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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0
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2answers
131 views

Nonexistence of Limit of Sum of Prime Factors

In trying to prove the following problem, I find great difficulty in proceeding to generalizing some results: Let $s(n)$ be the sum of prime factors of an integer $n$. Prove that $\lim_{n \to \infty} ...
0
votes
1answer
896 views

Exact area using limits and Riemann sum

the i need to find the exact area under tha curve of the function $f(x)=4+3x-x^2$ on the interval $[-1,3]$ using limits and a Riemann Sum. I have nothing started, because I am confused on where to ...
0
votes
2answers
545 views

How to solve the irrational inequality?

Solve the inequality $$\dfrac{2x^4+2x^2}{\sqrt{x+1}}+(x+2)\sqrt{x+1}>x ^3 + 2x^2 + 5x.$$ I tried. By putting $t = \sqrt{x+1}$, we have $$2t^8-t^7-8t^6+t^5+15t^4-4t^3-11t^2+4t+4>0.$$ Using Maple, ...
0
votes
1answer
79 views

Confusion regarding probability of microbe producing everlasting colony.

My question is about the given solution to problem 4 in Newman's book 'A Problem Seminar'. Note that the book is available online at Springer. Problem 4 A microbe either splits into two perfect ...
-1
votes
1answer
491 views

How to solve the problem that determines the age of Diophantus?

How to solve the problem that determines the age of Diophantus?$$$$"God gave him to be a boy for the sixth part of his life, and adding one to it twelfth part covered her cheeks fluff, He gave him the ...
-6
votes
5answers
650 views

How to solve the sequence: $87, 89, 95, 107, ?, 157$

This question appeared in a competitive exam. The question is: Q. Find the unknown term in $87,89,95,107,?,157$ 1)127 $\ \ \ \ \ \ \ \ $ 2)122 3)139 $\ \ \ \ \ \ \ \ $ ...
65
votes
23answers
9k views

An example of a problem which is difficult but is made easier when a diagram is drawn

I am writing a blog post related to problem solving and one of the main techniques used in problem solving is drawing a diagram. Essentially, I want to illustrate that some hard problems (for example, ...
45
votes
16answers
15k views

Interview riddle

On the Mathematics chat we were recently talking about the following problem @Chris'ssis had to solve during an interview : $$3\times 4=8$$ $$4\times 5=50$$ $$5\times 6=30$$ $$6\times 7=49$$ ...
59
votes
4answers
2k views

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer number of them. Is there a unified ...
54
votes
8answers
8k views

How to debug math?

May seem strange as I'm good in programming, but I just started diving into math. ATM I'm learning combinatorics at Khan Academy, and here's an example of a question that I struggled with (that's not ...
42
votes
14answers
4k views

Examples of famous problems resolved easily

Have there been examples of seemingly long standing hard problems, answered quite easily possibly with tools existing at the time the problems were made? More modern examples would be nice. An example ...
51
votes
5answers
2k views

Finding the value of $\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$

Is it possible to find the value of $$\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$$ Does it help if I set it equal to $x$? Or I mean what can I possibly do? ...
27
votes
7answers
1k 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 ...
20
votes
1answer
490 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?
16
votes
4answers
523 views

How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?

How do I integrate this guy? I've been stuck on this for hours.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
34
votes
1answer
705 views

Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers

For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
6
votes
5answers
567 views

How can I express the sum of $\sin a+\sin2a+\sin3a+\cdots+\sin(n-1)a$?

I want to sum up the partials of a harmonic series, how do I do it? If I was using the 'Lagrange trigonometric identity to solve this problem', how would I plot it on Wolfram mathematica (using which ...
31
votes
1answer
2k views

Is it possible to construct a sequence that ends in 1000000000?

Starting from the number $1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it or rearranging its digits (not allowing ...
25
votes
7answers
2k views

Sum of the sum of the sum of the first $n$ natural numbers

I have here another problem of mine, which I couldn't manage to solve. Given that: $$x_n = 1 + 2 + \dots + n \\ y_n = x_1 + x_2 + \dots + x_n \\ z_n = y_1 + y_2 + \dots + y_n $$ Find ...
19
votes
4answers
663 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for ...
12
votes
3answers
756 views

An Integral involving $e^{ax} +1$ and $e^{bx} + 1$

For fun, I was looking at the following Putnam-Style problem the other day on this page: (It is problem B2) Evaluate the integral ...
9
votes
5answers
627 views

Examples of open problems solved through short proof

Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with ...
10
votes
2answers
456 views

The $2013$th digit of $1234567891011213141516\ldots$

How do I find the $2013$th digit of the string $12345678910111213141516\ldots$ I still don't get it, how are you suppose to find the exact digit. How did you hint help at all?
10
votes
2answers
205 views

Proving that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$

How do I prove that: $$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$$ Do I use induction?
8
votes
5answers
5k views

Show me some pigeonhole problems [closed]

I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve. I would be thankful if ...
5
votes
2answers
151 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 ...
22
votes
4answers
455 views

Ways to fill a $n\times n$ square with $1\times 1$ squares and $1\times 2$ rectangles

I came up with this question when I'm actually starring at the wall of my dorm hall. I'm not sure if I'm asking it correctly, but that's what I roughly have: So, how many ways (pattern) that there ...
7
votes
2answers
179 views

Values of $a$ s.t. for all continuous $f$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)$

Determine all $a\in[0,1]$ such that for ${\it every}$ continuous function $f:[0,1]\to \mathbb{R}$ with $f(0)=f(1)$ there exists at least one $x$ where $f(x) = f(x+a)$. First of all, $a=0,1/2,1$ ...
4
votes
3answers
285 views

When asked to show that $X=Y$, is it reasonable to manipulate $X$ without thinking about $Y$?

When I'm asked something like "show X is equal to Y", I first try to manipulate what I know (X) into the result (Y). A lot of the time, I do not investigate the result I'm trying to conclude with. I ...
18
votes
3answers
594 views

A sum of fractional parts.

I am looking to evaluate the sum $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Using matlab, and experimenting around, it seems to be $\frac{(m-1)(n-1)}{4}$ when ...
15
votes
2answers
585 views

The Farmyard problem

Problem: There is a farmer who has a $1\text{ mile}\times 1\text{ mile}$ square piece of land. He knows that there is a completely straight pipe underneath some part of his property, but it could ...
12
votes
1answer
106 views

Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero coefficients than $f(x)$?

I saw this problem on a problem set and I have absolutely no idea how to proceed in a feasible way. Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero ...
12
votes
5answers
2k views

Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially

I have to give a combinatorial proof of $$\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}.$$ I find it difficult to solve such problems. I'm not a brilliant person and never will be so I need to have ...
10
votes
8answers
351 views

Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $

Evaluate the integral, $$ \int_{0}^{1} \ln(x)\ln(1-x)\,dx$$ I solved this problem, by writing power series and then calculating the series and found the answer to be $ 2 -\zeta(2) $, but I don't ...
5
votes
2answers
251 views

Let $k \geq 3$; prove $2^k$ can be written as $(2m+1)^2+7(2n+1)^2$

Prove: If $k \geq 3$, then $2^k$ can be written as $(2m+1)^2+7(2n+1)^2$, where $k, m, n \in \mathbb{N}$.
4
votes
2answers
212 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 ...
4
votes
4answers
340 views

Mathematical proof for long-term behavior of a sequence of integer vectors

There are some children sitting around a round table. Each child is given an even amount of $1$-cent coins ($0$ is even) by their teacher, all the children at once. A child will give half his money to ...
11
votes
4answers
605 views

The value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$?

How to find value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$ ? I've calculated it by MATLAB for some finite terms and I've got : $0.3001 - 0.4201i$, but I don't know how to ...
5
votes
1answer
148 views

What are the positive rational solutions of $x^{(x+y)} = (x+y)^y$?

I saw this problem in the Problem-Solving through Problems book by Larson (# 3.3.25b). I got to here: $$x \log(x) = y\log\left(1+ \frac yx\right)$$ But I can't seem to find a way to reduce this ...
4
votes
1answer
166 views

Additive function $f: \mathbb{Z}^\infty \rightarrow \mathbb{Z}$ is zero everywhere.

Let $f: \mathbb{Z}^\infty \rightarrow \mathbb{Z}$ be an additive function ($f(x+y)=f(x)+f(y)$ for every $x,y \in \mathbb{Z}^\infty$). In addition for every $x=(0,\dots, 0,1,0, \dots)$ we have ...
4
votes
3answers
1k views

Fractions in Ancient Egypt

In ancient Egypt, fractions were written as sums of fractions with numerator 1. For instance,$ \frac{3}{5}=\frac{1}{2}+\frac{1}{10}$. Consider the following algorithm for writing a fraction ...
4
votes
6answers
258 views

Repeating Decimals [duplicate]

I'm just wondering how do we simplify repeating decimals into a fraction in general? Like, for example, $$0.5656\dots$$ $$0.12424\dots$$ $$4.23777\dots$$ Thanks!
3
votes
0answers
110 views

Shortlist of problems in linear algebra

A while ago I remember seeing a very nice shortlist of problems in linear algebra. It was a list of about 40-50 problems. The idea was that if you solve them, you learn linear algebra very well and ...
3
votes
2answers
208 views

Least sum of distances

Problem: Let $A, B, C, D$ be points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of the distances $AX+ BX + CX + DX$. Context: During a course, I was assigned a ...
3
votes
2answers
390 views

Find the number of digits of $2013^{2013}$?

Is is possible to find the number of digits of $2013^{2013}$ without a calculator?
2
votes
2answers
249 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
1
vote
1answer
72 views

Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$

Given this diophantine equation: $$16r^4+112r^3+200r^2-112r+16=s^2$$ Wolfram alpha says the only solutions are $(r,s)=(0,\pm4)$ How would one prove these are the only solutions? Thanks.
7
votes
3answers
2k 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 ...
6
votes
3answers
243 views

“Long-division puzzles” can help middle-grade-level students become actual problem solvers, but what should solution look like?

This is my first post. I hope it's acceptable. EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and ...
6
votes
2answers
264 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, ...