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
30 views

Recurrence relation. Application to ternary sequences

The question is: How many ternary sequences have no double zero? For this I understand that our $n$-digit sequence either have $0,1,\dots,n$ zeroes, is this ok? If the answer of above is positive, ...
-5
votes
3answers
25 views

Probability: 52 cards in a deck [closed]

If you are dealt two cards successfully (with replacement of the first) from a standard 52-card deck, find the probability of getting a heart on the first card and a diamond on the second.
4
votes
1answer
129 views

Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$

Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$ I have no clue how to start. Any kind of help will be appreciated.
0
votes
0answers
24 views

Travelling salesman - organising a tour of any European destination based on the cheapest flights available.

I apologise if this has only a tenuous link to a mathematics forum I'm sure everyone is familiar with the £10 one-way flights by Ryanair and similar airlines in Europe. I was wondering whether there ...
1
vote
0answers
9 views

Sum two nearest function of two class are the nearest function of the sum class

Suppose $x,\mu:[0,1]\rightarrow \mathbb{R^2}$ two smooth function and $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (0) = 0, \gamma (1) = 1, \gamma$ is a diffeomorphism $\}$. Here $\Gamma$ ...
2
votes
2answers
51 views

How to solve this question in more time efficient way?

Q) if$$x\sin a=y\cos a=\frac{2z\tan a}{1-\tan^2 a}$$ then find $4z^2(x^2+y^2)$a)$(x^2+y^2)^{3}$b)$(x^2-y^2)^3$c)$(x^2-y^2)^2$d)$(x^2+y^2)^2$ Ans:c i solved this in a very long way: $$x\sin ...
1
vote
0answers
29 views

Linearity in quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Also let $\mathcal{C}$ is a linear manifold in the sense that $x_1,x_2\in \mathcal{C}$ implies that ...
0
votes
1answer
44 views

How can I prove the equation has unique positive real solution?

Without using derivative, prove that the equation $$x^5-2x^4-3x^3-4x^2-5x-6=0$$ has unique positive real solution. I tried, consider function $f: \mathbb{R} \rightarrow \mathbb{R}$ with ...
0
votes
0answers
25 views

How to create a custom scale for a range of values

I'm utterly sorry for the very non-specific question, but I'm not even sure what I am looking for. Any pointers and terminology so I can document myself would be helpful. Because I do not know how to ...
0
votes
1answer
47 views

Relationship of radius of sphere to an inscribed right circular cylinder for max and min values

I cannot seem to find the correlation between having an interval of a radius of a sphere with finding the greatest lateral surface area of a right circular cylinder inscribed in it. The question goes ...
7
votes
2answers
63 views

Three dimensional spherical excess formula

We all know the spherical excess formula: in a unit sphere, the area of a geodesic triangle is equal to the exceeding from $\pi$ of the sum of the three angles of the triangle. Is there a similar ...
2
votes
3answers
51 views

Ideas for solving this IVP

I am curious how to approach solving the initial value problem: $\begin{cases} y'(t) = 5t - 3\sqrt{y} \\ y(0) = 2 \end{cases}$. The equation isn't separable, and more generally it is not an exact ...
1
vote
1answer
29 views

Ideas for solving this nonlinear IVP

I am curious how to approach solving the initial value problem: $\begin{cases} y'(t) = 5t - 3\sqrt{y} \\ y(0) = 2 \end{cases}$. The equation isn't separable, and more generally it is not an exact ...
0
votes
2answers
48 views

Lambert W function with natural log

I need to solve the next equation x: $d-x+yln[\frac{d}{x}]=b$ I inserted this into Wolfram Alpha and it returned: $x = y \Bbb{W}[\frac{e^\frac{d-b}{y}d}{y})]$ y, d, b, and x are all real, ...
0
votes
0answers
19 views

Find the basis for the kernel, for a linear mapping

Let $T : R^4 \to R^3$ given by $$T(x, y, z, t) = (x−y+z+t, x+2z−t, x+y+3z−3t)$$ Using Gauss-array and reducing the system of equations to row echelon form I got: $\{(1,1,1),(-1,0,1)\}$ as basis for ...
0
votes
1answer
92 views

Probability Riddle

I was told a puzzle recently, and I can't figure out how to solve it. It went like this: You are a prisoner. You play a game with the guard many times a day. This game has a unique probability ...
1
vote
1answer
39 views

The Number of Two-digit Primes Which the Sum of their Digits is 6

Problem: Find the number of two-digit primes which the sum of their digits is six. We had this problem in a mathematic examination. The problem can be solved by testing all two-digit primes, but ...
0
votes
0answers
28 views

calculus book recommendations [duplicate]

i want to learn single variable calculus i completed schooling and i love calculus for now i am focusing on single variable calculus i tried many books like Calculus - "A Complete Course 7th ed - R. ...
0
votes
1answer
13 views

Question about invariants.

There is a list of $n$ numbers. We pick any two numbers, $u$ and $v$ and replace them by $uv + u + v$. Does the final answer after $n-1$ operations, depend on the initial choice. I noticed that if ...
3
votes
6answers
120 views

Why isn't $-2$ solution for $x$?

I came across an logarithm problem recently. I don't know why solution to this problem cannot be $-2$. Now, don't downvote now because you don't know why I'm asking this. I know that logarithms' ...
0
votes
1answer
22 views

Generating functions, problem solving. Distribute distinguishable balls to people

The problem reads: How many ways are there to distribute $26$ of $34$ distinguishable balls to $5$ people if Lucy gets at most $4$ balls? The generating function to distribute distinguishable balls ...
2
votes
1answer
68 views

How can you solve for s in this very complex problem?

I recently stumped across a problem, which I need to solve. Of course, I used an calculator and I got $s=3$, but I want to know how to do it step by step. The problem is kind of complex: ...
1
vote
2answers
32 views

Pink Kangaroo Maths Challenge: Ria Bakes Six Raspberry Pies

I have been doing some practice papers for an upcoming UKMT Maths Challenge. There's one question I can't seem to grasp. I'm not sure entirely sure where to start. I'm open to any ideas. Thank you ...
2
votes
1answer
43 views

Find the basis for the kernel and the image, for a linear mapping

Let $T : \Bbb{R}^3 → \Bbb{R}^3$ given by $$T(x, y, z) = (x + 2y − z, y + z, x + y − 2z).$$ Using Gauss-array and reducing the system of equations to row echelon form I got: $\{(3,-1,1)\}$ is a basis ...
2
votes
2answers
132 views

A unit square contains 1 million rectangles without any common points. Show that the total area of rectangles is less than 1.

"A unit square contains 1,000,000 rectangles without common points. Show that the total area of rectangles is less than 1." This statement is somewhat imprecise. Let's say that these are closed ...
2
votes
2answers
55 views

The Diophantine Equation: $x^3-3=k(x-3)$

I wish to know how to resolve the diophantine equation: $x^3-3=k(x-3)$ ? The problem is: Find all integers $x\ne3$ such that $x-3\mid x^3-3$. - From 250 Problem's in Elementary Number Theory, by ...
0
votes
1answer
29 views

One tap fills a pool. The other one empties it. It's a word problem.

In a pool there are two taps, one for filling and one for emptying. Once, when the pool was empty they opened the filling tap for $4$ hours. Afterwards, they opened by mistake the emptying tap and ...
3
votes
1answer
70 views

What type of functional equation is this?

I'm trying to solve the following functional equation $f\left(x\right)=A\mbox{ exp}\left\{ \int\frac{1}{f\left(x\right)x^{2}+Bx}dx\right\}$ where ...
0
votes
7answers
61 views

How do you work out the angle in this square?

I have labelled all the angles that I can work out. But I can't think of any other way to find the other angles without being 100% sure. Thank you! P.S. I have attached the official question - ...
2
votes
4answers
87 views

Coin flipping problem

Suppose that you are flipping a coin endless times. what's the expected round where you would get the same side $3$ consecutive times? I'm guessing it would take $7$ flips to see either ...
2
votes
1answer
75 views

Let $1 + 2^m = 3^n$. What the max value of $(m+n)$?

How do I determine the maximum value of $(m+n)$ if $m$ and $n$ are natural numbers if $1 + 2^m = 3^n$ holds? I have got $\text {max} (m+n)$ to be $5$ so far, but I do not know how to determine whether ...
0
votes
2answers
42 views

Work and time problem

I came up with this problem: $150$ workers were employed to do a particular work. On first day, $150$ workers worked. On second day, $146$.. and each subsequent day, workers kept on decreasing by 4. ...
1
vote
1answer
22 views

Smallest number of groups to sniff

The question given: The sniffer dog at the airport stops beside a trolley piled high with 60 suitcases. One of the suitcases contains contraband peanuts. The dog can tell whether peanuts are hidden in ...
0
votes
0answers
34 views

$n$-couples of people in a row.

For the following problem, I feel my reasoning is something wrong, so I would like if it is in the right direction or if it needs to be rephrased/corrected. The problem reads: How many ways are ...
-1
votes
3answers
42 views

Solving two variables with one equation

I have been trying to solve the following equation, but I am still stuck after trying many different methods. I have been given this equation to solve: $$(1 + z^{-1})^4 (a + bz^{-1} + az^{-2}) - (1 ...
0
votes
1answer
28 views

How does one establish a path of least resistance when solving equations?

For certain systems of equations, it is obvious what the easiest way to organize and manipulate the equations should be. For instance, $$y = 10x + 5$$ $$2x + y = 125$$ So you take the first ...
0
votes
1answer
37 views

solving an equation $x^x= c$ [duplicate]

I would like to find a solution $x$ for $x^x = c$ where $c$ is a positive constant. Firstly I'm looking for an approximative solution when $c$ tends to infinity. Thank you in advance
8
votes
2answers
82 views

Find a succinct problem whose solution requires methods from many sub-branches of mathematics

Some mathematical problems require solution techniques from a single branch (sub-discipline) of mathematics. For instance, most problems in formal logic can be addressed by the methods of formal ...
1
vote
1answer
89 views

What is the optimal strategy when playing `head or tail` per team

Introduction Once a week, we are playing head or tail in my favorite bar. There are $N$ people in the room and each person is guessing whether ...
0
votes
0answers
32 views

Solving a polynomial with modular arithmetic.

$x^5+ax^4+bx^3+cx^2+dx+e=0$ Assume that $a,b,c,d,e$ are not arbitrary and that they are known. I was wondering if it were possible to reduce or 'simplify' this using some modular arithmetic. It ...
0
votes
1answer
49 views

is A an even number?

Let $a,b,c,d$ be positive integers such that $(3a+5b)(7b+11c)(13c+17d)(19d+23a)=2001^{2001}$ hence, prove that $a$ is even. I tried to approach this problem reducting it modulo 6. From which we ...
2
votes
4answers
521 views

How long would it take Mustafa to do the job alone? [closed]

Murat and Mustafa can do a job together in fifteen days. After they have worked together for five days, Mustafa leaves the job. Murat completes the job in sixteen days. How long would it take ...
2
votes
2answers
39 views

Solving for $x$ in $x + \frac{s}{s^2+4} = \frac{2x}{s^2+4}$

How would I go about solving for $x$ in this equation? $$x + \frac{s}{s^2+4} = \frac{2x}{s^2+4}$$
1
vote
1answer
18 views

Is my intuition about this statistics problem sensible?

I'm trying to improve my knowledge of statistics and develop my intuition for solving statistical problems. While doing so I've worked on the following exercise: There are 20 players in a checkers ...
0
votes
1answer
55 views

Related Rate of Cylindrical Cone (Filling + Leaking)

So, we are only told of how to solve related rates with one underlying problem. Either a cone is leaking or Being filled up at some point $x$ but I never encountered both working at the same time. ...
1
vote
4answers
61 views

Explain the thought process to a given solution for $ \frac{1+n}{2^n} =\frac{3}{16} $ please

Part of the given solution to a question leaves me baffled: "...solve the equation $$ \frac{1+n}{2^n} =\frac{3}{16} $$ We can check (simply by plugging in values of $n$) that if $ \leq n \leq 5$, ...
0
votes
3answers
88 views

How can I prove this equation has no solution?

Solve the equation $$-x^3 + x + 2 =\sqrt{3x^2 + 4x + 5.}$$ I tried. The equation equavalent to $$\sqrt{3x^2 + 4x + 5} - 2 + x^3 - x=0.$$ $$\dfrac{3x^2+4x+1}{\sqrt{3x^2 + 4x + 5} + 2}+x^3 - x=0.$$ ...
0
votes
1answer
37 views

How this type of equation is solved? [closed]

I'm solving a relative and ends when the function (x²+y²)²-4x² derive out this equation, but not that I have to do to get resolve it $$f'x = 4x³+4y²x-8x$$ $$f'y = (4x²+4y⁴)y$$ How solve this ...
1
vote
0answers
30 views

Simple exercise in differential geometry

Problem: Prove the identity $V=\sum V[x_i]U_i$, where $x_1, x_2, x_3$ are the natural coordinate functions. (Hint: evaluate $V=\sum v_i U_i $ on $x_j$) Elementary differential geometry written by ...
1
vote
4answers
79 views

How would one solve the following equation?

This equation is giving me a hard time. $$e^x(x^2+2x+1)=2$$ Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph ...