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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32 views

Trying to use the deformation theorem to solve integral

I have this integral: $$\int_{|z|=2}\frac{\cosh z}{(z+1)^3(z-1)}dz$$ Both singularities $z=1,z=-1$ are inside the circle. I have already solve this using partial fractions, and I don't have much ...
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1answer
28 views

How does $2N_{h-2}$ become $2^{h/2}$?

I'm reading the Lecture 6 notes from MIT OCW Introduction to Algorithms, which discusses AVL trees, and I'm confused about one of the relations below: Balance: Worst when every node differs ...
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0answers
37 views

Proof Verification: If 650 points in a circle of radius 16, prove that some 10 must lie in a ring of inner radius 2 and outer radius 3.

If 650 points in a circle of radius 16, prove that some 10 must lie in a ring of inner radius 2 and outer radius 3. The area of any such ring is $5\pi$ and the area covered by the union of all rings ...
2
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2answers
72 views

Solve this equation. Can anybody do it?

There is this relation between x and y: $$x / y = a + b \log(y)$$ I have x. How do I ...
1
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1answer
25 views

How to calculate per unit costs for multiple items

I had a supplier give me a quote last week that seems very strange, can someone help me out? The quote is for IT hardware, but for simplicity (and anonymity) I'll use apples and oranges: ...
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2answers
62 views

I still forget concepts even after answering numerous math problems

Note: this is particularly aimed at high-school/entry level college problems When I'm learning a new topic: 1) I read the theory given in the textbook at the start of each topic 2) proceed to read ...
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2answers
113 views

Solving the exact differential equation y'=(x-y)/(x+y)

I need to solve the following exact equation: $y' = \frac{(x-y)}{(x+y)}$ I've been taught to put those in the form $M(x, y)dx + N(x, y)dy = 0$ and to make sure ${dM}\over{dy}$ = ${dN}\over{dx}$ So ...
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21 views

Explanation for a simple comparison

Ok, Yesterday I started to learn how to solve problems with comparisons, but I couldn't understand one thing of the "solve algotithm". Here is a part from a solve from a simple example problem ...
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1answer
36 views

Solving $Ae^x=Bx$ analytically, where $A$ and $B$ are constants?

This equation mixes both exponential terms and linear terms, something which I do not know how to deal with. Any pointers?
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1answer
204 views

equations solved with Newton's method by finding the zeros of functions?

I found this statement in one paper I read recently: This problem can be solved by finding the zero of functions: ...
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3answers
130 views

Find all real solutions of $6^x+1=8^x-27^{x-1}$

Find all real solutions of $6^x+1=8^x-27^{x-1}$. Things I tried: We want solutions of $$2^x3^x+1 = (2^x)^3-\frac{(3^x)^3}{27}.$$ Write $a=2^x$ and $b=3^x$. This gives $$ab+1 = a^3-\frac{b^3}{27}$$ or ...
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1answer
39 views

Solve for “lucky” numbers

A rational number is called "lucky" if it equals both $a+\frac{b}{c}$ and $a\times\frac{b}{c}$ for some positive integers $a,b,c$. How many lucky numbers are there between $5$ and $10$? Here's what I ...
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2answers
57 views

Question involving square equality between fractions and square roots [closed]

Find the values of the constants $p$ and $q$ such that $$\frac{\sqrt{p}}{\sqrt{p}+2p} = \frac{2\sqrt{p}-q}{3p+q} \tag{$p,q\ge0$}$$ How would you solve this? I've tried everything...
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0answers
54 views

Vieta jumping with non-monic polynomials

I have recently discovered Vieta jumping as a problem-solving technique. In order to teach myself about it, I have located most (all of?) the standard references, both here on MSE and "out there" (via ...
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2answers
24 views

Finding the number of solutions satisfying an equation?

Given one condition $x_1+x_2+x_3=n$ where n is known number. Given a set of data X={$a_1,a_2....a_n$}. Can you help me find all possible cases satisfying the above condition $x_1+x_2+x_3=n$ ???
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2answers
33 views

Solving two systems with two unknown?

Let's say if we are giving the following two equations: $$ 1= X/(X^2 +Y^2) $$ $$ 2= Y/(X^2 +Y^2) $$ How are we going to solve for X and Y [ by HAND ] ? Why would Summing the squares of the two ...
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4answers
47 views

Challenge: “Dividing” a number above 0 and ending up with the same, or a greater number (creative task)

Here's a question/challenge for those of you who know quite a bit about math, or enjoy to be creative with what you do know (just for reference: I'm virtually illiterate when it comes to any math more ...
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5answers
70 views

$|x| + |x-1| = 3$ how come its cases?

$$|x| + |x-1| = 3$$ in my textbook, they say that for this equation, there are 3 cases: $x\geq1$, $0 \leq x < 1$ and $ x < 0$ where do these come from and why? i thought, there are 4 cases ...
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2answers
29 views

$5-3|x-6|\leq 3x -7$

I have this inequation: $$5-3|x-6|\leq 3x -7$$ i solved this this way: i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - : 1) for $x\geq6$ ...
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1answer
23 views

Word problem with $p$

In the year 2000, there are $p$ penguins. After $t$ years, the number of penguins is given by $$ 2500 \times 1.02^t$$ Calculate the number of penguins in the year $2000$. I tried to substitute random ...
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1answer
29 views

Manipulating series

I have come across this in a solution for a BMO problem where you have to find $a_{2013}$ for: $a_n$ = $\frac{n+1}{n-1}$($a_1 + a_2 + ... + a_{n-1}$) where $a_1$ = 1. It says that you manipulate it ...
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1answer
38 views

Find a graph with at least two vertices and no self-loops in which all vertices have different degrees

I am an high-school senior interested in Graph Theory, on a web forum a CS teacher teased me with ("an easy but non-trivial") a terrific Graph Theory problem: Find a graph with at least two ...
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2answers
61 views

Olympiad Modulo Problem

I have begun preparing for the British Mathematical Olympiad and hope to do well. However, I have been working on the first problem in the book: A Mathematical Olympiad Primer by Geoff Smith, captain ...
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3answers
811 views

Improving concentration and stamina when solving difficult problems.

I am trying to improve my problem solving skills by solving olympiad problems (Putnam, IMO, etc). So far, I have discovered that problem solving is somewhat like panning for gold: you think of all the ...
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1answer
18 views

Vector force application problem

I'm having trouble starting off this question. Any help would be appreciated! "Lisa is trying to hold on to her toy car. Her sister Ruby is pulling with a force of 8 N on a bearing of 023° and her ...
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2answers
120 views

why is $\int_{\pi/2}^{5\pi/2}\frac{e^{\arctan(\sin x)}}{e^{\arctan(\sin x)}+e^{\arctan(\cos x)}}=\pi$?

I cannot make progress on the definite integral $$\int_{\pi/2}^{5\pi/2}\frac{e^{\arctan(\sin x)}}{e^{\arctan(\sin x)}+e^{\arctan(\cos x)}}\,dx=\pi$$ I know the result is $\pi$ from numerical ...
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1answer
67 views

Tricky Integral equation - where to start?

How would you go about solving this? $$p(x,t)=C\exp\left[-x+\int_0^t\int_0^\infty y\,p(y,\tau)\,\mathrm{d}y\,\mathrm{d}\tau\right]$$ Here $p(x,t)$ is the time-dependent probability distribution of a ...
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0answers
41 views

bounding the sum of squares of lengths of a quadrilateral inscribed in a unit square

Consider this nice little problem: if $ABCD$ is a quadrilateral inscribed in a unit square, then $$2\leq AB^2+BC^2+CD^2+DA^2\leq4$$ (Evidently this is problem 1 on paper 1 of the 1989 Irish ...
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2answers
75 views

How many $a$-nary sequences of length $b$ never have $c$ consecutive occurrences of a digit?

Let $S(a,b,c): = \#\{a$-nary sequences of length $b$ without $c$ consecutive occurrences of a digit$\}$. For example, $S(2,n,3)$ would be the number of binary sequences of length $n$ without $3$ ...
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2answers
48 views

How to solve this equation or system of equations?

I want to solve the equation $$(5 x-4) \cdot\sqrt{2 x-3}-(4 x-5)\cdot \sqrt{3 x-2}=2.$$ I tried. Put $a = \sqrt{2 x-3}\geqslant 0$ and $b =\sqrt{3 x-2}\geqslant 0 $. Suppose $$5x-4=m(2x-3)+n(3x-2)$$ ...
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0answers
36 views

Find the solution of this differential equation

I want to solve $\dot{\xi}(s)=\sqrt{\frac{(n-2)^2}{4}\xi(s)^2-\frac{n-2}{n}\xi(s)^{\frac{2n}{n-2}}}$ with the condition $\xi(0)=\biggl(\frac{n(n-2)}{4}\biggl)^{\frac{n-2}{4}}$. I know that ...
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2answers
163 views

Computing $\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$

I don't know how to compute: $$\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$$ I have tried substituting $t=\tan ^{2} x$ but got nothing out of it. I know there's some trick involved, but ...
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0answers
46 views

A problem related to Vectors.

A few days ago I posted an answer to a question on Phys.SE. The question is: Three particles $A,B$ and $C$ are at the vertices of an equilateral trinagle $ABC$. Each of the particle moves with ...
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2answers
36 views

How can I find $x$ such that $ax \equiv 1 \pmod{bx+c}$, given $a,b,c$?

Everything I've read about modular arithmetic generally concerns doing things in some "mod m" world where "m" is some constant. But I'm perplexed how to tackle modular arithmetic problems where the ...
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3answers
64 views

Equilateral triangle inscribed in a ellipse

"Given any point on a ellipse, is it always possible to inscribe an equilateral triangle, with a vertex coincident with that point, in the ellipse?" I thought I could use analytical geometry, but ...
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4answers
41 views

Find the minimum value of this expression with absolute values

The expression is $$|x-3| + |x-1| + |x| + |x+2| + |x+4|$$ I know that the minimum values for this expression is when x = 0 but is there any algebraic way to find this out? I did it on the ...
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1answer
66 views

Calculating completed percentage of a jigsaw puzzle

At first I thought the solution for this problem was simple…maybe to you it will be, but it evades me at present. I need to figure out how to calculate the completed percentage of a jigsaw puzzle. ...
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1answer
76 views

Acceptable Arrangements

A flagpole has spaces for seven colored flags arranged in a vertical line. Two of the flags are yellow, two are green, one is red, one is orange, and one is brown. Flags are to be placed on the pole ...
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1answer
55 views

Expected number of swaps required to get a palindrome out of a given string

Given a string, you keep swapping any two characters in the string randomly till the string becomes a palindrome. What is the expected number of swaps you will make? There will always be at least ...
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2answers
78 views

How do I evaluate this integral by hand?

TL;DR how do I evaluate $\int_0^{2 \pi } \frac{1}{\cos ^2(\theta )+1} \, d\theta$ by hand? I'm trying to solve this problem: Find the volume of the region defined by $x^2+xy+y^2+yz+z^2\le1$. ...
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4answers
599 views

Complicated but easy problem solving?

I'm going to be in the UKMT Team Challenge in a few days and revising some questions used in the previous year. The questions are really bugging my mind. I know it may seem like a lot and quite easy ...
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1answer
72 views

Soccer Team- Venn Diagram

If I could get help with this problem, it would be greatly appreciated. I have been trying using Venn diagram, but can't seem to understand it with four circles. On a soccer team there are four ...
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1answer
28 views

Probability problem regarding rooks on a chessboard

Eight rooks are placed in distinct squares of an 8 x 8 chessboard, with all possible replacements being equally likely. Find the probability that all the rooks are safe from one another.
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1answer
37 views

Finding/approximating 2 unknowns using one equation

I’m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
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1answer
56 views

Troubles with understanding the answer

I don't understand the proof. Where did they get the first line from? 21x11=1+5x46? Fermat's theorem in my view is a^46=_1mod47
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6answers
307 views

Which is larger $\sqrt[99]{99!}$ or $\sqrt[100]{100!}$

Which is larger $\sqrt[99]{99!}$ or $\sqrt[100]{100!}$ I know that it is the $\sqrt[100]{100!}$ but is there a formula to figure this out instead of doing it all out by hand?
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3answers
120 views

How exactly does the response “infintely many” answer the question of “how many”?

I admit that the level of this question is roughly about middle school, but this is what the question asks: The ratio of nickels to dimes to quarters is 3:8:1. If all the coins were dimes, the ...
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0answers
15 views

Can you find a method of moments of Gaussian AR(1)?

This is an exercise from Mathematical Statistics: Basic ideas and Selected topics, Bickel&Doksum, page 141. Gaussian AR(1) model; $X_i = \mu + e_i, i=1, \cdots,n$ $e_i = \beta e_{i-1} ...
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0answers
34 views

A question on combinations of a set of numbers

I have the set of the first $n$ primes $\{2,3,5,\ldots,p_n\}$. There are $n^n$ ways of selecting $n$ numbers from this set. Each combination has a number ($C_k$) associated with it and it is the ...
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1answer
62 views

Routes to a house

In this city, all the streets that run North and South have lettered names (A,B,C, etc.) and all the streets that run East-West have numbered names (1st, 2nd, 3rd, etc.). As you drive East, the ...