Found problems: 163
2007 IMC, 4
Let $ G$ be a finite group. For arbitrary sets $ U, V, W \subset G$, denote by $ N_{UVW}$ the number of triples $ (x, y, z) \in U \times V \times W$ for which $ xyz$ is the unity .
Suppose that $ G$ is partitioned into three sets $ A, B$ and $ C$ (i.e. sets $ A, B, C$ are pairwise disjoint and $ G = A \cup B \cup C$). Prove that $ N_{ABC}= N_{CBA}.$
2009 IMC, 1
Let $\ell$ be a line and $P$ be a point in $\mathbb{R}^3$. Let $S$ be the set of points $X$ such that the distance from $X$ to $\ell$ is greater than or equal to two times the distance from $X$ to $P$. If the distance from $P$ to $\ell$ is $d>0$, find $\text{Volume}(S)$.
2011 IMC, 2
An alien race has three genders: male, female and emale. A married triple consists of three persons, one from each gender who all like each other. Any person is allowed to belong to at most one married triple. The feelings are always mutual ( if $x$ likes $y$ then $y$ likes $x$).
The race wants to colonize a planet and sends $n$ males, $n$ females and $n$ emales. Every expedition member likes at least $k$ persons of each of the two other genders. The problem is to create as many married triples so that the colony could grow.
a) Prove that if $n$ is even and $k\geq 1/2$ then there might be no married triple.
b) Prove that if $k \geq 3n/4$ then there can be formed $n$ married triple ( i.e. everybody is in a triple).
2006 IMC, 1
Let $f: \mathbb{R}\to \mathbb{R}$ be a real function. Prove or disprove each of the following statements.
(a) If f is continuous and range(f)=$\mathbb{R}$ then f is monotonic
(b) If f is monotonic and range(f)=$\mathbb{R}$ then f is continuous
(c) If f is monotonic and f is continuous then range(f)=$\mathbb{R}$
2003 IMC, 3
Let $A$ be a closed subset of $\mathbb{R}^{n}$ and let $B$ be the set of all those points $b \in \mathbb{R}^{n}$ for which there exists exactly one point $a_{0}\in A $ such that $|a_{0}-b|= \inf_{a\in A}|a-b|$.
Prove that $B$ is dense in $\mathbb{R}^{n}$; that is, the closure of $B$ is $\mathbb{R}^{n}$
2014 Contests, 3
Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial
$$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$
has $n$ distinct real roots.
(Proposed by Stephan Neupert, TUM, München)
2008 IMC, 6
Let $ \mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$, and suppose that $ S$ is a set of points (not necessarily countable) in $ \mathcal{H}$ such that the distance between any two distinct points in $ S$ is equal to $ d$. Show that there is a point $ y\in\mathcal{H}$ such that
\[ \left\{\frac{\sqrt{2}}{d}(x\minus{}y): \ x\in S\right\}\]
is an orthonormal system of vectors in $ \mathcal{H}$.
2016 IMC, 4
Let $k$ be a positive integer. For each nonnegative integer $n$, let $f(n)$ be the number of solutions $(x_1,\ldots,x_k)\in\mathbb{Z}^k$ of the inequality $|x_1|+...+|x_k|\leq n$. Prove that for every $n\ge1$, we have $f(n-1)f(n+1)\leq f(n)^2$.
(Proposed by Esteban Arreaga, Renan Finder and José Madrid, IMPA, Rio de Janeiro)
2003 IMC, 5
a) Show that for each function $f:\mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{R}$, there exists a function $g:\mathbb{Q}\rightarrow \mathbb{R}$ with $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{Q}$.
b) Find a function $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, for which there is no function $g:\mathbb{Q}\rightarrow \mathbb{R}$ such that $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{R}$.
2012 IMC, 5
Let $c \ge 1$ be a real number. Let $G$ be an Abelian group and let $A \subset G$ be a finite set satisfying $|A+A| \le c|A|$, where $X+Y:= \{x+y| x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$. Prove that
\[|\underbrace{A+A+\dots+A}_k| \le c^k |A|\]
for every positive integer $k$.
[i]Proposed by Przemyslaw Mazur, Jagiellonian University.[/i]
2019 IMC, 9
Determine all positive integers $n$ for which there exist $n\times n$ real invertible matrices $A$ and $B$ that satisfy $AB-BA=B^2A$.
[i]Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan[/i]
2017 IMC, 3
For any positive integer $m$, denote by $P(m)$ the product of positive divisors of $m$ (e.g $P(6)=36$). For every positive integer $n$ define the sequence
$$a_1(n)=n,\qquad a_{k+1}(n)=P(a_k(n))\quad (k=1,2,\dots,2016)$$
Determine whether for every set $S\subset\{1,2,\dots,2017\}$, there exists a positive integer $n$ such that the following condition is satisfied:
For every $k$ with $1\leq k\leq 2017$, the number $a_k(n)$ is a perfect square if and only if $k\in S$.
1994 IMC, 6
Let $f\in C^2[0,N]$ and $|f'(x)|<1$, $f''(x)>0$ for every $x\in [0, N]$. Let $0\leq m_0\ <m_1 < \cdots < m_k\leq N$ be integers such that $n_i=f(m_i)$ are also integers for $i=0,1,\ldots, k$. Denote $b_i=n_i-n_{i-1}$ and $a_i=m_i-m_{i-1}$ for $i=1,2,\ldots, k$.
a) Prove that
$$-1<\frac{b_1}{a_1}<\frac{b_2}{a_2}<\cdots < \frac{b_k}{a_k}<1$$
b) Prove that for every choice of $A>1$ there are no more than $N / A$ indices $j$ such that $a_j>A$.
c) Prove that $k\leq 3N^{2/3}$ (i.e. there are no more than $3N^{2/3}$ integer points on the curve $y=f(x)$, $x\in [0,N]$).
2005 IMC, 2
2) all elements in {0,1,2}; B[n] = number of rows with no 2 sequent 0's; A[n] with no 3 sequent elements the same; prove |A[n+1]|=3.|B[n]|
2018 IMC, 6
Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exists $k$ nonzero vectors $v_1,v_2,…,v_k$ in $\mathbb{R}^n$ such that for every pair $i,j$ of indices with $|i-j|>1$ the vectors $v_i$ and $v_j$ are orthogonal.
[i]Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.[/i]
2012 IMC, 2
Let $n$ be a fixed positive integer. Determine the smallest possible rank of an $n\times n$ matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal.
[i]Proposed by Ilya Bogdanov and Grigoriy Chelnokov, MIPT, Moscow.[/i]
2012 IMC, 5
Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients.
[i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]
2005 IMC, 3
What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$, we have $ \text{trace}\left(AB\right) \equal{} 0$ ?
2008 IMC, 3
Let $ n$ be a positive integer. Prove that $ 2^{n\minus{}1}$ divides
\[ \sum_{0\leq k < n/2} \binom{n}{2k\plus{}1}5^k.\]
1994 IMC, 1
a) Let $A$ be a $n\times n$, $n\geq 2$, symmetric, invertible matrix with real positive elements. Show that $z_n\leq n^2-2n$, where $z_n$ is the number of zero elements in $A^{-1}$.
b) How many zero elements are there in the inverse of the $n\times n$ matrix
$$A=\begin{pmatrix} 1&1&1&1&\ldots&1\\
1&2&2&2&\ldots&2\\
1&2&1&1&\ldots&1\\
1&2&1&2&\ldots&2\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
1&2&1&2&\ldots&\ddots
\end{pmatrix}$$
2007 IMC, 3
Call a polynomial $ P(x_{1}, \ldots, x_{k})$ [i]good[/i] if there exist $ 2\times 2$ real matrices $ A_{1}, \ldots, A_{k}$ such that
$ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$
Find all values of $ k$ for which all homogeneous polynomials with $ k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)
2014 IMC, 4
We say that a subset of $\mathbb{R}^n$ is $k$-[i]almost contained[/i] by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-[i]generic[/i] if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $(k, n)$, find the minimal number of $d(k, n)$ such that every finite $k$-generic set in $\mathbb{R}^n$ contains a $k$-generic subset with at most $d(k, n)$ elements.
(Proposed by Shachar Carmeli, Weizmann Inst. and Lev Radzivilovsky, Tel Aviv Univ.)
2009 IMC, 2
Let $A,B,C$ be real square matrices of the same order, and suppose $A$ is invertible. Prove that
\[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]
2013 IMC, 4
Let $\displaystyle{n \geqslant 3}$ and let $\displaystyle{{x_1},{x_2},...,{x_n}}$ be nonnegative real numbers. Define $\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }$. Prove that:
\[\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.\]
[i]Proposed by Géza Kós, Eötvös University, Budapest.[/i]
2004 IMC, 3
Let $A_n$ be the set of all the sums $\displaystyle \sum_{k=1}^n \arcsin x_k $, where $n\geq 2$, $x_k \in [0,1]$, and $\displaystyle \sum^n_{k=1} x_k = 1$.
a) Prove that $A_n$ is an interval.
b) Let $a_n$ be the length of the interval $A_n$. Compute $\displaystyle \lim_{n\to \infty} a_n$.