MathDB
Problems
Contests
International Contests
Hungary-Israel Binational
2001 Hungary-Israel Binational
2001 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(6)
5
2
Hide problems
find $\angle {BAC}$
In a triangle
A
B
C
ABC
A
BC
,
B
1
B_{1}
B
1
and
C
1
C_{1}
C
1
are the midpoints of
A
C
AC
A
C
and
A
B
AB
A
B
respectively, and
I
I
I
is the incenter. The lines
B
1
I
B_{1}I
B
1
I
and
C
1
I
C_{1}I
C
1
I
meet
A
B
AB
A
B
and
A
C
AC
A
C
respectively at
C
2
C_{2}
C
2
and
B
2
B_{2}
B
2
. If the areas of
Δ
A
B
C
\Delta ABC
Δ
A
BC
and
Δ
A
B
2
C
2
\Delta AB_{2}C_{2}
Δ
A
B
2
C
2
are equal, find
∠
B
A
C
\angle{BAC}
∠
B
A
C
.
graph contain $C_4$ ?
Here
G
n
G_{n}
G
n
denotes a simple undirected graph with
n
n
n
vertices,
K
n
K_{n}
K
n
denotes the complete graph with
n
n
n
vertices,
K
n
,
m
K_{n,m}
K
n
,
m
the complete bipartite graph whose components have
m
m
m
and
n
n
n
vertices, and
C
n
C_{n}
C
n
a circuit with
n
n
n
vertices. The number of edges in the graph
G
n
G_{n}
G
n
is denoted
e
(
G
n
)
e(G_{n})
e
(
G
n
)
. (a) Let
p
p
p
be a prime. Consider the graph whose vertices are the ordered pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
with
x
,
y
∈
{
0
,
1
,
.
.
.
,
p
−
1
}
x, y \in\{0, 1, . . . , p-1\}
x
,
y
∈
{
0
,
1
,
...
,
p
−
1
}
and whose edges join vertices
(
x
,
y
)
(x, y)
(
x
,
y
)
and
(
x
′
,
y
′
)
(x' , y')
(
x
′
,
y
′
)
if and only if
x
x
′
+
y
y
′
≡
1
(
m
o
d
p
)
xx'+yy'\equiv 1 \pmod{p}
x
x
′
+
y
y
′
≡
1
(
mod
p
)
. Prove that this graph does not contain
C
4
C_{4}
C
4
. (b) Prove that for infinitely many values
n
n
n
there is a graph
G
n
G_{n}
G
n
with
e
(
G
n
)
≥
n
n
2
−
n
e(G_{n}) \geq \frac{n\sqrt{n}}{2}-n
e
(
G
n
)
≥
2
n
n
−
n
that does not contain
C
4
C_{4}
C
4
.
6
1
Hide problems
$32$ positive integers with the sum $120$
Let be given
32
32
32
positive integers with the sum
120
120
120
, none of which is greater than
60.
60.
60.
Prove that these integers can be divided into two disjoint subsets with the same sum of elements.
2
2
Hide problems
Find the locus
Points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
lie on a line
l
l
l
, in that order. Find the locus of points
P
P
P
in the plane for which
∠
A
P
B
=
∠
C
P
D
\angle{APB}=\angle{CPD}
∠
A
PB
=
∠
CP
D
.
$G_n$ contains two triangles that share exactly one vertex
Here
G
n
G_{n}
G
n
denotes a simple undirected graph with
n
n
n
vertices,
K
n
K_{n}
K
n
denotes the complete graph with
n
n
n
vertices,
K
n
,
m
K_{n,m}
K
n
,
m
the complete bipartite graph whose components have
m
m
m
and
n
n
n
vertices, and
C
n
C_{n}
C
n
a circuit with
n
n
n
vertices. The number of edges in the graph
G
n
G_{n}
G
n
is denoted
e
(
G
n
)
e(G_{n})
e
(
G
n
)
. If
n
≥
5
n \geq 5
n
≥
5
and
e
(
G
n
)
≥
n
2
4
+
2
e(G_{n}) \geq \frac{n^{2}}{4}+2
e
(
G
n
)
≥
4
n
2
+
2
, prove that
G
n
G_{n}
G
n
contains two triangles that share exactly one vertex.
3
2
Hide problems
f (f (x)) = f (x) + x(maybe is old)
Find all continuous functions
f
:
R
→
R
f : \mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that for all
x
∈
R
x \in\mathbb{ R}
x
∈
R
,
f
(
f
(
x
)
)
=
f
(
x
)
+
x
.
f (f (x)) = f (x)+x.
f
(
f
(
x
))
=
f
(
x
)
+
x
.
prove that $G_n$ contains $C_4$
Here
G
n
G_{n}
G
n
denotes a simple undirected graph with
n
n
n
vertices,
K
n
K_{n}
K
n
denotes the complete graph with
n
n
n
vertices,
K
n
,
m
K_{n,m}
K
n
,
m
the complete bipartite graph whose components have
m
m
m
and
n
n
n
vertices, and
C
n
C_{n}
C
n
a circuit with
n
n
n
vertices. The number of edges in the graph
G
n
G_{n}
G
n
is denoted
e
(
G
n
)
e(G_{n})
e
(
G
n
)
. If
e
(
G
n
)
≥
n
n
2
+
n
4
e(G_{n}) \geq\frac{n\sqrt{n}}{2}+\frac{n}{4}
e
(
G
n
)
≥
2
n
n
+
4
n
,prove that
G
n
G_{n}
G
n
contains
C
4
C_{4}
C
4
.
1
2
Hide problems
on equation $2000x^2 + y^2 = 2001z^2.$
Find positive integers
x
,
y
,
z
x, y, z
x
,
y
,
z
such that
x
>
z
>
1999
⋅
2000
⋅
2001
>
y
x > z > 1999 \cdot 2000 \cdot 2001 > y
x
>
z
>
1999
⋅
2000
⋅
2001
>
y
and
2000
x
2
+
y
2
=
2001
z
2
.
2000x^{2}+y^{2}= 2001z^{2}.
2000
x
2
+
y
2
=
2001
z
2
.
graph has got a triangle whose sides have di erent colors
Here
G
n
G_{n}
G
n
denotes a simple undirected graph with
n
n
n
vertices,
K
n
K_{n}
K
n
denotes the complete graph with
n
n
n
vertices,
K
n
,
m
K_{n,m}
K
n
,
m
the complete bipartite graph whose components have
m
m
m
and
n
n
n
vertices, and
C
n
C_{n}
C
n
a circuit with
n
n
n
vertices. The number of edges in the graph
G
n
G_{n}
G
n
is denoted
e
(
G
n
)
e(G_{n})
e
(
G
n
)
. The edges of
K
n
(
n
≥
3
)
K_{n}(n \geq 3)
K
n
(
n
≥
3
)
are colored with
n
n
n
colors, and every color is used. Show that there is a triangle whose sides have different colors.
4
2
Hide problems
$P (x) = x^3 − 3x + 1.$
Let
P
(
x
)
=
x
3
−
3
x
+
1.
P (x) = x^{3}-3x+1.
P
(
x
)
=
x
3
−
3
x
+
1.
Find the polynomial
Q
Q
Q
whose roots are the fifth powers of the roots of
P
P
P
.
use graph
Here
G
n
G_{n}
G
n
denotes a simple undirected graph with
n
n
n
vertices,
K
n
K_{n}
K
n
denotes the complete graph with
n
n
n
vertices,
K
n
,
m
K_{n,m}
K
n
,
m
the complete bipartite graph whose components have
m
m
m
and
n
n
n
vertices, and
C
n
C_{n}
C
n
a circuit with
n
n
n
vertices. The number of edges in the graph
G
n
G_{n}
G
n
is denoted
e
(
G
n
)
e(G_{n})
e
(
G
n
)
. (a) If
G
n
G_{n}
G
n
does not contain
K
2
,
3
K_{2,3}
K
2
,
3
, prove that
e
(
G
n
)
≤
n
n
2
+
n
e(G_{n}) \leq\frac{n\sqrt{n}}{\sqrt{2}}+n
e
(
G
n
)
≤
2
n
n
+
n
. (b) Given
n
≥
16
n \geq 16
n
≥
16
distinct points
P
1
,
.
.
.
,
P
n
P_{1}, . . . , P_{n}
P
1
,
...
,
P
n
in the plane, prove that at most
n
n
n\sqrt{n}
n
n
of the segments
P
i
P
j
P_{i}P_{j}
P
i
P
j
have unit length.