Difference between revisions of "2006 Romanian NMO Problems"
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[[2006 Romanian NMO Problems/Grade 9/Problem 1 | Solution]] | [[2006 Romanian NMO Problems/Grade 9/Problem 1 | Solution]] | ||
===Problem 2=== | ===Problem 2=== | ||
− | Let <math> | + | 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]] | [[2006 Romanian NMO Problems/Grade 9/Problem 2 | Solution]] | ||
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[[2006 Romanian NMO Problems/Grade 9/Problem 3 | Solution]] | [[2006 Romanian NMO Problems/Grade 9/Problem 3 | Solution]] | ||
===Problem 4=== | ===Problem 4=== | ||
− | <math> | + | <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 is only one winner; | ||
− | * there are <math> | + | * there are <math>3</math> students on the second place; |
− | * no student lost all <math> | + | * no student lost all <math>4</math> matches. |
− | How many students won only a single match and how many won exactly <math> | + | 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]] | [[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 13:57, 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
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
selected by Mircea Lascu
Grade 9
Problem 1
Find the maximal value of
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