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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2008 Czech and Slovak Olympiad III A
2008 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(3)
3
2
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czech republic,czech and slovakia MO(third round),2008 p3
Find all pairs of integers
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
a
2
+
a
b
+
1
∣
b
2
+
a
b
+
a
+
b
−
1
a^2+ab+1\mid b^2+ab+a+b-1
a
2
+
ab
+
1
∣
b
2
+
ab
+
a
+
b
−
1
.
czech republic,czech and slovakia MO(third round),2008 p6
Find the greatest value of
p
p
p
and the smallest value of
q
q
q
such that for any triangle in the plane, the inequality
p
<
a
+
m
b
+
n
<
q
p<\frac{a+m}{b+n}<q
p
<
b
+
n
a
+
m
<
q
holds, where
a
,
b
a,b
a
,
b
are it's two sides and
m
,
n
m,n
m
,
n
their corresponding medians.
2
2
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czech republic,czech and slovakia MO(third round),2008 p2
Two disjoint circles
W
1
(
S
1
,
r
1
)
W_1(S_1,r_1)
W
1
(
S
1
,
r
1
)
and
W
2
(
S
2
,
r
2
)
W_2(S_2,r_2)
W
2
(
S
2
,
r
2
)
are given in the plane. Point
A
A
A
is on circle
W
1
W_1
W
1
and
A
B
,
A
C
AB,AC
A
B
,
A
C
touch the circle
W
2
W_2
W
2
at
B
,
C
B,C
B
,
C
respectively. Find the loci of the incenter and orthocenter of triangle
A
B
C
ABC
A
BC
.
czech republic,czech and slovakia MO(third round),2008 p5
At one moment, a kid noticed that the end of the hour hand, the end of the minute hand and one of the twelve numbers (regarded as a point) of his watch formed an equilateral triangle. He also calculated that
t
t
t
hours would elapse for the next similar case. Suppose that the ratio of the lengths of the minute hand (whose length is equal to the distance from the center of the watch plate to any of the twelve numbers) and the hour hand is
k
>
1
k>1
k
>
1
. Find the maximal value of
t
t
t
.
1
2
Hide problems
czech republic,czech and slovakia MO(third round),2008 p1
Find all pairs of real numbers
(
x
,
y
)
(x,y)
(
x
,
y
)
satisfying:
x
+
y
2
=
y
3
,
x+y^2=y^3,
x
+
y
2
=
y
3
,
y
+
x
2
=
x
3
.
y+x^2=x^3.
y
+
x
2
=
x
3
.
czech republic,czech and slovakia MO(third round),2008 p4
In decimal representation, we call an integer
k
k
k
-carboxylic if and only if it can be represented as a sum of
k
k
k
distinct integers, all of them greater than
9
9
9
, whose digits are the same. For instance,
2008
2008
2008
is
5
5
5
-carboxylic because
2008
=
1111
+
666
+
99
+
88
+
44
2008=1111+666+99+88+44
2008
=
1111
+
666
+
99
+
88
+
44
. Find, with an example, the smallest integer
k
k
k
such that
8002
8002
8002
is
k
k
k
-carboxylic.