MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2007 Greece National Olympiad
2007 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
3
1
Hide problems
maximum circles
In a circular ring with radii
11
r
11r
11
r
and
9
r
9r
9
r
, we put circles of radius
r
r
r
which are tangent to the boundary circles and do not overlap. Determine the maximum number of circles that can be put this way. (You may use that
9.94
<
99
<
9.95
9.94<\sqrt{99}<9.95
9.94
<
99
<
9.95
)
4
1
Hide problems
sum of products not zero
Given a
2007
×
2007
2007\times 2007
2007
×
2007
array of numbers
1
1
1
and
−
1
-1
−
1
, let
A
i
A_{i}
A
i
denote the product of the entries in the
i
i
i
th row, and
B
j
B_{j}
B
j
denote the product of the entries in the
j
j
j
th column. Show that
A
1
+
A
2
+
⋯
+
A
2007
+
B
1
+
B
2
+
⋯
+
B
2007
≠
0.
A_{1}+A_{2}+\cdots +A_{2007}+B_{1}+B_{2}+\cdots +B_{2007}\neq 0.
A
1
+
A
2
+
⋯
+
A
2007
+
B
1
+
B
2
+
⋯
+
B
2007
=
0.
2
1
Hide problems
messy ineq with triangle
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be sides of a triangle, show that
(
c
+
a
−
b
)
4
a
(
a
+
b
−
c
)
+
(
a
+
b
−
c
)
4
b
(
b
+
c
−
a
)
+
(
b
+
c
−
a
)
4
c
(
c
+
a
−
b
)
≥
a
b
+
b
c
+
c
a
.
\frac{(c+a-b)^{4}}{a(a+b-c)}+\frac{(a+b-c)^{4}}{b(b+c-a)}+\frac{(b+c-a)^{4}}{c(c+a-b)}\geq ab+bc+ca.
a
(
a
+
b
−
c
)
(
c
+
a
−
b
)
4
+
b
(
b
+
c
−
a
)
(
a
+
b
−
c
)
4
+
c
(
c
+
a
−
b
)
(
b
+
c
−
a
)
4
≥
ab
+
b
c
+
c
a
.
1
1
Hide problems
fourth power becomes perfect square
Find all positive integers
n
n
n
such that
4
n
+
2007
4^{n}+2007
4
n
+
2007
is a perfect square.