The problem-solving tag has no wiki summary.
17
votes
9answers
581 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 ...
4
votes
2answers
79 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?
8
votes
1answer
51 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
31 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
41 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
103 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?
5
votes
3answers
100 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:
...
4
votes
6answers
83 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!
8
votes
2answers
117 views
Solving matrix equations of the form $X = AXA^T + C$
I'm trying to solve this matrix equation:
$$X = AXA^T + C$$
In particular,
$$
X = \begin{bmatrix} 1.5 & 1 \\ -0.7 & 0 \end{bmatrix}
X
\begin{bmatrix} 1.5 & -0.7 \\ 1 & 0 ...
0
votes
1answer
71 views
Puzzle identification and solving algorithm
I am trying to solve 8x8 puzzle (total 64 buttons). Similar to LightsOut, but in this rules are different. Goal is turn ON every button.
Example:
...
2
votes
0answers
43 views
Non-linear system of 4 unknowns
What are the non-zero solutions in $x,y,z$ and $t$ of the following system of equations
\begin{cases}
(1+ax+bz)(1-x)=1\\
(1+cy+bt)(1-y)=1 \\
(1+dx+bt)(1-z)=1\\
(1+fy+bz)(1-t)=1
\end{cases}
2
votes
1answer
137 views
Python numerical solution for a nonlinear second order ODE with two boundary conditions
I want to solve numerical the next equation, in Python
$$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$
it is a nonlinear second order $ODE$ with two $B.C$. ...
1
vote
1answer
48 views
Problems on submanifolds
I am learning differential geometry and a basis of the theory of smooth manifolds but i'm feeling a lack of practice in solving problems on submanifolds in $\mathbb{R}^n$ (problems like 'prove that ...
2
votes
2answers
80 views
Simultaneously solving of equations
I am trying to refresh some math skills and I am struggling over the following problem. I tried to solve it with the help of a number of sources (i.e. http://www.idomaths.com/simeq.php), but I haven't ...
0
votes
2answers
58 views
Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$
I am struggling to prove this map statement on sets.
The statement is:
Let $f:X \rightarrow Y$ be a map.
i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$
ii) $\forall_{A,B \subset X}: ...
-2
votes
0answers
31 views
Transforming a Joint PDF [duplicate]
I have a pdf $f(X,Y)=(\frac{1}{4})^2e^{−\frac{(|x|+|y|)}{2}}$. My goal is to find the joint PDF $f(W,Z)$ taking in consideration this $W=XY$ and $Z=Y/X$.
I know I can not use Jacobian because is a ...
0
votes
0answers
18 views
Solve the cubic equation for min $r$ and max $k$
I need to find the minimum $r$ and the maximum $k$ of the following cubic equation for which there does not exist three distinct real roots.
$rx^3-rkx^2+(r+k)x-rk=0$.
Is it possible to find such $r$ ...
1
vote
2answers
38 views
Show that one person must have x amount of dollars
A group of six friends discover they have a total of \$21.61 with them on a trip to the movies. Show that one or more of them must have at least \$3.61.
How should I approach this problem? I can see ...
1
vote
2answers
21 views
Simple recursive equation sub-solution.
I have tried to solve a very simple recursive equation:))), but I don't know what's wrong with my brain but I got other solution when I partially solve the equation.
Equation: $$T(n) = (n+2) + ...
3
votes
2answers
63 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
76 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
40 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.
2
votes
5answers
142 views
How to show $A-B \subseteq C \Rightarrow A\cup B \subseteq B\cup C$?
I really need help with this logical proof.
Show that $A-B \subseteq C \Rightarrow A\cup B \subseteq B\cup C$.
Please show the steps to the solution. Thank you!
1
vote
0answers
25 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
93 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
50 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
38 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 ...
58
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 ...
0
votes
4answers
139 views
1
vote
6answers
230 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
31 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 ...
30
votes
1answer
545 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?
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 $f(k)=2^k$ ...
-2
votes
1answer
115 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
82 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 ...
0
votes
1answer
91 views
I really need help on this question… Please help… [closed]
Mickey, as well as being a very clever mathematician, is also a keen gardener and has a fenced allotment which has a square shed placed in one of its corners.
In order to celebrate his wife Gilly’s ...
18
votes
4answers
103 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 ...
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 ...
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 ...
10
votes
4answers
194 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 ...
4
votes
5answers
82 views
Different ordered triples $(a,b,c)$ of non-negative integers
How many different ordered triples $(a,b,c)$ of non-negative integers are there such that $a+b+c=50$?
I tried to list the possibilities but the list is way too long, I know how to find the ordered ...
0
votes
3answers
48 views
system of equations with three equations.
We have to find all real solutions to this system of equations:
$$x=\frac{4z^2}{1+4z^2},y=\frac{4x^2}{1+4x^2},z=\frac{4y^2}{1+4y^2}$$
4
votes
1answer
51 views
Chessboard and Catalan numbers [duplicate]
Let's consider labeling the square of a $2\times n$ chessboard from $1$ to $2n$ such that the numbers increase from left-to-right and bottom-to-top.
Prove that the number of such labelings equals ...
-1
votes
2answers
356 views
In △ ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees?
In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because ...
9
votes
3answers
206 views
Solve the functional equation $f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)$ with $f : [0,\infty) \to \mathbb R$ continuous
Solve the functional equation
$$f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)\qquad \forall x\geq 0$$
with $f : [0,\infty) \to \mathbb R$ continuous.
I can't manage to get this one ...
0
votes
3answers
57 views
How to find remain factor of this trigonometic equation?
The equation $$3\sin^2 x - 3\cos x -6\sin x + 2\sin 2x + 3=0$$ has a solution $x = 0$. That is mean it has a factor $\cos x - 1$. I tried write the given equation has the form
$$(\cos x - 1)P(x)=0.$$ ...
