13th Internat.Math.Competition for Univ.Students,Odessa-2006

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13th International Mathematical Competition (IMC) for University Students, Odessa-2006
1st Day, 22 July 2006

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 $\text{range}(f)=\mathbb{R}$ then $f$ is monotonic. (b) If $f$ is monotonic and $\text{range}(f)=\mathbb{R}$ then $f$ is continuous. (c) If $f$ is monotonic and $f$ is continuous then $\text{range}(f)=\mathbb{R}$.

2. Find the number of positive integers $x$ satisfying the following two conditions: 1) $x<10^{2006}$, 2) $x^{2}-x$ is divisible by $10^{2006}$.

3. Let $A$ be an $n\times n$ matrix with integer entries and $b_{1},b_{2},...,b_{k}$ be integers satisfying $\det A=b_{1}\cdot b_{2}\cdot ...\cdot b_{k}$. Prove that there exist $n\times n$-matrices $B_{1},B_{2},...,B_{k}$ with integers entries such that $A=B_{1}\cdot B_{2}\cdot ...\cdot B_{k}$ and $\det B_{i}=b_{i}$ for all $i=1,...,k$.

4. Let $f$ be a rational function (i.e. the quotient of two real polynomials) and suppose that $f(n)$ is an integer for infinitely many integers $n$. Prove that $f$ is a polynomial.

5. Let $a, b, c, d, e$ be five strictly positive real numbers such that
\begin{gather*} a^{2}+b^{2}+c^{2}=d^{2}+e^{2}, \\ a^{4}+b^{4}+c^{4}=d^{4}+e^{4}. \end{gather*}
Compare

\[ a^{3}+b^{3}+c^{3} \]

with

\[ d^{3}+e^{3}. \]

6. Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true:
If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that $f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which

\[ a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0. \]

General info: http://www.imc-math.org/index.php?year=2006
Problem 1: http://www.mathlinks.ro/Forum/viewtopic.php?t=103086
Problem 2: http://www.mathlinks.ro/Forum/viewtopic.php?t=103090
Problem 3: http://www.mathlinks.ro/Forum/viewtopic.php?t=103092
Problem 4: http://www.mathlinks.ro/Forum/viewtopic.php?t=103093
Problem 5: http://www.mathlinks.ro/Forum/viewtopic.php?t=103089
Problem 6: http://www.mathlinks.ro/Forum/viewtopic.php?t=103094

[dm]

Условия 1-го дня в PDF-файле: http://rapidshare.de/files/26730854/imc ... a.pdf.html

[dm]

13th International Mathematical Competition (IMC) for University Students, Odessa-2006
2nd Day, 23 July 2006

1. Let $V$ be a convex polygon. (a) Show that if $V$ has $3k$ vertices, then $V$ can be triangulated such that each vertex is in an odd number of triangles. (b) Show that if the number of vertices is not divisible with 3, then $V$ can be triangulated such that exactly 2 vertices have an even number of triangles.

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

3. Compare $\tan(\sin x)$ with $\sin(\tan x)$, for $x\in (0,\frac{\pi}{2})$.

4. Let $v_{0}$ be the zero vector and let $v_{1},...,v_{n+1}\in\mathbb{R}^{n}$ such that the Euclidian norm $|v_{i}-v_{j}|$ is rational for all $0\le i,j\le n+1$. Prove that $v_{1},...,v_{n+1}$ are linearly dependent over the rationals.

5. Show that there are an infinity of integer numbers $m$, $n$, with $\gcd(m,n)=1$ such that the equation $(x+m)^{3}=nx$ has 3 different integer solutions.

6. Let $A_{i}$, $B_{i}$, $S_{i}$ ($i=1,2,3$) be invertible real $2\times 2$ matrices such that (1) not all $A_{i}$ have a common real eigenvector, (2) $A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i=1,2,3$, (3) $A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I$. Prove that there is an invertible $2\times 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1,2,3$.

Problem 1: http://www.mathlinks.ro/Forum/viewtopic.php?t=103464
Problem 2: http://www.mathlinks.ro/Forum/viewtopic.php?t=103466
Problem 3: http://www.mathlinks.ro/Forum/viewtopic.php?t=103467
Problem 4: http://www.mathlinks.ro/Forum/viewtopic.php?t=103663
Problem 5: http://www.mathlinks.ro/Forum/viewtopic.php?t=103468
Problem 6: http://www.mathlinks.ro/Forum/viewtopic.php?t=103665

[dm]

Появились тексты задач 4 и 6 второго дня.

Появились результаты: http://www.imc-math.org.uk/imc2006/results2006.htm

Условия двух дней в PDF-файле: http://rapidshare.de/files/27206493/imc ... a.pdf.html

[dm]

Появились авторские решения:
1-й день: http://www.imc-math.org.uk/imc2006/day1_solutions.pdf
2-й день: http://www.imc-math.org.uk/imc2006/day2_solutions.pdf
Условия: http://www.mathlinks.ro/Forum/viewtopic ... ch&id=4661

Еще выложили фотки:
http://www.imc-math.org.uk/index.php?ye ... tem=photos

Обсуждение на форуме: College Playground » Contests for Undergraduates » International Mathematics Competition @ MathLinks.Ro.