# 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.

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### A closed form for: $\int_{0}^{\infty} \frac{1}{(x-\log x)^2}dx$

Is it possible to find a closed-form expression for this integral? $$\int_{0}^{\infty} \frac{1}{(x-\log x)^2}dx$$ Generalization of the Integral: $$\int_{0}^{\infty} \frac{1}{(x-\log x)^{p}}dx$$...
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### Simple Solve $~[{y = 5 + 2 x + x^a}, {x}]~$ not working? [on hold]

I am trying to find a general expression for $~x~$, for the following equation $~y=5+2x+x^a~$, but Solve cannot accomplish this. I am using Solve$~[{y = 5 + 2 x + x^a}, {x}]~$ If I put any numerical ...
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### Combined variation with four variables: Is this even possible to solve?

I have this problem, and I'm quite confused on how to solve it. Mr. Plaridel owned a newspaper publication called Diaryong Tagalog Inc. He observed that when he used 3 printing presses, he can ...
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### Is there a polynomial $p(x)$ with integer coefficients such that $p(2013)=1789$ and $p(1515)=1830$?

Problem: Is there a polynomial $p(x)$ with integer coefficients such that $p(2013)=1789$ and $p(1515)=1830$? My attempt: After ruling out polynomials of degree 1, 2 and 3, and with further ...
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### dividing ordered set into subsets [closed]

Consider the set $N_9=\{ 1,2,3,...,9 \}$. can we divide this set into 3 subsets with 3 elements in each one and the sum of elements in each subset is constant?
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### Book recommendations for Problem Solving

I am looking for book recommendations that will teach me the art of problem solving. Learning theory is one thing but doing problems in limited time in a test is another. To increase these skills I ...
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### Chessboard problem [closed]

The 64 squares of an 8×8 chessboard are filled with positive integers in such a way that each integer is the average of the integers on the neighbouring squares. Show that in fact all the 64 entries ...
513 views

### Show divisibility by 7

I was stuck at this question: Suppose $a^2+b^2=c^2$ for $a,b,c \in \mathbb Z$, and neither $a$ nor $b$ is a multiple of 7. Show that $a^2-b^2$ is a multiple of 7 I tried to write $b^2$ as $c^2-a^2$...
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### Conceptual difficulty concerning factors with rational exponents

My level of perspective: I've been teaching myself mathematics mostly using Art of Problem Solving books, and various online resources. Have basically wrapped up the AoPS Prealgebra. Previously have ...
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### Solve $4 A (L^{3/4}) - wL (((24 - L) w)^{-3/4}) = -(((24 - L) w)^{1/4})$ for L? Using Mathematica

I'm trying to solve $$4 A (L^{3/4}) - wL (((24 - L) w)^{-3/4}) = -(((24 - L) w)^{1/4})$$ for $L$ using Mathematica, and it spits the following out: ...
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### How do I solve $y' = x^2 + y^2$ analytically? [duplicate]

I came across this equation: $$\frac{dy}{dx} = x^2 + y^2$$ $$y(0)=0$$ I found numerical solutions to it using Runge-Kutta methods, but I want to verify my answers by solving it analytically. At ...
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### Finding $\sin^2\alpha+\sin^2\beta+\sin^2\gamma$ given $\sin \alpha+\sin \beta+\sin\gamma=0=\cos\alpha+\cos\beta+\cos\gamma$

I am supposed to find the value of $\sin^2\alpha+\sin^2\beta+\sin^2\gamma$ and I have been provided with the information that $\sin \alpha+\sin \beta+\sin\gamma=0=\cos\alpha+\cos\beta+\cos\gamma$. I ...
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### Conjugacy classes of a group of order $8k$

Let G be a group of order $8k$, show that there are at least 5 different conjugacy classes. Hi everyone, I have this problem I think I had a solution involving stabilizers, however I feel there must ...
12k views

I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, ...
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### The 8 Queens Puzzle [closed]

The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, ...
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### Minimizing a project costs through dynamic programming

I have a project and I want to minimize the costs. I am responsible for the inspection of 1000 miles of sewer grid in Canada. My goal is to provide time high quality inspection reports. I tried to ...
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### Rewriting xyz as combination of squares

It is well-known that $xy=\frac{1}{4}[(x+y)^2-(x-y)^2]$. However for the problem I am currently working, I need to write $xyz$ as a lineal combination of squares or cubes of $\{\pm x\pm y \pm z\}$. ...
33 views

### Problem solving Q about number of houses in a street.

A row is marked with 1,2,3,4.... The marking continues down the other side - the largest mark opposite is 1. Each mark has another directly opposite it. If mark 17 is opposite number 56, how ...
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### Solving a inequality over the reals

I have the following inequality that I need to solve for real $z$: $$3\cdot\left(-d\left(d^3+4z^3-168\right)\right)\ge0\tag1$$ Where $d$ and $z$ are element of the real numbers (so they can be ...
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### Optimization problem for maximum profit

I have to write a linear program for the following problem: We have three products that are made in a factory: A, B and C, we are given the amount of energy every product needs (A ~ 1kWh, B ~ ...
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### Right Circular Cylinder: Distance between axis and plan

B is a point in the top circle of a right circular cylinder. C is a point in the bottom circle of the given cylinder. The angle between [BC] and the base's plan of the cylinder is 45 degrees. The ...
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### Understanding Intermediate steps in this Boundary value Problem

I have been trying to replicate a derivation from a scientific paper (for my understanding) which involves solving a Boundary value problem on the Stokes operator. The author jumps some steps while ...
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### intuition behind LCM method used to solve time/work aptitude problems

Below is a sample time/work problem. Rakesh alone can do a work in 10 days. Brijesh can do the same job in 15 days. If both Rakesh and Brijesh work together, then in how many days the work will get ...
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### Congruence equations in the form $x^k \equiv b \bmod m$

In my textbook its described as finding $k\;$th roots moduluo $m$. How do you solve equations of this form as I can't find any examples anywhere? I am just looking for a step by step example so I can ...
I am trying to solve the equaiton $n^3+2019 n=k^2$, where $n$ and $k$ be two positive integral numbers. I tried with Mathematica and get two solution $k = 78, n = 3$ and $k = 17498, n = 673$. How ...