# 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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### Simple Solve $~[{y = 5 + 2 x + x^a}, {x}]~$ not working?

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 ...
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 $x^2 - 4x \... 1answer 36 views ### What is the ratio of speed? [closed] If two trains start at same time from point A and B towards each other and after crossing they take |a| and |b| seconds in reaching B and A respectively. What is the ratio of speed of A and B. 1answer 63 views ### Given$\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, ... 4answers 747 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 ... 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 ... 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 ... 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$n \in \mathbb{Z}$, ... 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 ... 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? 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 ... 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 ... 3answers 280 views ### 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? $$\int_{0}^{\infty} \frac{1}{(x-\log x)^2}dx$$ where,$\log x$is a natural logarithm. The indefinite integral can not be ... 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 ... 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 ... 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, ... 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 ... 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$\{\pm x\pm y \pm z\}$. ... 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 ... 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$zare element of the real numbers (so they can be ... 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 ... 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)\... 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 1answer 62 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 ... 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.... 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. 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 ... 1answer 30 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}$$for that example: "For an ... 2answers 49 views ### Book recommendations for Problem Solving I am looking for book recommendations that will teach me the art of problem solving. Learning theory is one thing but doing problems in limited time in a test is another. To increase these skills I ... 0answers 24 views ### Understanding Intermediate steps in this Boundary value Problem I have been trying to replicate a derivation from a scientific paper (for my understanding) which involves solving a Boundary value problem on the Stokes operator. The author jumps some steps while ... 1answer 19 views ### intuition behind LCM method used to solve time/work aptitude problems Below is a sample time/work problem. Rakesh alone can do a work in 10 days. Brijesh can do the same job in 15 days. If both Rakesh and Brijesh work together, then in how many days the work will get ... 1answer 27 views ### Right Circular Cylinder: Distance between axis and plan B is a point in the top circle of a right circular cylinder. C is a point in the bottom circle of the given cylinder. The angle between [BC] and the base's plan of the cylinder is 45 degrees. The ... 1answer 67 views ### Solve using KKT conditions when number of variables is less than number of constraints$$\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 \\ & ... 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 ~ ... 2answers 47 views ### Probability of randomly choosing all elements fulfilling a certain condition Assume you have a bag containingm$marbles, of$c$different colors, where the number of marbles of each color is equal to$\frac mc$. If$n$marbles are drawn from the bag, without replacement, ... 1answer 12 views ### Determing power settings A company has three machines which output at different wattages(35, 45 and 60). The 35-watt machine runs best at a power level of 45%. The 60-watt machine runs best at a power level of 60%. What is ... 1answer 65 views ### How can I find all solutions of this equation? I am trying to solve the equaiton$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$. How ... 2answers 213 views ### List of not-especially famous problems in undergraduate level mathematics I know lists of problems like these have been compiled before, but most tend to collect either extremely difficult problems ( like Collatz conjecture in a question about number theory ) or ... 1answer 24 views ### Calculate the running and walking distance depending on speed and hours So here's my problem I'm having: Say someone runs or walks 40 kilometers in a day in 8 hours. We don't know how many kilometers in specific the person was walking or running, though that's what we'd ... 0answers 20 views ### Constructing numbers with a limited “inventory” of numbers and operations where numbers are consumed upon use. This might be way too specific, but I will give it a try posting it here nonetheless. There is a (multi)set of numbers that can contain the same number multiple times, and there is a set of "... 1answer 25 views ### Complicated transporting problem. In the following problem, goods will be transported from farmers to stores. I know how to minimise the transport costs in such problems. The complication comes because the goods have to go to some ... 2answers 71 views ### Solving system of 10 equations involving some degree 2 Recently I was doing a pipe network problem without Hardy Cross Method (approximate method ) I have obtained 10 desired equations with 10 unknowns as shown in this image below highlighted by red pen. ... 1answer 22 views ### Solving mathematical economics problem kindly I'm stuck in this problem instead of many attempts through net present value and other discounted cash flow methods, some one could give me a detailed information and answers on this problem : (... 6answers 137 views ### Find$a,b,c,d$such that$2^a + 2^b + 2^c = 4^d$Let$a,b,c,d$be whole numbers that satisfy $$2^a + 2^b + 2^c = 4^d$$ What values of$(a,b,c,d)$would make this equation true? Here is my work so far. Without loss of generality, assume$a\ge b\...
I'm trying to find an explicit solution $u(t)$ of $$\dot{u} = \frac{\beta}{\sqrt{\alpha t}-u},\quad u(0)=0$$ where $\alpha>0$ and $\beta\in\mathbb{R}$ are given constants. I did not know how to ...