The problem-solving tag has no wiki summary.
7
votes
6answers
374 views
Solve $\sqrt{x+4}-\sqrt{x+1}=1$ for $x$
Can someone give me some hints on how to start solving $\sqrt{x+4}-\sqrt{x+1}=1$ for x?
Like I tried to factor it expand it, or even multiplying both sides by its conjugate but nothing comes up ...
4
votes
6answers
118 views
Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$
Where do I start to solve a equation for x like the one below?
$$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$
After squaring it, it's too complicated; but there's nothing to factor or to ...
5
votes
1answer
71 views
Maximizing an unusual function (Putnam 1996)
“Fish," he said, "I love you and respect you very much. But I will kill you dead before this day ends.”
-- Ernest Hemingway, The Old Man and the Sea
I have, with varying degrees of concentration, ...
0
votes
1answer
36 views
About continuous functions and aritmethic progression
I've try solve this question, but I haven't sucess...
The problem is the following:
A continuous functions $f:[a,b]\rightarrow \mathbb{R}$ assume positive and negative values in its domain, show ...
1
vote
2answers
64 views
Solving a system of equation:
Solve for $x,y$:
\begin{align}
x^3 + y^3&=2\\
x^2 +x + 9y - 3y^2&=8 \\
\end{align}
I can find $x=y=1$ by guessing. Please help me solve it without using computer.
Thanks
Edited, sorry, I ...
3
votes
0answers
57 views
No idea how to solve this equation using two exponentials
The equation I have is:
$$A = B ( \exp(C x) - \exp(-Dx) )$$
How do I solve for $x$ given $A$, $B$, $C$, $D$?
I have no idea how. The only idea I have is that I could express these in terms of a ...
0
votes
1answer
22 views
Frequency determination from Dimension analysis
the time averaged total energy, $\bar E$,
has the following $\varepsilon$ expansion in $D$ dimension:
\begin{equation}
\bar{E}=\varepsilon^{2-D}\frac{E_0}{2\lambda}+ \varepsilon^{4-D}E_1
...
6
votes
1answer
115 views
Why substitution method does not work for $\int (x-\frac{1}{2x} )^2\, \mathrm dx$?
Why $$\int \ \left(x-\frac{1}{2x} \right)^2 \, \mathrm dx$$ is easy to integrate once $$\left(x-\frac{1}{2x} \right)^2$$ is expanded, but impossible using substitution method? (tried 5 different subs ...
0
votes
0answers
17 views
Creating simultaneous non-linear equations
Suppose, for each given $w_1 \in \mathbb{R}$, there exists unique $x_1,y_1,z_1 \in \mathbb{R}$ that satisfy the following system of equations:
$F_1(w_1,x_1,y_1,z_1)=0\\
F_2(x_1,y_1,z_1)=0\\
...
8
votes
2answers
114 views
Proving that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$
How do I prove that:
$$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$$
Do I use induction?
13
votes
2answers
107 views
Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$
How do I simplify:
$$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$
Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a ...
2
votes
1answer
36 views
Simplifying $\left|\left|\sqrt{-x^2}-1\right|-2\right|$
How do we simplify the expression $\left|\left|\sqrt{-x^2}-1\right|-2\right|$?
This is very confusing. Do they cancel out and become just simply $\sqrt{-x^2}-1-2$?
2
votes
1answer
44 views
Power series of $f(x)=\sqrt{\frac{1+x}{1-x}}$
How do I find the power series form of $\,f(x)\,$:
$$\displaystyle f(x)=\sqrt{\frac{1+x}{1-x}}$$
I tried to multiply the fraction by $\,\dfrac{1+x}{1+x}\,$ but it didn't help...
4
votes
7answers
111 views
Solving $x^{\log(x)}=\frac{x^3}{100}$
How do I find the solution to:
$$x^{\log(x)}=\frac{x^3}{100}$$
So I multiplied 100 both sides getting:
$$100x^{\log(x)}=x^3$$
Now what should I do?
4
votes
6answers
91 views
Simplify $\log_23\ \log_34\ \log_4 5\ \log_5 6\ \log_6 7\ \log_7 8$
How do I evaluate the product:
$$\log_23\ \log_34\ \log_4 5\ \log_5 6\ \log_6 7\ \log_7 8$$
I know that $$\log_ba=\frac{\log\ a}{\log\ b}$$
How can I apply it?
Thanks!
19
votes
11answers
876 views
Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$
Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)?
$$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$
Using the calculator I can see that the first one ...
5
votes
3answers
108 views
Using the hypothesis $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ to prove something else
Assuming that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$$
Is it possible to use this fact to prove something like:
...
0
votes
1answer
76 views
Puzzle identification and solving algorithm
I am trying to solve 8x8 puzzle (total 64 buttons). Similar to LightsOut, but in this rules are different. Goal is turn ON every button.
Example:
...
2
votes
0answers
43 views
Non-linear system of 4 unknowns
What are the non-zero solutions in $x,y,z$ and $t$ of the following system of equations
\begin{cases}
(1+ax+bz)(1-x)=1\\
(1+cy+bt)(1-y)=1 \\
(1+dx+bt)(1-z)=1\\
(1+fy+bz)(1-t)=1
\end{cases}
1
vote
1answer
50 views
Problems on submanifolds
I am learning differential geometry and a basis of the theory of smooth manifolds but i'm feeling a lack of practice in solving problems on submanifolds in $\mathbb{R}^n$ (problems like 'prove that ...
2
votes
2answers
80 views
Simultaneously solving of equations
I am trying to refresh some math skills and I am struggling over the following problem. I tried to solve it with the help of a number of sources (i.e. http://www.idomaths.com/simeq.php), but I haven't ...
-2
votes
0answers
31 views
Transforming a Joint PDF [duplicate]
I have a pdf $f(X,Y)=(\frac{1}{4})^2e^{−\frac{(|x|+|y|)}{2}}$. My goal is to find the joint PDF $f(W,Z)$ taking in consideration this $W=XY$ and $Z=Y/X$.
I know I can not use Jacobian because is a ...
0
votes
0answers
19 views
Solve the cubic equation for min $r$ and max $k$
I need to find the minimum $r$ and the maximum $k$ of the following cubic equation for which there does not exist three distinct real roots.
$rx^3-rkx^2+(r+k)x-rk=0$.
Is it possible to find such $r$ ...
1
vote
2answers
39 views
Show that one person must have x amount of dollars
A group of six friends discover they have a total of \$21.61 with them on a trip to the movies. Show that one or more of them must have at least \$3.61.
How should I approach this problem? I can see ...
1
vote
2answers
21 views
Simple recursive equation sub-solution.
I have tried to solve a very simple recursive equation:))), but I don't know what's wrong with my brain but I got other solution when I partially solve the equation.
Equation: $$T(n) = (n+2) + ...
3
votes
2answers
72 views
In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other?
In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other?
I tried to draw diagrams onto a $8\times8$ square but I'm only getting $16$ ...
4
votes
6answers
78 views
Repeating Decimals [duplicate]
I'm just wondering how do we simplify repeating decimals into a fraction in general?
Like, for example,
$$0.5656\dots$$
$$0.12424\dots$$
$$4.23777\dots$$
Thanks!
2
votes
0answers
26 views
How to find the point in a closed geometrical figure which maximizes the “direct-line-of-sight function”
To expand upon the title, and put it in clear terms, I phrase the problem thusly:
Consider the interior of any continuous, closed, non-self-intersecting curve in the plane. (I'm not sure if I'm ...
-2
votes
1answer
44 views
how do i solve this problem
I got this problem to solve in a non calculator paper test and didn't know how to solve it.
The mode of five numbers is 3.
The median is 7.
The mean is 6.
Work out the 5 numbers.
1
vote
0answers
26 views
Uniqueness of solution for boundary value problem
In the article : "Third order semipositone boundary value problems"
They assume that $w(t)$ is nondecreasing and $w(t)>0$ on $(q,1]$ .
and they found the Green's function ...
-2
votes
2answers
94 views
How do I work out this problem? [closed]
Three different numbers add up to 90.
All numbers are even.
The second number is twice the first number.
The first number multiplied by the second number is equal to the third number.
Work out the ...
1
vote
1answer
54 views
Sum over cosines = dirac delta - how to get the coefficients?
Given this formula:
$$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$
Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$?
I googled and searched all kinds of ...
2
votes
3answers
40 views
Finding the number of ordered pairs that satisfies an equation
How many number of ordered pairs of positive integer $(a,d)$ satisfies:
$$\frac{1}{\frac{1}{a+2d}-\frac{1}{a+3d}}-\frac{1}{\frac{1}{a}-\frac{1}{a+d}}=2012$$
This question is too complicated. Like I ...
0
votes
4answers
141 views
1
vote
6answers
231 views
Solve the equation $x-7=28$ [closed]
The question is $x-7=28$
But I'm not sure if when I subtract do I have to change the signs to negative?
3
votes
2answers
110 views
Irrational equation, How to solve it?
The equation
$$\sqrt[3]{x^2-1} + x = \sqrt{x^3-2}$$ has a solution $x = 3.$ How to solve this eqution?
1
vote
3answers
37 views
I need help with this word problem.
A pet store owner wants to mix together an high quality dry cat food costing 1.10
per pound with a lower quality dry cat food costing 0.85 per pound. How many
pounds of each should be mixed together ...
1
vote
1answer
28 views
Solving for $f(n+1)$ when $f(k)$ is known for $k=0,1,…,n$
I posted earlier about polynomials but this is different type of problem I think. I seem to have an answer but I mistrust it....
A polynomial $f(x)$ where deg[$f(x)$]$\le{n}$ satisfies $f(k)=2^k$ ...
30
votes
1answer
550 views
Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers
For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
-2
votes
1answer
119 views
What branch of mathematics improves logical thinking? [closed]
So, that's the question. I dare to generalize it even wider: what branch of mathematics improves the general thinking ability, intilligence, the way the person thinks, and makes it more logical? I'm ...
3
votes
0answers
85 views
Difference between two sets of data points
I'm making a simple calibration of a z-stage, by measuring a number of points in one direction with a constant $\Delta$Z between each sample. Then I reverse the direction and measure the same number ...
18
votes
4answers
105 views
Ways to fill a $n\times n$ square with $1\times 1$ squares and $1\times 2$ rectangles
I came up with this question when I'm actually starring at the wall of my dorm hall. I'm not sure if I'm asking it correctly, but that's what I roughly have:
So, how many ways (pattern) that there ...
4
votes
2answers
53 views
Find $x,y$ such that $x=4y$ and $1$-$9$ occur in $x$ or $y$ exactly once.
$x$ is a $5$-digits number, while $y$ is $4$-digits number. $x=4y$, and they used up all numbers from 1 to 9. Find $x,y$.
Can someone give me some ideas please? Thank you.
2
votes
1answer
35 views
How to compute a product of logarithms?
I've been reading through Stewart's Calculus textbook, and came across the following problem fairly early on -
What is $$\prod_{i = 2}^{31} \log_i (i + 1)\;?$$
I did some searching, and found ...
0
votes
1answer
43 views
Showing that $B_0=1$ and that $\sum^n_{k=0}{{n+1}\choose k}B_k =0$
Let $h(x)=x$ and $f(x)=e^x-1$ for $x\in \mathbb{R}$. It's known that $\frac{h(x)}{f(x)}$ has a Taylor series in $0$, which is written as follow:
$$\sum^\infty_{n=0}\frac{B_n}{n!}x^n$$
So, how do I ...
1
vote
1answer
53 views
Showing that $f$ is analytic in $\mathbb{R}$
Okay, let $y\in \mathbb{R}\;$ be fixed and consider the function $\;f(x)=e^{x+y}$.
How do I show that $\,f\,$ is analytic in $\,\mathbb{R}\,$ and use this fact and the Taylor series in $\,x=0\,$ of ...
59
votes
4answers
1k views
Probability that a stick randomly broken in five places can form a tetrahedron
Randomly break a stick in five places.
Question: What is the probability that the resulting six pieces can form a tetrahedron?
Clearly satisfying the triangle inequality on each face is a necessary ...
10
votes
4answers
199 views
Show that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.
The following problem was on a math competition that I participated in at my school about a month ago:
Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.
I will outline ...
2
votes
1answer
152 views
Python numerical solution for a nonlinear second order ODE with two boundary conditions
I want to solve numerical the next equation, in Python
$$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$
it is a nonlinear second order $ODE$ with two $B.C$. ...
0
votes
3answers
50 views
system of equations with three equations.
We have to find all real solutions to this system of equations:
$$x=\frac{4z^2}{1+4z^2},y=\frac{4x^2}{1+4x^2},z=\frac{4y^2}{1+4y^2}$$
