Olympiad problems

2023 IMC, 9

Tags: 3D geometry , Convexity , IMC , geometry , analytic geometry , IMC 2023

We say that a real number $V$ is [i]good[/i] if there exist two closed convex subsets $X$, $Y$ of the unit cube in $\mathbb{R}^3$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are disjoint. Find $\sup \{V\mid V\ \text{is good}\}$.

1994 IMC, 5

Tags: vector , induction , IMC , college contests

[b]problem 5.[/b] Let $x_1, x_2,\ldots, x_k$ be vectors of $m$-dimensional Euclidean space, such that $x_1+x_2+\ldots + x_k=0$. Show that there exists a permutation $\pi$ of the integers $\{ 1, 2, \ldots, k \}$ such that: $$\left\lVert \sum_{i=1}^n x_{\pi (i)}\right\rVert \leq \left( \sum_{i=1}^k \lVert x_i \rVert ^2\right)^{1/2}$$for each $n=1, 2, \ldots, k$. Note that $\lVert \cdot \rVert$ denotes the Euclidean norm. (18 points).

2009 IMC, 1

Tags: IMC , college contests

Suppose that $f,g:\mathbb{R}\to \mathbb{R}$ satisfying \[ f(r)\le g(r)\quad \forall r\in \mathbb{Q} \] Does this imply $f(x)\le g(x)\quad \forall x\in \mathbb{R}$ if [list] (a)$f$ and $g$ are non-decreasing ? (b)$f$ and $g$ are continuous?[/list]

1994 IMC, 3

Tags: IMC , real analysis , function , calculus , derivative

Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb R$. Show that for each pair of real numbers $a$, $b$, $a<b$, such that $$\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a$$ there is a number $c$ in the open interval $(a,b)$ for which $$f^{(n+1)}(c)=f(c)$$

2006 IMC, 2

Tags: function , IMC , college contests

Find all functions $f: \mathbb{R}\to{R}$ such that for any $a<b$, $f([a,b])$ is an interval of length $b-a$

2012 IMC, 2

Tags: limit , IMC , college contests

Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$, $a_1=\frac{1}{2}$, and \[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\] Show that the series $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value. [i]Proposed by Christophe Debry, KU Leuven, Belgium.[/i]

2013 IMC, 3

Tags: IMC , college contests

There are $\displaystyle{2n}$ students in a school $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$. Each week $\displaystyle{n}$ students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen? [i]Proposed by Oleksandr Rybak, Kiev, Ukraine.[/i]

2007 IMC, 2

Tags: modular arithmetic , algebra , polynomial , IMC , college contests

Let $ x$, $ y$ and $ z$ be integers such that $ S = x^{4}+y^{4}+z^{4}$ is divisible by 29. Show that $ S$ is divisible by $ 29^{4}$.

2017 IMC, 5

Tags: imc 2017 , IMC , polynomial , complex analysis

Let $k$ and $n$ be positive integers with $n\geq k^2-3k+4$, and let $$f(z)=z^{n-1}+c_{n-2}z^{n-2}+\dots+c_0$$ be a polynomial with complex coefficients such that $$c_0c_{n-2}=c_1c_{n-3}=\dots=c_{n-2}c_0=0$$ Prove that $f(z)$ and $z^n-1$ have at most $n-k$ common roots.

2020 IMC, 4

Tags: IMC , Polynomials , algebra , polynomial , college contests , IMC 2020

A polynomial $p$ with real coefficients satisfies $p(x+1)-p(x)=x^{100}$ for all $x \in \mathbb{R}.$ Prove that $p(1-t) \ge p(t)$ for $0 \le t \le 1/2.$

2011 IMC, 5

Tags: induction , geometry , perpendicular bisector , IMC , college contests

Let $F=A_0A_1...A_n$ be a convex polygon in the plane. Define for all $1 \leq k \leq n-1$ the operation $f_k$ which replaces $F$ with a new polygon $f_k(F)=A_0A_1..A_{k-1}A_k^\prime A_{k+1}...A_n$ where $A_k^\prime$ is the symmetric of $A_k$ with respect to the perpendicular bisector of $A_{k-1}A_{k+1}$. Prove that $(f_1\circ f_2 \circ f_3 \circ...\circ f_{n-1})^n(F)=F$.

2009 IMC, 3

Tags: IMC , college contests

In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from $1$ to $n$ and let $a_i$ be the number of friends of the $i^{\text{th}}$ resident. Suppose that \[ \sum_{i=1}^{n}a_i^2=n^2-n \] Let $k$ be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of $k.$

2009 IMC, 5

Tags: vector , linear algebra , matrix , IMC , college contests

Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S\subseteq \mathbb{M}$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$. Say that a vector subspace $T\subseteq \mathbb{M}$ is a $\emph{covering matrix space}$ if \[ \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p \] Such a $T$ is minimal if it doesn't contain a proper vector subspace $S\subset T$ such that $S$ is also a covering matrix space. [list] (a) (8 points) Let $T$ be a minimal covering matrix space and let $n=\dim (T)$ Prove that \[ \delta(T)\le \dbinom{n}{2} \] (b) (2 points) Prove that for every integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T=n$ and $\delta(T)=\dbinom{n}{2}$[/list]

2023 IMC, 4

Tags: algebra , polynomial , number theory , modular arithmetic , prime numbers , IMC , IMC 2023

Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?

2017 IMC, 10

Tags: college contests , IMC , imc 2017

Let $K$ be an equilateral triangle in the plane. Prove that for every $p>0$ there exists an $\varepsilon>0$ with the following property: If $n$ is a positive integer, and $T_1,\ldots,T_n$ are non-overlapping triangles inside $K$ such that each of them is homothetic to $K$ with a negative ratio, and $$ \sum_{\ell=1}^n \textrm{area}(T_\ell) > \textrm{area}(K)-\varepsilon, $$ then $$ \sum_{\ell=1}^n \textrm{perimeter}(T_\ell) > p. $$

2009 IMC, 3

Tags: linear algebra , matrix , algebra , polynomial , IMC , college contests

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that \[ A^2B+BA^2=2ABA \] Prove there exists $k\in \mathbb{N}$ such that \[ (AB-BA)^k=\mathbf{0}_n\] Here $\mathbf{0}_n$ is the null matrix of order $n$.

2017 IMC, 1

Tags: linear algebra , matrix , IMC , imc 2017

Determine all complex numbers $\lambda$ for which there exists a positive integer $n$ and a real $n\times n$ matrix $A$ such that $A^2=A^T$ and $\lambda$ is an eigenvalue of $A$.

2007 IMC, 5

Tags: function , modular arithmetic , algebra , polynomial , induction , IMC , college contests

Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.

2023 IMC, 10

Tags: number theory , asymptotic behaviour , IMC , college contests , IMC 2023

For every positive integer $n$, let $f(n)$, $g(n)$ be the minimal positive integers such that \[1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}.\] Determine whether there exists a positive integer $n$ for which $g(n)>n^{0.999n}$.

2008 IMC, 1

Tags: algebra , polynomial , modular arithmetic , ratio , number theory , relatively prime , IMC

Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.

2020 IMC, 8

Tags: IMC , real analysis , binomial coefficients , logarithms , IMC 2020

Compute $\lim\limits_{n \to \infty} \frac{1}{\log \log n} \sum\limits_{k=1}^n (-1)^k \binom{n}{k} \log k.$

2007 IMC, 1

Tags: geometry , geometric transformation , logarithms , function , rotation , IMC , college contests

Let $ f : \mathbb{R}\to \mathbb{R}$ be a continuous function. Suppose that for any $ c > 0$, the graph of $ f$ can be moved to the graph of $ cf$ using only a translation or a rotation. Does this imply that $ f(x) = ax+b$ for some real numbers $ a$ and $ b$?

2019 IMC, 6

Tags: IMC , differentiable function , function , real analysis , college contests

Let $f,g:\mathbb R\to\mathbb R$ be continuous functions such that $g$ is differentiable. Assume that $(f(0)-g'(0))(g'(1)-f(1))>0$. Show that there exists a point $c\in (0,1)$ such that $f(c)=g'(c)$. [i]Proposed by Fereshteh Malek, K. N. Toosi University of Technology[/i]

2005 IMC, 6

Tags: group theory , abstract algebra , IMC , college contests

6) $G$ group, $G_{m}$ and $G_{n}$ commutative subgroups being the $m$ and $n$ th powers of the elements in $G$. Prove $G_{gcd(m,n)}$ is commutative.

2012 IMC, 4

Tags: function , limit , integration , IMC , college contests

Let $f:\;\mathbb{R}\to\mathbb{R}$ be a continuously differentiable function that satisfies $f'(t)>f(f(t))$ for all $t\in\mathbb{R}$. Prove that $f(f(f(t)))\le0$ for all $t\ge0$. [i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]