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]] | ||
+ | |||
+ | ==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. | ||
+ | |||
+ | a) If the function is also continuous on <math>[0,1]</math> is it true that <math>f</math> is increasing? | ||
+ | |||
+ | b) The same question if the function is continuous on <math>\mathbb Q \cap [0, \infty)</math>. | ||
+ | |||
+ | [[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 be a triangle and the points
and
on the sides
respectively
, such that
. Let
be a point on the line
. Prove that the lines
and
are perpendicular if and only if
is the interior angle bisector of
.
Problem 2
A square of side is formed from
unit squares, each colored in red, yellow or green. Find minimal
, 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 we have
. The points
and
are the feet of the altitudes from
and
, and
is the orthocenter of the triangle. We consider the points
and
on the segments
and
such that
. Prove that
a) ;
b) .
Problem 4
Let be a set of positive integers with at least 2 elements. It is given that for any numbers
,
we have
, where by
we have denoted the least common multiple of
and
. Prove that the set
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 be a positive integer. Prove that there exists an integer
,
, and numbers
, such that
![$n = \sum_{1\leq i < j \leq k } a_ia_j$](http://latex.artofproblemsolving.com/c/3/f/c3f8748bf68501f3212d2a2fabd83d374f793ea2.png)
Problem 3
Let be a cube and
a variable point on the side
. The perpendicular plane on
which passes through
intersects the line
in
. Let
and
be the midpoints of the segments
and
respectively.
a) Prove that the lines and
are perpendicular if and only if
is the midpoint of
.
b) Find the minimal value of the angle between the lines and
.
Problem 4
Let . Prove that
![$2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3$](http://latex.artofproblemsolving.com/9/f/3/9f3b9cb0473c32de0b51a50dcdee707eb7fc00fd.png)
selected by Mircea Lascu
Grade 9
Problem 1
Find the maximal value of
![$\left( x^3+1 \right) \left( y^3 + 1\right)$](http://latex.artofproblemsolving.com/5/c/9/5c9771df93a5ab9e4e2c14fbf59265a888c74397.png)
where ,
.
Dan Schwarz
Problem 2
Let and
be isosceles triangle with the base
. We know that
. Let
be the midpoint of
. The points
are chosen such that
,
,
, and
. Prove that
is the midpoint of
and
.
Problem 3
We have a quadrilateral inscribed in a circle of radius
, for which there is a point
on
such that
.
(a) Prove that there are points which fulfill the above conditions.
(b) Prove that .
Virgil Nicula
Problem 4
students
participated at table tennis contest, which took
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
students on the second place;
- no student lost all
matches.
How many students won only a single match and how many won exactly matches? (In the above conditions)
10th Grade
Problem 1
Let be a set composed of
elements and let
be its power set. Find all functions
that have the properties
(a) , for
;
(b) , for all
, where
.
Problem 2
Prove that for all and
we have
Problem 3
Prove that among the elements of the sequence are an infinity of even numbers and an infinity of odd numbers.
Problem 4
Let ,
. Determine
sets
,
, from the plane, pairwise disjoint, such that:
(a) for every circle from the plane and for every
we have
;
(b) for all lines from the plane and every
, the projection of
on
is not
.
11th Grade
Problem 1
Let be a
matrix with complex elements and let
be the classical adjoint of
. Prove that if there exists a positive integer
such that
then
.
Marian Ionescu, Pitesti
Problem 2
We define a pseudo-inverse of a matrix
a matrix which fulfills the relations
a) Prove that any square matrix has at least a pseudo-inverse.
b) For which matrix is the pseudo-inverse unique?
Marius Cavachi
Problem 3
We have in the plane the system of points and
, which have different centers of mass. Prove that there is a point
such that
Problem 4
Let be a function such that for any
the sequence
is increasing.
a) If the function is also continuous on is it true that
is increasing?
b) The same question if the function is continuous on .
12th Grade
Problem 1
Let be a finite field. Prove that the following statements are equivalent:
(a) ;
(b) for all with
,
is reducible.
Problem 2
Prove that where
if
and
.
Dorin Andrica, Mihai Piticari
Problem 3
Let be a finite group of
elements
and
be the smallest prime factor of
. If
has only a subgroup
with
elements, then prove that
is in the center of
.
Note. The center of is the set
.
Problem 4
Let be a continuous function such that
Prove that there is
such that
Cezar Lupu, Tudorel Lupu