[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 301
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Alsu, the set of tetrahedrons with the lowest PH height is the set of tetrahedrons whose base area ABC is maximal among all four face areas. It does not look like the whole plane...
If we take a point P whose projection H is a kilometre away from ABC, the area of ABC itself is unlikely to be maximal.
It looks like the set is such a triangle for which the sides of the original ABC are the midlines (except for the boundaries of the big triangle). At the vertices of this large triangle there will be H points corresponding to equal in area to the four faces of the (almost lying) tetrahedron.
The sides of the large triangle are not included in this set since tetrahedrons must have volume. If the point H is on the sides of the large triangle, then only some of the faces are guaranteed to be at least as large as the area of the ABC face.
On the other hand, if we take at least one point H' outside this large triangle, the area of at least one face is guaranteed to be larger than the area of ABC, regardless of the position of vertex P'. That is, this point H' is not in our set.
Next (8th grade):
Видимо плохие в а) школьники были, раз б) солдаты =)
а в в) будет куб из инопланетян?
Самые высокие солдаты берутся за ручки и уходят с плаца. Что они там делают, неважно...
Score! :)
And one more to follow (9th):
Let a=2;
Тогда n=a*(a+1)*(a+2)*.....*(a+19)*(a+20);
Let a=2;
Тогда n=a*(a+1)*(a+2)*.....*(a+19)*(a+20);
it doesn't look like
In[7]:= GCD[30030,n+1]Out[7]= 1
No, because 22!+1 is reciprocal simply with 30030. This can easily be proved, even without matched sets.
It can't be that such problems are now being solved in the 9th or 10th grades. Am I that far behind?
Give me a link to a 9th or 10th grader who can do this, I want to meet him.
9441 % 3
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