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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writing numbers as sum of at least two consecutive odd positive integers

Since 24 = 3 + 5 + 7 +9, the number 24 can be written as the sum of at least two consecutive odd positive integers. (a) Can 2005 be written as the sum of at least two consecutive odd positive ...
0
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1answer
35 views

Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than $1$.

Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than $1$.For example, $22$ and $15$ are relatively prime, and thus $37 = 22+...
1
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1answer
19 views

Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors

Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors. (The proper divisors of a positive integer n are all positive integers ...
2
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0answers
6 views

How to find the average Kendall's distance between 2 rankings

Suppose I have 2 rankings: $1$, $2$, $3$ and $2, 1, 3$ then the Kendall's distance between the two is 1 since there is only one pairwise adjacent switch. My question is, suppose my 2 rankings each ...
2
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3answers
611 views

Solve $x^4+3x+20=0$ by Ferrari's method

Comparing the equation $$x^4+3x+20=0$$ With the equation $$(x^2+\lambda)^2-(mx+n)^2=0$$ we get $m^2=2\lambda,$ $-2mn=3,$ $n^2=\lambda^2-20$ Now, $4m^2n^2=9\Rightarrow 4(2\lambda)(\lambda^2-...
1
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1answer
402 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? $...
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0answers
27 views

Velocity time and length problem [closed]

A person walks from his house to the park and back in $2.5$ hours. On his way to the park, he walked at a rate of $6$ km/h, then on his way back he walked at a rate of $4$ km/h. Find the distance ...
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2answers
34 views

Solving an equation

I have the following equation: $x_1^3 = \hat{x}_1^3 + e_1\delta(x_1,e_1)$ I have to find the function $\delta(\cdot)$ for which this equation holds. By definition: $e_1 = \hat{x}_1 - x_1$ So I am ...
-1
votes
1answer
28 views

solving equation using square root

I have a question here... Usually, for $x^2 = 4$ $x=\sqrt{4}$ $x=±2$ But if the question is like this : $y^2 = (x+2)(x+2)$ $y^2 = (x+2)^2$ If I want to find $y$ in term of $x$,I will put square root ...
3
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0answers
67 views

Does this equation have no solutions?

The question is this : The source from where I got this question was devoid of any answers to it, so I came here, this is how I proceeded : LHS : $((((({(x)^x})^{2x})^{3x})^{....x^2})^2 = (((((x)^...
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0answers
54 views

Multiple choice excercise with more than one answer correct

Q. Consider the function $$F(z)=\int_{1}^{2} \frac {1}{(x-z)^2}dx, {\text {Im}(z) \gt 0}$$ Then there is a meromorphic function function $G(z)$ on $\Bbb C$ that agrees with $F(z)$ when ${\text {Im}(...
-1
votes
2answers
57 views

Probability problem

I created this problem based on the following probability riddle here. You're a king, and you were given two groups of people, and a certain information about them. First group has 2 people. One of ...
1
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1answer
53 views

Differential equation, Solution is a Bessel fucntion

this is my first post here. I knocked my head on a differential equation yesterday, this one: $$ \frac{12 \nu}{x^2} \frac{S(x)''}{S(x)} = -\lambda^2 $$ Where $nu$ is a constant. The book says the ...
0
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1answer
45 views

Minimizing a strictly convex function with inequality constraint

So we've been learning about the Kuhn Tucker conditions in my non-linear optimization course and I've been having trouble with this problem: QUestion: description here Question: a strictly convex ...
3
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2answers
63 views

Homeomorphism from $S^1\backslash(0,1)$ to $\mathbb{R}$

I am trying to derive a bijection between $S^1\backslash{(0,1)}$ and the real line, but I am stuck on using the most obvious way Let the top point of the circle be $(0,1)$, and the blue line hits ...
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0answers
33 views

Bayesian posterior probability [closed]

Let's say that you are in a casino and you have played on 3 different slot machines following this flow: Slot machine A, play 10 times, win 2 times for a total of 2$ Slot machine B, play 100 times, ...
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2answers
2k views

Problem Solving - Ways to add 6 even, positive, non-zero integers to get 26

I believe I have gotten all of the ways now - thanks for the hints below Yun, Andre Nicolas, and Gerry Myerson. If anyone could confirm my answer (I feel there should be more possibilities, but for ...
2
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1answer
28 views

Fox and Gemstone problem. Weighing stones to compare bags.

There is a problem called Fox and Gemstone at topcoder: https://community.topcoder.com/stat?c=problem_statement&pm=14296 Basically, you have the ability to weigh individual gemstones against ...
0
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1answer
65 views

Find the Smallest Value

Find the smallest value of $$a + \frac {1}{(a-b)b} $$ where a>b>0 I found this question in AM-GM inequality problems but I am stuck at this
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1answer
445 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 ...
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2answers
68 views

proving no real roots exist

Prove that $x^8-x^7+x^2-x+15$ has no real roots. I did it by first assuming it has real roots and then applying Descartes rule of signs. We find that if there are any real roots, they all must be ...
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1answer
30 views

find the number of tuples of positive integers [closed]

find the number of tuples (a,b,c,d) of positive integers \begin{array}{l} {a^3} = {b^2}\\ {c^3} = {d^2}\\ c - a = 64 \end{array} answer should be one of 0 , 1 , 2 , 4
0
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1answer
29 views

number theory problem finding triplets [closed]

Find number of triplets of positive integers satisfying $2^a-5^b\cdot 7^c=1$ Given options are $0 , 1 , 2$ or infinite.
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1answer
46 views

How to find the area shared by 4 quadrants inside a square?

I was to find the blue area in this question : As described about how it's a square with 4 quadrants of same radius intertwined with each other, now to find the blue part area I thought about ...
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0answers
31 views

How to study for open book exams?

I hope it's ok to ask about exams here? If not please tell me. This year was the first time I had open book exams, varying from three to six hours long. The six hour ones go fine because I have time ...
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2answers
107 views

What is the size of the angle $\angle AMC$? [duplicate]

Suppose we have a triangle $\triangle ABC$ where the size of two angles are given: $\angle B=15^\circ$ and $\angle C=30^\circ$. We draw the median $AM$, so now what is the size of angle $\angle AMC$? ...
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votes
2answers
52 views

Find the solution without “common sense” assumption [closed]

Is it possible to solve the following problem without make a "common sense" assumption? In the year 1887, one person's age was exactly the sum of the digits of his birth's year. What was the person'...
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0answers
68 views

Polynomial taking irrationals to irrationals

Problem: Find all polynomials from $\mathbb{R}\to \mathbb{R}$ $f$ with integer coefficients taking irrationals to irrationals. My attempt: It is clear that the problem statement is equivalent to ...
2
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3answers
71 views

Let $f$ be a differentiable function defined for all real $x$, where $f(x)\ge 0$ for all $x\in[0,a]$

Let $f$ to be a differentiable function defined for all real $x$, where $f(x)\ge 0$ for all $x\in[0,a]$.If $$\int_0^a f(x)\,dx = a, $$ then $$2\int_0^{5a}\left(f\left( \frac x 5 \right) + 3 \right)\,...
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1answer
36 views

Topology: What is a quick way to check whether a subset $D$ is dense in $(X, \mathcal{T})$?

Def $1$: Let $(X, \mathcal{T})$ be a topological space, then $D \subseteq X$ is dense if $\overline {D} = X$ Def $2$: $x \in \overline D$ iff for all $U \in \mathcal{T}, x \in U \implies D \cap U \...
0
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1answer
70 views

Simplifying $(m^3)^4(2x^3)^7(m^2)^5(3x)^2$

I am having trouble with simplifying the following algebraic expression: $$(m^3)^4(2x^3)^7(m^2)^5(3x)^2$$ I have been able to do the exponents and all the other equations I needed to simplify but ...
3
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1answer
91 views

Divide a 2D polygon with N vertices into triangles draw in a 3D space

I'm developing a C++ software and I have a problem with a polygon with N vertices. I have a set of N vertices unordered. This vertices describe an polygon. I'm developing a planetarium and I want ...
0
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1answer
15 views

How to find the seeding number for acheiving the minimum cost?

Consider there are n integers. I have to pick one Random integer 'R' and i have to subtract 'R' with each element of the 'N' integers. Result has to be added. Result that i will achieve should be the ...
2
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1answer
114 views

Please help me solve this $x(\sqrt{2x+5}+\sqrt[3]{7x+13}) = 3x+6$

Wolfram Alpha shows that the answer is $x=2\,$ and $x=-2\,$ but what would be the best way of simplifying this equation ? It has been many years since I was in school , and I just cannot wrap my head ...
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2answers
45 views

How do you deduce the integer whose multiplicative inverse decimal has a digit sequence or repetend length of 3 digits?

A positive integer's, n, reciprocal, $\frac{1}{n}$, in which the decimal's repetend has a length of three digits which starts at the decimal mark. e.g. 0.037037... of the integer, 27 ,reciprocal $\...
1
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1answer
57 views

How to predict the incidents of synchronization for multiple oscillations.

EDIT: I changed the title of this question and made this edit based on a conversation with a friend. While I am dealing with mechanical cams the plain fact is that what I have is an oscillation in ...
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2answers
76 views

Why we not check conditions while solving questions?

Note:Down ward problem is just an example to express my question(I know the both solution of problem are insufficient but the first solution is in my 10+2 book and second one is mine which is ...
3
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2answers
76 views

What type of functional equation is this?

I'm trying to solve the following functional equation $f\left(x\right)=A\mbox{ exp}\left\{ \int\frac{1}{f\left(x\right)x^{2}+Bx}dx\right\}$ where $f\left(x\right):\mathbb{R}_{+}\rightarrow\mathbb{R}...
0
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1answer
19 views

Two people running up and down escalators

Couldn't really figure out a good title for this post, I am sorry. But here's the problem: PREMISE: Two people, A and B, are running, with speed u, up and down separate escalators with length L. ...
1
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1answer
40 views

Models of Comprehension Schema

Let $M_\alpha$ for $\alpha\in ON$ be transitive sets and let $M=\bigcup_{\alpha\in ON}M_\alpha$. Suppose that (i) for every $\alpha<\beta$, we have $M_\alpha\in M_\beta$ and (ii) for every limit $\...
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2answers
94 views

How does one solve this kind of equation: $3^x=x+3$

How does one solve this kind of equation: $$3^x=x+3$$ I tried playing around with logs but it didn't get me anywhere. I plotted the two functions $f(x)=3^x$ and $g(x)=x+3$ on a graph to estimate the ...
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0answers
34 views

How often does theorem equivalence take place?

I remember reading once that it was found that two math theorems were essentially equivalent to each other, how often does this occur? ex. In two dimensions the divergence theorem is said to be ...
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1answer
18 views

Finding the set of values for k of a modulus function.

"Find the set of values of k for which |(x-4)(x+2)| = k has four solutions." EDIT: Ok so I thought I'd start with setting the modulus function equal to k and -k to get the two set of results. Doing ...
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1answer
15 views

Choosing pairs of numbers with distinct bounded sum

Stumbled across this problem in the list of examples for high school math exam, paraphrasing a little: ...
3
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1answer
148 views

Prove that $(a-b)^n\mid (a^n-b^n) \iff n=1$ under given conditions

Suppose that $a,b,(a-b)$ are pairwise co-prime (i.e. $a\perp b\perp (a-b)\perp a$), and that $\frac{a}{2}<b<a$, where $a$ and $b$ are both positive integers greater than $2$. Let $n$ be odd. ...
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0answers
48 views

Euclidean algorithm: THE GAME OF EUCLID

In his book "Elementary Number Theory:A Problem Solving Approach" (Euclidean algorithm/Derived Sets/ first chapter page:17,19) Joe Roberts describes a number: $$\tau=\frac{1+\sqrt{5}}{2}$$ further he ...
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2answers
2k views

Folding a rectangular paper sheet

You are given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of ...
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0answers
44 views

Numerically solve an equation using Python

How could I (numerically) solve this equation for $\alpha$ given $x_i$ (these are known) ? $\sum_{i=1}^N\frac{1}{x_i-\alpha} = \frac{2N}{\sum_{i=1}^{N}(x_i-\alpha)^2}\sum_{i=1}^{N}{(x_i-\alpha)}$ In ...
3
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1answer
35 views

Using the Weak Law of Large Numbers for a product or random variables?

I need to calculate the average of the following quantity: \begin{equation} S_n=\prod_{i=1}^nS(X_i) \tag{1} \label{eq:1} \end{equation} with $S(X_i):=o_{X_i}b_{X_i}$, where each $X_i\in \mathcal{X}=\...
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3answers
246 views

Trouble with Vakil's FOAG exercise 11.3.C

I'm having trouble with the exercise in the title, even with part (a), which asks to prove that if $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least 1 and $H$ is a non-empty ...