I think I get it, but it is a little difficult:
##
\begin{align*} \det B &= \sum_{\sigma\in S_n} \epsilon(\sigma) b_{\sigma(1)1}\cdots b_{\sigma(n)n}\\
&= \sum_{\sigma\in S_n} \epsilon(\sigma) a_{\sigma(1)1}\cdots a_{\sigma(n)n} (-1)^{\#S_\sigma}
\end{align*}##
where set ##S_\sigma =...
The problem with the determinants in the cofactors is that neighbor columns have the same sign and break the 'chessboard' structure.
That's why I reverse the sign of the ##k-1## first columns, in order to recover this structure, which allows me to apply ##{\cal P}(n-1)##.
Homework Statement
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Given a general triangle ABC, find the geometric locus of points such that the three orthoprojection onto the sides of the triangle are aligned.
Homework Equations
Let's call A', B', and C' the orthoprojection of a given point M onto (AB) , (BC) , and (AC).
M satisfies...
Homework Statement
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Given a rectangular parallelepiped ABCDEFGH, the diagonal [AG] crosses planes BDE and CFH in K and L. Show K and L are BDE's and CFH's centres of gravity.
I think I have understood the problem, could you verify my demo please ? Thanks
Homework Equations
The Attempt at...
Oh I know the culprit, it was a masouran (or plotosus lineatus), which is a sort of small catfish. It is cute and peaceful, but its sting is very painful
I spent my holidays in Mauritius island (indian ocean) and was fishing with cousins living here when I got stung several times by a poisonous fish in the foot. I felt a fast growing pain, was sweating and feeling dizzy, and by the time I reached my cousins on the beach, I started to have...
Number 10, the infinite product is finite and equal to ##\pi / 2##. For this I wrote
##\prod_{n = 1 }^N \frac{4n^2}{4n^2-1} = 4^N (N!)^2 \frac{2^N N!}{(2N)!} \frac{2^N N!}{(2N+1)!} = \frac{4^{2N} (N!)^4}{(2N+1) ((2N)!)^2 } ##
And Stirling's formula leads to the conclusion
The chief of the forty thieves has to devise a strategy that will give him the highest possible probability of survival.
Let us call strategy ##(\alpha,\beta)## the strategy that consists in placing ##\alpha## white and ##\beta## black balls in box 1, and let us define the events
##A##...
A function is a relation between two sets ##A## and ##B##, which might contain any type of object you like, but with the restriction that each object of ##A## is in relation with at most one object of ##B##, which is zero or one object. The definition set of a function is the subset of ##A##...
Ok, thank you everybody for your participation. I think that this game can end now as it does not interest the public it is for. I hope that you liked the problem as much as I did.
@BiGyElLoWhAt , if you still want to give away the solution, the last word belongs to you.
1- Any choice is allowed, knowing that the only thing that will save him is to pick a white ball. An empty box leads to his execution
2- No handling possible after the distribution
Homework Statement
This is a problem that I really liked and that I want to share with you. Firstly because of the story around it, secondly because of the unexpected solution, and finally because it can be investigated with a computer for those who are the least comfortable with maths...
Yes, this is the way I've done it, and I see no content in this kind of exercise. As you say, it is almost without words!
Furthermore, a 13yo who doesn't know or forgot that the median line splits the area of a triangle in two is done without trigonometry. He will never find, unless someone sees...
Homework Statement
On the picture, compare the area of triangle ABC to the area of A'B'C'.
This problem was shown to me by a 13 years old. Trigonometry forbidden. It seems to me that this is the kind of problem you either solve in 2 minutes, or never solve. In both cases, you don't learn...
Lol, this is not a lonely question in a problem set, it is the conclusion of a lengthy problem on sequences, and all the steps needed to answer this question were worked in previous questions. There was almost no work involved here, but honestly, this equality looks too good, and I found it hard...
Yes, we had the analytical answer, as the limit of two converging sequences, but the way the question was put, I suddenly had a doubt and needed to see it work on a computer program. But thanks to @phyzguy 's program I am convinced now :-)
Homework Statement
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I'm helping a 12th grader with his homework, and he is asked to prove the following equality as the conclusion of a problem :
## 1 + \frac{1}{1+ \frac{1}{1+...}} = \sqrt{1+ \sqrt{1+ \sqrt{1+...}}}##
Written like this, the formula is intimidating while it really...
If it is true that many of the best scientists are also good musicians, I'm not sure there is a causal link between the 2 but rather an incidental link.
Many of the best scientists also attended the best schools, and the people attending the best schools are mostly but not only the offsprings...
Homework Statement
We fit the plane with a coordinate system, and we consider the set of points with coordinates in ##\mathbb{N}\times\mathbb{N} ##. To link two points in this coordinate system, we only allow unit displacements, and only increasing displacements.
In how many ways can one...
You really can't compare. Try to take these exams and you'll see that the expectations are very different. You should ask ' I'm chinese, what are the best books to prepare university in China ? '.
It's not an easy question, because every school system prepares the students for a specific final exam which can be very different from a country to another (graduation in the USA, gaokao in China, abitur in Germany, baccalaureat in France, etc ... ). It would not surprise me that an excellent...
The problem statement states that all three points belong to the unit circle, so ##z_2## and ##z_3## can be deduced from ##z_1## by a rotation centered at the origin. That is to say there exists angles ##\theta## and ##\rho## to determine such that ##z_2 = e^{i\theta} z_1 ## and ## z_3 =...
Imagine that you had to prove the reciprocal, that is, if ##ABC## is equilateral, then ##z_1 + z_2 + z_3 = 0##. This can be solved by an elementary geometric argument : ##\angle AOB = \angle BOC = \angle COA = 2\pi / 3##. So ## z_2 = e^{2i\pi/3} z_1 ## and ## z_3 = e^{2i\pi/3} z_2 ##. The cubic...
Consider points ##A(z_1)##, ##B(z_2)##, and ##C(z_3)##. Triangle ##ABC## is equilateral iff ##C## is deduced from ##B## by a rotation of center ##A## and angle ##\pi / 3##. How do you write such transformation ?
It's interesting, but what is exactly that 'force' that can force a ~50 years old man, overweight furthermore, to act like a kangaroo eating fictitious peanuts thrown at him for 15 minutes ? Have you ever tried to jump for 15 minutes ? It's hard ! Do you know how it works ?
Yes I asked, she said she knew what she was doing but that she could not resist from doing it. But the first thing we noticed is that she had difficulties 'waking up'. She was very tired after it.
For christmas, my wife offered a collective gift to my niece, my nephew, and myself: a ticket for a famous hypnosis show we desperatly wanted to see. The show in itself was absolutely incredible, fantastic. At the end, the artist, who knows that the public is split between absolute wonder and...
I suggest middle school + high school textbooks. The fundamental set of knowledge is here (except for series which is a university level topic). I recently read an excellent middle school level textbook (not in english unfortunately) and I really enjoyed myself. In particular, I did learn things...
But why would one want to prove integrability of a step function defined on a segment ? Such functions are integrable by definition. It is like if someone said 'Let ABCD a rectangle. Show that angle ABC equals 90 degrees'. I don't get the subtlety.
I don't understand the problem statement. The integral of a step function defined on a segment is a fundamental definition rather than a property. It seems to me that you are asked to prove a definition, which seems to be nonsense to me.
Spanking is not meant to physically harm children. It is meant to teach/remind the limits not to cross (ex: in the park, your kid bites another kid. Are you going to 'positively reinforce' him or spank him ? ). It must be incredibly difficult to raise a kid who feels his parents are afraid to...
I thought the riddle would last a little longer, but @andrewkirk came up with a written answer in 15 minutes ! :woot:
An alternative answer is the following: there are 60 equal angular sections on the watch. Every hour, the hour needle crosses exactly five angular sections, and between two...
A teenager is looking at his father's beautiful mechanical swiss made watch, and asks him: 'Dad, is there a time of the day, beside noon and midnight, where the hour and minutes needles are aligned and pointing in the same direction ? '. What would you answer at the nearest second ?
@theBin: Honestly you are attracting attention for nothing and I don't like it.
As I said, it is a high school problem, and anyone that knows two or three things about line intersections can get to the same result I got to, in minutes, following the process described in a previous post. So...
@theBin : it is just a high school problem that I tried to brush up on my geometry, so it doesn't need advanced geometry to be solved.
However, you say that the 'or parallel' statement is wrong, but it's not what I find (by the means I used).
Intuititively, I understand the idea, but I don't immediately see how it simplifies the problem. In my mind, it amounts to replace the word 'parallelogram' by the word 'rectangle' in the problem statement.