Use this tag when you want to determine the thinking that is needed to solve a certain type of problem, as opposed to looking for a specific answer to a question.

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1
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1answer
124 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
419 views

In how many ways can he form a rectangle using all the tiles each time?

A contractor has exactly $1088$ square tiles. In how many ways can he form a rectangle using all the tiles each time?
1
vote
3answers
83 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
241 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
360 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 + ...
19
votes
6answers
2k 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
383 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
204 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
224 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, ...
12
votes
1answer
198 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
256 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
60 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
95 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
575 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
1k 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 ...
16
votes
3answers
742 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
95 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
361 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
394 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 ...
18
votes
4answers
783 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
64 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
158 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
197 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
199 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
354 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
72 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
313 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
176 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
544 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 ...
16
votes
4answers
482 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
3answers
179 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
281 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
76 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 ...
3
votes
2answers
214 views

What area of mathematics is this problem asking about?

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
37 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 ...
7
votes
2answers
102 views

Show $\lim_{n \to \infty} \min\{a_{n},b_{n}\} = \min\{a,b\}$

If $\lim_{n \to \infty} a_{n} = a$ and $\lim_{n \to \infty} b_{n} = b$, how can we show that $\lim_{n \to \infty} \min\{a_{n},b_{n}\} = \min\{a,b\}$? I say $\min\{a_{n},b_{n}\} $ has two cases: ...
0
votes
1answer
40 views

neighborhood - simple but i need help

I have this set of complex numbers: $\{ 1-i , 2-i , 3-i \}$, and another set $$B:= \{ w \in \Bbb{C} \mid 0 \leq \mathrm{Re}(w)\leq 4 \land -2 < \mathrm{Im}(w)\leq0 \} \setminus \{ a+bi \in \Bbb{C} ...
1
vote
1answer
56 views

elementary neighborhood problem

I am to find the proper number from $x \in \{2,3,4\}$ for which this following set is a neighborhood in $\mathbb{R}$ or in $\mathbb{C}$, $$A:= \left] 1,4 \right[ \cap \left[ 2,5 \right]$$ Firstly, I ...
1
vote
3answers
57 views

need help - $\lim_{n \to \infty} i^{3n} $

it seems first easy to me, but now i am tossing my head against wall not being able to solve the problem. i need to check for convergence of this sequence below. i dont know how to start although it ...
2
votes
2answers
47 views

simple convergence test $\lim_{n \to \infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}$

i am pulling my hair out in solving this problem. i know, it is a stupid question but i am not that good at maths, and many thanks for any help $\lim_{n \to \infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}$ ...
1
vote
2answers
54 views

solve $\lim_{n \to \infty} \frac{(-2)^n}{3^{2n}} $

i am eating myself not being able to solve this problem. i somehow feel that the sequence converges to $0$, but once i calculate, it is not coming to that result. or am i making stupid mistake on the ...
0
votes
1answer
53 views

solve $\lim_{n \to \infty}\left( \frac{3+2n}{\sqrt{3}+\sqrt{2}n} + i\,n\right)$

I am checking this sequence for convergence but i am not sure whether i am on the right path in calculations, these steps are what i am doing now. $\infty + n$ will go to infinity, right? $$\lim_{n ...
8
votes
1answer
162 views

Proving a number defined by a sequence is a square number

I found this problem in a math magazine: Given the sequence $(x_n)_{n \in \mathbb{N}}$ defined by: $$ x_0 = 0\\ x_1 = 1\\ x_{n+2}+x_{n+1}+2x_{n}=0 $$ Prove that $s_n = 2^{n+1}-7x_{n-1}^2, n ...
-1
votes
1answer
991 views

Commercial Mathematics. A question on payment.

Please help me out with this problem. Charlie buys a car for \$120,000. He pays half of the amount in cash and agrees to pay the balance in 12 annual instalments of \$5000 each. If the rate of ...
1
vote
0answers
37 views

Strategies for deriving properties of an expression

For a given $c^*$, suppose that the following system of non-linear equations in $x$ and $y$, $f(x,y;c)=0\\ g(x,y;c)=0$ possesses a unique solution $(x^*,y^*)$. The equations are such that I do not ...
10
votes
5answers
508 views

Prove $|a+b|+|a-b| \geq |a|+|b|$

I am fighting with this proof-writing problem for a while. The statement says $$|a+b|+|a-b| \geq |a|+|b|.$$ I know the triangle inequality which says$$|a+b| \leq |a|+|b|.$$ How can I use this ...
0
votes
2answers
192 views

Help solving probability problem!

Can you help solve this problem? Place contains $N$ coupons $(N>3)$. $3$ tickets have prizes $ \$10000, \$1000, \$500$ respectively. $X$ is the number of coupons drawn before a prize-containing ...
3
votes
1answer
112 views

greatest common divisor is 7 and the least common multiple is 16940

How many such number-pairs are there for which the greatest common divisor is 7 and the least common multiple is 16940?
0
votes
2answers
102 views

Finding roots of product of two polynomials

Let $P$ and $Q$ be polynomials of degree $2$ and $3$ respectively. If we know the roots of both $P$ and $Q$, is there an easier way of finding the roots of the product $PQ$? Do we really have to ...
-1
votes
1answer
88 views

simple arithmetic problem but..

i am having problem in this proof. i need to find the certain coefficients of this statement on the right side. given: $P: \mathbb{C} \Longrightarrow \mathbb{C}, \quad P(x) := 12 − 7x + x^2$ $Q : ...