Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

4
votes
2answers
315 views

Proving $ \frac{\csc x + \cot x}{\tan x + \sin x} = \cot x\csc x $

I am currently working on understanding trig identities. A question has me stumped, and no matter how I look at it, it never leads to the proof. I believe I am making a mistake when dividing multiple ...
0
votes
1answer
136 views

Petersen graph edge chromatic number

Hi I keep on getting 3 for the edge chromatic number of the Petersen graph. But the Petersen graph has edge chromatic number of 4 and I don’t know how to do that. Can someone please show this by ...
5
votes
2answers
75 views

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$ I don't know where to start as in is there a ...
1
vote
1answer
144 views

Inscribe an equilateral triangle inside a triangle

Given a triangle ΔABC, how to draw all possible inscribed equilateral triangles with given side whose vertices lie on different sides of ΔABC?
1
vote
1answer
36 views

How to make a kid understand geometry and help him solve problems?

I'm tutoring a 13 year-old boy, a middle school student. He has almost no problem with elementary algebra: he just applies the rules and everything falls into place. However he does struggle with ...
-3
votes
1answer
40 views

how to simplify and solve this equation :

Solve this equation for $x$: $$\left|x \sqrt{1-x^2} +x \right| = \sqrt{1+x^2}$$ I'm having a problem getting rid of the square root!
3
votes
1answer
78 views

Is there anything that non-newtonian calculus can do, which newtonian calculus cannot?

As well, are there problems where non-newtonian calculus leads to a more elegant or simple solution than regular calculus?
1
vote
2answers
42 views

19 people on 8 benches with 1, 2 or 3 people. How many with only 2?

I found a math problem for 2nd graders in Norway. The article only presented drawing a table of everything and simply looking at it, instead of a mathematical approach, like a formula. I'm not sure ...
0
votes
1answer
15 views

You see a row of 5 trees, each one 20 meters from the next. How long is the row?

I'm thinking of this problem in 2 ways. There's 5 trees, each the same width $X$. This leaves 4 inner "gaps" of $20$ meters in length? So, my answer would be $(20 \times 4) + (5\times X)$ or rather $...
2
votes
1answer
43 views

Mathematic problem

I'm stuck on a problem that maybe most of you will be able to solve instantly. Here is the problem: A publisher needs to bind $4500$ books. One print shop can bind these books in $30$ days, another ...
2
votes
0answers
46 views

Show that a certain set of points lies on a straight line [duplicate]

Let $S$ be an infinite set of points in the plane. The distance between two points of $S$ is integral. Prove that $S$ is a subset of a straight line. Here is my attempt: Suppose not, then for each ...
2
votes
2answers
153 views

Integer solutions of a variable coefficient polynomial

I have many equations to solve similar to this one: $$2 a b^3 - a b^2 + a b - 2 a - b^4 + b^3 - 2 b^2 + 2 b = 0$$ Here, b is a base and a is a non-zero digit in a b-adic number, so $1 \leq a \leq b-...
-1
votes
2answers
58 views

Solving $2y=\sqrt{3+\frac{1}{2y}}$

Any way to solve this irrational equation in $\mathbb{R}$? I think it has some artifice, but I do not see it $$2y=\sqrt{3+\frac{1}{2y}}$$
3
votes
2answers
56 views

Trig and Triangle Math Club Question: Finding Side Length

I recently had a math club competition, and I was unsure of how to approach one of the problems on the test: In $\triangle ABC$, $\ \ \ \ \ \ \ \ \cos(2A-B) + \sin(A + B) = 2$ $\ \ \ \ \ \ \ \ \...
2
votes
1answer
94 views

Set of sets such that any three of them has nonempty intersection. Show all of them have nonempty intersection. [closed]

Given $2^{n-1}$ subsets of a set with $n$ elements with the property that any three have nonempty intersection, prove that the intersection of all the sets is nonempty. My attempt: Let $|X|=n$ and $...
9
votes
0answers
223 views

$m^{5}n+7$ and $n^{5}m+7$ are cubes of integer. [closed]

Find integer numbers $m$, $n$ such that $m^{5}n+7$ and $n^{5}m+7$ are cubes of integer. I guess the only pair is $(1;1)$, but I don't know what way we should follow to solve this problem. Help me ...
8
votes
1answer
227 views

Why is the maximum of i.i.d. Gaussians asymptotically $\sqrt{2 \log n}$?

Assuming that $\xi$ is bounded (as a function of $x$?), the claim is that given the equation: $$\xi \frac{\sqrt{2\pi}}{n} = \frac{1}{x} e^{-\frac{x^2}{2}} \left( 1 + O\left(\frac{1}{x^2} \right) \...
0
votes
0answers
47 views

On Geometry problem solving methods

Till now, among all types of Geometry problem solving methods I have found these:. 1. Euclidean geometry (including trigonometry). 2. Analytical geometry (including all kinds of coordinates, ...
-5
votes
1answer
43 views

Is $(-1)^{1/3} = -1$ , but $(-1)^{2/6} = 1$. Why aren't these the same? [duplicate]

So if you try to solve $(-1)^{1/3}$ you can do $(-1)^{1/3} = \sqrt[3]{-1} = -1$ (cubic root of $-1$) But what if I write $1/3$ as $2/6$? $$(-1)^{1/3} = (-1)^{2/6}$$ So $(-1)^{2/6} = \sqrt[6]{(-...
0
votes
1answer
59 views

Proving - set theory

Is x∈ ((A\C) ∩ B) ∪ ((A\B) ∩ C) same as x∈ (AΔC)\B ? I need to prove that $(A\backslash (B \cup C))\cup (C\backslash (A\cup B))$ $=$ $(A\Delta C)\backslash B $ and using 'Let x∈... ...' method I get ...
1
vote
2answers
76 views

Proof and problem solving - set theory

Prove that $(A\Delta C)\backslash B = (A\backslash (B \cup C))\cup (C\backslash (A\cup B))$. I tried with an $x$ that can be in $(A\Delta C)\backslash B$, so $x$ is in $A \Delta C$ but not in $B$. If ...
0
votes
0answers
43 views

Find vectors forming a closed path

Let us consider a function $f: [a,b]\times S^3 \rightarrow \mathbb{R}^2$ where $a,b$ are reals and $S^3$ is the sphere defined by $x^2+y^2+z^2+w^2=1$. I will denote $\vec{u} = (k,\vec{r})$ where $k\...
-1
votes
4answers
73 views

$x\cdot y = \sqrt2$. What can be said about $x$ and $y$ [closed]

Given, $x,y\in\Bbb R$ and $x\cdot y=\sqrt{2}$ .Can $x$ and $y$ be taken as $\sqrt[4]{2}$?
1
vote
1answer
60 views

What is the time in the afternoon problem

I have been reading Ray's Intellectual Arithmetic and I stumbled upon the following problem: What is the time in the afternoon, when the time past noon is equal to $1/5$ of the time past ...
0
votes
1answer
59 views

How to solve a 1D spring system?

I have a problem which reduces to a simplified 1D spring-mass system, in which I do not care about spring constants nor masses (they can assumed to be the same everywhere). I'm pretty sure it boils to ...
0
votes
2answers
68 views

Is Problem Solving In Maths reasoning from previously found results ? Or is there something else I'm missing?

Don't get me wrong, I don't like to see hard to answer, ambiguous, subjective questions brought up either, but I could really use some advice. My problem is the following: If I can't solve problems ...
1
vote
0answers
114 views

Probability of two persons to sit next to each other in a plane

Two weeks ago I came up with the following probability problem (maybe a similar one already exists): A plane has $60$ seats, disposed by groups of $3$ seats, $2$ groups per row ($10$ rows): $$ \text{...
0
votes
1answer
30 views

A random trigonometry problem

I am a graduate student, I was solving some high school problems for a student of mine. I am stuck on this one for quite a long time. (Any high school problem solving technique is usable) $\tan^2 (Z) ...
6
votes
2answers
792 views

Method for finding efficient algorithms?

TL;DR What can you recommend to get better at finding efficient solutions to math problems? Background The first challenge on Project Euler says: Find the sum of all the multiples of 3 or 5 ...
0
votes
2answers
52 views

How do you interpret the negative solution?

A rectangle is such that its area is $308 \,m^2$. Its length is $8\, m$ longer than its width. What is its length? Let $x$ denote its width. Its length is then $x+8$ and $x$ satisfies \begin{...
1
vote
0answers
67 views

Vakil FOAG Exercise 11.3.c part b

I am trying to do exercise 11.3.c part b from Vakil's note. It asks to show that if $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension $r$ and $Y$ is a closed subset of codimension $r$, then they ...
2
votes
2answers
99 views

Finding the last 4 digits of a huge power [duplicate]

I know this is more of a 'aops' type of question but here we go, I went to this math competition last year and there was this one problem that clearly I didn't solve but it recently came back to my ...
-5
votes
2answers
69 views

Apply Law of Exponents

Given $a^4+a^3+a^2+a+1=0,$ find the value of $a^{2016}+a^{2015}+1.$ Can anybody help me find the answer and teach me how to get it?
0
votes
1answer
32 views

Interpreting solutions to spring-mass ODEs.

I have the following spring-mass ODE solutions: $$1:\;\;\;x(t)=-3\sin(2t)+4\cos(2t)+12t\sin(t)$$ $$2:\;\;\;x(t)=6e^{-t}\cos(3t)-3e^{-t}\sin(2t)+40\sin(7t)$$ How is is possible to figure if each one is ...
7
votes
1answer
103 views

When does $x^{x+1}= (x+1)^x$?

After seeing the problem of which is bigger out of $9^{10}$ or $10^9$ (and eventually working out a few ways to answer that) I got interested in where it switches. I.e. $2^3$ < $3^2$ But: $3^4$ >...
1
vote
1answer
42 views

How many ways to get the terms of sequence 1-100 sum up to 9?

I stumbled upon this in a binomial expansion problem where I was supposed to find the coefficient of $x^9$ in $$\prod_{i=1}^{100}{(1+x^i)}$$ The problem is simple enough, all I need to do is to find ...
0
votes
1answer
32 views

How to use algebra to solve this?

Use Algebra to solve for $x$: $8(9^x)+3(6^x)-81(4^x)=0$ My attempt was to decompose the exponential functions into $2^x$ and $3^x$ where possible and then substitute them for some $p$ and $q$. I was ...
-1
votes
1answer
58 views

How to guarantee entry into cave [closed]

Ali Baba is trying to enter a cave. At the entrance, there is a drum with four openings, in each of which there is a pot with a herring inside. The herring may be lying with its tail up or down. Ali ...
0
votes
1answer
28 views

Find how many payements will I need to do and what is going to be the final amount.

This is a financial mathematics problem: On time $0$ we have to pay $297505.48$ of a loan. Assuming a capitalization interest of $8\%$ and a fixed annual payment of $49623.55$, how many payments will ...
0
votes
0answers
32 views

A question about profit and loss

A shopkeeper allows a discount of 20% on the marked price but charges 5% sales tax on the marked price and 5% service tax on the discounted price. If the customer pays Rs. 2670 as price including ...
5
votes
0answers
59 views

Network Connectivity Problem

Assume a network (a complete undirected graph) comprised of $N$ participants (vertices). Each person holds a different piece of information, unknown to all other members. The participants can ...
0
votes
0answers
93 views

how to solve this profit and loss problem

Xavier buys an apple from Yahoo for Rs.5 and sells it to Zoe for Rs.10. He later buys the apple back from Zoe for Rs.12 and sells back to Yahoo for Rs.15. What is his profit over venture?
1
vote
0answers
34 views

Problem for elementary school -find letter a number such that - $SEND+MORE=MONEY$ [duplicate]

Problem: Find the number for every letter(different letter is a different number) such that this equality holds. $$SEND+MORE=MONEY$$ My solution: Because idea of this problem isn't to solve it's to ...
1
vote
0answers
35 views

Find the argument

Find the argument $$\begin{align} &A. \left(\frac{\sqrt{3}}2+\frac i2 \right)^7 \\ &B. (11 − i 11\sqrt{3})^9 \end{align}$$ I get $\frac{7\pi}6 +2kπ$ for A. I just want to make sure that ...
0
votes
1answer
29 views

Regarding algebric manipulation in order to find $a_0^2 - a_1^2 + a_2^2 + … a_{2n}^2$

In the problem below I am unsure about the validity of the manipulation in Step 3 of the solution. Tldr : How is $(1/x^2+1+x^2)^n = 1/x^{2n}(1 + x^2 + x^4)^n$? (I plugged in small numbers and it ...
1
vote
1answer
49 views

$ (5+2\sqrt6)^{\sin x} +(5-2\sqrt6)^{\sin x} = 2\sqrt3 $ , where $ 0 ≤ x≤ 360 $

There is something I haven't picked up on, a hint would be appreciated Given that $(\sqrt3+\sqrt2)^2 = (5+2\sqrt6)$ and $ (\sqrt3-\sqrt2)^2 = (5-2\sqrt6)$ Find the values of x for which$ (5+2\sqrt6)...
0
votes
1answer
41 views

Given a vector between points $A$ and $B$, how determine the coordinates of point $C$ when $AC$ is collinear to $AB$ and we know the length of $AC$?

Say we have a vector $\vec{v}$ that defines the chemical bond between two atoms, and whose components are known $$ \vec{v} = \begin{bmatrix}v_1 \\ v_2 \\ v_3 \end{bmatrix} $$ Lets define another ...
11
votes
0answers
365 views

Every 4-regular simple graph contains a 3-regular subgraph

The following result was conjectured by Berge & Sauer, and proved by Tashkinov [T]. Theorem A. Every 4-regular simple graph contains a 3-regular subgraph. A simple graph is the one with no ...
0
votes
0answers
50 views

How to solve a $10$ degree Polynomial? (without using the normal tricks)

The problem specifically is a $x^{10}$ polynomial over $x^9$ polynomial that we have to graph. To graph, we need to find all the vertical intercepts so both polynomial need to be factored. Is there ...
-4
votes
1answer
38 views

Express the equation $\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$ as a rational number. [closed]

$$\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$$ My first thought was to find the sum of this equation using induction to find a general solution that could bring this sum to a rational ...