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Difference between revisions of "2006 Romanian NMO Problems"

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[[2006 Romanian NMO Problems/Grade 8/Problem 4|Solution]]
 
[[2006 Romanian NMO Problems/Grade 8/Problem 4|Solution]]
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==Grade 9==
 +
===Problem 1===
 +
Find the maximal value of
 +
 +
<center><math>\left( x^3+1 \right) \left( y^3 + 1\right)</math>,</center>
 +
 +
where <math>x,y \in \mathbb R</math>, <math>x+y=1</math>.
 +
 +
''Dan Schwarz''
 +
 +
[[2006 Romanian NMO Problems/Grade 9/Problem 1 | Solution]]
 +
===Problem 2===
 +
Let <math>ABC</math> and <math>DBC</math> be isosceles triangle with the base <math>BC</math>. We know that <math>\angle ABD = \frac{\pi}{2}</math>. Let <math>M</math> be the midpoint of <math>BC</math>. The points <math>E,F,P</math> are chosen such that <math>E \in (AB)</math>, <math>P \in (MC)</math>, <math>C \in (AF)</math>, and <math>\angle BDE = \angle ADP = \angle CDF</math>. Prove that <math>P</math> is the midpoint of <math>EF</math> and <math>DP \perp EF</math>.
 +
 +
[[2006 Romanian NMO Problems/Grade 9/Problem 2 | Solution]]
 +
===Problem 3===
 +
We have a quadrilateral <math>ABCD</math> inscribed in a circle of radius <math>r</math>, for which there is a point <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</math>.
 +
 +
(a) Prove that there are points <math>A,B,C,D,P</math> which fulfill the above conditions.
 +
 +
(b) Prove that <math>PD=r</math>.
 +
 +
''Virgil Nicula''
 +
 +
[[2006 Romanian NMO Problems/Grade 9/Problem 3 | Solution]]
 +
===Problem 4===
 +
<math>2n</math> students <math>(n \geq 5)</math> participated at table tennis contest, which took <math>4</math> 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 <math>3</math> students on the second place;
 +
 +
* no student lost all <math>4</math> matches.
 +
 +
How many students won only a single match and how many won exactly <math>2</math> matches? (In the above conditions)
 +
 +
[[2006 Romanian NMO Problems/Grade 9/Problem 4 | Solution]]
 +
 +
== 10th Grade ==
 +
===Problem 1===
 +
Let <math>\displaystyle M</math> be a set composed of <math>\displaystyle n</math> elements and let <math>\displaystyle \mathcal P (M)</math> be its power set. Find all functions <math>\displaystyle f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \}</math> that have the properties
 +
 +
(a) <math>\displaystyle f(A) \neq 0</math>, for <math>\displaystyle A \neq \phi</math>;
 +
 +
(b) <math>\displaystyle f \left( A \cup B \right) = f \left( A \cap B \right) + f \left( A \Delta B \right)</math>, for all <math>\displaystyle A,B \in \mathcal P (M)</math>, where <math>\displaystyle A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right)</math>.
 +
 +
[[2006 Romanian NMO Problems/Grade 10/Problem 1 | Solution]]
 +
===Problem 2===
 +
Prove that for all <math>\displaystyle a,b \in \left( 0 ,\frac{\pi}{4} \right)</math> and <math>\displaystyle n \in \mathbb N^\ast</math> we have
 +
<cmath>\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} . </cmath>
 +
 +
[[2006 Romanian NMO Problems/Grade 10/Problem 2 | Solution]]
 +
===Problem 3===
 +
Prove that among the elements of the sequence <math>\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}</math> are an infinity of even numbers and an infinity of odd numbers.
 +
 +
[[2006 Romanian NMO Problems/Grade 10/Problem 3 | Solution]]
 +
===Problem 4===
 +
Let <math>\displaystyle n \in \mathbb N</math>, <math>\displaystyle n \geq 2</math>. Determine <math>\displaystyle n</math> sets <math>\displaystyle A_i</math>, <math>\displaystyle 1 \leq i \leq n</math>, from the plane, pairwise disjoint, such that:
 +
 +
(a) for every circle <math>\displaystyle \mathcal C</math> from the plane and for every <math>\displaystyle i \in \left\{ 1,2,\ldots,n \right\}</math> we have <math>\displaystyle A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi</math>;
 +
 +
(b) for all lines <math>\displaystyle d</math> from the plane and every <math>\displaystyle i \in \left\{ 1,2,\ldots,n \right\}</math>, the projection of <math>\displaystyle A_i</math> on <math>\displaystyle d</math> is not <math>\displaystyle d</math>.
 +
 +
[[2006 Romanian NMO Problems/Grade 10/Problem 4 | Solution]]
 +
 +
== 11th Grade ==
 +
===Problem 1===
 +
Let <math>A</math> be a <math>n\times n</math> matrix with complex elements and let <math>A^\star</math> be the classical adjoint of <math>A</math>. Prove that if there exists a positive integer <math>m</math> such that <math>(A^\star)^m = 0_n</math> then <math>(A^\star)^2 = 0_n</math>.
 +
 +
''Marian Ionescu, Pitesti''
 +
 +
[[2006 Romanian NMO Problems/Grade 11/Problem 1 | Solution]]
 +
===Problem 2===
 +
We define a ''pseudo-inverse'' <math>B\in \mathcal M_n(\mathbb C)</math>  of a matrix <math>A\in\mathcal M_n(\mathbb C)</math> a matrix which fulfills the relations
 +
<cmath> A = ABA \quad \text{ and } \quad B=BAB. </cmath>
 +
a) Prove that any square matrix has at least a pseudo-inverse.
 +
 +
b) For which matrix <math>A</math> is the pseudo-inverse unique?
 +
 +
''Marius Cavachi''
 +
 +
[[2006 Romanian NMO Problems/Grade 11/Problem 2 | Solution]]
 +
===Problem 3===
 +
We have in the plane the system of points <math>A_1,A_2,\ldots,A_n</math> and <math>B_1,B_2,\ldots,B_n</math>, which have different centers of mass. Prove that there is a point <math>P</math> such that
 +
<cmath>PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n .</cmath>
 +
 +
 +
[[2006 Romanian NMO Problems/Grade 11/Problem 3 | Solution]]
 +
===Problem 4===
 +
Let <math>f: [0,\infty)\to\mathbb R</math> be a function such that for any <math>x>0</math> the sequence <math>\{f(nx)\}_{n\geq 0}</math> is increasing.
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 +
a) If the function is also continuous on <math>[0,1]</math> is it true that <math>f</math> is increasing?
 +
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b) The same question if the function is continuous on <math>\mathbb Q \cap [0, \infty)</math>.
 +
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[[2006 Romanian NMO Problems/Grade 11/Problem 4 | Solution]]
 +
 +
== 12th Grade ==
 +
===Problem 1===
 +
Let <math>\displaystyle \mathcal K</math> be a finite field. Prove that the following statements are equivalent:
 +
 +
(a) <math>\displaystyle 1+1=0</math>;
 +
 +
(b) for all <math>\displaystyle f \in \mathcal K \left[ X \right]</math> with <math>\displaystyle \textrm{deg} \, f \geq 1</math>, <math>\displaystyle f \left( X^2 \right)</math> is reducible.
 +
 +
[[2006 Romanian NMO Problems/Grade 12/Problem 1 | Solution]]
 +
===Problem 2===
 +
Prove that <cmath> \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 ,  </cmath> where <math>f(x) = \frac{\arctan x}{x}</math> if <math>x \in \left( 0,1 \right]</math> and <math>f(0)=1</math>.
 +
 +
''Dorin Andrica, Mihai Piticari''
 +
 +
[[2006 Romanian NMO Problems/Grade 12/Problem 2 | Solution]]
 +
===Problem 3===
 +
Let <math>\displaystyle G</math> be a finite group of <math>\displaystyle n</math> elements <math>\displaystyle ( n \geq 2 )</math> and <math>\displaystyle p</math> be the smallest prime factor of <math>\displaystyle n</math>. If <math>\displaystyle G</math> has only a subgroup <math>\displaystyle H</math> with <math>\displaystyle p</math> elements, then prove that <math>\displaystyle H</math> is in the center of <math>\displaystyle G</math>.
 +
 +
''Note.'' The center of <math>\displaystyle G</math> is the set <math>\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}</math>.
 +
 +
[[2006 Romanian NMO Problems/Grade 12/Problem 3 | Solution]]
 +
===Problem 4===
 +
Let <math>f: [0,1]\to\mathbb{R}</math> be a continuous function such that
 +
<cmath> \int_{0}^{1}f(x)dx=0.  </cmath>
 +
Prove that there is <math>c\in (0,1)</math> such that
 +
<cmath> \int_{0}^{c}xf(x)dx=0.  </cmath>
 +
 +
''Cezar Lupu, Tudorel Lupu''
 +
 +
[[2006 Romanian NMO Problems/Grade 12/Problem 4 | Solution]]

Latest revision as of 14:57, 7 May 2012

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 $\angle MPC$.

Solution

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).

Solution

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) $CH=DE$.

Solution

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

Solution

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.

Solution

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

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

Solution

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 $BC'$.

Solution

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

Solution

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

Solution

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 $DP \perp EF$.

Solution

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

Solution

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)

Solution

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 $\displaystyle A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right)$.

Solution

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 \[\frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} .\]

Solution

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.

Solution

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 $\displaystyle d$.

Solution

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

Solution

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

Solution

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 \[PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n .\]


Solution

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 $\mathbb Q \cap [0, \infty)$.

Solution

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.

Solution

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

Solution

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 $\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}$.

Solution

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

Solution

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