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Problems
Contests
National and Regional Contests
Israel Contests
Grosman Mathematical Olympiad
2001 Grosman Memorial Mathematical Olympiad
2001 Grosman Memorial Mathematical Olympiad
Part of
Grosman Mathematical Olympiad
Subcontests
(6)
2
1
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max of (1/2001 \sum x_n^2)-(1/2001 \sum x_n)^2 , n=1 to 2001
If
x
1
,
x
2
,
.
.
.
,
x
2001
x_1,x_2,...,x_{2001}
x
1
,
x
2
,
...
,
x
2001
are real numbers with
0
≤
x
n
≤
1
0 \le x_n \le 1
0
≤
x
n
≤
1
for
n
=
1
,
2
,
.
.
.
,
2001
n = 1,2,...,2001
n
=
1
,
2
,
...
,
2001
, find the maximum value of
(
1
2001
∑
n
=
1
2001
x
n
2
)
−
(
1
2001
∑
n
=
1
2001
x
n
)
2
\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n^2\right)-\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n\right)^2
(
2001
1
n
=
1
∑
2001
x
n
2
)
−
(
2001
1
n
=
1
∑
2001
x
n
)
2
Where is this maximum attained?
6
1
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15x^2 + y^2 = 2^{2000} diophantine
(a) Find a pair of integers (x,y) such that
15
x
2
+
y
2
=
2
2000
15x^2 +y^2 = 2^{2000}
15
x
2
+
y
2
=
2
2000
(b) Does there exist a pair of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
15
x
2
+
y
2
=
2
2000
15x^2 + y^2 = 2^{2000}
15
x
2
+
y
2
=
2
2000
and
x
x
x
is odd?
5
1
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good triangles in a plane, triangle construction related
Triangle
A
B
C
ABC
A
BC
in the plane
Π
\Pi
Π
is called good if it has the following property: For any point
D
D
D
in space outside the plane
Π
\Pi
Π
, it is possible to construct a triangle with sides of lengths
C
D
,
B
D
,
A
D
CD,BD,AD
C
D
,
B
D
,
A
D
. Find all good triangles
4
1
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min distance of projections in a triangle 4-5-6
The lengths of the sides of triangle
A
B
C
ABC
A
BC
are
4
,
5
,
6
4,5,6
4
,
5
,
6
. For any point
D
D
D
on one of the sides, draw the perpendiculars
D
P
,
D
Q
DP, DQ
D
P
,
D
Q
on the other two sides. What is the minimum value of
P
Q
PQ
PQ
?
3
1
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2001 lines in the plane
We are given
2001
2001
2001
lines in the plane, no two of which are parallel and no three of which are concurrent. These lines partition the plane into regions (not necessarily finite) bounded by segments of these lines. These segments are called sides, and the collection of the regions is called a map. Intersection points of the lines are called vertices. Two regions are neighbors if they share a side, and two vertices are neighbors if they lie on the same side. A legal coloring of the regions (resp. vertices) is a coloring in which each region (resp. vertex) receives one color, such that any two neighboring regions (vertices) have different colors. (a) What is the minimum number of colors required for a legal coloring of the regions? (b) What is the minimum number of colors required for a legal coloring of the vertices?
1
1
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x_1 +x_2 +...+x_{2000} = 2000, x_1^4 +...+x_{2000}^4= x_1^3 +...+x_{2000}^3
Find all real solutions of the system
{
x
1
+
x
2
+
.
.
.
+
x
2000
=
2000
x
1
4
+
x
2
4
+
.
.
.
+
x
2000
4
=
x
1
3
+
x
2
3
+
.
.
.
+
x
2000
3
\begin{cases} x_1 +x_2 +...+x_{2000} = 2000 \\ x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases}
{
x
1
+
x
2
+
...
+
x
2000
=
2000
x
1
4
+
x
2
4
+
...
+
x
2000
4
=
x
1
3
+
x
2
3
+
...
+
x
2000
3