MathDB
Problems
Contests
International Contests
Hungary-Israel Binational
2002 Hungary-Israel Binational
2002 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(3)
3
2
Hide problems
$r_n = p(r_{n+1} ) $, p is a polynomial
Let
p
(
x
)
p(x)
p
(
x
)
be a polynomial with rational coefficients, of degree at least
2
2
2
. Suppose that a sequence
(
r
n
)
(r_{n})
(
r
n
)
of rational numbers satisfies
r
n
=
p
(
r
n
+
1
)
r_{n}= p(r_{n+1})
r
n
=
p
(
r
n
+
1
)
for every
n
≥
1
n\geq 1
n
≥
1
. Prove that the sequence
(
r
n
)
(r_{n})
(
r
n
)
is periodic.
neither of a^{p−1} - 1 & (a + 1)^{p−1}-1 is divisible p^2
Let
p
≥
5
p \geq 5
p
≥
5
be a prime number. Prove that there exists a positive integer
a
<
p
−
1
a < p-1
a
<
p
−
1
such that neither of
a
p
−
1
−
1
a^{p-1}-1
a
p
−
1
−
1
and
(
a
+
1
)
p
−
1
−
1
(a+1)^{p-1}-1
(
a
+
1
)
p
−
1
−
1
is divisible by
p
2
p^{2}
p
2
.
2
2
Hide problems
p(M ) = \frac{MA′ . MB′ . MC ′ }{MA. MB.MC}
Let
A
′
,
B
′
,
C
′
A', B' , C'
A
′
,
B
′
,
C
′
be the projections of a point
M
M
M
inside a triangle
A
B
C
ABC
A
BC
onto the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively. Define
p
(
M
)
=
M
A
′
⋅
M
B
′
⋅
M
C
′
M
A
⋅
M
B
⋅
M
C
p(M ) = \frac{MA'\cdot MB'\cdot MC'}{MA \cdot MB \cdot MC}
p
(
M
)
=
M
A
⋅
MB
⋅
MC
M
A
′
⋅
M
B
′
⋅
M
C
′
. Find the position of point
M
M
M
that maximizes
p
(
M
)
p(M )
p
(
M
)
.
Find the angles of a triangle
Points
A
1
,
B
1
,
C
1
A_{1}, B_{1}, C_{1}
A
1
,
B
1
,
C
1
are given inside an equilateral triangle
A
B
C
ABC
A
BC
such that
B
1
A
B
^
=
A
1
B
A
^
=
1
5
0
,
C
1
B
C
^
=
B
1
C
B
^
=
2
0
0
,
A
1
C
A
^
=
C
1
A
C
^
=
2
5
0
\widehat{B_{1}AB}= \widehat{A1BA}= 15^{0}, \widehat{C_{1}BC}= \widehat{B_{1}CB}= 20^{0}, \widehat{A_{1}CA}= \widehat{C_{1}AC}= 25^{0}
B
1
A
B
=
A
1
B
A
=
1
5
0
,
C
1
BC
=
B
1
CB
=
2
0
0
,
A
1
C
A
=
C
1
A
C
=
2
5
0
. Find the angles of triangle
A
1
B
1
C
1
A_{1}B_{1}C_{1}
A
1
B
1
C
1
.
1
2
Hide problems
$x^3 + y^4\leq x^2 + y^3$
Suppose that positive numbers
x
x
x
and
y
y
y
satisfy
x
3
+
y
4
≤
x
2
+
y
3
x^{3}+y^{4}\leq x^{2}+y^{3}
x
3
+
y
4
≤
x
2
+
y
3
. Prove that
x
3
+
y
3
≤
2.
x^{3}+y^{3}\leq 2.
x
3
+
y
3
≤
2.
Find the greatest exponent $k$
Find the greatest exponent
k
k
k
for which
200
1
k
2001^{k}
200
1
k
divides
200
0
200
1
2002
+
200
2
200
1
2000
2000^{2001^{2002}}+2002^{2001^{2000}}
200
0
200
1
2002
+
200
2
200
1
2000
.