| Author | Maths Problems |
|---|
for ULTRA_XEROX:
1} seems easy. Take n=1 and a,b,c are naturals such that a+b=c |
for ULTRA_XEROX:
Sry, didnt understand questions 2,3. maybe much above my level |
1. http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem (solved) :(
2. http://en.wikipedia.org/wiki/Squaring_the_circle (solved) :(
3. http://en.wikipedia.org/wiki/P_versus_NP_problem (not solved)
biggest problem of theoretical informatics. you solve it - you'll become famous, millionair and you'll make lot of theoretical information scientists happy :) |
| You forgot to mention that n has to be greater than zero in 1st questiion |
| You forgot to mention that n has to be greater than two in 1st questiion |
| i'll try to post later some another interesting math problems, which can be solved also by usual players =) |
| #105, yeah sorry, my fault... =( |
No The Answer for Post 97 : 30
How did you get that,Lord STB? |
Problem #1
Prove that:
x/sqrt(x^2 + 8*z*y) + y/sqrt(y^2 + 8*z*x) + z/sqrt(z^2 + 8*x*y) >= 1
for all positive real numbers a,b,c
http://dcdn.lordswm.com/photo-catalog/0001305/648.jpg
________________________________________
Problem #2
http://dcdn.lordswm.com/photo-catalog/0001305/649.jpg
is this triangle possible? give a proof if yes or not.
Player banned by moderator ElfPride until 2013-12-24 17:12:36 // FR#4.2.1//Abuse of redundant characters are forbidden,including delimitation of several parts of a large post (warning) |
for ULTRA_XEROX: For Q2, the answer must be no.. I am not good proofs, but i am gonna go with :
1. The given Triangle is an acute triangle................Assumption on the basis of diagram
2. all the three central angles will be obtuse angles.
3. Therefore, the central angles will be bigger than angles of the big triangle
4. Therefore, The existence of three isosceles triangles inside is not possible |
#110
1. The given Triangle is an acute triangle................Assumption on the basis of diagram
1. do you mean equilateral? it is not necessarily that a=b=c. a,b,c are arbitrarily selected.
it is only a drawning, which can't give correct information about angles.
*hint_1: 3 isosceles triangles
http://mathworld.wolfram.com/IsoscelesTriangle.html
try to make a proof with angles under assumption that there are 3 isosceles triangles...
*hint_2: contradiction proof
example:
ALPHA = <a,a // "<" is a sign for angle
BETA = <b,b
GAMMA = <c,c
ALPHA + BETA + GAMMA = xx°
...
etc |
ALPHA = <a,a // "<" is a sign for angle
alpha is equal to angle between side "a" and "a" (because there are 2 "a" sides)
etc... |
Problem #2
http://dcdn.lordswm.com/photo-catalog/0001305/649.jpg
is this triangle possible? give a proof if yes or not.
! it is an 2D-triangle ! flat as piece of paper :) |
btw, here one nice citate:
another roof another proof
Paul Erdos, very interesting person, great mathematician :)
i think this citate is right here because this finally a math corner and he was a person who solved math problems....lots of them :) |
#109:
Answer to problem #1:
First assumption: x,y,z are positive, real numbers. (You wrote a,b,c)
Derivate the function considering x to be the variable. (I will not bother to write it down because I'm lazy.) It is now obvious that the function f(x) have a minimum in x=1. Since the functions of both y and z are identical to f(x), we realize that the minimum for f(x,y,z) is when x=y=z=1. Thus, it can not be lower than 1.
Answer to problem #2:
First look at one third of the triangle: a,a,b. It is obviously isosceles.
Now look at the next third of the triangle: b,b,c. It is obviously isosceles too.
It is now impossible to draw a line between the two corners where c is supposed to be and still have the "center point" inside the the greater triangle. |
#109
- Answer to problem #1:
First assumption: x,y,z are positive, real numbers. (You wrote a,b,c)
yes it should be x,y,z.
Derivate the function considering x to be the variable. (I will not bother to write it down because I'm lazy.) It is now obvious that the function f(x) have a minimum in x=1.
...partial differential equation... yeah it is right and will of course work, but it is a little bit "too complex route" here.
there exists also a simpler solution: inequation =)
- Answer to problem #2:
First look at one third of the triangle: a,a,b. It is obviously isosceles.
Now look at the next third of the triangle: b,b,c. It is obviously isosceles too.
i agree.
It is now impossible to draw a line between the two corners where c is supposed to be and still have the "center point" inside the the greater triangle.
nice proof sketch ;) |
#116
- Comments to problem #1:
...partial differential equation... yeah it is right and will of course work, but it is a little bit "too complex route" here.
there exists also a simpler solution: inequation =)
I tried that first but the answer didn't appear for a few minutes so I resorted to the bulletproof way. Since the formula itself almost screamed that it had a minimum in x=y=z=1, it was easy to replace y and z with 1 and it became easy to derivate:
f'(x) = 1/sqrt(x^2+8) - x^2/(x^2+8)^(3/2) - 8/(8x+1)^(3/2)
With this simplification, it isn't bulletproof anymore, but it still holds up decently.
I'm very much looking forward to seeing your solution.
- Comments to problem #2:
I also considered calculating the areas of the great triangle and compare it to the sum of areas of the 3 small triangles. It turns out that the sum of the 3 small ones are larger. This proof is more analytical, but takes a lot more effort to type the formulas. |
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| [Post deleted by moderator Lord STB // Asking for illegal script] |
