# Tagged Questions

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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### 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 ...
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### 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. ...
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### $\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-…}}}}$, 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 ...
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### 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....
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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. ...
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### 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 ...
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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, $$\... 1answer 569 views ### Multiplication Table with a frame and picture of equal sum Is there an n \times n multiplication table such that if you form a border of width k ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ... 2answers 137 views ### Not able to solve ({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0. I'm not able to solve$$({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$$If you put values of p (like \frac{1}{2} or 2) back in the equation it doesn't satisfy! So please ... 3answers 811 views ### Penguin Brainteaser : 321-avoiding permutations There are k penguins, k\ge 3. They are all different heights. How many ways are there to order the penguins in a line, left to right, so that we cannot find any three that are arranged tallest to ... 0answers 276 views ### Vieta jumping with non-monic polynomials I have recently discovered Vieta jumping as a problem-solving technique. In order to teach myself about it, I have located most (all of?) the standard references, both here on MSE and "out there" (via ... 3answers 39 views ### Finding Matrix A from Eigenvalues and Eigenvectors (Diagonalization) Question: Let A be a 3 \times 3 Matrix such that [-3,4,1] is the eigenvector corresponding to eigenvalue 3, and [6,-3,2] is an eigenvector corresponding to the eigenvalue 2. If v = ... 1answer 80 views ### Confusion regarding probability of microbe producing everlasting colony. My question is about the given solution to problem 4 in Newman's book 'A Problem Seminar'. Note that the book is available online at Springer. Problem 4 A microbe either splits into two perfect ... 2answers 567 views ### How to solve the irrational inequality? Solve the inequality$$\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, ...
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 ...