

Finding prime solutions to $100q+80 = p^3 + q^2$ Does them being prime imply some patterns on division modulo 3 or some other integer? How is this done?
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### Remainders questions help

If we divide a number by 3, 4 ,5 , 6 , we have the remainders 2, 3 , 4 , 5. Is there any way to get a pattern without guessing so many numbers and checking by 3, 4 ,5 ,6?
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### How to compute $\prod_0^n( 1- { 2 \over (2+k)(3+k)}=$?

I have spent quite some time to solve this question, before I asked Wolfram Alpha and got this: $$\prod_0^n \left(1- {2\over(2+k)(3+k)}\right) = { n+4 \over 3(n+2)}.$$ Now that I know that this ...
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### Logarithmic inequality for a>1

Is $\log_{\sqrt a}(a+1)+\log_{a+1}\sqrt a\ge \sqrt6$ always true for $a>1$? What is the approach? Do we check the first a's and then form a induction hypothesis?
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### Show that if x,y,z are not divisible by 53, then $x^{26}+4y^{26} \neq\ z^{26}$

Show that if x,y,z are not divisible by 53, then $x^{26}+4y^{26} \neq\ z^{26}$ I've got that $x,y,z$ to the 52nd power are congruent to 1 modulo 53 from Fermat's. How is it continued? Help would be ...
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### Arithmetic progression with common difference 2061

If there are 30 consequent members of an arithmetic progression with CD of 2061, show that among them are at most 20 squares of natural numbers. I wrote out $a_1$ through $a_{30}$ and tried to find ...
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### Show that if x,y are and $x^4y^2+x^2+2x^3y+6x^2y+8 \leq 0$ then $x \geq -1/6$

Show that if x,y are real and $x^4y^2+x^2+2x^3y+6x^2y+8 \leq 0$ then $x \geq -1/6$ So far I've tried factoring $x^2$ and throwing the 8 on the LHS, but can't get to the needed result. Help would ...
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### Some algebra trouble

How do I show that $$\frac{sa_0-a_1}{s-r} r +\frac{a_1-ra_0}{s-r} s$$ equals $a_1$?
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### Issue on proving quadratic formula

I have come across a stage of the proof: $$\left(x+\frac b{2a}\right)^2=\frac{b^2-4ac}{4a^2}$$ How does $\left(x+\frac b{2a}\right)^2$ not equal $\pm x\pm \frac b{2a}$ when taking the square root?
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### How to determine the derivative of $f$ at $x=2$ by looking at the graph only?

How to determine the derivative of $f$ at $x=2$ (i.e., $f^\prime(2)$) by looking at the graph only ? I am well aware of the theory of the derivative and how to compute it. But how can I ...
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### Solved to be 7 after arithmetic

I recently made a blunder while trying to explain a question asked to me in an interview, The question was Think of $X$ Add $X$ to itself ($X+X = y$) Times the result by $3$ ($y\times 3 = z$) ...
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### Finding all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors.

Find all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors. I've tried checking the first 6-7 $n$'s on wolframalpha, but I don't see any patterns for even nor odd $n$'s. At first ...
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### Two ships leaving a port at different times and different speeds. When do they meet?

Can someone please show me the working out to this word problem I have the answer but have no clue how to do the working out. At noon ship A leaves port steaming at 8 knots 2 hours later ship B ...
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### divide clock into halfs

John has special clocks one hands do 1 turn per minute, second do 1 turn per 3 minutes and third do 1 turn per 15 minutes. how many times and when the first divide clockface into three equal parts in ...
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### Prove that for all positive integers $x$, $\left\lfloor \frac{x^2 +2x + 2}{4}\right\rfloor =\left\lfloor \frac{x^2 + 2x + 1}{4}\right\rfloor$.

Title says it all, basically. I believe it to be true that $$\left\lfloor \dfrac{x^2 + 2x + 2}{4} \right\rfloor=\left\lfloor \dfrac{x^2 + 2x + 1}{4} \right\rfloor$$ for all positive integers $x$. I ...
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### Trouble understanding factorial algebra

I am having trouble understanding some of the algebraic concepts used here. In fact, the entire thing to me makes sense, except for the second red line. I don't understand how the diagonal swap ...
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### Solving Polynomial Equations and Inequalities

The distance, in km, of a ship from its harbour is modeled by the function $d(t)= -3t^3 + 3t^2 + 18t$ where $t$ is the time elapsed in hours since departure from the harbour. a) When does the ...
### Sum of roots of an equation $\sqrt{x-1}+\sqrt{2x-1}=x$
Find the sum of the roots of the equation $\sqrt{x-1}+\sqrt{2x-1}=x$ My attempt: Squaring the equation: $(x-1)+(2x-1) +2\sqrt{(x-1)(2x-1)}=x^2$ $\implies x^2-3x+2=2\sqrt{(x-1)(2x-1)}$ \$\implies (x-...