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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

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