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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3
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0answers
40 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
68 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
75 views

Olympic problem on irreducible fraction

Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$.
0
votes
1answer
13 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
240 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 ...
5
votes
2answers
174 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
2answers
52 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
36 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
43 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
38 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
35 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
30 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
88 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
19 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
46 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
44 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
56 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
43 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?
0
votes
3answers
41 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
62 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
40 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
107 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
votes
1answer
32 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
44 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
26 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
31 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
54 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
28 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 ...
62
votes
13answers
17k 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 'quality'. The totality ...
1
vote
1answer
211 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
46 views

Problem involving line segment comparisons

I came across this question, and I find myself having no clue how to proceed. ...
2
votes
2answers
28 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
0answers
25 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
27 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 ...
1
vote
2answers
52 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
5answers
231 views

First Order Logic

Is it possible to represent the english sentence with numerical value in First order Logic. For example if the sentence is: Nobody has more than one mother. I am wandering who can i show the ...
1
vote
4answers
41 views

How do I work with equations with more than two variables?

I was trying to rewrite this equation in terms of $s$: $$ p = 4s \frac{(s - 1)}{2} + s (2r + 1) $$ After failing at that, I tried with Wolfram Alpha, and got the answer I wanted. But, how did it get ...
0
votes
1answer
41 views

An unusual inequality

Problem: $(x_i)_{i=1}^n$ is a finite sequence of positive integers. Define $f\big(S\big)=\displaystyle \sum_{i\,\in\, S\,\subseteq\, [n]}x_i$, and suppose $f$ is injective. Prove that: ...
0
votes
2answers
246 views

Finding the Reading Rate

Part 1: If you read 15 minutes per day every day and end up reading 12 books of 200 pages each in 1 year, what is your reading rate in pages per minute? Part 2: If you increase your reading speed so ...
23
votes
7answers
723 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 ...
2
votes
3answers
1k views

Crossword puzzle- Crossnumber puzzle

The puzzle below is a cross number puzzle, similar to a crossword puzzle except that the entries are numbers. Enter one digit per square. The thick heavy line is a separator. ACROSS: a. A prime ...
0
votes
2answers
59 views

Solving an equation with both linear and exponential terms

Can I find an algebraic solution for the equation below? Thank you. $$ x+e^{x}(x+a)=b $$
2
votes
1answer
43 views

Interesting question about functions

I saw the following question and I would like to share. I don't know the answer. Suppose that the function $f:\Bbb{N}\to\Bbb{N}$ has the property $f(f(n))<f(n+1)$ for any $n\in\Bbb{N}$. Prove that ...
7
votes
1answer
241 views

How to suceed in mathematical olympiads and competitions?

This question may be slightly off-topic but it still relates to Maths so I hope that it does not get taken off. I am a student who is 16 years old, is generally good in Maths and enjoys it and I ...
4
votes
1answer
104 views

Can someone please clarify combinations vs permutations?

I see similar questions asked on here and obviously I did some research and read my book, but it seems like every explanation contradicts another in some way. There are basically infinite scenarios ...
0
votes
2answers
56 views

Ways to add a number using just 1's, 5's, 10's, 25's, 50's

given the set $\{1, 5, 10, 25, 50\}$, in how many ways, can you combine this numbers to get a specific number. For example, 11 can be shaped as $1\cdot11$, or $5\cdot112 + 1\cdot111$, or $10\cdot111 + ...
1
vote
4answers
70 views

Problem of the month. Thinking problem?

Cherise scored 85 on her last math exam of 100 questions. Her teacher has an unusual way of scoring this test. He calculated her score by subtracting 2 times the number of wrong answers from the ...
1
vote
1answer
87 views

How deep would the water be?

The lake created by Hoover Dam on the Arizona-Nevada border has a capacity of 31,250,000 acre-feet. If the whole lake suddenly flooded the Mojave Desert (area 15,000 sq. miles) how deep would the ...
0
votes
1answer
76 views

Sliding Question

At the local playground, three boys like to go down the a very wide slide. After going down the slide, they turn around and climb back up the slide rather than walking around to climb the ladder. The ...
0
votes
2answers
50 views

Finding the distance between the point $(3,7)$ and $x = 0$

In the paper of maths, there was an M.C.Q.S.: The distance of the point $(3,7)$ from $x = 0$ is $3$ $7$ $10$ $8$ So can anyone tell me what was the correct answer and how to ...