## Descartes' rules of signs

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 ...

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- Shen-Fu Tsai

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 ...

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

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

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

read more »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...

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

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

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

read more »A simple yet surprisingly interesting result.

read more »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...

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

read more »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 ...

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

read more »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 ...

read more »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...

read more »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...

read more »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...

read more »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...

read more »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...

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

read more »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 ...

read more »International Mathematical Olympiad (IMO) 2014 Problem 6

read more »Asian Pacific Mathematics Olympiad (APMO) 2015 Problem 4

read more »International Mathematical Olympiad (IMO) 2010 Problem 6

read more »International Mathematical Olympiad (IMO) 2015 Problem 6

read more »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 …

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

read more »The infamous Grasshopper problem

read more »Insane DFS

read more »Swap and Sum

read more »Robot

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