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Questions tagged [problem-solving]

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.

3
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1answer
39 views

Three dimensional Laplace equation with constant Temp. on one face. [Solution not satisfying BC]

The governing differential equation is $$\nabla^2 T=0 \tag A$$ The boundary conditions for this problem are as foll0ws: $$T(0,y,z)=T_{hi} \tag {1A}$$ $$T(L,y,z) = T(x,0,z) = T(x,l,z) = T(x,y,0)= ...
0
votes
0answers
10 views

Calculating density of individuals within an area (tree stand density)

Any help greatly appreciated! I want to find the density of trees surrounding a sample point (measured in trees/m$^2$). The distance from the sample point to the three nearest trees has been ...
0
votes
3answers
36 views

Easier way to solve a LES (only pen/paper, no calculator)

I want to solve the following Linear Equation System with only pen and paper; $$ 470 = x_A - \frac{3}{10}x_B \tag{1} $$ $$ 940 = x_B - \frac{2}{10}x_A \tag{2} $$ I attempt to solve for $x_A$ by ...
1
vote
0answers
11 views

Prove that there exists a $m×m$ lattice square in the $x-y$ plane such that none of its coordinates are visible [duplicate]

Call a lattice point 'visible' if the $gcd$ of its coordinates is 1. Then there exists a $m×m$ square in the $x-y$ plane such that none of its coordinates are visible. You can actually define such ...
2
votes
2answers
37 views

Soft question: solution strategies, how to attack the problem when using pen and paper?

Whenever I solve a problem (textbook or exams), I always copy key parts of the question to my paper where I'm doing my calculations, even though I have the problem in text-format right next to me on ...
1
vote
1answer
31 views

Restoring permutation from differences of adjacent elements

Suppose a permutation $\pi \in S_n$ is encoded by a list of integers $P=(p_1, p_2, ... p_{n-1})$, where $p_i = \pi(i+1) - \pi(i)$, i.e. $P$ is the list of differences of adjacent elements. Now, given $...
1
vote
0answers
41 views

Odds of winning a prize in a weighted, random raffle

There is a raffle a state holds annually to assign salmon fishing licenses to fisherman; in a specific stetch of river, to control harvest and limit access, to ensure resource management and a high ...
2
votes
3answers
52 views

Some questions on the intersection of three cones.

I have three cones in $\mathbb{R}^3$, explicitly defined by the equations: $$ (x-\alpha_x)^2+(y-\alpha_y)^2=(z-r_1)^2 \,, \\ (x-\beta_x)^2+(y-\beta_y)^2=(z-r_2)^2 \,, \\ (x-\gamma_x)^2+(y-\gamma_y)^2=(...
1
vote
2answers
61 views

Use of big mysterious shortcuts in academic papers, example integration by parts

I read a paper where the author did a very strange but valid integration by parts: What I thought was unusual is the repeating occurrence of $(q_T - q_t)$ in essentially all the terms (ignore $h$). ...
0
votes
0answers
11 views

REFEREENCE REQUEST for Non-Local Boundary Value problems

It would be really helpful if someone could suggest me any reference (Books or Papers) where I would find worked-out examples of Elliptic Boundary value problems (especially Laplace equation) with non-...
0
votes
0answers
19 views

Solve equation for $x$, why not a closed format?

I am trying to solve for $x$ the following: $\big(\frac{x-m}{d*x+f}+1\big)^s = \big(\frac{n-m}{d*x+f}+1\big)$ I have tried with Wolfram Alpha functionality ...
0
votes
0answers
21 views

How to keep the rod in equilibrium?

What would be the force F to keep the rod AB in equilibrium? I got confused while using Lami's theorem. $$\frac{F}{\sin160^\circ}=\frac{F_1}{\sin60^\circ}=\frac{W}{\sin140^\circ}$$ If F is keeping it ...
0
votes
0answers
25 views

Laplace equation problem with numerous non-homogeneous BC(s) [Linear Superposition]

I need to solve the three-dimensional Laplace equation ($\nabla^2T = 0$) where $\nabla^2=\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$ in the domain ...
2
votes
3answers
45 views

7 coins in bag which add up to $1.25

So I came across this question, there are 7 coins in bag of only nickels, dimes and quarters and we know it totals to $1.25. So, how many of each coin are there? Just by looking, one solution is 4 ...
1
vote
0answers
14 views

solving PDE in spherical coordinate in d-dimension

I want to solve the equation below $$\partial_t F(r,t)= \frac{a}{r^{d-1}}\partial_r\big(r^{d-1} \partial_r F(r,t)\big)$$ where $r$ denotes the radius in spherical coordinates, and $a$ is a constant. ...
0
votes
1answer
19 views

Polydivisble numbers using base b

Picked the definition from wikipedia: A polydivisible number is a number with digits abcde... that has the following properties : Its first digit a is not 0. The ...
1
vote
0answers
32 views

Are being man and their birthday independent for two persons? [duplicate]

I'm given this problem: Consider there are two applicants for a job. One of the applicants is a man, and it is known that he was born on a Wednesday. What is the probability that a second applicant ...
5
votes
1answer
123 views

A pre-calculus problem about a quadratic function

This is from a test (I'm not a high school student) given to rising high school juniors. The problem was designed to take less than 20 minutes, preferably 5-15. Judging from its source, this is not a ...
0
votes
0answers
40 views

A square dissection into squares of distinct size (aka “perfect squaring of a square”)

I am wondering if anyone can give an example of a square dissection into squares of distinct size. By using the distinctness we can see that the smallest square must be on the interior of the square, ...
1
vote
1answer
73 views

About IMC (Invitational international mathematics competition)

Last year I was invited to the international mathematics competition, however, I wasn't well prepared. Can you guys please recommend some books for me to read, in order to get better, at solving IMC ...
0
votes
1answer
50 views

Question from an IMC(International mathematics competition) key statge III selection exam

A positive integer n does not have any 9 digits, it has four 8 digits, three 7 digits, two 6 digits and some other digits. If the sum of the digits of the numbeer n is 104 andthe sum of the digits of ...
-1
votes
1answer
59 views

How to find the radius of the circle after adding the area of segment?

I am making a game where a circle can hit a wall an interact with it to simulate a ball hitting a wall. When the circle is 1/4 the radius through the wall the area of the part in the wall will be ...
-1
votes
2answers
23 views

Basic “Punctuation” and “words” used in basic Mathematics

I live in a place where solving math problems is taught perfectly, but Punctuation and correct words are overlooked. I don't know to use comma, period, dash, paragraph change, etc and words like ...
6
votes
3answers
897 views

Soft question- The Bashing Technique and Other powerful techniques for Olympiads

I am relatively new to the Olympiad style maths problems and I find the word "bashing" being thrown around a lot in the community and I haven't really understood it. I do know about "co-ordinate ...
1
vote
1answer
29 views

How do I find the probability for this circuit to run current?

The probability of the closing of the ith relay in the circuit below is given by $p_i$, $i$ = 1,2,3,4,5. If all the relays function independently, what is the probability that a current flows between $...
2
votes
1answer
164 views

IMC 2011 question involving six tangent circles

Two circles $A$ and $B$, both with radius $1$, touch each other externally. Four circles $P$, $Q$, $R$, and $S$, all with the same radius $r$, are such that $P$ touches $A$, $B$, $Q$, $S$ externally; $...
2
votes
0answers
11 views

Path / Graph problem with X nodes looking for Y paths with the most similar length.

I have the following graph / path problem: There is exactly 1 start node and 1 end node. There are also X (in this case 7) nodes, each connected to all other nodes and the start and end node with ...
3
votes
0answers
40 views

What is the probability of two even numbers given one is even?

Your friend has generated two random numbers from $1...10$, independently of each other. What is the probability that both numbers are even given the information that there is an even number among the ...
0
votes
1answer
14 views

Solving recurrence equation for lossy duplication

Suppose we have a word of length $L$ from a two-letter alphabet (say, $\mathcal{A} = \{A,B\}$), and we duplicate it. However, our duplication is fallible: each element of the result is incorrect ($A$ ...
0
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3answers
51 views

A problem about expected value

Some students have a list of processes to execute sequentially but the running of a process may fail during execution so the probability of a process x to be executed successfully is Px. Assume that ...
0
votes
2answers
89 views

Question 10 from the 2011 IMC (international mathematics competition) key stage III paper, about combinatorics [on hold]

Find the number of positive integers which satisfy the following conditions: 1/ It contains 8 digits each of which is 0 or 1. 2/ The first digit is 1. 3/ The sum of the digits on the even places ...
1
vote
1answer
38 views

Question 11 from the 2011 IMC (international mathematics competition) key stage III paper, about the evaluation of an integer k

A checker is placed on a square of an infinite checkerboard, where each square is 1cm by 1 cm. It moves according to the following rules: 1/ In the first move, the checker moves 1 square North 2/ ...
2
votes
1answer
38 views

Question from the 2011 IMC (international mathematics competition) key stage III paper, about a logical sequence

There are two kinds of students in a cerain class, those who always lie and those who never lie. Each student know what kinde each of the other students is. In a meeting today, each student tells what ...
0
votes
4answers
95 views

Prove that if $x$, $y$, and $z$ are real numbers such that $x^2(y-z)+y^2(z-x)+z^2(x-y)=0,$ then at least two of them are equal

The question expressed in the title was asked in a past international math exam. My idea to solve it was to factorize it and then end up showing that the polynomial that was formed would directly show ...
0
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0answers
25 views

Sampling subsets of positive numbers with sum close to a given number

Given a set of $N$ positive numbers $a_1 \ldots a_N$, I am trying to generate random subsets of this set such that the sum of numbers in subset is close to a given number $M$. Suppose I can only ...
5
votes
0answers
222 views

Evaluating Coefficients for a Fourier Series when Exponential terms are present [Approach needed]

On the last step of solving a three-dimensional Laplace equation,($\nabla^2T=0$) with BC(s) as $T(0,y,z) = T(L,y,z) = T_a$, $T(x,0,z) = T(x,l,z) = T_a$, $\frac{\partial T(x,y,0)}{\partial z} = p_c\...
1
vote
2answers
63 views

Question from the 2011 IMC (International Mathematics Competition) Key Stage III paper, about the evaluation of a quadratic equation

When $a=1, 2, 3, ..., 2010, 2011$, the roots of the equation $x^2-2x-a^2-a=0$ are $(a_1, b_1), (a_2, b_2), (a_3, b_3),\cdots, (a_{2010}, b_{2010}), (a_{2011}, b_{2011})$ respectively. Evaluate: $...
0
votes
2answers
62 views

Geometric argument showing that the sum of the series $\sum_{n = 1}^{\infty} n^{-1}(n + 1)^{-1}$ is $1$

I can't see the relationship between this telescopic series and the area under the curve for functions $f(x)=x^n$.
1
vote
3answers
114 views

Is it possible to find $\limsup\limits_{n\to\infty} \frac{2×3^n-3×2^n}{2^{\alpha(n)}-3^n}$?

I need help to solve this problem: $$\lim_{n\to\infty}\frac{2\cdot 3^n-3\cdot 2^n}{2^{\alpha(n)}-3^n}$$ or $$\limsup_{n\to\infty}\frac{2\cdot 3^n-3\cdot 2^n } {2^{\alpha(n)}-3^n}$$ where, $\...
0
votes
1answer
39 views

Sidestepping Factorial Calculations?

Please forgive me if I ramble or use weird terms to try and describe what I’m after; I’m a programmer not a mathematician. The scenario: A large number of players, playing a real-time game in 3d ...
0
votes
1answer
26 views

How can I calculate the probability of a certain arrangement of chess pieces? [duplicate]

I'm trying to calculate the probability that if 8 rooks are placed on a chessboard, they are placed in a fashion such that none of the rooks can capture another rook. So, no row or column can contain ...
-1
votes
1answer
25 views

Solutions for $n$? Use Stirling approximations if needed

$$(2n)! = a^{2n}$$ where $a \in \mathbb R$, and $n \in \mathbb N$. This is relevant because of a research question I'd asked and received an answer to by Sotiris here
12
votes
1answer
302 views

Two fluids flowing perpendicular in thermal contact with a Wall [Help to mathematically model]

I will try to describe briefly how I am modelling the problem. (Please bear with the length). The governing equation describing temperature for a block at steady state is $$\nabla^2 T = 0$$ where $\...
3
votes
1answer
58 views

T shaped tetris figures on a plane

I am just wondering how many (and by how many I mean countably or uncountably many) T shaped figures can we place on a XY plane. I assume that that T consists of 2 perpendicular lines and has 0 area. ...
0
votes
0answers
17 views

Programming Equivalent PBL (Project Based Learning) in Math

I have been doing programming for about 4 years now and I have learned alot by working on projects such as: 2D Games, Websites, Web Templates etc... and I worked on these projects with my own will, ...
0
votes
0answers
28 views

How to form an equation based off $x$ and $f(x)$ values

So I'm making a program (and I know, this isn't stack overflow) but I need help on ONE algorithm in order to finish it for good. Here's the pattern I'm dealing with: When $x = 8, f(x) = 1$ When $x =...
-2
votes
1answer
47 views

Combinations with Piano Keys

How many sound combinations can be created by the $10$ selected piano keys if each sound combination contains from $3$ to $10$ keys? Can anyone throw at least a hint? I am having difficulties with ...
39
votes
7answers
5k views

How to attack “if true, prove it; if not true, give a counterexample” question?

I am taking a basic analysis course. This is a general question that I often encounter in weekly homework. How should we start to attack this type of question: if the statement is true, prove it; if ...
1
vote
0answers
40 views

How to express y from $\dfrac{y-2x}{y}=1$?

I need to express y from $\dfrac{y-2x}{y}=1$, I mean get the $y=...$ equation. But I'm getting 0 instead of y: $\dfrac{y-2x}{y}=1$ $y=y-2x$ $y-y=-2x$ $0=-2x$
2
votes
1answer
127 views

Laplacian with Integral BC(s)

I want to solve the three-dimensional laplacian $$\nabla^{2} T = 0$$ where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ defined on $...