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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0
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1answer
80 views

Infinetly many primes of form $4k+3$ [duplicate]

Prove that there are infinitely many primes of the form $4k + 3$ (where $k$ is an integer). Note that it is a special case of "Theorem 6 (Dirichlet). Let a and b be positive coprime integers. Then ...
0
votes
1answer
72 views

Greatest common divisor of $3$ numbers

Let $a,b, c$ belong to $\mathbb Z$ such that $(a,b,c) \neq (0,0,0)$. Define the [highest common factor] greatest common divisor ${\rm gcd}(a, b, c)$ to be the largest positive integer that divides $a, ...
8
votes
3answers
190 views

Existence of a certain subset of $\mathbb{R}$

To every real $x$ assign a finite set $\mathcal{A}(x)\subset \mathbb{R}$ where $x\not\in \mathcal{A}(x)$. Does there exist $\mathcal{W}\subset \mathbb{R}$ such that: $$1.\;\;\mathcal{W}\cap ...
6
votes
4answers
632 views

Some Questions regarding preparing for Math Olympiads (searched but didn't get answers)

Many questions have been asked on this site regarding preparation for olympiads like the Putnam. I've read those questions and accordingly decided to start with Engel's "Problem Solving" but I have a ...
0
votes
1answer
52 views

Solving $\int\sqrt{1+(-2ax+b)^2}\;dx$

List item What solution $$\int\sqrt{1+(-2ax+b)^2}\;dx$$Unable to develop anything ...$~$:'( I tried completing squares, but can not move much.
4
votes
3answers
67 views

Elementary set theory problem - I get an incorrect result

The problem given is this: $\bigcap_{i \in I}(A_i \cup B_i)$ and $(\bigcap_{i \in I}A_i) \cup(\bigcap_{i \in I}B_i)$ I am asked if they are the same. Here is the reasoning I used: for the first ...
0
votes
1answer
40 views

How many smallest number moves need to measure 6 liter of water?

You are given two (unmarked) containers of capacity 9 liter and 4 liter and a huge tank of water. Need is to get measure of exactly 6 liter, of water. A move is either filling a container completely ...
-2
votes
3answers
85 views

How to know number of eggs in this problem?

A women took a certain number of eggs to the market and sold some of them. The next day through her poultry industry, the number left over had been doubled, and she sold the same number as the ...
0
votes
1answer
40 views

Combinations with multiple kids

In a certain country, it has been found over many years that $55$% of the babies born there are males. For a family in that country with five children, what is the probability that (i) the two ...
1
vote
0answers
61 views

When does a ball in a game of brick breaker never hit the remaining breaks?

I have a block size 2N*2N and some squares are filled with bricks and some aren't. I have a ball that travels distance 1 in the x and y direction and bounces off with perpendicular direction if it ...
0
votes
2answers
99 views

Using plant to find depth of water (triangles)

John and Chris were out in their row boat one day, and Chris spied a water lily. Knowing that Pat liked a mathematical challenge, Chris announced that, with the help of the plant, it was possible to ...
2
votes
2answers
55 views

2 states, 2 interarrival distribution Renewal Process.

Karlin and Taylor (1975): 18. Consider a stochastic process $X(t)$, $t \geq 0$, which alternates in 2 states $A$ and $B$. Denote by $\xi_1, \eta_1, \xi_2, \eta_2, \ldots,$ the successive ...
0
votes
3answers
78 views

Proving this inequality

I am having trouble with proving an inequality. Assume we have two positive real numbers $a$ and $b$ such that $a+b=1$ and numbers $x > 0$ and $y > 0$. Prove: $$\frac{2}{\frac{a}{x} + ...
1
vote
1answer
28 views

Finding F(x) given any reals x and y

I have a problem and I think I know how to solve it so here it is: Determine $F(x)$ if, for all real $x$ and $y$, $F(x)F(y)-F(xy) = x+y$. I tried a couple of cases and these were my results: When ...
0
votes
1answer
38 views

Linking 5 communities by plane, train, and bus

Every pair of communities in a country is linked directly by exactly one mode of transportation: bus, train or airplane. All three modes of transportation are used in the country; no community is ...
2
votes
2answers
54 views

Based on census data and namm data, how many musicians are there?

There are $115,226,802$ households in the US. $58$% of households have at least $1$ musician. $43$% have $2$ or more musicians. How many musicians are there? Any help would be appreciated...
20
votes
12answers
8k views

MENSA IQ Test and rules of maths

In a Mensa calendar, A daily challenge - your daily brain workout. I got this and put a challenge up at work. The Challenge starts with.. ...
0
votes
3answers
59 views

Problem Solving

Here the question on my daughter's 4th grade homework assignment: Sarah is decorating a mosaic board it is 12 inches wide and 16 inches long. Sarah is using square tiles all the same size. What size ...
1
vote
3answers
54 views

Permutations of numbers

Given the five digits $1,3,4,6,$ and $7$. In the following question, it should be understood that repition of a digit is not allowed. (i) How many three-digit numbers can be formed from the ...
2
votes
1answer
138 views

Every point of the interval $(0,1)$ is an interior point of that interval. Thus $(0,1)^0 = (0,1)$.

This is a question I found here (on pg.2 example 1.2.2): http://web.pdx.edu/~erdman/PTAC/problemtext_pdf.pdf Definition of interior and interior point: Let $A \subseteq \mathbb{R}$. The point $a$ is ...
0
votes
1answer
37 views

Problem solving question with average

Johnny had to take a test a day late. His 96 raised the class average from 71 to 72. How many students, including Johnny, took the test? I tried to do trial and error to see how many students there ...
1
vote
1answer
208 views

How many ordered triples $(x,y,z)$ of positive integers satisfy $xyz=4000$

How would I find this out? Is there an equation or summation?
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votes
3answers
58 views

no positive roots for $(y+1)^{7}-2(y+1)^{5}+10(y+1)^{2}-1$

Hint for showing that $(y+1)^{7}-2(y+1)^{5}+10(y+1)^{2}-1$ has no positive roots thanks
0
votes
1answer
210 views

Show that the union and the intersection of any two $\epsilon$-neighborhoods which overlap is itself an $\epsilon$ neighborhood.

I want to check if my reasoning and mathematical language is correct here. There's only two cases to check for. One where one of the neighborhoods is inside the other, and the other where the two ...
0
votes
1answer
51 views

Solve equation of form $(d_B - 0.32)^{0.8} (d_B + 1.45)^{1.1} = exp(0.8)$ for the term $d_B$

I have the following equation: $$ \left(\frac{\sqrt{d_B}-\sqrt{d_{Beq}}}{\sqrt{d_{Bmin}}-\sqrt{d_{Beq}}} \right)^{1-\frac{c1}{c2}}\left(\frac{\sqrt{d_B}+\sqrt{c3}}{\sqrt{d_{Bmin}}+\sqrt{c3}} ...
4
votes
8answers
304 views

Evaluating $\int \frac{1}{\sqrt{x^2 + a^2}}\, dx$ without resorting to trigonometric $u$-substitution

I am looking for a quick and intuitive way to evaluate this indefinite integral without resorting to any trigonometric functions. I'm not sure if it is at all possible to do so, but I was just ...
1
vote
2answers
75 views

Can someone help out with this geometry problem plase?

I have a pyramid $P$ that has a square pyramid and each of its 4 triangles is equilateral.I also have cuboid $C$ with height 25 and width 50, something like this: So the volume of $C$ is $25 . 50 . ...
0
votes
2answers
28 views

Define variable value - General Math

I met a little problem in one of my Math - tasks. Its quite simple: I get to cases = case 1: $(476\cdot x)+220$ case 2: $(278\cdot x)+675$ The variable x have to be a value, so case 1 and case 2 ...
0
votes
1answer
51 views

What kind of formula would I use to get all possible outcomes?

I am into a CCG, and I got a question come to mind "how many possible out comes are there for deck combinations?" The game is broken into three: Main Character (6 cards available, only one deck), ...
0
votes
1answer
190 views

How many number of buses that the car encounter?

A car travels from B at a speed of 20 km/hr. The bus travel starts from A at a time of 6 A.M. There is a bus for every half an hour interval. The car starts at 12 noon. Each bus travels at a speed of ...
0
votes
1answer
28 views

Solving the following equation

What is the easiest way to solve the following equation for g in terms of x? I seem to be going in circles, and don't know how exactly to deal with the $\pm$ symbol. Equation: $$\pm \sqrt{2}x + ...
1
vote
2answers
220 views

Prove that for any integer $k>1$ and any positive integer $n$, there exist $n$ consecutive odd integers whose sum is $n^k$

Found these problems in a problem book and got stuck. The book doesn't have solutions to I've come here for help. (1) Prove that for any integer $k>1$ and any positive integer $n$, there exist $n$ ...
2
votes
3answers
268 views

Where is the lost dollar?

Somebody explained me this problem, but I am not sure to understand what is wrong. ...
0
votes
1answer
39 views

Proof of transformations to find an approximate value

I have no idea what this questions is asking, or how to go about solving.. can someone please help? Answer:
0
votes
1answer
77 views

formula to apportion cost of transport among three people in a liftshare

I share lifts with Sed and Awk to work every day. We tally journeys owed on a spreadsheet. A week might look like this: ...
0
votes
2answers
239 views

Linear Algebra and a cube

I am currently working no a linear algebra question and do not understand how to solve it. The questions gives: ...
2
votes
2answers
78 views

Solving Abstract Problems

I'm doing this Solving Abstract Problem but I'm not sure which one it is. I mean from the Series I can see there's a pattern but in the Options I don't see images that link with the Series. Do you ...
2
votes
2answers
390 views

Question about the “master theorem” of recurrences - no “$b$” term

I'm using the master theorem to find the asymptotic run time of recurrences. For example, for a $T(n) = 4 T(n/5) + n^1$ I find that $T(n)$ is $\Theta(n^1)$, or, simply, constant time, via the set of ...
2
votes
0answers
73 views

Is this graph problem already solved

I would like to solve the following: Let $G=(V,E)$ be a directed graph such as $\forall (x,y) \in E, x \neq y$. Find any (all would be even better) graphs $S$ such that: $S \subset V$ $\#\{(x,y) ...
0
votes
1answer
171 views

Velocity and distance problem

From city A, car a sets on it's road towards city B. 9 Hours later, car b, sets from city B towards city A. The two cars met along the way. At the point of the meeting, car a, passed 240 km more than ...
0
votes
4answers
63 views

How can I solve these equations? [closed]

I can't figure it out, please help :S $$\begin{array}{rcl} 2x + 2xz &=& 0\\ -2y + 2yz &=& 0\\ x^2 + y^2 &=& 4 \end{array}$$ Thanks in advance!
3
votes
3answers
2k views

Magic Trick to Read your Mind

I am a student in High School. My math professor made a magic trick the other day in my class and he read our minds. I knew a similar trick which was based on mathematics, that's why I am asking here. ...
0
votes
1answer
173 views

Solving a tough system of linear equations

I have three equations and have to solve for $x, y, z$. $$ l_1l_2 + m_1m_2 + n_1n_2 = 0 $$ $$ xl_1 + ym_1 + zn_1 = 0 $$ $$ xl_2 + ym_2 + zn_2 = 0 $$ After eliminating a variable (from the last two), ...
2
votes
2answers
157 views

Show that if $z = e^{i\theta}$ is a solution to $0 = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0$, …

Show that if $z = e^{i\theta}$ is a solution to $0 = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0$ [1] where all $a_i$ are real, then $0 = a_{n-1}\sin\theta + a_{n-2}\sin2\theta + \cdots + a_0 \sin ...
0
votes
2answers
68 views

Define S as a set of primes such that if a, b are in S, ab+4 is in S. Show that S must be empty.

Define $S$ as a set of primes such that $(a \in S) \land (b \in S) \implies (ab + 4) \in S$ [$a$ and $b$ can be the same number]. Show that $S$ must be empty. A hint is given ... "work modulo 7." ...
2
votes
2answers
48 views

Searching for an explicit functional form

Let $f:\mathbb N \to \mathbb R$ be a strictly decreasing function. Suppose $$\frac{f(x)}{f(x+1)}=2^x,\qquad \forall x\in\mathbb N.$$ Is it possible to find an explicit functional form of $f$? Any hint ...
2
votes
1answer
45 views

Simplify $y^\top x -\log(\sum_i e^{x_i})$

Simplify $\sup_x y^\top x -\log(\sum_i e^{x_i})$ The first order conditions yield $y_i=\frac{e^{x_i}}{\sum_i e^{x_i}}$. How do I eliminate $x_i$ from the equation? I know the answer to be $\sum ...
2
votes
3answers
97 views

Separating $3n$ points on the plane by a line

I am trying to solve a problem in geometry (a contest-type question), and I wondering if the following result is true. (If it is true, then it makes life much easier!) Suppose there are $3n$ ...
1
vote
1answer
50 views

General solution of a second order PDE

I have the second order PDE $t^5u_{xx}-tu_{tt}+2u_t=0$ and need to find it's general solution. My problem is that since it's a second order PDE the method for first order quasi-linear PDE doesn't seem ...
5
votes
1answer
81 views

An integration question.

An help in the following problem: Let $f:[-1,1] \longrightarrow \mathbb{R}$ a $C^1$ function, i.e., continuously differentiable. Suppose that we have $$\int_{-1}^{1} f(x)\;dx = \pi ...