RO-NMO 2007

In a triangle $ABC$ , where $a = BC$ , $b = CA$ and $c = AB$ , it is known that: $a + b - c = 2$ and $2ab - c^2 = 4$ . Prove that $ABC$ is an equilateral triangle.

Consider the triangle $ABC$ with $m(\angle BAC = 90^\circ)$ and $AC = 2AB$ . Let $P$ and $Q$ be the midpoints of $AB$ and $AC$ ,respectively. Let $M$ and $N$ be two points found on the side $BC$ such that $CM = BN = x$ . It is also known that $2S[MNPQ] = S[ABC]$ . Determine $x$ in function of $AB$ .

Consider the triangle $ABC$ with $m(\angle BAC) = 90^\circ$ and $AB < AC$ .Let a point $D$ on the side $AC$ such that: $m(\angle ACB) = m(\angle DBA)$ .Let $E$ be a point on the side $BC$ such that $DE\perp BC$ .It is known that $BD + DE = AC$ . Find the measures of the angles in the triangle $ABC$ .

Let $m,n$ be two natural numbers with $m > 1$ and $2^{2m + 1} - n^2\geq 0$ . Prove that: $$2^{2m + 1} - n^2\geq 7 .$$

Prove that the number $10^{10}$ can't be written as the product of two natural numbers which do not contain the digit " $0$ " in their decimal representation.

In a building there are 6018 desks in 2007 rooms, and in every room there is at least one desk. Every room can be cleared dividing the desks in the oher rooms such that in every room is the same number of desks. Find out what methods can be used for dividing the desks initially.

a) In a triangle $MNP$ , the lenghts of the sides are less than $2$ . Prove that the lenght of the altitude corresponding to the side $MN$ is less than $\sqrt {4 - \frac {MN^2}{4}}$ .

b) In a tetrahedron $ABCD$ , at least $5$ edges have their lenghts less than $2$ .Prove that the volume of the tetrahedron is less than $1$ .

Let $ABCD$ be a tetrahedron.Prove that if a point $M$ in a space satisfies the relation: \begin{align*} MA^2 + MB^2 + CD^2 &= MB^2 + MC^2 + DA^2 &= MC^2 + MD^2 + AB^2 &= MD^2 + MA^2 + BC^2 . \end{align*} then it is found on the common perpendicular of the lines $AC$ and $BD$ .

The plane is divided into strips of width $1$ by parallel lines (a strip - the region between two parallel lines). The points from the interior of each strip are coloured with red or white, such that in each strip only one color is used (the points of a strip are coloured with the same color). The points on the lines are not coloured. Show that there is an equilateral triangle of side-length $100$ , with all vertices of the same colour.

Given a set $A$ and a function $f: A\rightarrow A$ , denote by $f_{1}(A)=f(A)$ , $f_{2}(A)=f(f_{1}(A))$ , $f_{3}(A)=f(f_{2}(A))$ , and so on, ( $f_{n}(A)=f(f_{n-1}(A))$ , where the notation $f(B)$ means the set $\{ f(x) \ : \ x\in B\}$ of images of points from $B$ ). Denote also by $f_{\infty}(A)=f_{1}(A)\cap f_{2}(A)\cap \ldots = \bigcap_{n\geq 1}f_{n}(A)$ .

a) Show that if $A$ is finite, then $f(f_{\infty}(A))=f_{\infty}(A)$ .

b) Determine if the above is true for $A=\mathbb{N}\times \mathbb{N}$ and the function $$f\big((m,n)\big)=\begin{cases}(m+1,n) & \text{if }n\geq m\geq 1 (0,0) & \text{if }m>n (0,n+1) & \text{if }n=0. \end{cases}$$

Show that the equation $z^{n}+z+1=0$ has a solution with $|z|=1$ if and only if $n-2$ is divisble by $3$ .

For which integers $n\geq 2$ , the number $(n-1)^{n^{n+1}}+(n+1)^{n^{n-1}}$ is divisible by $n^{n}$ ?

a) For a finite set of natural numbers $S$ , denote by $S+S$ the set of numbers $z=x+y$ , where $x,y\in S$ . Let $m=|S|$ . Show that $|S+S|\leq \frac{m(m+1)}{2}$ .

b) Let $m$ be a fixed positive integer. Denote by $C(m)$ the greatest integer $k\geq 1$ for which there exists a set $S$ of $m$ integers, such that $\{1,2,\ldots,k\}\subseteq S\cup(S+S)$ . For example, $C(3)=8$ , with $S=\{1,3,4\}$ . Show that $\frac{m(m+6)}{4}\leq C(m) \leq \frac{m(m+3)}{2}$ .

Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$ . Prove that $(AB-BA)^{2}=O_{2}$ .

Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$ ).

Show that there is an interval $I$ such that $f(I)=[a,b]$ .

Let $n\geq 2$ be an integer and denote by $H_{n}$ the set of column vectors $^{T}(x_{1},\ x_{2},\ \ldots, x_{n})\in\mathbb{R}^{n}$ , such that $\sum |x_{i}|=1$ .

Prove that there exist only a finite number of matrices $A\in\mathcal{M}_{n}(\mathbb{R})$ such that the linear map $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ given by $f(x)=Ax$ has the property $f(H_{n})=H_{n}$ .

a) Prove the above for $n=2$ ;

b) Prove the above for $n\geq 3$ as well.

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$ . Let's call this property $(P)$ .

a) Show that if $f$ is a function with property $(P)$ , then there exists a real $x$ such that $f'(x)=0$ .

b) Give an example of a non-constant function $f$ with property $(P)$ .

c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$ , $f(1)=1$ . Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$ ) and find all functions $f\in\mathcal{F}$ for which this minimum is attained.

Let $f: [0,1]\rightarrow(0,+\infty)$ be a continuous function.

a) Show that for any integer $n\geq 1$ , there is a unique division $0=a_{0}<a_{1}<\ldots<a_{n}=1$ such that $\int_{a_{k}}^{a_{k+1}}f(x) dx=\frac{1}{n}\int_{0}^{1}f(x) dx$ holds for all $k=0,1,\ldots,n-1$ .

b) For each $n$ , consider the $a_{i}$ above (that depend on $n$ ) and define $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$ . Show that the sequence $(b_{n})$ is convergent and compute it's limit.

Let $n\geq 1$ be an integer. Find all rings $(A,+,\cdot)$ such that all $x\in A\setminus\{0\}$ satisfy $x^{2^{n}+1}=1$ .

Let $n\geq 3$ be an integer and $S_{n}$ the permutation group. $G$ is a subgroup of $S_{n}$ , generated by $n-2$ transpositions. For all $k\in\{1,2,\ldots,n\}$ , denote by $S(k)$ the set $\{\sigma(k) \ : \ \sigma\in G\}$ .

Show that for any $k$ , $|S(k)|\leq n-1$ .