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Problems
Contests
National and Regional Contests
Israel Contests
Grosman Mathematical Olympiad
1997 Israel Grosman Mathematical Olympiad
1997 Israel Grosman Mathematical Olympiad
Part of
Grosman Mathematical Olympiad
Subcontests
(6)
6
1
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exists a monochromatic simple path of length n
In the plane are given
n
2
+
1
n^2 + 1
n
2
+
1
points, no three of which lie on a line. Each line segment connecting a pair of these points is colored by either red or blue. A path of length
k
k
k
is a sequence of
k
k
k
segments where the end of each segment (except for the last one) is the beginning of the next one. A path is simple if it does not intersect itself. Prove that there exists a monochromatic simple path of length
n
n
n
.
5
1
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max no of partitions of an n x n square
Consider partitions of an
n
×
n
n \times n
n
×
n
square (composed of
n
2
n^2
n
2
unit squares) into rectangles with one integer side and the other side equal to
1
1
1
. What is the largest possible number of such partitions among which no two have an identical rectangle at the same place?
4
1
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if 2 altitudes of a tetrahedron intersect, then so do the other two altitudes
Prove that if two altitudes of a tetrahedron intersect, then so do the other two altitudes.
3
1
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\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3
Find all real solutions of
13
+
x
4
+
14
−
x
4
=
3
\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3
4
13
+
x
+
4
14
−
x
=
3
.
2
1
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polygon whose vertices have integer coordinates and whose area is 1/2
Is there a planar polygon whose vertices have integer coordinates and whose area is
1
/
2
1/2
1/2
, such that this polygon is (a) a triangle with at least two sides longer than
1000
1000
1000
? (b) a triangle whose sides are all longer than
1000
1000
1000
? (c) a quadrangle?
1
1
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at max 3 primes between 10 and 10^P10} all of whose decimal digits are 1
Prove that there are at most three primes between
10
10
10
and
1
0
10
10^{10}
1
0
10
all of whose decimal digits are
1
1
1
.