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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18
votes
1answer
413 views

Is $\sum_{k=1}^{n} k^k / \sum_{k=1}^{n} k \in \mathbb{N}$ for some $n > 1$?

Let $ A = \sum_{k=1}^{n} k^k $ and $ B = \sum_{k=1}^{n} k$, where $n >1 $ is a positive integer. Is $A/B$ ever an integer?
0
votes
0answers
83 views

markov chain - transition matrix - why is this?

I am stuck in this trivial issue: i am given a state for which i need to give a markov chain transition matrix. i couldnot do that and i saw the solution, now i dont understand why solution this is. ...
0
votes
0answers
19 views

List all equations for straight line! [on hold]

Can someone list all the equations for a straight line geometry? Thank You.
1
vote
3answers
83 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 ...
0
votes
0answers
13 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
votes
0answers
29 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 ...
21
votes
7answers
467 views

“Here's a cool problem”: a collection of short questions with clever solutions

This game will be familiar to many mathematicians, and it is always good fun to play. I am looking to find a list of good questions with short, when-you-see-it solutions. The kind of question one ...
1
vote
1answer
47 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 ...
1
vote
2answers
47 views

Olympic problem on irreducible fraction

Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$.
4
votes
1answer
80 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 ...
0
votes
1answer
11 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 ...
5
votes
2answers
137 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, ...
1
vote
1answer
317 views

Algebra: Finding rate

Here Is The Problem. Julia drove from her home to her aunt's house in 3 hours and 30 minutes. The distance between the houses is 175 miles. Knowing that distance = rate X time, find that car's ...
4
votes
2answers
516 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 ...
1
vote
2answers
41 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
votes
2answers
29 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)? ...
1
vote
0answers
18 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 ...
1
vote
3answers
26 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 ...
3
votes
1answer
25 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 ...
0
votes
2answers
19 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
votes
4answers
67 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
votes
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
votes
1answer
24 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
32 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
29 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
39 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?
1
vote
2answers
355 views

Proving Holder's inequality using Jensen's inequality

Let $p$ and $q$ be positive reals such that $\frac{1}{p}+\frac{1}{q} = 1$, so that $p,q$ in $(1,\infty)$. For $\vec a$ and $\vec b \in \mathbb{R}^2$ prove that $|\vec a \cdot \vec b | \leq ||\vec ...
-2
votes
1answer
41 views

What's the result of this?

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?
1
vote
3answers
26 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. $2(4-X)(4-X)+X = 3$
59
votes
13answers
12k 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 ...
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
33 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
60 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 ?
8
votes
3answers
169 views

Existence of a certain subset of $\mathbb{R}$

To every real $x$ assign a finite set $\mathcal{A}(x)\subset \mathbb{R}$ where $x\not\in \mathcal{A}(x)$. Does there exist $\mathcal{W}\subset \mathbb{R}$ such that: $$1.\;\;\mathcal{W}\cap ...
0
votes
1answer
23 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
30 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
votes
0answers
20 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
19 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
29 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
votes
1answer
20 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 ...
0
votes
1answer
49 views

For a poker hand, five cards are chosen from an ordinary deck of playing cards. How to find the probability to get the following hands

How would you find the of probabilities: (a) a hand with 1 heart, 2 diamonds, 2 clubs (b) a hand with no face cards (c) a hand with at least 3 queens
6
votes
3answers
242 views

Calculating Non-Integer Exponent

I just wanted to directly calculate the value of the number $2^{3.1}$ as I was wondering how a computer would do it. I've done some higher mathematics, but I'm very unsure of what I would do to solve ...
0
votes
2answers
70 views

Problem solving with letter sequences

I've been through some passed exam questions and came across this one: (1) How to find out how many 5 letter sequences are possible that use the letters m, a, t, h, s once each? (2) How to find out ...
-1
votes
1answer
48 views

How to calculate the diameter of a rubix

What are the possible formulas to do calculation?
1
vote
1answer
60 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
votes
0answers
19 views

Problem involving line segment comparisons

I came across this question, and I find myself having no clue how to proceed. ...
2
votes
2answers
24 views

School Play and Ticket problem.

I may be missing something obvious here, but cant seem to see what. Can anyone give me some insight on how to solve this. 100 tickets are sold for a school play. Tickets for a child cost £1.50 each. ...
1
vote
2answers
48 views

Amount of Miles

Gary has two cars: His fairly new van and his old clunker station wagon. the wagon currently has 16 times as many miles on it as the van had when the wagon had 3 times as many miles as the van has ...
1
vote
0answers
18 views

Diffeomorphism/Problem/Euclidean spaces

Problem: Let $f$ : $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ be a $C^{1}$ function such that |$f'(t)$| $\leq$ $k$ < $1$, $\forall$ $t$ $\in$ $\mathbb{R}$. Define $\phi$ : $\mathbb{R}^{2}$ ...
0
votes
1answer
24 views

Quick Bounds Question [ Sets]

1) Let $S =\{(−1)^n\; \mid\; n \in \mathbb{Z}\}$ . What is the greatest lower bound of $S$? -1 is the Lower bound. But is it also the greatest lower bound? Or does it not exist? Thanks. And also ...