# 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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### Books or website about solving IMO problems

Hey I want to solve IMO problems like the problem in the image below, but I cannot solve the problem or any of the problems in the IMO, so do you guys have some good website or books that teach how to ...
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### Parameterization which is closed under addition

Suppose $\beta_1(t)$ and $\beta_2(t)$ are two parametric curves defined on $[0,1]$. Let $\beta_1^*(t)$ and $\beta_2^*(t)$ are two re-parametrized of the above curves. Now, I looking for a re-...
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### Why is the solution to $\sqrt{6-5x}=x$ only $x=1$ and not $x=-6$? [duplicate]

I solved the equation $\sqrt{6-5x}=x$ as follows: $$(\sqrt{6-5x})^2=x^2$$ $$6-5x=x^2$$ $$0=x^2+5x-6=(x+6)(x-1)$$ $$x=-6 \quad \text{or} \quad x=1$$ If I plug in $x=-6$ into the original equation, I ...
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### I need some help with Geometry. Is this a correct answer to this problem?

Good day, I have a question regarding geometry. I don't know whether my answer is correct because the answer in my book uses a totally different method for solving this particular problem. Here's ...
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### Find a thousand natural numbers such that their sum equals their product

The question is to find a thousand natural numbers such that their sum equals their product. Here's my approach : I worked on this question for lesser cases : \begin{align} &2 \times 2 = 2 + 2\\ ...
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### Variation of the opaque forest problem (a.k.a farmyard problem)

I was wondering about the following variation of the opaque forest problem (see here and there for previous questions) : What is the least length set of segments that will intersect every straight ...
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### Combinatorics Question with bridges and inability to cross over each other

Several small villages are situated on the banks of a straight river. On one side, there are $20$ villages in a row, and on the other there are $15$ villages in a row. I would like to build bridges, ...
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### How can the lagrange multipliers in a simple constrained cost minimization problem be calculated? (for binding constraints)

Is there a simple algebric way to calculate the shadow prices (lambda) of the binding constraints given below? This is a cost minimization problem dependent on the generation output. The cost of ...
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### Which is larger, $e^\pi$ or $\pi^e$? [duplicate]

I don't know how to approach this. I tried expanding $e^{\pi}$ using the power series but that was a dead end since I didn't know what to do with it. I tried estimating if $e \log({\pi})$ was ...
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### What is the probability that the upturned faces of three fair dice are all of different numbers?

Three fair dice are rolled ($6$ sides). What is the probability that the upturned faces of the three dice are all of different numbers? I got that the number of possible outcomes total is $6^3$ and ...
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### How to solve this equation using logs

How do solve this equation for x using logarithms? $$4^x = 6^x-3$$ If it is not possible using logarithms, please provide another way. Thank you in advance
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### Is there any easy way to solve two equations with three unknowns?

Is there a way to solve the below simultaneous equations? One possible solution is $a_1=20.0948$, $a_2=10.0948$, $a_3=6.3448$. The variables are actually dual variables of the binding constraints. ...
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### Solving modulo equations with one variable

Given the following equation: $$10 = 4^x \pmod {18}$$ How can one know what are the correct values for $x$ ?
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### Why do people say that some problem is hard when they do not actually prove it?

I have read many times in different papers something like the following (I do not remember the exact words though): "The problem is nonlinear non-convex programming problem which is hard to ...
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### Launching a Plaintext Attack against Affine Cipher

Update 2 Being new to the world of Stack Exchange I did not realize that there exists a site solely devoted to cryptography. In light of this, I hope someone could help me migrate this question to ...
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### Integration of the product of Hermite Polynomial and exponential function

how to proceed with these two integration.. $$\int^0_{−∞}e^{−ax2}H_{2k}(x)dx=?$$ $$\int^∞_{0}e^{−ax2}H_{2k}(x)dx=?$$ where $$H_n(x)$$ is the Hermite Polynomial (physicist's convention).
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### how to find $t$ from $2t^2-0.01t^4=100$?

how to find $t$, from $2t^2-0.01t^4=100$? I was guessing may be I can take $t^2$ common but if it is so so why cannot we take $t$ common in other cases? I mean, for example: $t^2+4t=-4$ why can we ...
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### Where did two dollars go?? [duplicate]

One of my school friends gave me this sum.Its basically a story formated into a sum STORY There were 3 friends. They each gave 20 dollars to buy a radio. They bought the radio for 60 dollar. Later ...
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### How to solve even/odd divide-and-conquer problems?

I am looking into something called the Josephus problem, which seems to be popular, so I am sure there are lots of explanations online, but I want to do the work myself, but I do need a small push to ...
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### Sum of squares using generating functions

I tried using generating functions to solve the sum of squares formula based on the recurrence $a_n = a_{n-1} + n^2$ with $a_0 = 0$. G(x) = \sum_{n=0}^{\infty} a_n x^n \\ G(x) - 0 = \sum_{n=1}^{\...
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### Need to solve for t but can not work out how to get t on one side

I have a object in free fall with $g$ = acceleration, $y$ is the position above the ground and $t$ = time. I worked out that to find the speed at and $t$ is $dy = g . t$ So to get the position $py$ ...
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### Find numbers that fit each riddle look for more than one answer [closed]

There are two $2$ digits numbers. The first number is greater than $50$ and ends in $0$. When you subtract one number from the other number the difference is $29$
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### A quicker generalized method to finding a curve tangent to another curve?

Let's say we have a curve of $\sin(x)$ and we have to find a curve tangent to this in form of $c(x-d)^{1/3}$. This curve should have the same tangent line as $\sin(x)$ at any point around $(a,\sin(a))$...
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### Solving a Chessboard problem using the Invariance principle

Problem Statement There is an integer in each square of an 8 x 8 chessboard. In one move, you may choose any 4 x 4 or ...
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### Optimization-like question

Let's say I have a formula like $ax + by + cz = N$. $a, b, c$, and $N$ are known and cannot be changed. $x, y$, and $z$ are known and can be changed. The problem is that the equation is not true! My ...
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### What could be examples at calculus or introductory analysis level for the idea contained in the statement by David Hilbert?

I read the following quote in the book "As opposed to abstraction the art of doing mathematics consists in finding special cases which contain all the germs of generality. --David Hilbert", however ...
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### How is a general case equivalent to a special case and how showing a special case demonstrates a general case, in the proof of pythagoras theorem?

"The general theorem expressed by $\lambda a^2 = \lambda b^2 + \lambda c^2$ is equivalent not only to the special case $a^2 = b^2 + c^2$ but to any other special case. Therefore, if any such ...
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### Where can I find a lot of good exercises on the wave equation?

I find myself in the situation of needing to understand the wave equation inside and out -- I've studied it, obviously, and have been looking for resources for some time. So far in my search I'm ...
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### Solving problems of the form $x^c - c^x = d$ in the complex plane.

Is there a known procedure for solving for $x$ in $x^c - c^x = d$ with known $c, d \in \mathbb C$?