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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0answers
37 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 ...
9
votes
2answers
192 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
48 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 ...
2
votes
3answers
65 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 ...
1
vote
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 ...
1
vote
1answer
74 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. ...
0
votes
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 ...
0
votes
1answer
84 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 ...
3
votes
2answers
80 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 $x^2+xy+y^2+yz+z^2\le1$. ...
5
votes
4answers
609 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 ...
0
votes
1answer
73 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 ...
0
votes
1answer
29 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.
0
votes
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 ...
0
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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
6
votes
6answers
316 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?
0
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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} ...
3
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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 ...
1
vote
1answer
63 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 ...
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2answers
64 views

Olympic problem on irreducible fraction

Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$.
0
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1answer
12 views

Multiplying non-decreasing sequences

Let $(a_n)$ and $(b_n)$ be non-decreasing sequences of positive terms (i.e. $a_n\gt0$ and $b_n\gt0$ for all $n\ge1$). Prove that the sequence $(c_n)$ is non-decreasing, where $c_n=a_nb_n$ for all ...
4
votes
1answer
178 views

Math Olympiads: GCD of terms in a sequence equals GCD of terms in other sequence

Recently, someone asked for a proof of a problem from the Russian Mathematical Olympiad, 1995. Math Olympiads: GCD of terms in a sequence equals GCD of their indices. The problem was to show that if ...
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2answers
161 views

Math Olympiads: GCD of terms in a sequence equals GCD of their indices.

The sequence $a_1 ,a_2 ,a_3 ,...$ of positive integers satisfies $\text{gcd}(a_i ,a_j ) = \text{gcd} (i, j)$ for $i \neq j$. Prove that $a_i = i$ for all $i$. Source: Russian Mathematical Olympiad, ...
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2answers
48 views

Real Life Rounding Phenomena When Solving for Variables

I have a question that I've been thinking a long time about without being able to come up with an answer and would appreciate some help: I am attempting to subtract two distinct fees from a total ...
0
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2answers
34 views

Deck of playing cards

Been going through an previous exam question and came across this: 5 cards are drawn from a deck of playing cards. What is the probability of drawing 3 aces? How do you calculate it using the C(n,r)? ...
2
votes
3answers
36 views

Programming Help - Solving for e(n)

I've been wrestling with this issue for a week and I just need some guidance on the math part of it. If I could just understand the math behind it I could piece together the functions to make it ...
1
vote
0answers
37 views

How to prove the relation of coefficents of a system of equations?

Consider the system of equations $$\begin{cases} a_1x^2+b_1y^2 + c_1xy+d_1x + e_1y+f_1=0,\\ a_2x^2+b_2y^2 + c_2xy+d_2x + e_2y+f_2=0. \end{cases}$$ I want to find the Real number $k$ so that the ...
3
votes
1answer
30 views

Maximum likelihood to throw exactly two 6s

One throws a dice $n$ times. For which value of $n$ is maximum the probability to obtain exactly two 6s? I get $$n=11 \text{ or } n=12.$$ My solution: the probability to obtain exactly two 6s in ...
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2answers
29 views

Simple Word problem question with boxes and bottles

Bottles are either packed in boxes of 6 *OR* 12. The number of small boxes must atleast be half the number of big boxes. If 240 bottles need to be packed, what's the minimum mumber of boxes needed? ...
2
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4answers
80 views

Contest problem involving primes and factorization

Prove that for any nonnegative integer $n$, the number $$5^{5^{n+1}}+ 5^{5^{n}}+1$$ is not prime. I want only some hints and the method to follow, but I don't need the full solution. Thanks.
0
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1answer
16 views

Competion Problem in graph theory

How can I prove that every graph has two vertices which are endpoints of the same number of edges? Any hints?
0
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1answer
34 views

Problem solving involving time

You have from 10 pm to 11:30 pm to do a project. At 10:34 what fraction of the project remains? I keep getting stuck and I don't know why. There is an hour and a half to do the project and at 10:34 ...
2
votes
2answers
39 views

Solve an equation in positive integers

Does $$x^2+y^2=3(z^2+ u^2)$$ have solutions in positive integers? I was assigned this problem, but I am struggling to find a solution. I guess that a proof by contradiction is required.
1
vote
1answer
45 views

If hexagon + triangle = 8, what is the value of a trapezoid? [closed]

If hexagon + triangle = 8, what is the value of a trapezoid? (using blocks) easy problem I'm having a difficult time figuring this out. I know that there are 6 triangles inside of a hexagon so I'm ...
2
votes
1answer
40 views

Competition problem (unknown source)

For what positive $x$ does the series $$(x-1)+( \sqrt[2]{x}-1)+ ( \sqrt[3]{x}-1)+ … + ( \sqrt[n]{x}-1) + …$$ converge?
-2
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1answer
44 views

What's the result of this? [closed]

I would like to ask you guys for the result of this equation: $2/2+6(2/2*3)-(9/(8+1)*2)*(2*7+1) = x$ What is x?
0
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3answers
40 views

Basic Algebraic Manipulation

How would I solve for $X$ in this instance? I can't figure out how to get the $X$ variables by themselves and the known values on the other side by themselves. $2(4-X)(4-X)+X = 3$
2
votes
4answers
60 views

Calculate the integral using another integral

Need help with this integration: Let $$A = \int_0^\pi \frac{\cos x}{(x+2)^2}dx$$ Compute $$\int_0^{\frac{\pi}{2}} \frac{\sin x \cos x}{x+1}dx$$ In terms of $A$. I tried to do some algebraic ...
0
votes
1answer
37 views

Evaluating the following sum

I have no idea how to solve evaluate this integral: $$\lim_{n\to\infty} \frac{1^a + 2^a + \cdots + n^a}{n^{1+a}}, a > -1$$ I want to set this up as some sort of integration since it is a ...
0
votes
1answer
79 views

probability that a year has 53 mondays

We have the years from 2001, 2002, 2003,... to 2010. Say, a year is chosen at random from the listed years. What is the probability that the chosen year has 53 Mondays ?
0
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1answer
27 views

Showing a function is not monotonic.

I need help with what this question is asking. Define $f$ by: $$f(x) = \begin{cases}x^2\sin\frac{1}{x}, & \mbox{if }x\neq 0 \\ 0 & \mbox{if }x=0\end{cases}$$ Let $g(x) = x + 2f(x)$. Show ...
3
votes
1answer
38 views

Prove the following trigonometric polynomial has $2n$ zeros

I am having a lot of trouble with this problem, any help would be greatly appreciated! Prove that the that the trigonometric polynomial $$a_0 + a_1\cos(x)+\cdots+a_n\cos(nx), $$ where the ...
0
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0answers
25 views

Given two sets, how can I say statistically if they are similar/different

This is a very open ended question. Suppose I have two sets of data samples of the same form, say [item, rating]. Rating is a value on the interval [0,100] and item is a unique identifier given to a ...
0
votes
1answer
24 views

Radius of Convergence Problem solving

I did this questions using the Ratio Test which showed that the radius of convergence is the same. I'm not sure if that is correct. (I am having my doubts about c_n becoming c_n+1 for the ratio test ...
0
votes
1answer
43 views

Sequences and Series ( Power Series ) question.

I know that the sum from 0 to infinity of part A is the same as the sum to infinity from 1 if you decrease the power by 1. So I'm guessing the series will converge, but I don't know how to find the ...
0
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1answer
24 views

Calculating probabilities of events

Was going through past previous exam questions and came across this one: A manufacturer of lie detectors is testing its newest design. It asks 300 people to lie deliberately and another 500 people ...
59
votes
12answers
13k views

Dividing 100% by 3 without any left

In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quantity'. The totality ...
1
vote
1answer
134 views

How to solve this trigonometric equation / geometric problem

Is there any way to solve this type of equation exactly for x, where a-h are precalculated constants: $a\cos(g x)+b \sin(g x)+c\cos(h x)+d\sin(hx)+ex+f=0$ Or is my only/best option some sort of ...
0
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0answers
28 views

Problem involving line segment comparisons

I came across this question, and I find myself having no clue how to proceed. ...