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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What is the largest possible area that $PQRS$ could have?

In a $14\times 18$ rectangle $ABCD$, points $P,Q,R$ and $S$ are chosen, one on each side $ABCD$ as pictured. The lengths $AP, PB, BQ, QC, CR, RD, DS$ and $SA$ are all positive integers and $PQRS$ is a ...
4
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0answers
33 views

Division of a square and value of a disk

I cam across this problem and I really don't know how to solve it. So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the ...
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2answers
43 views

Check my work: Evaluating $\tan\frac{7\pi}{8}$ using a half-angle formula

I am doing a trig problem involving half-angle identities, and I am not sure if my solution is correct. Can someone please check my work? The question: Find the exact value of $\tan\frac{7\pi}{8}...
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0answers
31 views

Solve Equation with max integer [on hold]

Solve please $\dfrac{\left[\sqrt{x-[x ]}\right]}{(x+3)(x+4)}\ \geq0$ edit
0
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2answers
44 views

Determining values satisfying an inequality

I have the following inequality: $$\left\lceil \frac{\log((n-1)/6000)}{\log(3)} \right\rceil < \left\lceil \frac{\log((n-1)/3000)}{\log(3)} \right\rceil,$$ where $n$ is a positive integer, and I ...
1
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1answer
30 views

How do I know if this equation can be solved symbolically?

Can these equations be solved symbolically for $x$? $$ \begin{align} x &= \frac{p - p_m(x)}{p_m(x) - p_m(x)^2} \\ \\ p_m(x) &= \frac{e^x}{e^x + e^y} \\ \end{align} $$ If not ...
0
votes
2answers
62 views

How to solve the given problem of simple interest?

The problem statement is: What annual instalment will discharge a debt of 1092 due in 3 years at 12% simple interest? Now, what I know is Simple interest =( principal* Rate per annum*Time in ...
4
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1answer
67 views

Studying for grad school qualifying exams; need a little help on how to effectively study higher math. [on hold]

This is entirely embarrassing to admit, but I'm realizing, one year into my doctorate program, I don't know how to effectively study math. I feel like a failure and a fraud for even having to come ...
1
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0answers
34 views

Isosceles Triangle With Height Limiting To Zero, part 2

The figure shows an isosceles triangle ABC with ∠B=∠C . The bisector of angle B intersects the side AC at the point P. Suppose that the base BC remains fixed but the altitude |AM| of the triangle ...
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2answers
36 views

Grade 10, waiting for the train.

At Berracan station, northbound trains arrive every three minutes starting at noon and finishing at midnight while southbound trains arrive every five minutes starting at noon and finishing at ...
0
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2answers
47 views

What is the smallest number of coins I could have? [closed]

A country has 6 coins of the following denominations: 1 cent, 2 cents, 4 cents, 10 cents, 20 cents and 40 cents. Using the coins I have, I can pay exactly for any amount up to and including 200 cents. ...
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0answers
32 views

Clearance in a semi-circular tunnel [closed]

A single-lane street 10ft. wide goes through a semi circular tunnel radius 9ft. How high is the tunnel at the edge of each lane?(Round off to 2 decimal places) This is our coming new lesson and I ...
2
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3answers
44 views

Find all the integer pairs $(r,s)$ that satisfy $s= (r^2 +3r +8) / (r^2 +r -2)$?

I have been trying to solve this question but struggling to see where to start. Examples I've seen that works are the pairs: $(-3,2) , (4,2), (0,-4)$
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vote
2answers
39 views

How would you work out these combinations?

If there are 16 different ice-cream flavours, how many combinations are there for a two scoop? If there are still 16 different ice-cream flavours, how many combinations are there for a three scoop? ...
0
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1answer
41 views

Probability Average amount of rolls

I have a question regarding probability. I'll start by saying I've never taken a statistics or other similar course and was trying to work out this for a game. On average how many attempts will it ...
1
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1answer
64 views

Can someone suggest books on mathematics and problem solving which nurtures the reader? [closed]

Can someone suggest books on mathematics and problem solving which nurtures the reader like Alexander Soifer's books? Thanks in advance
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votes
1answer
32 views

Using induction on modified inequalities.

Here's the original problem: Prove by induction that $\left(\frac{1}{2}\right) \left(\frac{3}{4}\right) \cdots \left(\frac{2n-1}{2n} \right) \leq \frac{1}{\sqrt{n+1}}$ for all $n \in \mathbb{N}$. ...
1
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1answer
33 views

Text problem with workers

I'm having an entrance examination in two days and I'm having problems with this math problem here. A group of workers works on two jobs in two days. The second job is 2 times smaller in volume ...
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2answers
42 views

Problem: Using Cauchy's integral formula show…

I've stumbled upon a problem I can not solve in the book Mathematical Methods for Electrical Engineers by Thomas B.A. Senior(Page 171). The book gives the following instruction: Using Cauchy's ...
0
votes
1answer
50 views

Special Numbers [closed]

Q . Suppose that we state that a positive integer number 𝑛𝑛 is called “special” if the set {1,2,3, . . . ,2016} can be split into 𝑛 subsets, all of them with the same number of elements and the ...
9
votes
2answers
100 views

Favourite problem books at university level

As background let me start by stating what I perceive to be the point of problem books, or to put the matter in perhaps more acceptable way, how I define problem books. A large majority of textbooks ...
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3answers
54 views

Find m. $y=e^{mx},m\in\mathbb R,\frac{d^2y}{dx^2}-3\frac{dy}{dx}-4y=0$

If: $$y=e^{mx},m\in\mathbb R$$ Find m if: $$\frac{d^2y}{dx^2}-3\frac{dy}{dx}-4y=0$$ Differentiating and substituting gives: $$m^2e^{mx}-3me^{mx}-4e^{mx}=0$$ Dividing across by $e^{mx}$ and solving ...
3
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2answers
104 views

How many positive integers from set $\{1,2…,10^{30}\}$ can't be represented as 2nd, 3rd, or 5th power of some positive integer?

An interesting problem I ran across. My guess is that it can be solved somehow using inclusion-exclusion principle. It would be a fun thing to learn how to do this, so I could use that knowledge in ...
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0answers
24 views

Solving for the growth rate in a growing annuity formula

Firstly, If there is a better forum for this question, please help direct me. I looked in the quantitative finance forum, but someone already asked this and the question was closed because "its too ...
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1answer
60 views

Checkers Board Problem

Here we consider a checkerboard expanded to size 12 × 12 instead of the ordinary 8 × 8 checkerboard. a) How many squares on this board contain more than a third of the total number of dark small ...
4
votes
2answers
55 views

How can I solve this nice rational equation

I am trying solve this equation $$\dfrac{3x^2 + 4x + 5}{\sqrt{5x^2 + 4x +3}}+\dfrac{8x^2 + 9x + 10}{\sqrt{10x^2 + 9x +8}} = 5.$$ Where $x \in \mathbb{R}$. I knew that $x=-1$ is a given solution. But I ...
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votes
1answer
81 views

How to solve $\log_2(x) +3 = \log_3(x+2)$ [closed]

Hi Math Stack Exchange Communities, I am new here. I have a question regarding logarithm solving. Let's say I have this equation: $$\log_2 (x) +3 = \log_3 (x+2)$$ How can I solve this kind of ...
0
votes
2answers
34 views

primitive polynomials and their factorisation

A polynomial with integer coefficients is called primitive if its coefficients are relatively prime. For example, $$3{x^2} + 7x + 9$$ is primitive while $$10{x^2} + 5x + 15$$ is not. (a) Prove that ...
-1
votes
1answer
36 views

writing numbers as sum of at least two consecutive odd positive integers [closed]

Since 24 = 3 + 5 + 7 +9, the number 24 can be written as the sum of at least two consecutive odd positive integers. (a) Can 2005 be written as the sum of at least two consecutive odd positive ...
0
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1answer
54 views

Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than $1$.

Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than $1$.For example, $22$ and $15$ are relatively prime, and thus $37 = 22+...
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1answer
25 views

Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors

Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors. (The proper divisors of a positive integer n are all positive integers ...
4
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2answers
53 views

How to find the average Kendall's distance between 2 rankings

Suppose I have 2 rankings: $1$, $2$, $3$ and $2, 1, 3$ then the Kendall's distance between the two is 1 since there is only one pairwise adjacent switch. My question is, suppose my 2 rankings each ...
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2answers
58 views

Probability problem

I created this problem based on the following probability riddle here. You're a king, and you were given two groups of people, and a certain information about them. First group has 2 people. One of ...
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0answers
55 views

Multiple choice excercise with more than one answer correct

Q. Consider the function $$F(z)=\int_{1}^{2} \frac {1}{(x-z)^2}dx, {\text {Im}(z) \gt 0}$$ Then there is a meromorphic function function $G(z)$ on $\Bbb C$ that agrees with $F(z)$ when ${\text {Im}(...
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votes
1answer
31 views

solving equation using square root

I have a question here... Usually, for $x^2 = 4$ $x=\sqrt{4}$ $x=±2$ But if the question is like this : $y^2 = (x+2)(x+2)$ $y^2 = (x+2)^2$ If I want to find $y$ in term of $x$,I will put square root ...
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1answer
55 views

Differential equation, Solution is a Bessel fucntion

this is my first post here. I knocked my head on a differential equation yesterday, this one: $$ \frac{12 \nu}{x^2} \frac{S(x)''}{S(x)} = -\lambda^2 $$ Where $nu$ is a constant. The book says the ...
3
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0answers
67 views

Does this equation have no solutions?

The question is this : The source from where I got this question was devoid of any answers to it, so I came here, this is how I proceeded : LHS : $((((({(x)^x})^{2x})^{3x})^{....x^2})^2 = (((((x)^...
2
votes
1answer
29 views

Fox and Gemstone problem. Weighing stones to compare bags.

There is a problem called Fox and Gemstone at topcoder: https://community.topcoder.com/stat?c=problem_statement&pm=14296 Basically, you have the ability to weigh individual gemstones against ...
0
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1answer
48 views

Minimizing a strictly convex function with inequality constraint

So we've been learning about the Kuhn Tucker conditions in my non-linear optimization course and I've been having trouble with this problem: QUestion: description here Question: a strictly convex ...
0
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1answer
78 views

Find the Smallest Value

Find the smallest value of $$a + \frac {1}{(a-b)b} $$ where a>b>0 I found this question in AM-GM inequality problems but I am stuck at this
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votes
1answer
30 views

find the number of tuples of positive integers [closed]

find the number of tuples (a,b,c,d) of positive integers \begin{array}{l} {a^3} = {b^2}\\ {c^3} = {d^2}\\ c - a = 64 \end{array} answer should be one of 0 , 1 , 2 , 4
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2answers
68 views

proving no real roots exist

Prove that $x^8-x^7+x^2-x+15$ has no real roots. I did it by first assuming it has real roots and then applying Descartes rule of signs. We find that if there are any real roots, they all must be ...
0
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1answer
30 views

number theory problem finding triplets [closed]

Find number of triplets of positive integers satisfying $2^a-5^b\cdot 7^c=1$ Given options are $0 , 1 , 2$ or infinite.
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1answer
51 views

How to find the area shared by 4 quadrants inside a square?

I was to find the blue area in this question : As described about how it's a square with 4 quadrants of same radius intertwined with each other, now to find the blue part area I thought about ...
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votes
2answers
55 views

Find the solution without “common sense” assumption [closed]

Is it possible to solve the following problem without make a "common sense" assumption? In the year 1887, one person's age was exactly the sum of the digits of his birth's year. What was the person'...
3
votes
2answers
67 views

Homeomorphism from $S^1\backslash(0,1)$ to $\mathbb{R}$

I am trying to derive a bijection between $S^1\backslash{(0,1)}$ and the real line, but I am stuck on using the most obvious way Let the top point of the circle be $(0,1)$, and the blue line hits ...
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0answers
74 views

Polynomial taking irrationals to irrationals

Problem: Find all polynomials from $\mathbb{R}\to \mathbb{R}$ $f$ with integer coefficients taking irrationals to irrationals. My attempt: It is clear that the problem statement is equivalent to ...
0
votes
2answers
112 views

What is the size of the angle $\angle AMC$? [duplicate]

Suppose we have a triangle $\triangle ABC$ where the size of two angles are given: $\angle B=15^\circ$ and $\angle C=30^\circ$. We draw the median $AM$, so now what is the size of angle $\angle AMC$? ...
2
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3answers
73 views

Let $f$ be a differentiable function defined for all real $x$, where $f(x)\ge 0$ for all $x\in[0,a]$

Let $f$ to be a differentiable function defined for all real $x$, where $f(x)\ge 0$ for all $x\in[0,a]$.If $$\int_0^a f(x)\,dx = a, $$ then $$2\int_0^{5a}\left(f\left( \frac x 5 \right) + 3 \right)\,...
0
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1answer
36 views

Topology: What is a quick way to check whether a subset $D$ is dense in $(X, \mathcal{T})$?

Def $1$: Let $(X, \mathcal{T})$ be a topological space, then $D \subseteq X$ is dense if $\overline {D} = X$ Def $2$: $x \in \overline D$ iff for all $U \in \mathcal{T}, x \in U \implies D \cap U \...