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Hungary-Israel Binational
1997 Hungary-Israel Binational
1997 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(3)
3
2
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Hungary-Israel Binational 1997_3
Let
A
B
C
ABC
A
BC
be an acute angled triangle whose circumcenter is
O
O
O
. The three diameters of the circumcircle that pass through
A
A
A
,
B
B
B
, and
C
C
C
, meet the opposite sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
at the points
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
, respectively. The circumradius of
A
B
C
ABC
A
BC
is of length
2
P
2P
2
P
, where
P
P
P
is a prime number. The lengths of
O
A
1
OA_1
O
A
1
,
O
B
1
OB_1
O
B
1
,
O
C
1
OC_1
O
C
1
are integers. What are the lengths of the sides of the triangle?
Hungary-Israel Binational 1997_6
Can a closed disk can be decomposed into a union of two congruent parts having no common point?
1
2
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Hungary-Israel Binational 1997_1
Is there an integer
N
N
N
such that \left(\sqrt{1997}\minus{}\sqrt{1996}\right)^{1998}\equal{}\sqrt{N}\minus{}\sqrt{N\minus{}1}?
An odd number of letters
Determine the number of distinct sequences of letters of length 1997 which use each of the letters
A
A
A
,
B
B
B
,
C
C
C
(and no others) an odd number of times.
2
2
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Hungary-Israel Binational 1997_2
Find all the real numbers
α
\alpha
α
satisfy the following property: for any positive integer
n
n
n
there exists an integer
m
m
m
such that \left |\alpha\minus{}\frac{m}{n}\right|<\frac{1}{3n}.
Concurrent lines when squares erected on sides of ABC
The three squares
A
C
C
1
A
′
′
ACC_{1}A''
A
C
C
1
A
′′
,
A
B
B
1
′
A
′
ABB_{1}'A'
A
B
B
1
′
A
′
,
B
C
D
E
BCDE
BC
D
E
are constructed externally on the sides of a triangle
A
B
C
ABC
A
BC
. Let
P
P
P
be the center of the square
B
C
D
E
BCDE
BC
D
E
. Prove that the lines
A
′
C
A'C
A
′
C
,
A
′
′
B
A''B
A
′′
B
,
P
A
PA
P
A
are concurrent.