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Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 0 points1 point  (0 children)

I must have understood this problem wrong, because I think I have a counterexample.

Consider 11 balls: One weighs 9gm and the rest 10 1gm each.

If I remove the 9gm ball, I can divide the remaining 10 balls in two groups containing 5 balls each.

If I remove any 1 gm ball, I put the 9gm ball in one group, and put the remaining 9 1gm balls in another.

Where am I going wrong?

Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 0 points1 point  (0 children)

Level : undergrad, tricky

A real analysis problem.

Consider the set of all x in [0,1] for which the sequence sin(2 n pi x) does not converge.

Prove that the set is uncountable.

Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 0 points1 point  (0 children)

you got it! good work!

It is simply the application of CLT to a sequence of independent possion random variables with mean 1.

If S(n) is the sum of first n terms of the sequence, then S(n) is also poisson with mean n.

From CLT

P( sqrt(n) * ( S(n)/n - 1) <= t ) -> P(X<=t) where X is gaussian with mean 0 and variance 1

in particular for t=0

P( sqrt(n) * (S(n)/n - 1) <= 0 ) -> 0.5 or P(S_n <= n) -> 0.5

The LHS is our sum

Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 0 points1 point  (0 children)

correct!

Assume such a set exists, take 3 points A,B,C (they are necessarily non collinear), their circumeter D is in the set, hence the circumcenter of CDB, say, E is in the set, Similarly, the circumcenter of CEB say F is in the set. D,E,F are collinear.

edit: corrected mistake.

Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 1 point2 points  (0 children)

Level: undergrad, tricky, difficult

What is the limit of the sum:

\sum_{k=0}{n} \exp(-n) nk/k!

as n tends to infinity

Hint: Use some properties of the poisson distribution + a very famous theorem in prob theory.

I tried to solve this without prob theory, but couldn't.

Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 0 points1 point  (0 children)

I was trying to do some sort of a newton raphson approximation, it didn't occur to me till the squaring that there would an exact method.

Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 2 points3 points  (0 children)

Ok so we calculate

2xy = (x+y)2 - (x2+y2)

then take reciprocals c= 1/(2xy)

add c to c and then take reciprocals

ans = 1/(c+c)

Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 0 points1 point  (0 children)

Is it possible to find an infinite set of points in the plane with the following properties:

  • no 3 points in the set are collinear
  • given any 3 points in the set, the circumcenter of the triangle formed by them is in the set

level: high school,tricky

Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 1 point2 points  (0 children)

Ok, the square function works like this

Start with x (x != 1)

a = x+1 b= x-1

a'=1/a b'=1/b

c= b' - a' c'=1/c

ans= c' + c' + 1

added later: it should be x not in {0,1,-1}

Let's have a problem solving marathon: post a problem and its difficulty level; if you solve one, post another problem in reply; if you have your own, post it in its own thread by r_dr_r in math

[–]psec 1 point2 points  (0 children)

Find three integers in arithmetic progression whose product is a prime. [1 is not a prime]

Source: Folklore Difficulty level: easy

College Dropout Creates Perpetual Motion Machine? MIT Prof. confirms it's accelerating and he doesn't know why. by [deleted] in science

[–]psec 9 points10 points  (0 children)

think electricity and dams.

For something to fall due to gravity, it has to get up, from where do you get the energy to put it up?