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Difference between revisions of "2006 Romanian NMO Problems"
(→‎8th Grade)
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==8th Grade==
 
==8th Grade==
 
===Problem 1===
 
===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.  
+
We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.
 +
 
 +
[[2006 Romanian NMO Problems/Grade 8/Problem 1|Solution]]
 
===Problem 2===
 
===Problem 2===
 
Let <math>n</math> be a positive integer. Prove that there exists an integer <math>k</math>, <math>k\geq 2</math>, and numbers <math>a_i \in \{ -1, 1 \}</math>, such that <center><math>n = \sum_{1\leq i < j \leq k } a_ia_j</math>.</center>
 
Let <math>n</math> be a positive integer. Prove that there exists an integer <math>k</math>, <math>k\geq 2</math>, and numbers <math>a_i \in \{ -1, 1 \}</math>, such that <center><math>n = \sum_{1\leq i < j \leq k } a_ia_j</math>.</center>
 +
 +
[[2006 Romanian NMO Problems/Grade 8/Problem 2|Solution]]
 +
===Problem 3===
 +
Let <math>ABCDA_1B_1C_1D_1</math> be a cube and <math>P</math> a variable point on the side <math>[AB]</math>. The perpendicular plane on <math>AB</math> which passes through <math>P</math> intersects the line <math>AC'</math> in <math>Q</math>. Let <math>M</math> and <math>N</math> be the midpoints of the segments <math>A'P</math> and <math>BQ</math> respectively.
 +
 +
[[2006 Romanian NMO Problems/Grade 8/Problem 3|Solution]]
 +
a) Prove that the lines <math>MN</math> and <math>BC'</math> are perpendicular if and only if <math>P</math> is the midpoint of <math>AB</math>.
 +
 +
b) Find the minimal value of the angle between the lines <math>MN</math> and <math>BC'</math>.
 +
===Problem 4===
 +
Let <math>a,b,c \in \left[ \frac 12, 1 \right]</math>. Prove that <center><math>2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3</math>.</center>
 +
 +
''selected by Mircea Lascu''
 +
 +
[[2006 Romanian NMO Problems/Grade 8/Problem 4|Solution]]

Revision as of 09:32, 27 July 2006

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.

Solution 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

Solution

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