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2006 Romanian NMO Problems

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7th Grade

Problem 1

Let $ABC$ be a triangle and the points $M$ and $N$ on the sides $AB$ respectively $BC$ , such that $2 \cdot \frac{CN}{BC} = \frac{AM}{AB}$ . Let $P$ be a point on the line $AC$ . Prove that the lines $MN$ and $NP$ are perpendicular if and only if $PN$ is the interior angle bisector of $\angle MPC$ .

Problem 2

A square of side $n$ is formed from $n^2$ unit squares, each colored in red, yellow or green. Find minimal $n$ , such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).

Problem 3

In the acute-angle triangle $ABC$ we have $\angle ACB = 45^\circ$ . The points $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ , and $H$ is the orthocenter of the triangle. We consider the points $D$ and $E$ on the segments $AA_1$ and $BC$ such that $A_1D = A_1E = A_1B_1$ . Prove that

a) $A_1B_1 = \sqrt{ \frac{A_1B^2+A_1C^2}{2} }$ ;

b) $CH=DE$ .

Problem 4

Let $A$ be a set of positive integers with at least 2 elements. It is given that for any numbers $a>b$ , $a,b \in A$ we have $\frac{ [a,b] }{ a- b } \in A$ , where by $[a,b]$ we have denoted the least common multiple of $a$ and $b$ . Prove that the set $A$ has exactly two elements.

Marius Gherghu, Slatina

8th Grade

Problem 1

We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.

Problem 2

Let $n$ be a positive integer. Prove that there exists an integer $k$ , $k\geq 2$ , and numbers $a_i \in \{ -1, 1 \}$ , such that

$n = \sum_{1\leq i < j \leq k } a_ia_j$ .

Problem 3

Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$ . The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$ . Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively.

a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$ .

b) Find the minimal value of the angle between the lines $MN$ and $BC'$ .

Problem 4

Let $a,b,c \in \left[ \frac 12, 1 \right]$ . Prove that

$2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3$ .

selected by Mircea Lascu

Grade 9

Problem 1

Find the maximal value of

$\left( x^3+1 \right) \left( y^3 + 1\right)$ ,

where $x,y \in \mathbb R$ , $x+y=1$ .

Dan Schwarz

Problem 2

Let $ABC$ and $DBC$ be isosceles triangle with the base $BC$ . We know that $\angle ABD = \frac{\pi}{2}$ . Let $M$ be the midpoint of $BC$ . The points $E,F,P$ are chosen such that $E \in (AB)$ , $P \in (MC)$ , $C \in (AF)$ , and $\angle BDE = \angle ADP = \angle CDF$ . Prove that $P$ is the midpoint of $EF$ and $DP \perp EF$ .

Problem 3

We have a quadrilateral $ABCD$ inscribed in a circle of radius $r$ , for which there is a point $P$ on $CD$ such that $CB=BP=PA=AB$ .

(a) Prove that there are points $A,B,C,D,P$ which fulfill the above conditions.

(b) Prove that $PD=r$ .

Virgil Nicula

Problem 4

$2n$ students $(n \geq 5)$ participated at table tennis contest, which took $4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this:

there is only one winner;

there are $3$ students on the second place;

no student lost all $4$ matches.

How many students won only a single match and how many won exactly $2$ matches? (In the above conditions)

10th Grade

Problem 1

Let $\displaystyle M$ be a set composed of $\displaystyle n$ elements and let $\displaystyle \mathcal P (M)$ be its power set. Find all functions $\displaystyle f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \}$ that have the properties

(a) $\displaystyle f(A) \neq 0$ , for $\displaystyle A \neq \phi$ ;

(b) $\displaystyle f \left( A \cup B \right) = f \left( A \cap B \right) + f \left( A \Delta B \right)$ , for all $\displaystyle A,B \in \mathcal P (M)$ , where $\displaystyle A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right)$ .

Problem 2

Prove that for all $\displaystyle a,b \in \left( 0 ,\frac{\pi}{4} \right)$ and $\displaystyle n \in \mathbb N^\ast$ we have $\frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} .$

Problem 3

Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.

Problem 4

Let $\displaystyle n \in \mathbb N$ , $\displaystyle n \geq 2$ . Determine $\displaystyle n$ sets $\displaystyle A_i$ , $\displaystyle 1 \leq i \leq n$ , from the plane, pairwise disjoint, such that:

(a) for every circle $\displaystyle \mathcal C$ from the plane and for every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$ we have $\displaystyle A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi$ ;

(b) for all lines $\displaystyle d$ from the plane and every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$ , the projection of $\displaystyle A_i$ on $\displaystyle d$ is not $\displaystyle d$ .

11th Grade

Problem 1

Let $A$ be a $n\times n$ matrix with complex elements and let $A^\star$ be the classical adjoint of $A$ . Prove that if there exists a positive integer $m$ such that $(A^\star)^m = 0_n$ then $(A^\star)^2 = 0_n$ .

Marian Ionescu, Pitesti

Problem 2

We define a pseudo-inverse $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations $A = ABA \quad \text{ and } \quad B=BAB.$ a) Prove that any square matrix has at least a pseudo-inverse.

b) For which matrix $A$ is the pseudo-inverse unique?

Marius Cavachi

Problem 3

We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$ , which have different centers of mass. Prove that there is a point $P$ such that $PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n .$

Problem 4

Let $f: [0,\infty)\to\mathbb R$ be a function such that for any $x>0$ the sequence $\{f(nx)\}_{n\geq 0}$ is increasing.

a) If the function is also continuous on $[0,1]$ is it true that $f$ is increasing?

b) The same question if the function is continuous on $\mathbb Q \cap [0, \infty)$ .

12th Grade

Problem 1

Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent:

(a) $\displaystyle 1+1=0$ ;

(b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$ , $\displaystyle f \left( X^2 \right)$ is reducible.

Problem 2

Prove that $\lim_{n \to \infty} n \left( \frac{\pi}{4} - n \int_0^1 \frac{x^n}{1+x^{2n}} \, dx \right) = \int_0^1 f(x) \, dx ,$ where $f(x) = \frac{\arctan x}{x}$ if $x \in \left( 0,1 \right]$ and $f(0)=1$ .

Dorin Andrica, Mihai Piticari

Problem 3

Let $\displaystyle G$ be a finite group of $\displaystyle n$ elements $\displaystyle ( n \geq 2 )$ and $\displaystyle p$ be the smallest prime factor of $\displaystyle n$ . If $\displaystyle G$ has only a subgroup $\displaystyle H$ with $\displaystyle p$ elements, then prove that $\displaystyle H$ is in the center of $\displaystyle G$ .

Note. The center of $\displaystyle G$ is the set $\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}$ .

Problem 4

Let $f: [0,1]\to\mathbb{R}$ be a continuous function such that $\int_{0}^{1}f(x)dx=0.$ Prove that there is $c\in (0,1)$ such that $\int_{0}^{c}xf(x)dx=0.$

Cezar Lupu, Tudorel Lupu

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