# International Mathematics Competition for University Students 2024

Select Year:

IMC 2024
 Information Schedule Problems & Solutions Results Contact Travel

## IMC2024: Day 1, Problem 5

Problem 5. Let $\displaystyle n>d$ be positive integers. Choose $\displaystyle n$ independent, uniformly distributed random points $\displaystyle x_1,\ldots,x_n$ in the unit ball $\displaystyle B\subset \mathbb{R}^d$ centered at the origin. For a point $\displaystyle p\in B$ denote by $\displaystyle f(p)$ the probability that the convex hull of $\displaystyle x_1,\ldots,x_n$ contains $\displaystyle p$. Prove that if $\displaystyle p,q\in B$ and the distance of $\displaystyle p$ from the origin is smaller than the distance of $\displaystyle q$ from the origin, then $\displaystyle f(p)\geq f(q)$.

Fedor Petrov, St Petersburg State University

 IMC1994 IMC1995 IMC1996 IMC1997 IMC1998 IMC1999 IMC2000 IMC2001 IMC2002 IMC2003 IMC2004 IMC2005 IMC2006 IMC2007 IMC2008 IMC2009 IMC2010 IMC2011 IMC2012 IMC2013 IMC2014 IMC2015 IMC2016 IMC2017 IMC2018 IMC2019 IMC2020 IMC2021 IMC2022 IMC2023

© IMC