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Problems
Contests
National and Regional Contests
Israel Contests
Grosman Mathematical Olympiad
1999 Israel Grosman Mathematical Olympiad
1999 Israel Grosman Mathematical Olympiad
Part of
Grosman Mathematical Olympiad
Subcontests
(6)
3
1
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sequence of incircle's triangles
For every triangle
A
B
C
ABC
A
BC
, denote by
D
(
A
B
C
)
D(ABC)
D
(
A
BC
)
the triangle whose vertices are the tangency points of the incircle of
△
A
B
C
\vartriangle ABC
△
A
BC
with the sides. Assume that
△
A
B
C
\vartriangle ABC
△
A
BC
is not equilateral. (a) Prove that
D
(
A
B
C
)
D(ABC)
D
(
A
BC
)
is also not equilateral. (b) Find in the sequence
T
1
=
△
A
B
C
,
T
k
+
1
=
D
(
T
k
)
T_1 = \vartriangle ABC, T_{k+1} = D(T_k)
T
1
=
△
A
BC
,
T
k
+
1
=
D
(
T
k
)
for
k
∈
N
k \in N
k
∈
N
a triangle whose largest angle
α
\alpha
α
satisfies
0
<
α
−
6
0
o
<
0.000
1
o
0 < \alpha -60^o < 0.0001^o
0
<
α
−
6
0
o
<
0.000
1
o
6
1
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6 midpoints are coplanar, given 3 parallelograms in space
Let
A
,
B
,
C
,
D
,
E
,
F
A,B,C,D,E,F
A
,
B
,
C
,
D
,
E
,
F
be points in space such that the quadrilaterals
A
B
D
E
,
B
C
E
F
,
C
D
F
A
ABDE,BCEF, CDFA
A
B
D
E
,
BCEF
,
C
D
F
A
are parallelograms. Prove that the six midpoints of the sides
A
B
,
B
C
,
C
D
,
D
E
,
E
F
,
F
A
AB,BC,CD,DE,EF,FA
A
B
,
BC
,
C
D
,
D
E
,
EF
,
F
A
are coplanar
4
1
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integer f(x) = x^4 +ax^3 +bx^2 +cx+d with exactly one real root
Consider a polynomial
f
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
f(x) = x^4 +ax^3 +bx^2 +cx+d
f
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
with integer coefficients. Prove that if
f
(
x
)
f(x)
f
(
x
)
has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients
5
1
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subsequence of 1999 terms is either monotonically increasing or decreasing
An infinite sequence of distinct real numbers is given. Prove that it contains a subsequence of
1999
1999
1999
terms which is either monotonically increasing or monotonically decreasing.
2
1
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0 <\sqrt[4]{n}- [\sqrt[4]{n}]< 10^{-5}
Find the smallest positive integer
n
n
n
for which
0
<
n
4
−
[
n
4
]
<
1
0
−
5
0 <\sqrt[4]{n}- [\sqrt[4]{n}]< 10^{-5}
0
<
4
n
−
[
4
n
]
<
1
0
−
5
.
1
1
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T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}
For any
16
16
16
positive integers
n
,
a
1
,
a
2
,
.
.
.
,
a
15
n,a_1,a_2,...,a_{15}
n
,
a
1
,
a
2
,
...
,
a
15
we define
T
(
n
,
a
1
,
a
2
,
.
.
.
,
a
15
)
=
(
a
1
n
+
a
2
n
+
.
.
.
+
a
15
n
)
a
1
a
2
.
.
.
a
15
T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}
T
(
n
,
a
1
,
a
2
,
...
,
a
15
)
=
(
a
1
n
+
a
2
n
+
...
+
a
15
n
)
a
1
a
2
...
a
15
. Find the smallest
n
n
n
such that
T
(
n
,
a
1
,
a
2
,
.
.
.
,
a
15
)
T(n,a_1,a_2,...,a_{15})
T
(
n
,
a
1
,
a
2
,
...
,
a
15
)
is divisible by
15
15
15
for any choice of
a
1
,
a
2
,
.
.
.
,
a
15
a_1,a_2,...,a_{15}
a
1
,
a
2
,
...
,
a
15
.