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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4
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0answers
130 views

Puzzle - zero knowledge proof

I am solving the following problem : I have edge-matching puzzles, where all pieces are squares and the grid has $n$*$n$ format. There is no global image to guide a puzzle solver. Despite the puzzles ...
1
vote
1answer
72 views

Solving an equation involving floor/ceiling as a summation bound

Is it possible to solve the following equation for $\alpha$? $$ M = \lfloor \alpha \rfloor + \alpha \sum_{k=\lfloor \alpha \rfloor +1}^N \frac{1}{k}$$ where $\alpha \geq 1$. Intuitively, $M$ is a ...
0
votes
2answers
57 views

What are the ages?

A man taking the census walks up to the apartment of a mathematician and asks him if he has any children and how old they are. The mathematician says: "${\it\mbox{I have three daughters and the ...
1
vote
1answer
77 views

Show that there is a number between 1 and 1000 such that there is a perfect square

Show that there exists an integer $n \in S = \{1,2, \ldots, 1000\}$ such that $$\prod_{i\in S-\{n\}}i!$$ is a perfect square. I was thinking in trying to prove it by contradiction using the ...
0
votes
2answers
42 views

Find the value of $\theta$ in a trigonometeric problem

$2 \sin{2^\theta} = 1-\cot^\theta$ $2 \sin^2 \theta = 1 - \cot \theta$. Find the value of $\theta$.
3
votes
1answer
43 views

Residues computation when we need power series

I'm trying to compute residues in situation where we need to manipulate power series to get it, but I can't find a good way. Indeed for the sake of example, consider the residue of the following ...
1
vote
0answers
77 views

100th degree polynomial $P(2^k)=k$ for $k=0,1,…100$

I sometimes see this kind of question, but I completely forgot how to solve it. Could anyone solve it for me?
0
votes
1answer
225 views

Finding dy/dx as a function of x for a dog-walker dragged by a dog travelling in a straight line

Hello. I was wondering if anyone could provide some insight into how to solve the following Calculus word problem: Max is walking his dog Beau in the Cartesian plane, with the leash between them at ...
1
vote
1answer
39 views

Function with invariant area under curve

I'm trying to find a function $f$ that fulfills the following property: The area under the curve starting at some point $x_0$ with a width of $x_0$ should always be the same for all $x_0$. In other ...
1
vote
2answers
55 views

Putnam-Style Sequences Problem

Let $S_1$ denote the sequence of positive integers $1,2,3,4,5,6,\ldots,$ and define the sequence $S_{n+1}$ in terms of $S_n$ by adding $1$ to those integers in $S_n$ which are divisible by $n$. Thus, ...
2
votes
1answer
58 views

Fair Division: Making the Differences in Players' Valuations Believable

When teaching basic fair division algorithms, the students always propose some simple and (at the first glance) correct solutions for $n$ players, which unfortunately are not correct! The only way I ...
7
votes
3answers
171 views

If $x_1, \ldots, x_6$ are positive real numbers that add up to $2$. Show that:

If $x_1,x_2,x_3,x_4,x_5$ and $x_6$ are positive real numbers that add up to $2$, then: $$2^{12} \leq \left(1+\dfrac{1}{x_1}\right) ...
2
votes
1answer
52 views

Prove that $a=2,b=2,c=2$ for a system of three equations.

Let us consider a system equations : \begin{gather} a^3+b=3a+4\tag{i}\\ 2b^3+c=6b+6 \tag{ii}\\ 3c^3+a=9c+8\tag{iii} \end{gather} I have tried with the following steps : ...
0
votes
1answer
43 views

Norm, limit, and max norm

For any $\vec{u} \in \mathbb{R}^2$ prove that $$\lim_{p\to \infty} \|\vec{u}\|_p = \max (|u_1|, |u_2|)$$ Then, when $p\to\infty$ we get $||\vec{u}||_{\infty}$. And we get the max norm, ...
0
votes
1answer
36 views

Dynamic problems

A natural number n represents the initial position in the game. When it is a players turn he/she is allowed to ...
1
vote
2answers
54 views

A non-equality and an inequality involving $y$ and $y_0$ from Spivak Calculus 4th ed.

It's Problem 22. from Chapter 1. I'm given: $y_0 \neq 0$ $|y - y_0| < \frac{|y_0|}{2}$ $|y - y_0| < \frac{\epsilon|y_0|^2}{2}$ and I must use them to prove that: $y \neq 0$ $|\frac{1}{y} - ...
4
votes
3answers
1k views

What does it mean to solve a math problem analytically?

I'm reading a Calculus book for my own edification and at the beginning the pre-calculus introduction has the problem, $3x+y=7$ They talk about solving the problem graphically, analytically, and ...
0
votes
2answers
41 views

How to solve this nonstandard equation?

How to solve the equation $$\mathrm{e}^x + (x^3-x)\cdot \ln(x^2+x+2) - \mathrm{e}^{\sqrt[3]{x}}=0?$$ I tried. We have $x=1$ is a root of the equation. If $x>1$, $x > \sqrt[3]{x}$, therefore ...
0
votes
1answer
50 views

Help solving a problem with inequalities with absolute values

I have these statements presented: $|x - x_0| < \frac{\epsilon}{2(|y_0| + 1)}$ , $|x - x_0| < 1$ , $|y - y_0| < \frac{\epsilon}{2(|x_0| + 1)}$ And I must prove that: $|xy - x_0y_0| < ...
3
votes
3answers
1k views

how does the word math represent a four digit number

I just don't get it had it on a test can you help me? It was a test for math olympiads this is how the question looked exactly: ...
0
votes
1answer
81 views

In a party with 2000 persons, determine # of people who know everyone

In a party with 2000 persons, among any set of four there is at least one person who knows each of the other three. There are three people who are not mutually acquainted with each other. How many ...
0
votes
1answer
116 views

Prove using Jensen's Inequality

Let $\alpha_1, \alpha_2, . . . , \alpha_n$ be the interior angles of a convex (but not necessarily regular) n-gon. Prove, that for all integers $n\geq3$: $$\cos \alpha_1 + \cos \alpha_2 + \cdots + ...
7
votes
2answers
137 views

Continuous functions on $\mathbb{R}^2$ with special property

The following problem is from Miklos Schweitzer competition (Year 1983, Problem 7): Prove that if the function $f: \mathbb{R}^{2}\to [0, 1]$ is continuous, and its average on every circle of ...
0
votes
2answers
69 views

Prove completing the square

Prove that: $x + y + xy - x^2 - y^2 \leq 1$ If I use $-x^2 + xy - y^2$ to start completing the square, I get: $x + y -((x+y)(x-y)) - xy$ I am confused on how to keep going.
0
votes
1answer
45 views

Help understand Schwarz Inequality soqlution from Spivak's Calculus

I'm given this inequality from Spivak's Calculus book: And I need to do this: Here is the answer from the answer book: I am completely unsure what to do.I have $x_1, x_2, y_1, y_2$ in the ...
0
votes
2answers
40 views

Problem with finding “x” in triangle

I have got a problem with finding the x. I think the question isn't true or there should more informations on it.
1
vote
2answers
281 views

Funções/Sequências (Functions/Sequence)

Em Português: Seja $n$ um natural fixado. Dizemos que uma sequência $(x_1 , ..., x_n)$ tal que $x_j \in \{ 0,1\}$ para $1 \leq j \leq n$ é aperiódica se não existir divisor $0 < d < n$ tal que ...
0
votes
1answer
55 views

How to solve $\text{ constant} = \sin(2*\theta)\;?$

What would you do to solve $0.587 = \sin(2\theta)$? I know that this question is rather basic, but I've had no luck trying to find answers online. I was wondering if $\sin$ could be replaced by ...
1
vote
1answer
48 views

The Average Speed of an object

I'm pretty sure this has more to do with fundamental Math than Physics and that is why I'm asking this here rather than Physics.SE Imagine some object travelling along a straight path from point $A$ ...
3
votes
2answers
270 views

Expected number of points on circle to form an acute angled triangle

This problem was asked to me in an interview. We keep on adding points on a circle uniformly until there exist three points on the circle which form an acute angled triangle. What is the expected ...
0
votes
3answers
48 views

I reach a half-dead end when trying to find the minimum possible value of an equation

I have the equation $x^2 + 4xy + 5y^2 - 4x - 6y +7$ and I'm supposed to transform it to look like this: $[x + 2(y - 1)]^2 + (y + 1)^2 + 2$ First I transformed it into: $x^2 + 4x(y - 1) + 5y^2 - 6y ...
0
votes
2answers
103 views

How do you find the smallest possible value of aan equation with two unknowns?

I'm solving a list of problems where I'm given an equation and I find the smallest possible value by comparing the equation to a quadratic equation and completing the square, however the next one ...
0
votes
1answer
40 views

Extraneous solutions to simple equations

I had an interesting thought during my procrastination: is it legal to take an equation, say $3 = a * b * c$ and do the following: $3 = abc$ $0 = abc - 3$ $0 / a = bc - 3/a$ $0 / b = ...
0
votes
2answers
107 views

Solving a system of equation

Solve the following system of equations: $\left\{ \begin{align} & 2{{x}^{2}}-5xy-5{{y}^{2}}+x+10y-35=0 \\ & {{x}^{3}}+5x{{y}^{2}}+42=0 \\ \end{align} \right.$ By using a computer, I have ...
1
vote
0answers
117 views

solving a system of 2 non-linear equations with 2 unknowns

i'm trying to find a solution for these two equations, $p$ & $q$ are variables and $c$ is known constant (it's given randomly) : $$ \begin{align} ...
0
votes
1answer
44 views

Question about one of the first problems in Spivak's Calculus

It's about Chapter I, Problem 21 from Spivak's Calculus: Prove that if: |x - x$_0$| < $\frac{\epsilon}{2}$ and |y - y$_0$| < $\frac{\epsilon}{2}$ then |(x + y) - (x$_0$ + y$_0$)| < ...
0
votes
0answers
124 views

Gauss' Summation Trick; Applications and Generalizations

I'm going to write an article about the summation trick attributed to Guass and its applications and generalizations. I'm sure you know what is the trick I mean: $1+2+\cdots+100=101+101+\cdots+101$ ...
1
vote
4answers
230 views

$xy=22$ and $yz=26$: What is $x+y+z $ equal to?

Given the following: $$xy=22,\qquad yz=26,$$ where $x,y,z\in\mathbb{N}$. Which of the following is a possible value of $ x + y + z $? $ \textbf {(A) } 22 \qquad \textbf {(B) } 24 \qquad \textbf {(C) ...
1
vote
1answer
167 views

Proving Goldbach's conjecture (hypothetically)

Part $1$. If $\pi(n) \sim \frac{n}{\ln(n)}$ by the prime number theorem, can we treat $\frac{1}{\ln(n)} $ as the probability that a number less than $n$ is a prime number? Say we have some operation ...
0
votes
1answer
198 views

If we draw infinitely many lines on a table, can we find a triangle somewhere? [closed]

If we draw infinitely many lines on a table, can we find a triangle somewhere? We prove that there is a subgraph $C_3$ in $C_n$, which will be called a triangle. Suppose we have an infinite ...
1
vote
2answers
77 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
0
votes
2answers
88 views

How to show the existence of a number with certain divisibility conditions between two multiples?

How can we show that between two even natural numbers they're exists a natural number that isn't even? How can we show that they're exists a natural number that is odd and not divisible by 3, between ...
6
votes
5answers
213 views

How can I express the sum of $\sin a+\sin2a+\sin3a+\cdots+\sin(n-1)a$?

I want to sum up the partials of a harmonic series, how do I do it? If I was using the 'Lagrange trigonometric identity to solve this problem', how would I plot it on Wolfram mathematica (using which ...
1
vote
1answer
64 views

English question regarding pigeonhole principle classic question.

Mr. and Mrs. Smith invited four couples to their home. Some guests were friends of Mr. Smith, and some others were friends of Mrs. Smith. When the guests arrived, people who knew each other ...
0
votes
1answer
41 views

conditional probability Pc(B)

I am looking for the probability of Pc(B) where the event of B={no two people are born in the same month} and event C= {exactly three people were born in the summer of june, july august} and there are ...
1
vote
0answers
33 views

probability of an event question

Find the probability of the event B= { NO two people are born in the same month}. There are 9 people involved by the way. Is it 1/12 * 1/11 * 1/10 * 1/9 * 1/8 * 1/7 * 1/6 * 1/5 * 1/4 * 1/3 * 1/2 * 1/1 ...
0
votes
1answer
42 views

How to best solve a system of equations like this one

I need some help trying to solve a system of two equations with two unknowns. Background: This is not homework, just hobby. I have some LEDs that I needed to identify, but the manufacturer datasheet ...
1
vote
1answer
72 views

Question about the matrix representation of the differentiation map on the subspace generated by $\{1, t, e^{t}, e^{2t}\}$

As mentioned in a previous post (I think), I've been trying to learn some linear algebra, and so I've begun to post little questions whose answers I'm sure are obvious to most here; this is just a way ...
0
votes
1answer
53 views

Pigeonhole Principle to solve question straightforward

A store wants to celebrate its anniversary and will give a $200 shopping certi cate to the first customer to enter the store whose birthday is the same as that of two other previously admitted ...
0
votes
0answers
27 views

The product of $i$ consecutive natural numbers is divisible for $i!$ [duplicate]

There is a theorem that I've used it a few times, and never saw a demo of it, and when I tried, I could not, commenting with a teacher, it would not give me much attention and said it would use the ...