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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1995 Kurschak Competition
1995 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Three Thales circles
Points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
are such that no three of them are collinear. Let
E
=
A
B
∩
C
D
E=AB\cap CD
E
=
A
B
∩
C
D
and
F
=
B
C
∩
D
A
F=BC\cap DA
F
=
BC
∩
D
A
. Let
k
1
k_1
k
1
,
k
2
k_2
k
2
and
k
3
k_3
k
3
denote the circles with diameter
A
C
‾
\overline{AC}
A
C
,
B
D
‾
\overline{BD}
B
D
and
E
F
‾
\overline{EF}
EF
, respectively. Prove that either
k
1
,
k
2
,
k
3
k_1,k_2,k_3
k
1
,
k
2
,
k
3
pass through one point, or no two of them intersect.
2
1
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n-var polynomial, sign condition at points {-1,+1}^n
Consider a polynomial in
n
n
n
variables with real coefficients. We know that if every variable is
±
1
\pm1
±
1
, the value of the polynomial is positive, or negative if the number of
−
1
-1
−
1
's is even, or odd, respectively. Prove that the degree of this polynomial is at least
n
n
n
.
1
1
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Grid rectangle divided into grid triangles with area 1/2
Given in the plane is a lattice and a grid rectangle with sides parallel to the coordinate axes. We divide the rectangle into grid triangles with area
1
2
\frac12
2
1
. Prove that the number of right angled triangles is at least twice as much as the shorter side of the rectangle.(A grid polygon is a polygon such that both coordinates of each vertex is an integer.)