2002 Romanian NMO Problems
7th Grade
Problem 1
Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours have altogether an odd number of winning cards. Show that any player has at that moment at least one winning card.
Problem 2
Prove that any real number can be written as a difference of two positive and less than
irrational numbers.
Problem 3
Let be a trapezium and
and
be it's parallel edges. Find, with proof, the set of interior points
of the trapezium which have the property that
belongs to at least two lines each intersecting the segments
and
and each dividing the trapezium in two other trapezoids with equal areas.
Problem 4
An equilateral triangle of sides
is given and a triangle
is constructed under the following conditions:
, such that
and
. Find the length of the segment
.
Show that for any acute triangle
one can find points
such that
and
.
8th Grade
Problem 1
For any number , denote by
the number of pairs
whose elements are of positive integers such that
Calculate
.
Find
such that
.
Problem 2
Given real numbers show that there exists at most one function
which satisfies:
Problem 3
Let be a frustum of a regular pyramid. Let
and
be the centroids of bases
and
respectively. It is known that
and
.
Prove that the planes
have a common point
, and the planes
have a common point
, both situated on
.
Find the length of the segment
.
Problem 4
The right prism , has a convex polygon as its base. It is known that
. Show that:
;
the prism is regular.
Grade 9
Problem 1
Let . Show that
Problem 2
Let be a right triangle where
and
such that
. It is known that the symmetric point of
with respect to the line
lies on
. Find the measure of
.
Problem 3
Let and
be positive integers with
. Show that the equation:
has no positive integer solutions.
Problem 4
Find all functions which satisfy the inequality:
for all non-negative integers
.
10th Grade
Problem 1
Let be four points in the plane. The segments
and
are said to be connected, if there is some point
in the plane such that the triangles
and
are right-angled at
and isosceles.
Let be a convex hexagon such that the pairs of segments
and
are connected. Show that the points
and
are the vertices of a parallelogram and
and
are connected.
Problem 2
Find all real polynomials and
, such that:
for all
.
Problem 3
Find all real numbers in the interval
, that satisfy:
\begin{align*}a+b+c+d+e &= 0\\ a^3+b^3+c^3+d^3+e^3&= 0\\ a^5+b^5+c^5+d^5+e^5&=10 (Error compiling LaTeX. Unknown error_msg)
Problem 4
Let be an interval and
a function such that:
Show that
is monotonic on
if and only if, for any
, either
or
11th Grade
Problem 1
In the Cartesian plane consider the hyperbola
\[\Gamma=\left\{M(x,y)\in\mathbb{R}^2 \left\vert \frac{x^2}{4}-y^2=1\right\}\] (Error compiling LaTeX. Unknown error_msg)
and a conic , disjoint from
. Let
be the maximal number of pairs of points
such that
, for any
For each , find the equation of
for which
. Justify the answer.
Problem 2
Let be a function that has limits at any point and has no local extrema. Show that:
is continuous;
is strictly monotone.
Problem 3
Let be a non-zero matrix.
If
, prove the existence of two invertible matrices
, such that:
where
is the
-unit matrix.
Show that if
and
have the same rank
, then the matrix
has rank
, for any
.
Problem 4
Let be a continuous and bijective function. Describe the set:
Note: You are given the result that there is no one-to-one function between the irrational numbers and
.
12th Grade
Problem 1
Let be a ring.
Show that the set
is a subring of the ring
.
Prove that, if any commutative subring of
is a field, then
is a field.
Problem 2
Let be an integrable function such that:
Show that there exists
, such that:
Problem 3
Let be a continuous and bounded function such that
Prove that
is a constant function.
Problem 4
Let be a field having
elements, where
is a prime and
is an arbitrary integer number. For any
, one defines the polynomial
. Show that:
is divisible by
;
has at least
essentially different irreducible factors
.