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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8
votes
5answers
771 views

How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers?

This was Problem 3 (first day) of the 1990 IMO. A full solution can be found here. How many rationals of the form $\large \frac{2^n+1}{n^2},$ $(n \in \mathbb{N} )$ are integers? The possible ...
6
votes
3answers
105 views

What should I do if I don't know where to start?

Sometimes getting started on a problem seems to be the hardest part. Once you find something to sink your teeth into, everything goes smoothly. What are some good things to try when you're staring at ...
3
votes
1answer
310 views

Evaluating the time average over energy

For more info see the article equations 37 Edit: The $\varepsilon ^3 $ has vanished due to time average. But how to get the 4th order? Let us define some function for scalar field $$\phi= ...
0
votes
3answers
320 views

Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$

I am struggling to prove this map statement on sets. The statement is: Let $f:X \rightarrow Y$ be a map. i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$ ii) $\forall_{A,B \subset X}: ...
11
votes
6answers
2k views

List of problem books in undergraduate and graduate mathematics

I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, ...
4
votes
3answers
289 views

$\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-…}}}} $, what is the nearest value of $y^2 - y$?

I found this question somewhere and have been unable to solve it. It is a modification of a very common algebra question. $\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} $, what is the ...
3
votes
6answers
327 views

Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$

Where do I start to solve a equation for x like the one below? $$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$ After squaring it, it's too complicated; but there's nothing to factor or to ...
3
votes
2answers
130 views

Evaluating decay rate with trigonometric explanation

According to the equation 4, $$\phi(0,t)= \frac{A_0}{(1+\frac{2t^2}{R^4})^{3/4}}\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 t}{R^2}\right]\right)\tag{1}$$ what conditions makes, ...
9
votes
0answers
217 views

Pólya and Szegő, Part I, Ch. 4, 174.

The following is a problem proposed in Pólya and Szegő's book "Problems and Theorems in Analysis" Assume that $0<f(x)<x$ and $$f(x)=x-ax^k+bx^\ell+x^\ell \varepsilon(x),\,\;\;\;\lim_{x\to ...
172
votes
4answers
10k views

The Mathematics of Tetris

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...
22
votes
10answers
1k views

Puzzles or short exercises illustrating mathematical problem solving to freshman students

At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ...
12
votes
4answers
3k views

How to Improve Mathematical Thinking and General Problem Solving Skills?

I'm a sophomore in university and seriously feel that I'm bad at solving mathematical and algorithmic problems (be it discrete math, calculus or just puzzles). I noticed that I'm only good at solving ...
5
votes
1answer
240 views

Mathematical problem with square numbers in the decimal system

Moderator Note: this is a question from the Federal Mathematics Competition 2013. Good morning, here's another (pretty difficult) mathematical problem... The task may sound a little strange (I'm ...
5
votes
3answers
490 views

When chessboards meet dominoes

You probably have heard about the following brainteaser : Consider a 8×8 chessboard. Remove two extreme squares (top-left and bottom-right e.g.). Can you fill the remaining chessboard with 1×2 ...
3
votes
1answer
174 views

Permutation of 1…9 with no ascending or descending subsequence of length 4

Arrange the numbers $1,2,...,9$ in such an order that no four of them appear (adjacently or otherwise) in ascending or descending order. Show that there is no arrangement of the numbers $1,2,...,10$ ...
1
vote
2answers
122 views

Not able to solve $({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$

I'm not able to solve $$({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$$ If you put values of $p$ (like $\frac{1}{2}$ or 2) back in the equation it doesn't satisfy! So please ...
0
votes
3answers
785 views

Smallest multiple whose digits are only ones and zeros

I have a collection of typewritten pages that formed the basis of a third year problem solving course offered about 25 years ago at U. Waterloo. I've been slowly working through the problems and have ...
-1
votes
2answers
501 views

In △ ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees?

In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because ...
11
votes
3answers
368 views

Penguin Brainteaser : 321-avoiding permutations

There are $k$ penguins, $k\ge 3$. They are all different heights. How many ways are there to order the penguins in a line, left to right, so that we cannot find any three that are arranged tallest to ...
2
votes
2answers
125 views

how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$

I am given this equation: $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$ I want to prove it: what i did is I take any $a \in f^{-1}(B_1 \cap B_2)$, then there is $b \in (B_1 \cap B_2)$ so ...
0
votes
2answers
101 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
208 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
220 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
38 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
2answers
132 views

Need help on this problem about maximizing a function

"As an epidemic spreads through a population, the number of infected people, $I$, is expressed as a function of the number of susceptible people, $S$, by: $$I(S)=k \ln(S/S_0)-S+S_0+I_0$$ For $k$, ...
31
votes
13answers
3k 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 ...
41
votes
5answers
1k 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? ...
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 ...
17
votes
0answers
396 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?
13
votes
4answers
440 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.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
11
votes
3answers
596 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 ...
15
votes
3answers
588 views

prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit

i am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all ...
6
votes
5answers
327 views

For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .

If $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$.
5
votes
2answers
238 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 $k, m, n \in \mathbb{N}$.
20
votes
4answers
320 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 ...
4
votes
4answers
324 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 ...
10
votes
4answers
508 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
5answers
1k 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 ...
4
votes
6answers
136 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!
16
votes
3answers
345 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 ...
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, ...
4
votes
2answers
301 views

How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I'm looking for a ...
3
votes
2answers
188 views

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

Is is possible to find the number of digits of $2013^{2013}$ without a calculator?
3
votes
3answers
360 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 ...
3
votes
1answer
114 views

$0 = \left(\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right)^2 -k^2-m^2$ solving for $k$

This question is related to $\delta(f(k))$ concerning the Dirac-delta. OK I know this might seem trivial but the result is very very important to me so I want to check with you if my logic seems ...
3
votes
1answer
1k views

Putnam 2012 B3 - Tournament combinatorics

A round-robin tournament among $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ ...
3
votes
2answers
362 views

$W$ white balls, $B$ black balls, adding $K$ of the resultant color each iteration

The problem is stated as follows. We have a box with $W$ white balls and $B$ black ones. Repeat $N$ times: each iteration a ball is taken out (uniformly), and put back along with $K$ (constant) more ...
2
votes
2answers
526 views

calculate the limit of this sequence $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1..}}}}$ [duplicate]

Possible Duplicate: $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$ i am trying to calculate the limit of ...
2
votes
1answer
225 views

Expected value of randomly distributed projection

Problem: Find the mathematical expectation of the area of the projection of a cube with edge of length 1 onto a plane with an isotropically distributed random direction of projection. Source: ...
1
vote
2answers
77 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: ...