The problem-solving tag has no wiki summary.
1
vote
1answer
21 views
Counting Card hands with various restrictions
I would like to know if my solutions are correct for the following three combinatorial card questions. In each question, assume we have a standard deck of cards (13 ranks, and 4 suits).
How many ...
1
vote
2answers
170 views
Need help with an integration word problem. This appears to be unsolvable due to lack of information.
I'm not sure I understand what to do with what's given to me to solve this. I know it has to do with the relationship between velocity, acceleration and time.
At a distance of 45m from a traffic ...
8
votes
1answer
170 views
Finding the general term of a sequence
I would like to find an expression for the sequence $\;\{a_n\},\;n=0,\,1,\,2,\,3,\,\dots,\;$ $$-\frac{1}{6},\,\frac{2}{7},\,\frac{5}{8},\,\frac{8}{9},\,\frac{11}{10},\,\ldots$$
So by trial and ...
11
votes
3answers
430 views
Arc sums for a circle of $k$ positive integers whose total sum is $n$
This problem got me thinking about the following more general scenario:
Suppose you have $k$ positive integers with total sum $n$, and you arrange them in a circle.
Given such an arrangement, you ...
-1
votes
1answer
65 views
How many students like all the three subjects?
Among 64 students, 28 of them like Science, 41 like Mathematics and 20 like English. 24 of them like both Math and English. 12 students like both Science and English. 10 students like both science and ...
0
votes
3answers
56 views
what is the value of the digit in the ones place of the following?
1×3×5×7×9×11×13×...×2007×2009
what is the value of the digit in the ones place of the following?
I can't find the solution for this problem.
Please give me some hints
1
vote
3answers
31 views
At what speed should it be traveling if the driver aims to arrive at Town B at 2.00 pm?
A car will travel from Town A to Town B. If it travels at a constant speed of 60 km/h, it will arrive at 3.00 pm. If travels at a constant speed of 80kh/h, it will arrive at 1.00 pm. At what speed ...
1
vote
1answer
80 views
Modular arithmetic word problem
We are buying a total of 12 fruits (apples and bananas) for 132 dollars.
If the apples are 3 dollars more expensive than the bananas, and we bought more apples than bananas, how many bananas we ...
-2
votes
2answers
96 views
Change parabolic equation to canonical form
I have equation $y = -x^2 + 2x + 7$.
How can I change it to canonical form, which looks like $y^2 = 2px$ ?
($p$ will be parameter)
What i ve tried so far:
$$\begin{align}
y &= -x^2 + 2x + 7\\
y ...
0
votes
2answers
73 views
Area of a function is the same as the area of the inverse function
The area of between the function $f(x)=x^2$ and the $x$-axis from $1\to a$ is the same as the area between $f^{-1}(x)$ and the $y$-axis from $1 \to b$ when $f(a)=b$
It says write two equations of $a$ ...
4
votes
2answers
180 views
prove the divergence of cauchy product of convergent series $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$
i am given these series which converge. $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$ i solved this with quotient test and came to $-1$, which is obviously wrong. because it must be $0<\theta<1$ so ...
0
votes
1answer
114 views
How many eggs are there in the basket?
There is a basket of an eggs. The remainder is 1 When we put the eggs into the group of 2. The remainder is 2 when put the eggs in the group of 3. The remainder is 3 when we put eggs in the group of ...
1
vote
1answer
88 views
How can I solve for $x\,$? $:\;\;x-x_r=(x-x_1)e^{\large -(x-x_1)^2}$
I want to find $x$ for given values of $x_r$ and $x_1\,$ (domain $\mathbb{R}$):
$$x-x_r=(x-x_1)e^{\large -(x-x_1)^2}$$
Thanks
1
vote
0answers
52 views
Find total number of sets of integers which satisfy a given equality and inequality
Compute the total number of different sets of integers a1, a2,..,an which satisfy the following equality and constraints:
$$
...
0
votes
1answer
21 views
How far could the plane travel before it made its return?
A fighter plane had enough fuel to last a 6-hour flight. The speed of the wind and the speed of the plane made up a total of 1500km/h when the plane was flying in the direction of the wind during its ...
1
vote
1answer
71 views
what is the ratio of their speed?
Jonathan and Cindy run on a circular track where AB is the diameter of the track, as shown below.
If Jonathan and Cindy run towards each other at the same time from Point A and Point B respectively, ...
1
vote
4answers
108 views
In how many ways can he form a rectangle using all the tiles each time?
A contractor has 1088 square tiles. In how many ways can he form a rectangle using all the tiles each time?
1
vote
3answers
78 views
How old was the teacher in 2008?
Julie ask her teacher, "how old were you in 2008?" "My age in 2008 was the sum of all the digits of my year of birth,' replied the teacher. How old was the teacher in 2008?
8
votes
5answers
173 views
Solving an equation with a logarithm in the exponent
I try to solve the following equation:
$$ (N+1)^{\log_N{125}} = 216 $$
I know the answer is 5 here but how could I rewrite the equations so I can solve it?
I tried to take the log of both sides but ...
5
votes
0answers
221 views
A system of equations of Vietnamese Mathematical Olympiad 2013
This is a system of equation of Vietnamese Mathematical Olympiad 2013, the first day. Solve the system of equations
$$\begin{cases}
\sqrt{\sin^2 x + \dfrac{1}{\sin^2 x}} + \sqrt{\cos^2 y + ...
18
votes
6answers
1k views
Helping my daughter with her homework: solving an algebra word problem.
Three bags of apples and two bags of oranges weigh $32$ pounds.
Four bags of apples and three bags of oranges weigh $44$ pounds.
All bags of apples weigh the same. All bags of oranges weigh the ...
15
votes
2answers
357 views
Why are braid numbers of the form $Q_h^2$ or $2 \times Q_h^2$?
Consider two piles of $h$ playing cards each, all distinct. Repeatedly take one of the cards on top of one of these two piles and move it on top of one of two new piles, until both of the new piles ...
1
vote
2answers
126 views
Prove $\sqrt[n]{n}\leq 1+\frac{2}{\sqrt{n}}$
I am given this statement and I need to prove it. the statement is for all $n\ge 1$: $$\sqrt[n]{n}\leq 1+\frac{2}{\sqrt{n}}$$
I am trying to prove with induction. But I am stuck for step n=k+1, how ...
5
votes
4answers
182 views
I need a lot of questions for mathematics. Algebra to calculus so that I learn by solving.
One huge problem I have with learning mathematics is that I have not got enough problems to solve, with answers. Is there a resource that I can get hundreds of mathematical questions, small questions, ...
11
votes
1answer
103 views
Request for a proof of the following continued-fraction identity
I have been poring over many texts about continued fractions, but none of them seem to be helping me to prove the following beautiful continued-fraction identity (I am nowhere close):
$$
...
5
votes
1answer
215 views
Mathematical problem with square numbers in the decimal system
Moderator Note: this is a question from the Federal Mathematics Competition 2013.
Good morning,
here's another (pretty difficult) mathematical problem... The task may sound a little strange (I'm ...
0
votes
0answers
36 views
Define a problem using chernoff bounds
We are preparing this for an exam.
Given the division of a plane into a number of regions of different sizes. We would like to find, or guess, which is the biggest region, by doing the following.
We ...
1
vote
3answers
91 views
How to find the best point?
I want to find x in the figure below where it must be as close as possible to 0, but not near to red points (the minimum distance of x to each red point must be at least 1).
4
votes
3answers
257 views
Solving $xe^{-x}+2e^{-x}=0$
While I was studying my maths book, I came across this equation:
$$
xe^{-x}+2e^{-x}=0
$$
I tried to solve it in different ways, but each time I break up some rule. My best try was this:
Let's ...
2
votes
2answers
157 views
calculate the limit of this sequence $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1..}}}}$ [duplicate]
Possible Duplicate:
$\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$
i am trying to calculate the limit of ...
10
votes
3answers
372 views
prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit
i am given this problem:
let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all ...
1
vote
3answers
50 views
Ratio Problem Technique
Suppose $8$ people can paint $6$ houses in $3$ hours. How many houses can $3$ people paint in $4$ hours?
So it seems that $1$ person can paint $3/4$ of a house in $3/8$ of an hour. Then this implies ...
1
vote
3answers
97 views
Polar Curve and maximum width
Find the maximum width of the petal of the four-leaved rose $r = \cos2\theta$, which lies along the x-axis
Here is the solution
Can someone tell me how on earth did the solution come up with ...
0
votes
2answers
139 views
faster and unconventional way of doing maths problem?
I just wanted to know is there any book or resource or perhaps online resource which could help me to do faster problem solving problems like below(so that i could do it mentally an perhaps a lot ...
14
votes
2answers
456 views
Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$
The question is to find all complex roots of
$$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$
and it is meant to be solved by hand.
Is there any quick way to solve this using some trick that I'm not ...
1
vote
2answers
63 views
why $\sum_{k=0}^{\infty}(10^{-2})^k = \frac{1}{1-10^{-2}}$
i was reading a book and suddenly saw this step: $\sum_{k=0}^{\infty}(10^{-2})^k = \frac{1}{1-10^{-2}}$
i am actually not bad at calculation and also i am okay in precalculus, but i am really stuck ...
6
votes
4answers
133 views
prove $\lceil{x}\rceil=-\lfloor-x\rfloor$
i am trying to prove that $\lceil{x}\rceil=-\lfloor-x\rfloor$, but having difficulties to prove. the definitions are:
$\lceil{x}\rceil:=m-1<x\leq m$ and $\lfloor{x}\rfloor:=n\leq x<n+1$.
how ...
6
votes
5answers
184 views
how to prove $\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$
i am asked to prove this statement:
$$\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$$
Now after several attempts, i am lost not knowing where and how to start. if I use induction, i am stuck on ...
0
votes
3answers
60 views
Displacement from variable forces
Displacement from a singular force over time is given by the equation
$${1\over2}{F\over m}t^2 $$
Where F is force, m is mass, and t is time.
But what if F is variable over time?
My best guess is to ...
2
votes
1answer
130 views
prove with $\epsilon$-$\delta$-argument: $x\rightarrow |-2x+3|$ is continuous
i am asked to prove with $\epsilon$-$\delta$-argument that $x\rightarrow |-2x+3|$ is continuous
my steps:
Definition of $\epsilon-\delta$-argument:
$\forall \epsilon >0 \exists \delta>0$ with ...
1
vote
1answer
52 views
check for Continuity $x \rightarrow 2x^4-2$
Given the function: $f:\Bbb{R}\rightarrow \Bbb{R},\quad x \rightarrow 2x^4-2$
i am asked to check for Continuity for all values of $x$. i am now overasked how to do this since $\Bbb{R}$ is not a ...
4
votes
4answers
191 views
Proof of $n^2 \leq 2^n$.
I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$.
My steps are:
induction base case: $n=0:$ $0² \leq 2⁰$ which is okay.
inductive step: $n \rightarrow n+1:$ ...
3
votes
0answers
103 views
Cluster point of $a_{n}:=n+(-1)^{n}n$
I am trying to find the cluster point of the sequence $a_{n}:=n+(-1)^nn$. Can you please check my solution?
The subsequence diverges for increasing even $n$ since $2n$ grows infinitely.
The ...
2
votes
3answers
123 views
how to find a cluster point of $a_{n}:=(2+(-1)^n)\frac{n}{n+1}$
i am tryint to find a cluster point of this sequence, but i am having difficulties in definitions.
the sequence is this: $(a_{n})_{n \in \Bbb{N}}$ with $a_{n}:=(2+(-1)^n)\frac{n}{n+1}$
the ...
11
votes
4answers
339 views
How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?
How do I integrate this guy? I've been stuck on this for hours..
$$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
3
votes
2answers
113 views
How to solve system of equations?
I want to sove the system of equations $$\begin{cases}
x^3 y-y^4=7,\\
x^2 y+2 xy^2+y^3=9.
\end{cases}
$$
I tried divide these two equations we obtain
$$\dfrac{x^3 - y^3}{(x+y)^2 } = \dfrac{7}{9}$$
...
16
votes
3answers
192 views
Find all polynomials that fix $\mathbb Q$ and the irrationals
Problem: Describe all polynomials $\mathbb{R}\rightarrow\mathbb{R}$ with coefficients in $\mathbb C$ which send rational numbers to rational numbers and irrational numbers to irrational numbers.
2
votes
1answer
62 views
What is the greatest amount of postage you would not be able to pay…
What is the greatest amount of postage you would not be able to pay using only a combination of seven cent and seventeen cent stamps?
I have done a similar problem and got it correct but I am just ...
2
votes
2answers
134 views
What area of mathematics is this problem asking about? [closed]
A colleague posted this on a whiteboard (as a brain-teaser I guess):
A $\rightarrow$ B;
B $\rightarrow$ C;
AD $\rightarrow$ E;
BE $\rightarrow$ C;
BF $\rightarrow$ D;
AC $\rightarrow$ F
What is ...
1
vote
1answer
35 views
Ratio isn't answering correct for this problem
Assume there are 2 products A and B made by different companies. Product A costs 1.2006 USD and Product B costs 2.8298 USD. They decided to exchange their products equally without using Money as a ...
