# International Mathematics Competition for University Students 2016

Select Year:

IMC 2019
 Information Results/Prizes Problems & Solutions Photos

## IMC2016: Problems on Day 1

1. Let $f\colon [a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Suppose that $f$ has infinitely many zeros, but there is no $x\in(a,b)$ with $f(x)=f'(x)=0$.

(a) Prove that $f(a)f(b)=0$.

(b) Give an example of such a function on $[0,1]$.

Proposed by Alexandr Bolbot, Novosibirsk State University

2. Let $k$ and $n$ be positive integers. A sequence $(A_1,\ldots,A_k)$ of $n\times n$ real matrices is preferred by Ivan the Confessor if $A_i^2\ne 0$ for $1\le i \le k$, but $A_iA_j=0$ for $1\le i,j\le k$ with $i\ne j$. Show that $k\le n$ in all preferred sequences, and give an example of a preferred sequence with $k=n$ for each $n$.

Proposed by Fedor Petrov, St. Petersburg State University

3. Let $n$ be a positive integer. Also let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be real numbers such that $a_i+b_i>0$ for $i=1,2,\ldots,n$. Prove that $$\sum_{i=1}^n \frac{a_ib_i-b_i^2}{a_i+b_i}\leq \frac{\sum\limits_{i=1}^n a_i \cdot \sum\limits_{i=1}^n b_i -\left(\sum\limits_{i=1}^n b_i\right)^2}{\sum\limits_{i=1}^n (a_i+b_i)}.$$

Proposed by Daniel Strzelecki, Nicolaus Copernicus University in TorĂșn, Poland

4. Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties:

(i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements;

(ii) for any two sets $A,B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$.

Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements.

Proposed by Fedor Petrov, St. Petersburg State University

5. Let $S_n$ denote the set of permutations of the sequence $(1,2,\dots,n)$. For every permutation $\pi=(\pi_1,\dots,\pi_n)\in S_n$, let $\mathrm{inv}(\pi)$ be the number of pairs $1\le i\lt j\le n$ with $\pi_i\gt \pi_j$; i.e. the number of inversions in $\pi$. Denote by $f(n)$ the number of permutations $\pi\in S_n$ for which $\mathrm{inv}(\pi)$ is divisible by $n+1$.

Prove that there exist infinitely many primes $p$ such that $f(p-1)\gt \dfrac{(p-1)!}p$, and infinitely many primes $p$ such that $f(p-1)\lt \dfrac{(p-1)!}p$.

Proposed by 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 IMC 2016 IMC2017 IMC2018 IMC2019