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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9
votes
4answers
338 views
+50

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: $$\int_{0}^{\infty} \frac{1}{(x-\log x)^{p}}dx$$...
0
votes
0answers
28 views

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 ...
0
votes
1answer
16 views

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 ...
2
votes
1answer
24 views

How to get the value of the expression?

I have this statement: If $\frac{a}{b+c+d} + \frac{b}{a+c+d} + \frac{c}{a+b+d} + \frac{d}{a+b+c} = 1$ Find the value of $\frac{a^2}{b+c+d} + \frac{b^2}{a+c+d} + \frac{c^2}{a+b+d} + \frac{d^2}{...
-1
votes
0answers
18 views

Convert space to transfer function

Need elaboration in solving steps of... Given The state space X(k)= [ 0.5 -0.5; 0.5 0.5] X(k-1) + [0;1] u(k) and out y(k) = [ 1 0] x(k) It is requested to get the tansfer function, Stability... ...
4
votes
1answer
96 views

Please Explain How to Solve this Equation in Reals

When I solve the equation $(x-1) \cdot \sqrt{x^2 - 4}=0$ in the set of all real numbers (I have not known about complex numbers). I do following steps. First step. I solve the inequality $x^2 - 4x \...
3
votes
1answer
89 views

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 ...
-7
votes
0answers
47 views

Why doesn't the rule of three work with age problems? [closed]

why doesn't the rule of three work with age problems? For example: When my sister was $7$, I was twice her age. Now my sister is $13$ years old. How old am I? If you do $7$ $\enspace 14$ $13$ $\...
7
votes
2answers
214 views

List of not-especially famous problems in undergraduate level mathematics

I know lists of problems like these have been compiled before, but most tend to collect either extremely difficult problems ( like Collatz conjecture in a question about number theory ) or ...
-1
votes
1answer
36 views

What is the ratio of speed? [closed]

If two trains start at same time from point A and B towards each other and after crossing they take |a| and |b| seconds in reaching B and A respectively. What is the ratio of speed of A and B.
2
votes
1answer
64 views

Given $\triangle ABC$ with $C=60^\circ$, show that $\frac{1}{a+c}+\frac{1}{b+c}=\frac{3}{a+b+c}$

Given that $C=60^\circ$ on a triangle $ABC$, prove the following relation: $$\frac{1}{a+c}+\frac{1}{b+c}=\frac{3}{a+b+c}$$ P.S. Maybe this info could be of help: I used the cosine rule of triangles, ...
19
votes
4answers
755 views

Motivation for this solution to a British olympiad problem

I was doing question 6 from this BMO1 paper: https://bmos.ukmt.org.uk/home/bmo1-2019.pdf and I didn't manage to get it. Then I looked at the solution and found the solution. I can see how the solution ...
0
votes
0answers
21 views

Correct way of multiplying/dividing both parts of an equation by a polynomial $P(x)$

It is perfectly "legal" to multiply both parts of any equation, say $f(x) = g(x)$, by any number $a\ne0$ Now I want to formulate a perfectly correct algorithm to multiply both parts of an equation by ...
9
votes
2answers
139 views

How to maximize $\int_{0}^{1} f(x)^5 dx$ given $\int_{0}^{1} f(x)^3 dx= 0$ , $\int_{0}^{1} f(x) dx= 0$ and $-1 \le f(x) \le 1$?

How to maximize $\int_{0}^{1} f(x)^5 dx$ given $\int_{0}^{1} f(x)^3 dx= 0$ , $\int_{0}^{1} f(x) dx= 0$ and $-1 \le f(x) \le 1$? In not even sure where to start with this problem any hints would be ...
1
vote
0answers
19 views

Carmichael number equal to order of an element

Is there always a number like d (mod n) such that ord d (mod n) is equal to the carmichael number of(n) For example Carmichael number of 12 is 2 And order 5 mod(12) is 2 I tried to handle it using ...
5
votes
1answer
58 views

Partition the integers into three subsets such that for any $n$, the three integers $n, n+p$ and $n+q$ belong to different subsets

Question from Engel's book problem solving strategies. Let $p$ and $q$ be fixed integers. The set of integers are to be partitioned into three subsets $A,B,C$ such that for any $n \in \mathbb{Z}$, ...
1
vote
1answer
744 views

Solving system if equations containing trigonometric functions with Ti-Nspire

In trying to solve the following system of equation: $20000\times9.81+a\cos b=0$ $a\sin b=6.17\times20000$ Find $a$ and $b$ . It gives me something containing "n2" in bold and I don't know why? $...
-1
votes
1answer
15 views

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?
0
votes
2answers
50 views

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 ...
0
votes
2answers
35 views

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 ...
12
votes
2answers
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 $c^2-a^2$...
0
votes
1answer
47 views

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 ...
0
votes
2answers
767 views

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: ...
1
vote
2answers
96 views

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 ...
3
votes
3answers
68 views

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 ...
1
vote
2answers
82 views

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 ...
42
votes
9answers
12k views

List of problem books in undergraduate and graduate mathematics

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, ...
1
vote
0answers
47 views

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, ...
5
votes
0answers
61 views

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 ...
0
votes
1answer
33 views

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 $\{\pm x\pm y \pm z\}$. ...
0
votes
2answers
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 ...
2
votes
2answers
27 views

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 ...
1
vote
1answer
29 views

Solving a recursive relationship

I was wondering whether the following relationship could be solved. $$S_0= 100, C=(0.05, 0.15,...), P = (0.03, 0.05, ...), D = 120, R = 0.08\\ S_1 = S_0(1-C_1)\left(1-P_1-\frac{1}{D + 1}-(1-R)\right)\...
0
votes
2answers
46 views

How to solve $x+\sqrt{1+x^2}=3\sqrt{3}$ for $x$?

How do I solve the following equation for x: $x+\sqrt{1+x^2}=3\sqrt{3}$ I'm failing miserably in isolating the $x$
1
vote
2answers
987 views

Finding Y coordinate of third triangle point when X coordinate and two other points are already known

Suppose you know the coordinates for points A and B of a triangle. We can refer to those coordinates as (Ay,Ax) and (By,Bx). Also, suppose you know the X coordinate for point C (Cx) but do not know ...
0
votes
1answer
64 views

What branch of maths helps develop an effective problem solving mindset

I have started doing maths about 5 years ago , i didnt have a math science orientation , but i have developped a study program specially for elementary algebra and geometry as well as basic statistics ...
2
votes
5answers
57 views

What is the solution to this equation: $e^x+2x=0$

I cant find any program that actually solves this type of equations and I cant find anything helpful about this type. What is the name of these equations and how do I solve this one? Thanks.
0
votes
1answer
43 views

Probability of each outcome, when Flipping a Hemisphere

Say I have a hemisphere (A solid sphere, cut exactly in half), and I have an experement in which, when flipped, it can land either as: Flat Face Up Flat Face Down How would one go about calculating ...
1
vote
1answer
29 views

Finding the root of an equation involving digamma functions

Is it possible to get an analytic solution for the equation \begin{align} \frac{1}{x} + 2\psi(2x) + \pi \cot(\pi x) = 0 \end{align} for $x\in(0,1)$ (using the Newton-Raphson method I get $x\approx 0....
0
votes
1answer
39 views

How to find the solution of a Numerical Reasoning Problem

There is a girl who likes number 100 but doesn't like 99, she likes 900 but doesn't like 850, she likes 2500 but doesn't like 2600. I have to find out which number she is going to like. The proposed ...
0
votes
2answers
47 views

Probability of randomly choosing all elements fulfilling a certain condition

Assume you have a bag containing $m$ marbles, of $c$ different colors, where the number of marbles of each color is equal to $\frac mc$. If $n$ marbles are drawn from the bag, without replacement, ...
0
votes
1answer
32 views

Problems to read/write the polynomial equation

Can't figure out how the i could get from that equation $$\tag{1} p(x)=\sum_{l=0}^n \left(\sum_{j=0}^n a_j\binom{j}{l}\tilde x^{j-l}\right)(x - \tilde x)^{j-l}$$ for that example: "For an ...
0
votes
1answer
67 views

Solve using KKT conditions when number of variables is less than number of constraints

$$\begin{align} \text{minimize} \quad & f(x) = x_1^2 + x_2^2 +x_3^2 \\ \text{subject to} \quad & 2x_1+ x_2-5\leq 0 \\ & x_1+x_3-2\leq 0 \\ & 1-x_1 \leq 0 \\ & 2-x_2 \leq 0 \\ & ...
0
votes
1answer
33 views

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 ~ ...
0
votes
1answer
27 views

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 ...
0
votes
0answers
24 views

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 ...
-2
votes
1answer
19 views

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 ...
3
votes
3answers
97 views

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 ...
0
votes
1answer
65 views

How can I find all solutions of this equation?

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 ...
0
votes
1answer
12 views

Determing power settings

A company has three machines which output at different wattages(35, 45 and 60). The 35-watt machine runs best at a power level of 45%. The 60-watt machine runs best at a power level of 60%. What is ...