Mathematical Olympiad: International Mathematics Competition for University Students 2011

Saturday, September 24, 2011

International Mathematics Competition for University Students 2011

IMC 2011: Problems of 1st Day

Problem 1. Let $f: \Bbb{R} \to \Bbb{R}$ be a continuous function. A point $x$ is called a shadow point if there is a point $y \in \Bbb{R}$ with $y>x$ such that $f(y)>f(x)$ . Let $a<b$ be real numbers and suppose that

all points in $(a,b)$ are shadow points;

$a,b$ are not shadow points.

Prove that

a) $f(x) \leq f(b),\ \forall a<x<b$ ;

b) $f(a)=f(b)$ .

Problem 2. Does there exist a real $3 \times 3$ matrix $A$ such that $\text{tr}(A)=0$ and $A^2+A^t=I$ ?

Problem 3. Let $p$ be a prime number. Call a positive integer $n$ interesting if
$x^n-1=(x^p-x+1)f(x)+pg(x)$ for some polynomials $f,g \in \Bbb{Z}[X]$ .

a) Prove that the number $p^p-1$ is interesting.

b) For which $p$ is $p^p-1$ the minimal interesting number?

Problem 4. Let $A_1,A_2,...,A_n$ be finite, nonempty sets. Define the function

$\displaystyle f(t)=\sum_{k=1}^n \sum_{1\leq i_1 < i_2 <...< i_k\leq n} (-1)^{k-1} t^{|A_{i_1}\cup A_{i_2}\cup ... \cup A_{i_k}|}$

Prove that $f$ is nondecreasing on $[0,1]$ .

Problem 5. Let $n$ be a positive integer and let $V$ be a $(2n-1)$ -dimensional vector space over the field with two elements. Prove that for arbitrary vectors $v_1,v_2,...,v_{4n-1} \in V$ , there exists a sequence $1\leq i_1 <...< i_{2n} \leq 4n-1$ of indices such that $v_{i_1}+...+v_{i_{2n}}=0$ .

IMC 2011: Problems of 2st Day

Problem 1. Let $(a_n)\subset (\frac{1}{2},1)$ . Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$ . Is this sequence convergent? If yes find the limit.

Problem 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 $k\geq n/2$ 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).

Problem 3. Calculate $\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \left( 1+\frac{1}{2n}\right)\left( 1+\frac{1}{2n+1}\right)$ .

Problem 4. Let $f$ be a polynomial with real coefficients of degree $n$ . Suppose that $\displaystyle \frac{f(x)-f(y)}{x-y}$ is an integer for all $0 \leq x<y \leq n$ . Prove that $a-b | f(a)-f(b)$ for all distinct integers $a,b$ .

Problem 5. 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 $(f_1\circ f_2 \circ f_3 \circ...\circ f_{n-1})^n(F)=F$ .

Mathematical Olympiad

Saturday, September 24, 2011

International Mathematics Competition for University Students 2011

IMC 2011: Problems of 1st Day

IMC 2011: Problems of 2st Day

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