## Search found 176 matches

Thu Mar 30, 2017 1:21 am
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 22563

### Re: Beginner's Marathon

Problem 8:

Suppose that 5 points lie on a sphere. Prove that there exists a closed semi-sphere (half a sphere including boundary), which contains 4 of the points.
Thu Mar 30, 2017 1:15 am
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 22563

### Re: Beginner's Marathon

P7. There are 2000 points on a circle and each point is given a number which is equal to the average of the two numbers which are its nearest neighbors. Show that all the numbers must be equal. An alternate Solution: We use extremal principal. Since there a finite number of numbers, there must exis...
Mon Mar 06, 2017 6:13 pm
Forum: Junior Level
Replies: 7
Views: 2969

### Re: Don't Intersect Please :D

Given $2n$ points in the plane with no three collinear, show that it is possible to pair them up in such a way that the $n$ line segments joining paired points do not intersect. Here is my take. Take all possible pairings of the points, and select the one with minimum sum of the length of the segme...
Wed Mar 01, 2017 5:44 pm
Forum: Combinatorics
Topic: Combi Marathon
Replies: 48
Views: 27707

### Re: Combi Marathon

$\text{Problem 14:}$ There are $n + 1$ cells in a row labeled from $0$ to $n$ and $n + 1$ cards labeled from $0$ to $n$. The cards are arbitrarily placed in the cells, one per cell. The objective is to get card $i$ into cell $i$ for each $i$. The allowed move is to find the smallest $h$ su...
Wed Mar 01, 2017 5:25 pm
Forum: Combinatorics
Topic: Combi Marathon
Replies: 48
Views: 27707

### Re: Combi Marathon

$\text{Solution to Problem 13}$ Lemma: Let $S$ be a set of $m$ integers. Let $T$ be the set of integers that can be written as a sum of $t$ distinct elements of $S$. Then $|T|\ge t(m-t)+1$. Proof: Let $a_1<a_2<a_3<\ldots <a_m$ be the elements of $S$. Now, take the sum $a_1+a_2+\ldots +a_t$. Now, we ...
Tue Feb 28, 2017 2:32 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63045