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.

learn more… | top users | synonyms

8
votes
5answers
892 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 ...
19
votes
4answers
473 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
2
votes
3answers
2k 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}: f(...
3
votes
2answers
788 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 ...
12
votes
4answers
2k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
6
votes
2answers
522 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
2answers
5k views

In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other?

In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other? I tried to draw diagrams onto a $8\times8$ square but I'm only getting $16$ ways....
5
votes
4answers
717 views

Proof of $n^2 \leq 2^n$.

I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$. My steps are: induction base case: $n=0:$ $0² \leq 2⁰$ which is okay. inductive step: $n \rightarrow n+1:$ $(n+1)²\...
4
votes
3answers
10k views

Missing dollar problem

This sounds silly but I saw this and I couldn't figure it out so I thought you could help. The below is what I saw. You see a top you want to buy for $\$97$, but you don't have any money so you ...
4
votes
4answers
4k views

Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K

A similar question to mine was answered here on stackexchange: Probability of winning the game "1-2-3" However, I am unable to follow the formulas so perhaps someone could show the ...
2
votes
3answers
9k views

Smallest multiple whose digits are only ones and zeros [duplicate]

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
vote
2answers
114 views

How can I solve this problem without having to do it by hand?

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
82
votes
11answers
3k views

Problems that become easier in a more general form

When solving a problem, we often look at some special cases first, then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, i.e. ...
25
votes
11answers
3k 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. ...
28
votes
14answers
951 views

Examples where it is easier to prove more than less

Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ (e.g. since the induction hypothesis gives you more ...
20
votes
3answers
1k 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 $n\...
7
votes
2answers
370 views

Prove Divisibility In Fibonacci Sequence Over A Prime Number

In The Fibonacci sequence which is defined as $$ F_n=F_{n-1}+F_{n-2}, $$ lets say we have the number $p$ which is an odd prime. Prove that: $F_{p-1} + F_{p+1} -1$ Is divisible by $p$. Prove that ...
12
votes
1answer
18k views

Expected Ratio of Coin Flips

If you flip a coin until you decide to stop and you want to maximize the ratio of heads to total flips, what is that expected ratio? Assuming that you want to maximize the ratio, meaning whether ...
3
votes
6answers
725 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 expand?...
6
votes
3answers
996 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 ...
4
votes
1answer
3k views

Induction Proof: Formula for Sum of n Fibonacci Numbers

I am stuck though on the way to prove this statement of fibonacci numbers by induction : my steps: definition: $F_{0}:=0, F_{1}:=1 $ and $F_{n}:=F_{n-1}+F_{n-2}$ The Hypothesis is: $\sum_{i=0}^{n} ...
14
votes
0answers
311 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 0}\...
2
votes
2answers
51 views

simple convergence test $\lim_{n \to \infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}$

i am pulling my hair out in solving this problem. i know, it is a stupid question but i am not that good at maths, and many thanks for any help $\lim_{n \to \infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}$ ...
2
votes
3answers
315 views

Where is the lost dollar?

Somebody explained me this problem, but I am not sure to understand what is wrong. ...
0
votes
1answer
2k views

Can this crate have even numbers in all rows and columns?

A milk crate holds 24 bottles in four rows and six columns. Can you put 18 bottles of milk in the crate so that each row and each column of the crate have an even number of bottles in it?
227
votes
4answers
23k 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 ...
51
votes
5answers
3k 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? $$x=\sqrt{1+2\sqrt{2+3\sqrt{3+4\...
21
votes
5answers
12k 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 ...
12
votes
1answer
405 views

Differentiable functions satisfying $f'(f(x))=f(f'(x))$

I am wondering whether or not there is a reasonable characterization of differentiable functions $f: \mathbb{R}\to \mathbb{R}$ such that $f'(f(x))=f(f'(x))$ for each $x\in\mathbb{R}$. (Or, if you like ...
6
votes
1answer
196 views

If $(n_k)$ is strictly increasing and $\lim_{n \to \infty} n_k^{1/2^k} = \infty$ show that $\sum_{k=1}^{\infty} 1/n_k$ is irrational

Prove that for a strictly increasing natural sequence $(n_k) $ satisfying $\lim_{n \to \infty} n_k^{1/2^k}=\infty$, $\sum_{k=1}^{\infty} 1/n_k$ is irrational. This is another problem "problems in ...
18
votes
5answers
2k views

Show that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.

The following problem was on a math competition that I participated in at my school about a month ago: Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions. I will outline ...
10
votes
2answers
311 views

Real Induction Over Multiple Variables?

I've seen in several different places* that one can use normal mathematical induction to prove the truth of a statement that relies not on just one variable (say, $x$,) but multiple variables (for ...
6
votes
5answers
420 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 $x+y$.
4
votes
2answers
205 views

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
4
votes
3answers
2k 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 a||...
4
votes
3answers
364 views

Strategies to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy a given functional equation

My question is as follows: What methods can be used to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ satisfying a certain functional equation. An example of a case where this applies is the ...
3
votes
1answer
493 views

Mathematics of Tetris 2.0

Based on the question The Mathematics of Tetris, I was wondering if it is possible to have a series of tetris blocks that is impossible to clear. For example, getting the string TTTSS.. forces the ...
3
votes
1answer
140 views

Exponential Diophantine: $2^{3x}+17=y^2$

Is there a way of solving the following equation, in integers $(x,y)$, by hand? : $2^{3x}+17=y^2$. You can also try: $2^{2x}+17=y^2$ or more generally $2^x+17=y^2$; each of these has at least 1 ...
8
votes
2answers
98 views

Solution to $e^{e^x}=x$ and other applications of iterated functions?

While trying to solve $e^{e^x}=x$, I ran into the simple solution $x=-W(-1)$. I found it by using the equation $$e^x=x$$Then powering both sides with a base $e$.$$e^{e^x}=e^x$$Now note that the left ...
6
votes
3answers
126 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 ...
4
votes
2answers
175 views

If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $

Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$ \frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $$ for each ...
3
votes
2answers
154 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, $$\...
3
votes
1answer
437 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= \sum_{k=1}...
2
votes
1answer
569 views

Multiplication Table with a frame and picture of equal sum

Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ...
1
vote
2answers
137 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 ...
12
votes
3answers
811 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 ...
1
vote
0answers
276 views

Vieta jumping with non-monic polynomials

I have recently discovered Vieta jumping as a problem-solving technique. In order to teach myself about it, I have located most (all of?) the standard references, both here on MSE and "out there" (via ...
1
vote
3answers
39 views

Finding Matrix A from Eigenvalues and Eigenvectors (Diagonalization)

Question: Let $A$ be a $3 \times 3$ Matrix such that $[-3,4,1]$ is the eigenvector corresponding to eigenvalue $3$, and $[6,-3,2]$ is an eigenvector corresponding to the eigenvalue $2$. If $v$ = ...
0
votes
1answer
80 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 ...
0
votes
2answers
568 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, ...