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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4
votes
5answers
82 views

How to solve a convoluted absolute value inequality?

$$ \lvert \lvert x-2\rvert -3\rvert \lt 5 $$ How can I attack this the best way? I see that both sides are positive. Squaring yields: $$ \lvert x-2\rvert ^2 -6 \lvert x-2\rvert +9\lt 25 $$ $$ x^2-...
0
votes
0answers
85 views

Help in writing a nasty expression in nice closed form

This question is abouting re-writing a product in nice closed form. I have the following $$f(v_1) = \left(\sum_{i=1}^K \pi \lambda_i \delta_1 v_1^{\delta_1-1} P_i^{\delta_1} e^{-\beta_i (v_1P_i)^{\...
0
votes
1answer
40 views

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, ...
2
votes
0answers
63 views

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 ...
2
votes
2answers
81 views

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, ...
0
votes
1answer
42 views

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
0
votes
0answers
51 views

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.
1
vote
2answers
73 views

$\text{lcm} (a, b)=\text{lcm} (a+c, b+c)$

Can $\text{lcm} (a, b)=\text{lcm} (a+c, b+c)$ for natural $a, b, c$? I've tried writing out all divisors of $a, b, c$ like $p_1 p_2$ etc. And tried that maybe if $a+c> a$ and $b+c> b$ the $\...
4
votes
3answers
355 views

Beautiful cyclic inequality

Prove that cyclic sum of $\displaystyle \sum_{\text{cyclic}} \dfrac{a^3}{a^2+ab+b^2} \geq \dfrac{a+b+c}{3}$ , if $a, b, c > 0$ I'm really stuck on this one. Tried some stuff involving QM> AM(...
7
votes
2answers
677 views

Beautiful problem on a progression

$\{x_n\}$ is a sequence defined as follows: $x_1=20,\quad x_2=14,\quad x_{n+2}=x_n - \frac{1}{x_{n+1}}$. Prove that $0$ is among the members of this sequence. Find its number. I tried some stuff ...
1
vote
2answers
54 views

Algebra Logical Pythagorean theorem help

A wire is attached to the top of a pole. The pole is 2 feet shorter than the wire, and the distance from the wire on the ground to the bottom of the pole is 9 feet less than the length of the wire. ...
2
votes
2answers
109 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer? ...
0
votes
2answers
180 views

A boy's father is 25 years older than him. The sum of their ages is 31. How old is the boy?

Q.) A boy is $y$ years old. His father is 25 years older than he is. The sum of their ages is 31. How old is he? In class we wrote the answer as $\displaystyle 2y+25=31 \rightarrow 2y=31-25=6 \...
2
votes
2answers
106 views

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?
1
vote
3answers
32 views

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?
1
vote
1answer
32 views

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 ...
3
votes
3answers
70 views

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?
1
vote
1answer
52 views

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 ...
2
votes
1answer
69 views

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 ...
3
votes
2answers
54 views

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 ...
1
vote
2answers
55 views

Some help with sin and cos

I'm having trouble to understand the following equalities in these two equations, i.e. how to apply the addition formulas. Firstly: $$ \frac {1- \frac {sin^2(\frac x2)} {cos^2(\frac x2)}} {1+ \frac {...
0
votes
1answer
67 views

Combining two liquids with different weights to achieve a desired volume and weight

I have two liquids - water and alcohol, each liquid has a different mass Water weighs 1 gram per ML Alcohol weighs 0.5 gram per ML (just for the sake of the example) I wish to combine these ...
-1
votes
1answer
25 views

How many milliliters of liquid to fill [duplicate]

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more millilitres of liquid need to be added in order to fill the ...
0
votes
1answer
45 views

Simplify the algebraic expression

Can someone please explain to me how the algebraic expression in the picture is simplified. To be more specific, how (1) becomes (2). $3x^2(6x-4)^4 + x^3(6\times 4\times (6x-4)^3)$ $3x^2(6x-4)^3(6x-...
1
vote
4answers
683 views

How to find the 4th degree polynomial with given values at $0,1,2,3,4$?

Determine a fourth degree polynomial p that has $p(0), p(1), p(2), p(3), p(4)$ equal to $7, 1, 3, 1, 7$, respectively. Using my ideas, I first write out the points on the polynomial as $(0,7), (1, 1),(...
1
vote
2answers
85 views

Find all almost lower bounds and almost upper bounds of $\{\frac 1n: n\in \Bbb N\}$

A number $x$ is called an almost upper bound for $A$ if there are only finitely many numbers $y$ in $A$ with $y\ge x$. An almost lower bound is defined similarly. (a) Find all almost lower bounds ...
0
votes
4answers
63 views

How to solve this inequality? $\sqrt x\geqslant x-6.$

How to solve this inequality? $$\sqrt x\geqslant x-6.$$ My answer is $[4,9]$, but it must be $[0,9]$, I don't understand what's wrong. Could you give me solution?
1
vote
1answer
55 views

Strong Induction Proof / Algebra

Alright, I pretty much have the proof done, now just trying to do the algebra on it. This is the question... The information I have is: $$a_k = C_1 r^k + C_2 s^k$$ $$a_{k-1} = C_1 r^{k-1} + C_2 ...
1
vote
1answer
37 views

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 $a_1$?
0
votes
1answer
56 views

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?
1
vote
5answers
70 views

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 ...
1
vote
4answers
833 views

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 $3$ ($y\times 3 = z$) ...
2
votes
3answers
81 views

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 ...
1
vote
2answers
57 views

Proof by induction of sum

My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 40. Exercise 10. Prove by induction, that for $n\ge1$ we have $$\sum_{k=n+1}^{2n}\frac{1}{...
3
votes
2answers
66 views

If $f(x) = \sqrt{x}$, what is the domain of $f^4(x)$?

I am unclear if I should consider the function's domain before or after raising it to the power. My textbook gives the following definition of raising a function to a power: By $f^n$, we mean the ...
0
votes
2answers
65 views

How many milliliters to fill cone

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more milliliters of liquid need to be added in order to fill the ...
0
votes
2answers
77 views

Completely factor a polynomial using the rational root theorem and synthetic division

I am currently seriously confused. My problem, as stated above, is about completely factoring a polynomial. My question is, once you get your possible factors, how do you then simplify it down? Ill ...
0
votes
1answer
89 views

Linear equation with 3 equal signs

Here is my linear equation: Solve for p: $\frac{5}{6} = \frac{n}{72} = \frac{m+n}{84} = \frac{p-m}{120}$ How am I supposed to solve for this 1 variable when there are multiple equal signs and 3 ...
3
votes
3answers
106 views

Proof of: if $x^2+y^2=2xy$ then $x=y$

I am trying to prove $x^2+y^2=2xy$ then $x=y$ What I have done is suppose $x^2+y^2=2xy$ then $x^2+y^2+(-2xy)=0\iff x^2+(-xy)+(-xy)+y^2=0 \iff (x+(-y))\cdot x+(x+(-y))\cdot-y=0 \iff (x+(-y))^2=0$ i ...
1
vote
5answers
102 views

Solve $\sqrt{3x}+\sqrt{2x}=17$

This is what I did: $$\sqrt{3x}+\sqrt{2x}=17$$ $$\implies\sqrt{3x}+\sqrt{2x}=17$$ $$\implies\sqrt{3}\sqrt{x}+\sqrt{2}\sqrt{x}=17$$ $$\implies\sqrt{x}(\sqrt{3}+\sqrt{2})=17$$ $$\implies x(5+2\sqrt{6})=...
0
votes
2answers
169 views

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 ...
1
vote
1answer
25 views

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 ...
0
votes
5answers
108 views

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 ...
2
votes
1answer
107 views

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 ...
0
votes
1answer
207 views

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 ...
1
vote
3answers
80 views

Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $

How can I prove that, for $a,b \in \mathbb{Z}$ we have $$ 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 \, ? $$ Here, $\left \lfloor\,\right \...
6
votes
3answers
206 views

Why is Division harder than Multiplication?

Both conceptually and computationally it feels easier to see that: $ 6 \cdot 3.7 = 22.2$ than it is to see that $ 22.2 \div 6 = 3.7 $. Thoughts about the roots of this asymmetry? An analogous ...
2
votes
3answers
57 views

Positive values of $x$ that satisfy the inequality $\frac{1}{x}-\frac{1}{x-1}>\frac{1}{x-2}$

Determine the set of positive values of $x$ that satisfy the inequality $$\frac{1}{x}-\frac{1}{x-1}>\frac{1}{x-2}.$$ My attempt: \begin{align} \frac{-1}{x(x-1)} & >\frac{1}{(x-2)} \\[0.1in] ...
1
vote
1answer
193 views

Mortgage payment calculation without annuty.

I have been asked the following problem by a student of mine and there is a specific method that he requested. A mortgage of $\$450,000$ is loaned for a monthly payment for $30$ years with nominal ...
1
vote
2answers
58 views

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-...