2002 Romanian NMO Problems

7th Grade

Problem 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.

Solution

Problem 2

Prove that any real number $0<x<1$ can be written as a difference of two positive and less than $1$ irrational numbers.

Solution

Problem 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.

Solution

Problem 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 $PN\perp AC$ .

Solution

8th Grade

Problem 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 $P(n)=2002$ .

Solution

Problem 2

Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies: $f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}$

Solution

Problem 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 $[PP']$ .

Solution

Problem 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.

Solution

Grade 9

Problem 1

Let $ab+bc+ca=1$ . Show that $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}$

Solution

Problem 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 $\measuredangle B$ .

Solution

Problem 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.

Solution

Problem 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 $x,y$ .

Solution

10th Grade

Problem 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.

Solution

Problem 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 $x\in\mathbb{R}$ .

Solution

Problem 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 (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 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 $f(y)\le f\left(\frac{x+y}{2}\right)\le f(x)$

Solution

11th Grade

Problem 1

In the Cartesian plane consider the hyperbola

\[\Gamma=\left\{M(x,y)\in\mathbb{R}^2 \left\vert \frac{x^2}{4}-y^2=1\right\}\] (Error compiling LaTeX. Unknown error_msg)

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.

Solution

Problem 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.

Solution

Problem 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 $n\ge 3$ .

Solution

Problem 4

Let $f:[0,1]\rightarrow [0,1]$ be a continuous and bijective function. Describe the set: $A=\{f(x)-f(y)\mid x,y\in[0,1]\backslash\mathbb{Q}\}$ Note: You are given the result that there is no one-to-one function between the irrational numbers and $\mathbb{Q}$ .

Solution

12th Grade

Problem 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.

Solution

Problem 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: $\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}$

Solution

Problem 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.

Solution

Problem 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: