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Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2000 Greece JBMO TST
2000 Greece JBMO TST
Part of
Greece JBMO TST
Subcontests
(4)
2
1
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constant sum PM+PN inside a convex ABCD with AB=CD, MN<=BD
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
A
B
=
C
D
AB=CD
A
B
=
C
D
. From a random point
P
P
P
of it's diagonal
B
D
BD
B
D
, we draw a line parallel to
A
B
AB
A
B
that intersects
A
D
AD
A
D
at point
M
M
M
and a line parallel to
C
D
CD
C
D
that intersects
B
C
BC
BC
at point
N
N
N
. Prove that: a) The sum
P
M
+
P
N
PM+PN
PM
+
PN
is constant, independent of the position of
P
P
P
on the diagonal
B
D
BD
B
D
. b)
M
N
≤
B
D
MN\le BD
MN
≤
B
D
. When the equality holds?
4
1
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(s-c)/\sqrt{a}+(s-b)/\sqrt{c}+(s-a)/\sqrt{b}\ge (s-b)/\sqrt{a}+(s-c)/\sqrt{b}+
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be sidelengths with
a
≥
b
≥
c
a\ge b\ge c
a
≥
b
≥
c
and
s
≥
a
+
1
s\ge a+1
s
≥
a
+
1
where
s
s
s
be the semiperimeter of the triangle. Prove that
s
−
c
a
+
s
−
b
c
+
s
−
a
b
≥
s
−
b
a
+
s
−
c
b
+
s
−
a
c
\frac{s-c}{\sqrt{a}}+\frac{s-b}{\sqrt{c}}+\frac{s-a}{\sqrt{b}}\ge \frac{s-b}{\sqrt{a}}+\frac{s-c}{\sqrt{b}}+\frac{s-a}{\sqrt{c}}
a
s
−
c
+
c
s
−
b
+
b
s
−
a
≥
a
s
−
b
+
b
s
−
c
+
c
s
−
a
3
1
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integer solutions of 2x^2+2ax+a-1=0, where a \in Z
Find
a
∈
Z
a\in Z
a
∈
Z
such that the equation
2
x
2
+
2
a
x
+
a
−
1
=
0
2x^2+2ax+a-1=0
2
x
2
+
2
a
x
+
a
−
1
=
0
has integer solutions, which should be found.
1
1
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3n+5/2n+3 irreducible,( \overline{xy}+12)(\overline{xy}-3) \in N => x+y=9
a) Prove that the fraction
3
n
+
5
2
n
+
3
\frac{3n+5}{2n+3}
2
n
+
3
3
n
+
5
is irreducible for every
n
∈
N
n \in N
n
∈
N
b) Let
x
,
y
x,y
x
,
y
be digits of decimal representation system with
x
>
0
x>0
x
>
0
, and
x
y
‾
+
12
x
y
‾
−
3
∈
N
\frac{\overline{xy}+12}{\overline{xy}-3}\in N
x
y
−
3
x
y
+
12
∈
N
, prove that
x
+
y
=
9
x+y=9
x
+
y
=
9
. Is the converse true?