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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2010 Czech And Slovak Olympiad III A
2010 Czech And Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
3
1
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kindapping members from 4 parties
Rumburak kidnapped
31
31
31
members of party
A
A
A
,
28
28
28
members of party
B
B
B
,
23
23
23
members of party
C
C
C
,
19
19
19
members of Party
D
D
D
and each of them in a separate cell. After work out occasionally they could walk in the yard and talk. Once three people started to talk to each other members of three different parties, Rumburak re-registered them to the fourth party as a punishment.(They never talked to each other more than three kidnapped.) a) Could it be that after some time all were abducted by members of one party? Which? b) Determine all four positive integers of which the sum is
101
101
101
and which as the numbers of kidnapped members of the four parties allow the Rumburaks all of them became members of one party over time.
4
1
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locus of intersections of non parallels isosceles trapezoids
A circle
k
k
k
is given with a non-diameter chord
A
C
AC
A
C
. On the tangent at point
A
A
A
select point
X
≠
A
X \ne A
X
=
A
and mark
D
D
D
the intersection of the circle
k
k
k
with the interior of the line
X
C
XC
XC
(if any). Let
B
B
B
a point in circle
k
k
k
such that quadrilateral
A
B
C
D
ABCD
A
BC
D
is a trapezoid . Determine the set of intersections of lines
B
C
BC
BC
and
A
D
AD
A
D
belonging to all such trapezoids.
5
1
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on the board are written numbers 1, 2,. . . , 33 , product, square root, game
On the board are written numbers
1
,
2
,
.
.
.
,
33
1, 2,. . . , 33
1
,
2
,
...
,
33
. In one step we select two numbers written on the product of which is the square of the natural number, we wipe off the two chosen numbers and write the square root of their product on the board. This way we continue to the board only the numbers remain so that the product of neither of them is a square. (In one we can also wipe out two identical numbers and replace them with the same number.) Prove that at least
16
16
16
numbers remain on the board.
6
1
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min of \frac{a+b+c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c} , LCM
Find the minimum of the expression
a
+
b
+
c
2
−
[
a
,
b
]
+
[
b
,
c
]
+
[
c
,
a
]
a
+
b
+
c
\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}
2
a
+
b
+
c
−
a
+
b
+
c
[
a
,
b
]
+
[
b
,
c
]
+
[
c
,
a
]
where the variables
a
,
b
,
c
a, b, c
a
,
b
,
c
are any integers greater than
1
1
1
and
[
x
,
y
]
[x, y]
[
x
,
y
]
denotes the least common multiple of numbers
x
,
y
x, y
x
,
y
.
2
1
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distance of 2 from 19 points in a circle or radius 12 less than 7
A circular target with a radius of
12
12
12
cm was hit by
19
19
19
shots. Prove that the distance between two hits is less than
7
7
7
cm.
1
1
Hide problems
4^a + 4a^2 + 4 = b^2 diophantine
Determine all pairs of integers
a
,
b
a, b
a
,
b
for which they apply
4
a
+
4
a
2
+
4
=
b
2
4^a + 4a^2 + 4 = b^2
4
a
+
4
a
2
+
4
=
b
2
.