[Solutions] Romanian Mathematical Competitions 2002

Romania National Olympiad 2002

Grade level 7

1. Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours have altogether an odd number of winning cards. Show that any player has at that moment at least one winning card.
2. Prove that any real number $0<x<1$ can be written as a difference of two positive and less than $1$ irrational numbers.
3. Let $ABCD$ be a trapezium and $AB$ and $CD$ be it's parallel edges. Find, with proof, the set of interior points $P$ of the trapezium which have the property that $P$ belongs to at least two lines each intersecting the segments $AB$ and $CD$ and each dividing the trapezium in two other trapezoids with equal areas.
4. a) An equilateral triangle of sides $a$ is given and a triangle $MNP$ is constructed under the following conditions: $P\in (AB)$, $M\in (BC)$, $N\in (AC)$, such that $MP\perp AB$, $NM\perp BC$ and $PN\perp AC$. Find the length of the segment $MP$.
   b) Show that for any acute triangle $ABC$ one can find points $P\in (AB)$, $M\in (BC)$, $N\in (AC)$ such that $MP\perp AB$, $NM\perp BC$ and $PN\perp AC$.

Grade level 8

1. For any number $n\in\mathbb{N},n\ge 2$, denote by $P(n)$ the number of pairs $(a,b)$ whose elements are of positive integers such that \[\frac{n}{a}\in (0,1),\quad \frac{a}{b}\in (1,2)\quad \text{and}\quad \frac{b}{n}\in (2,3). \] a) Calculate $P(3)$.
   b) Find $n$ such that $P(n)=2002$.
2. Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies: \[f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}\]
3. Let $[ABCDEF]$ be a frustum of a regular pyramid. Let $G$ and $G'$ be the centroids of bases $ABC$ and $DEF$ respectively. It is known that $AB=36,DE=12$ and $GG'=35$.
   a) Prove that the planes $(ABF)$, $(BCD)$, $(CAE)$ have a common point $P$, and the planes $(DEC)$, $(EFA)$, $(FDB)$ have a common point $P'$, both situated on $GG'$.
   b) Find the length of the segment $[PP']$.
4. The right prism $[A_1A_2A_3\ldots A_nA_1'A_2'A_3'\ldots A_n']$, $n\in\mathbb{N}$, $n\ge 3$, has a convex polygon as its base. It is known that $A_1A_2'\perp A_2A_3'$, $A_2A_3'\perp A_3A_4'$, $\ldots$, $A_{n-1}A_n'\perp A_nA_1'$, $A_nA_1'\perp A_1A_2'$. Show that
   a) $n=3$;
   b) the prism is regular.

Grade level 9

1. Let $ab+bc+ca=1$. Show that \[\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}.\]
2. Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.
3. Let $k$ and $n$ be positive integers with $n>2$. Show that the equation \[x^n-y^n=2^k\] has no positive integer solutions.
4. Find all functions $f: \mathbb{N}\to\mathbb{N}$ which satisfy the inequality: \[f(3x+2y)=f(x)f(y)\] for all non-negative integers $x,y$.

Grade level 10

1. Let $X,Y,Z,T$ be four points in the plane. The segments $[XY]$ and $[ZT]$ are said to be connected, if there is some point $O$ in the plane such that the triangles $OXY$ and $OZT$ are right-angled at $O$ and isosceles. Let $ABCDEF$ be a convex hexagon such that the pairs of segments $[AB],[CE],$ and $[BD],[EF]$ are connected. Show that the points $A,C,D$ and $F$ are the vertices of a parallelogram and $[BC]$ and $[AE]$ are connected.
2. Find all real polynomials $f$ and $g$, such that: \[(x^2+x+1)\cdot f(x^2-x+1)=(x^2-x+1)\cdot g(x^2+x+1), \] for all $x\in\mathbb{R}$.
3. Find all real numbers $a,b,c,d,e$ in the interval $[-2,2]$, that satisfy:
   \begin{align*}a+b+c+d+e &= 0\\ a^3+b^3+c^3+d^3+e^3&= 0\\ a^5+b^5+c^5+d^5+e^5&=10 \end{align*}
4. Let $I\subseteq \mathbb{R}$ be an interval and $f:I\rightarrow\mathbb{R}$ a function such that: \[|f(x)-f(y)|\le |x-y|,\quad\text{for all}\ x,y\in I. \] Show that $f$ is monotonic on $I$ if and only if, for any $x,y\in I$, either $$f(x)\le f\left(\frac{x+y}{2}\right)\le f(y)$$ or $$f(y)\le f\left(\frac{x+y}{2}\right)\le f(x).$$

Grade level 11

1. In the Cartesian plane consider the hyperbola \[\Gamma=\{M(x,y)\in\mathbb{R}^2 \vert \frac{x^2}{4}-y^2=1\} \] and a conic $\Gamma '$, disjoint from $\Gamma$. Let $n(\Gamma ,\Gamma ')$ be the maximal number of pairs of points $(A,A')\in\Gamma\times\Gamma '$ such that $AA'\le BB'$, for any $(B,B')$. For each $p\in\{0,1,2,4\}$, find the equation of $\Gamma'$ for which $n(\Gamma ,\Gamma ')=p$. Justify the answer.
2. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has limits at any point and has no local extrema. Show that:
   a) $f$ is continuous;
   b) $f$ is strictly monotone.
3. Let $A\in M_4(C)$ be a non-zero matrix.
   a) If $\text{rank}(A)=r<4$, prove the existence of two invertible matrices $U,V\in M_4(C)$, such that: \[UAV=\begin{pmatrix}I_r&0\\0&0\end{pmatrix}\] where $I_r$ is the $r$-unit matrix.
   b) Show that if $A$ and $A^2$ have the same rank $k$, then the matrix $A^n$ has rank $k$, for any $n\ge 3$.
4. Let $f:[0,1]\rightarrow [0,1]$ be a continuous and bijective function. Describe the set \[A=\{f(x)-f(y)\mid x,y\in[0,1]\backslash\mathbb{Q}\}\]

Grade level 12

1. Let $A$ be a ring.
   a) Show that the set $Z(A)=\{a\in A|ax=xa,\ \text{for all}\ x\in A\}$ is a subring of the ring $A$.
   b) Prove that, if any commutative subring of $A$ is a field, then $A$ is a field.
2. Let $f:[0,1]\rightarrow\mathbb{R}$ be an integrable function such that: \[0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.\] Show that there exists $x_1\not= x_2, x_1,x_2\in [0,1]$, such that: \[\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}\]
3. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that
   \[x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.\] Prove that $f$ is a constant function.
4. Let $K$ be a field having $q=p^n$ elements, where $p$ is a prime and $n\ge 2$ is an arbitrary integer number. For any $a\in K$, one defines the polynomial $f_a=X^q-X+a$. Show that:
   a) $f=(X^q-X)^q-(X^q-X)$ is divisible by $f_1$;
   b) $f_a$ has at least $p^{n-1}$ essentially different irreducible factors $K[X]$.

Romania Team Selection Test 2002

1. Find all sets $A$ and $B$ that satisfy the following conditions:
   a) $A \cup B= \mathbb{Z}$;
   b) if $x \in A$ then $x-1 \in B$;
   c) if $x,y \in B$ then $x+y \in A$.
2. The sequence $ (a_n)$ is defined by: $ a_0=a_1=1$ and $ a_{n+1}=14a_n-a_{n-1}$ for all $ n\ge 1$. Prove that $ 2a_n-1$ is a perfect square for any $ n\ge 0$.
3. Let $M$ and $N$ be the midpoints of the respective sides $AB$ and $AC$ of an acute-angled triangle $ABC$. Let $P$ be the foot of the perpendicular from $N$ onto $BC$ and let $A_1$ be the midpoint of $MP$. Points $B_1$ and $C_1$ are obtained similarly. If $AA_1$, $BB_1$ and $CC_1$ are concurrent, show that the triangle $ABC$ is isosceles.
4. For any positive integer $n$, let $f(n)$ be the number of possible choices of signs $+\ \text{or}\ - $ in the algebraic expression $\pm 1\pm 2\ldots \pm n$, such that the obtained sum is zero. Show that $f(n)$ satisfies the following conditions:
   a) $f(n)=0$ for $n=1\pmod{4}$ or $n=2\pmod{4}$.
   b) $2^{\frac{n}{2}-1}\le f(n)\le 2^n-2^{\lfloor\frac{n}{2}\rfloor+1}$, for $n=0\pmod{4}$ or $n=3\pmod{4}$.
5. Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$.
6. Let $P(x)$ and $Q(x)$ be integer polynomials of degree $p$ and $q$ respectively. Assume that $P(x)$ divides $Q(x)$ and all their coefficients are either $1$ or $2002$. Show that $p+1$ is a divisor of $q+1$.
7. Let $a,b$ be positive real numbers. For any positive integer $n$, denote by $x_n$ the sum of digits of the number $[an+b]$ in it's decimal representation. Show that the sequence $(x_n)_{n\ge 1}$ contains a constant subsequence.
8. At an international conference there are four official languages. Any two participants can speak in one of these languages. Show that at least $60\%$ of the participants can speak the same language.
9. Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$.
10. Let $n\geq 4$ be an integer, and let $a_1,a_2,\ldots,a_n$ be positive real numbers such that \[ a_1^2+a_2^2+\cdots +a_n^2=1 . \] Prove that the following inequality takes place \[ \frac{a_1}{a_2^2+1}+\cdots +\frac{a_n}{a_1^2+1} \geq \frac{4}{5}\left( a_1 \sqrt{a_1}+\cdots +a_n \sqrt{a_n} \right)^2 . \]
11. Let $n$ be a positive integer. $S$ is the set of nonnegative integers $a$ such that $1<a<n$ and $a^{a-1}-1$ is divisible by $n$. Prove that if $S=\{ n-1 \}$ then $n=2p$ where $p$ is a prime number. Mihai Cipu and Nicolae Ciprian Bonciocat nhat view topic 4 Let $f:\mathbb{Z}\rightarrow\{ 1,2,\ldots ,n\}$ be a function such that $f(x)\not= f(y)$, for all $x,y\in\mathbb{Z}$ such that $|x-y|\in\{2,3,5\}$. Prove that $n\ge 4$. Ioan Tomescu WakeUp view topic Day 4 1 Let $(a_n)_{n\ge 1}$ be a sequence of positive integers defined as $a_1,a_2>0$ and $a_{n+1}$ is the least prime divisor of $a_{n-1}+a_{n}$, for all $n\ge 2$.
12. Prove that a real number $x$ whose decimals are digits of the numbers $a_1,a_2,\ldots a_n,\ldots $ written in order, is a rational number.
13. Find the least positive real number $r$ with the following property:
14. Whatever four disks are considered, each with centre on the edges of a unit square and the sum of their radii equals $r$, there exists an equilateral triangle which has its edges in three of the disks.
15. After elections, every parliament member (PM), has his own absolute rating. When the parliament set up, he enters in a group and gets a relative rating. The relative rating is the ratio of its own absolute rating to the sum of all absolute ratings of the PMs in the group. A PM can move from one group to another only if in his new group his relative rating is greater. In a given day, only one PM can change the group. Show that only a finite number of group moves is possible.
    (A rating is positive real number.)
16. Let $m,n$ be positive integers of distinct parities and such that $m<n<5m$. Show that there exists a partition with two element subsets of the set $\{ 1,2,3,\ldots ,4mn\}$ such that the sum of numbers in each set is a perfect square.
17. Let $ABC$ be a triangle such that $AC\not= BC,AB<AC$ and let $K$ be it's circumcircle. The tangent to $K$ at the point $A$ intersects the line $BC$ at the point $D$. Let $K_1$ be the circle tangent to $K$ and to the segments $(AD),(BD)$. We denote by $M$ the point where $K_1$ touches $(BD)$. Show that $AC=MC$ if and only if $AM$ is the bisector of the $\angle DAB$.
18. There are $n$ players, $n\ge 2$, which are playing a card game with $np$ cards in $p$ rounds. The cards are coloured in $n$ colours and each colour is labelled with the numbers $1,2,\ldots ,p$. The game submits to the following rules: each player receives $p$ cards. The player who begins the first round throws a card and each player has to discard a card of the same colour, if he has one; otherwise they can give an arbitrary card. The winner of the round is the player who has put the greatest card of the same colour as the first one. the winner of the round starts the next round with a card that he selects and the play continues with the same rules. The played cards are out of the game. Show that if all cards labelled with number $1$ are winners, then $p\ge 2n$.

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