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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1994 Greece National Olympiad
1994 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(5)
5
1
Hide problems
r=\sqrt{\r_1r_2r_3/(r_1+r_2+r_3)}, 3 tangent circles ext. in pairs
Three circles
O
1
,
O
2
,
O
3
O_1, \ O_2, \ O_3
O
1
,
O
2
,
O
3
with radiii
r
1
,
r
2
,
r
3
r_1, \ r_2, \ r_3
r
1
,
r
2
,
r
3
respectively are tangent extarnally in pairs. Let r be the radius of the inscrined circle of triangle
O
1
O
2
O
3
O_1O_2O_3
O
1
O
2
O
3
. Prove that
r
=
r
1
r
2
r
3
r
1
+
r
2
+
r
3
.
r=\sqrt{\dfrac{r_1r_2r_3}{r_1+r_2+r_3}}.
r
=
r
1
+
r
2
+
r
3
r
1
r
2
r
3
.
4
1
Hide problems
x_1+x_2+x_3, multiples of 3 for 1<= x_j <= 300
How many sums
x
1
+
x
2
+
x
3
,
1
≤
x
j
≤
300
,
j
=
1
,
2
,
3
x_1+x_2+x_3, \ \ 1\leq x_j\leq 300, \ j=1,2,3
x
1
+
x
2
+
x
3
,
1
≤
x
j
≤
300
,
j
=
1
,
2
,
3
are multiples of
3
3
3
;
3
1
Hide problems
(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2<= 4
If
a
2
+
b
2
+
c
2
+
d
2
=
1
a^2+b^2+c^2+d^2=1
a
2
+
b
2
+
c
2
+
d
2
=
1
, prove that
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
d
)
2
+
(
a
−
c
)
2
+
(
a
−
d
)
2
+
(
b
−
d
)
2
≤
4
(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2\leq 4
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
d
)
2
+
(
a
−
c
)
2
+
(
a
−
d
)
2
+
(
b
−
d
)
2
≤
4
When does equality holds?
2
1
Hide problems
2 integer roots for x^3+1995x^2-1994x+m
Fow which real values of
m
m
m
does the polynomial
x
3
+
1995
x
2
−
1994
x
+
m
x^3+1995x^2-1994x+m
x
3
+
1995
x
2
−
1994
x
+
m
have all three roots integers?
1
1
Hide problems
2(1991m^2+1993mn+1995n^2) newver perfect square
Prove that number
2
(
1991
m
2
+
1993
m
n
+
1995
n
2
)
2(1991m^2+1993mn+1995n^2)
2
(
1991
m
2
+
1993
mn
+
1995
n
2
)
where
m
,
n
m,n
m
,
n
are poitive integers, cannot be a square of an integer.