MathDB
Problems
Contests
International Contests
Hungary-Israel Binational
2003 Hungary-Israel Binational
2003 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(3)
3
2
Hide problems
integers between $n^2$ and $n^2 + n + 3\sqrt{n}$
Let
n
n
n
be a positive integer. Show that there exist three distinct integers between
n
2
n^{2}
n
2
and
n
2
+
n
+
3
n
n^{2}+n+3\sqrt{n}
n
2
+
n
+
3
n
, such that one of them divides the product of the other two.
set {s=\sum\frac{1}{x_i}|x_i\in\mathbb{N},s<d}
Let
d
>
0
d > 0
d
>
0
be an arbitrary real number. Consider the set
S
n
(
d
)
=
{
s
=
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
∣
x
i
∈
N
,
s
<
d
}
S_{n}(d)=\{s=\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}|x_{i}\in\mathbb{N},s<d\}
S
n
(
d
)
=
{
s
=
x
1
1
+
x
2
1
+
...
+
x
n
1
∣
x
i
∈
N
,
s
<
d
}
. Prove that
S
n
(
d
)
S_{n}(d)
S
n
(
d
)
has a maximum element.
2
2
Hide problems
$CC_1$ bisects $\widehat{QC_1P}$ .
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. The tangents to its circumcircle at
A
,
B
,
C
A, B, C
A
,
B
,
C
form a triangle
P
Q
R
PQR
PQR
with
C
∈
P
Q
C \in PQ
C
∈
PQ
and
B
∈
P
R
B \in PR
B
∈
PR
. Let
C
1
C_{1}
C
1
be the foot of the altitude from
C
C
C
in
Δ
A
B
C
\Delta ABC
Δ
A
BC
. Prove that
C
C
1
CC_{1}
C
C
1
bisects
Q
C
1
P
^
\widehat{QC_{1}P}
Q
C
1
P
.
one of the lines $AA_1 , BB_1, CC_1$ is a median of ABC
Let
M
M
M
be a point inside a triangle
A
B
C
ABC
A
BC
. The lines
A
M
,
B
M
,
C
M
AM , BM , CM
A
M
,
BM
,
CM
intersect
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. Assume that
S
M
A
C
1
+
S
M
B
A
1
+
S
M
C
B
1
=
S
M
A
1
C
+
S
M
B
1
A
+
S
M
C
1
B
S_{MAC_{1}}+S_{MBA_{1}}+S_{MCB_{1}}= S_{MA_{1}C}+S_{MB_{1}A}+S_{MC_{1}B}
S
M
A
C
1
+
S
MB
A
1
+
S
MC
B
1
=
S
M
A
1
C
+
S
M
B
1
A
+
S
M
C
1
B
. Prove that one of the lines
A
A
1
,
B
B
1
,
C
C
1
AA_{1}, BB_{1}, CC_{1}
A
A
1
,
B
B
1
,
C
C
1
is a median of the triangle
A
B
C
.
ABC.
A
BC
.
1
2
Hide problems
n positive reals
If
x
1
,
x
2
,
.
.
.
,
x
n
x_{1}, x_{2}, . . . , x_{n}
x
1
,
x
2
,
...
,
x
n
are positive numbers, prove the inequality
x
1
3
x
1
2
+
x
1
x
2
+
x
2
2
+
x
2
3
x
2
2
+
x
2
x
3
+
x
3
2
+
.
.
.
+
x
n
3
x
n
2
+
x
n
x
1
+
x
1
2
≥
x
1
+
x
2
+
.
.
.
+
x
n
3
\frac{x_{1}^{3}}{x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}}+\frac{x_{2}^{3}}{x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}}+...+\frac{x_{n}^{3}}{x_{n}^{2}+x_{n}x_{1}+x_{1}^{2}}\geq\frac{x_{1}+x_{2}+...+x_{n}}{3}
x
1
2
+
x
1
x
2
+
x
2
2
x
1
3
+
x
2
2
+
x
2
x
3
+
x
3
2
x
2
3
+
...
+
x
n
2
+
x
n
x
1
+
x
1
2
x
n
3
≥
3
x
1
+
x
2
+
...
+
x
n
.
alternately write divisors of 100! on the blackboard
Two players play the following game. They alternately write divisors of
100
!
100!
100
!
on the blackboard, not repeating any of the numbers written before. The player after whose move the greatest common divisor of the written numbers equals
1
,
1,
1
,
loses the game. Which player has a winning strategy?