MathDB
Problems
Contests
International Contests
Hungary-Israel Binational
2000 Hungary-Israel Binational
2000 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(3)
3
2
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locus of the orthocenter of the touchpoints of the incircle
Let
A
B
C
{ABC}
A
BC
be a non-equilateral triangle. The incircle is tangent to the sides
B
C
,
C
A
,
A
B
{BC,CA,AB}
BC
,
C
A
,
A
B
at
A
1
,
B
1
,
C
1
{A_1,B_1,C_1}
A
1
,
B
1
,
C
1
, respectively, and M is the orthocenter of triangle
A
1
B
1
C
1
{A_1B_1C_1}
A
1
B
1
C
1
. Prove that
M
{M}
M
lies on the line through the incenter and circumcenter of
△
A
B
C
{\vartriangle ABC}
△
A
BC
.
kl positive integers
Let
k
k
k
and
l
l
l
be two given positive integers and
a
i
j
(
1
≤
i
≤
k
,
1
≤
j
≤
l
)
a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)
a
ij
(
1
≤
i
≤
k
,
1
≤
j
≤
l
)
be
k
l
kl
k
l
positive integers. Show that if
q
≥
p
>
0
q \geq p > 0
q
≥
p
>
0
, then
(
∑
j
=
1
l
(
∑
i
=
1
k
a
i
j
p
)
q
/
p
)
1
/
q
≤
(
∑
i
=
1
k
(
∑
j
=
1
l
a
i
j
q
)
p
/
q
)
1
/
p
.
(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.
(
j
=
1
∑
l
(
i
=
1
∑
k
a
ij
p
)
q
/
p
)
1/
q
≤
(
i
=
1
∑
k
(
j
=
1
∑
l
a
ij
q
)
p
/
q
)
1/
p
.
2
2
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binomial coeffcient and relation divisible
Prove or disprove: For any positive integer
k
k
k
there exists an integer
n
>
1
n > 1
n
>
1
such that the binomial coeffcient
(
n
i
)
\binom{n}{i}
(
i
n
)
is divisible by
k
k
k
for any
1
≤
i
≤
n
−
1.
1 \leq i \leq n-1.
1
≤
i
≤
n
−
1.
on $S = \{m^2 + dn^2 \}
For a given integer
d
d
d
, let us define
S
=
{
m
2
+
d
n
2
∣
m
,
n
∈
Z
}
S = \{m^{2}+dn^{2}| m, n \in\mathbb{Z}\}
S
=
{
m
2
+
d
n
2
∣
m
,
n
∈
Z
}
. Suppose that
p
,
q
p, q
p
,
q
are two elements of
S
S
S
, where
p
p
p
is prime and
p
∣
q
p | q
p
∣
q
. Prove that
r
=
q
/
p
r = q/p
r
=
q
/
p
also belongs to
S
S
S
.
1
2
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$(A − A) \cap (B − B)$ is nonempty
Let
A
A
A
and
B
B
B
be two subsets of
S
=
{
1
,
2
,
.
.
.
,
2000
}
S = \{1, 2, . . . , 2000\}
S
=
{
1
,
2
,
...
,
2000
}
with
∣
A
∣
⋅
∣
B
∣
≥
3999
|A| \cdot |B| \geq 3999
∣
A
∣
⋅
∣
B
∣
≥
3999
. For a set
X
X
X
, let
X
−
X
X-X
X
−
X
denotes the set
{
s
−
t
∣
s
,
t
∈
X
,
s
≠
t
}
\{s-t | s, t \in X, s \not = t\}
{
s
−
t
∣
s
,
t
∈
X
,
s
=
t
}
. Prove that
(
A
−
A
)
∩
(
B
−
B
)
(A-A) \cap (B-B)
(
A
−
A
)
∩
(
B
−
B
)
is nonempty.
partitions of $2000$ (in a sum of positive integers).
Let
S
S
S
be the set of all partitions of
2000
2000
2000
(in a sum of positive integers). For every such partition
p
p
p
, we define
f
(
p
)
f (p)
f
(
p
)
to be the sum of the number of summands in
p
p
p
and the maximal summand in
p
p
p
. Compute the minimum of
f
(
p
)
f (p)
f
(
p
)
when
p
∈
S
.
p \in S .
p
∈
S
.