Hard to believe that I don't remember seeing Descartes' rules of signs before -- you'd think this is impossible given the time I spent on ...

This is the last of a small series of similar and basic results in convex geometry.

Previous I used Caratheodory's theorem to prove Radon's theorem, now I'm going to prove the former.

I came across Radon's theorem the other day, and found this proof. It must have been discovered before.

This is not hard but still interesting. A spread of an array of numbers is the difference between its maximum and minimum. Given an array...

I heard this is classic, but turns out not too hard.

This is also an interesting result, although it's quite simple.

For (iin{1,2,3}), there are (2n+i) integers consisting of (n+i) unique ones.

A simple yet surprisingly interesting result.

There are 2n+1 coins each associated with a weight. When we remove any coin, we can split the rest into two piles each with n coins...

Given a sequence of (n) integers (a_1,ldots,a_n), we map it to ...

On the 2D plane there are (n) blue and (n) red points, no three of them are co-linear. Then we can always pair a blue point with a red ...

It is said to be well-known, but perhaps I've never seen it before:

Let (A(n)) denote the number of sequences (a_1ge a_2gecdots{}ge a_k) of positive integers for which (a_1+cdots+a_k = n) and each ...

We call a (5)-tuple of integers arrangeable if its elements can be labeled (a, b, c, d, e) in some order so that (a-b+c-d+e=29). Deter...

I found a different solution to the second part of this problem. We define a chessboard polygon to be a polygon whose edges are situate...

Let (m_1, m_2, ldots, m_n) be a collection of (n) positive integers, not necessarily distinct. For any sequence of integers (A = (a_1...

The problem is rephrased. Q: A diamond of size n consists of 2n rows of nodes with lengths from top to bottom being 1, 3, 5, ..., 2n-1...

Let (P_1, P_2, dots, P_{2n}) be (2n) distinct points on the unit circle (x^2+y^2=1), other than ((1,0)). Each point is colored eit...

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of...

There are n>=2 line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an ...

International Mathematical Olympiad (IMO) 2014 Problem 6

Asian Pacific Mathematics Olympiad (APMO) 2015 Problem 4

International Mathematical Olympiad (IMO) 2010 Problem 6

International Mathematical Olympiad (IMO) 2015 Problem 6

In a recent algorithmic coding contest which I didn't do well, the hardest problem killed me. It distinguished me from other superior coders. But it's still an interesting one. The problem is essentially to solve a 0/1 knapsack problem on a tree where each node is associated with an …

United States of America Mathematical Olympiad (USAMO) 2015 Problem 3

The infamous Grasshopper problem

Insane DFS

Swap and Sum

Robot