MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Northern Summer Camp Of Mathematics
2011 Northern Summer Camp Of Mathematics
2011 Northern Summer Camp Of Mathematics
Part of
Northern Summer Camp Of Mathematics
Subcontests
(5)
5
1
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Groups of scientists [Vietnam Northern Camp, 2011 - P5]
In a meeting, there are
2011
2011
2011
scientists attending. We know that, every scientist know at least
1509
1509
1509
other ones. Prove that a group of five scientists can be formed so that each one in this group knows
4
4
4
people in his group.
4
1
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7^n+147 is a perfect square
Find all positive integers
n
n
n
such that
7
n
+
147
7^n+147
7
n
+
147
is a perfect square.
3
1
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Q is on the smaller arc AC of circumcircle of ABC
Given an acute triangle
A
B
C
ABC
A
BC
such that
∠
C
<
∠
B
<
∠
A
\angle C< \angle B< \angle A
∠
C
<
∠
B
<
∠
A
. Let
I
I
I
be the incenter of
A
B
C
ABC
A
BC
. Let
M
M
M
be the midpoint of the smaller arc
B
C
BC
BC
,
N
N
N
be the midpoint of the segment
B
C
BC
BC
and let
E
E
E
be a point such that
N
E
=
N
I
NE=NI
NE
=
N
I
. The line
M
E
ME
ME
intersects circumcircle of
A
B
C
ABC
A
BC
at
Q
Q
Q
(different from
A
,
B
A, B
A
,
B
, and
C
C
C
). Prove that(i) The point
Q
Q
Q
is on the smaller arc
A
C
AC
A
C
of circumcircle of
A
B
C
ABC
A
BC
.(ii)
B
Q
=
A
Q
+
C
Q
BQ=AQ+CQ
BQ
=
A
Q
+
CQ
2
1
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All functions with f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2
Find all functions
f
:
N
∪
{
0
}
→
N
∪
{
0
}
f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}
f
:
N
∪
{
0
}
→
N
∪
{
0
}
such that
f
(
1
)
>
0
f(1)>0
f
(
1
)
>
0
and f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \forall m,n \in \mathbb N\cup \{0\}.
1
1
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System of equations [Vietnam Northern Camp, 2011 - P1]
Solve the system of equations
(
x
+
x
2
+
1
)
(
y
+
y
2
+
1
)
=
1
,
(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=1,
(
x
+
x
2
+
1
)
(
y
+
y
2
+
1
)
=
1
,
y
+
y
x
2
−
1
+
35
12
=
0.
y+\frac{y}{\sqrt{x^2-1}}+\frac{35}{12}=0.
y
+
x
2
−
1
y
+
12
35
=
0.