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Contests
International Contests
Czech-Polish-Slovak Junior Match
2013 Czech-Polish-Slovak Junior Match
2013 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
6
1
Hide problems
min radius such 3 equal circles cover a square
There is a square
A
B
C
D
ABCD
A
BC
D
in the plane with
∣
A
B
∣
=
a
|AB|=a
∣
A
B
∣
=
a
. Determine the smallest possible radius value of three equal circles to cover a given square.
5
2
Hide problems
midpoint lies on perpendicular bisectors, circumcenters related
Point
M
M
M
is the midpoint of the side
A
B
AB
A
B
of an acute triangle
A
B
C
ABC
A
BC
. Point
P
P
P
lies on the segment
A
B
AB
A
B
, and points
S
1
S_1
S
1
and
S
2
S_2
S
2
are the centers of the circumcircles of
A
P
C
APC
A
PC
and
B
P
C
BPC
BPC
, respectively. Show that the midpoint of segment
S
1
S
2
S_1S_2
S
1
S
2
lies on the perpendicular bisector of segment
C
M
CM
CM
.
a+b+c>= 3 if ab+ac+bc >= a +b +c and a,b,c>0
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers for which
a
b
+
a
c
+
b
c
≥
a
+
b
+
c
ab + ac + bc \ge a + b + c
ab
+
a
c
+
b
c
≥
a
+
b
+
c
. Prove that
a
+
b
+
c
≥
3
a + b + c \ge 3
a
+
b
+
c
≥
3
.
4
2
Hide problems
d | \overline{aabbcc} iff d | \overline{abc} where d is two digit number
Determine the largest two-digit number
d
d
d
with the following property: for any six-digit number
a
a
b
b
c
c
‾
\overline{aabbcc}
aabb
cc
number
d
d
d
is a divisor of the number
a
a
b
b
c
c
‾
\overline{aabbcc}
aabb
cc
if and only if the number
d
d
d
is a divisor of the corresponding three-digit number
a
b
c
‾
\overline{abc}
ab
c
.Note The numbers
a
≠
0
,
b
a \ne 0, b
a
=
0
,
b
and
c
c
c
need not be different.
angle chasing candidate, <ADB =< PDC wanted, circumcircle related
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
D
A
B
=
∠
A
B
C
=
∠
B
C
D
>
9
0
o
\angle DAB =\angle ABC =\angle BCD > 90^o
∠
D
A
B
=
∠
A
BC
=
∠
BC
D
>
9
0
o
. The circle circumscribed around the triangle
A
B
C
ABC
A
BC
intersects the sides
A
D
AD
A
D
and
C
D
CD
C
D
at points
K
K
K
and
L
L
L
, respectively, different from any vertex of the quadrilateral
A
B
C
D
ABCD
A
BC
D
. Segments
A
L
AL
A
L
and
C
K
CK
C
K
intersect at point
P
P
P
. Prove that
∠
A
D
B
=
∠
P
D
C
\angle ADB =\angle PDC
∠
A
D
B
=
∠
P
D
C
.
3
2
Hide problems
AK = KL inside a cyclic pentagon ABCDE with AB = BC = CD
The
A
B
C
D
E
ABCDE
A
BC
D
E
pentagon is inscribed in a circle and
A
B
=
B
C
=
C
D
AB = BC = CD
A
B
=
BC
=
C
D
. Segments
A
C
AC
A
C
and
B
E
BE
BE
intersect at
K
K
K
, and Segments
A
D
AD
A
D
and
C
E
CE
CE
intersect at point
L
L
L
. Prove that
A
K
=
K
L
AK = KL
A
K
=
K
L
.
5 of n people may sit at a circle on a table among friends / strangers
In a certain group there are
n
≥
5
n \ge 5
n
≥
5
people, with every two people who do not know each other exactly having one mutual friend and no one knows everyone else. Prove
5
5
5
of
n
n
n
people, may sit at a circle around the table so that each of them sits between a) friends, b) strangers.
2
2
Hide problems
coloring all postitive integers red or green
Each positive integer should be colored red or green in such a way that the following two conditions are met: - Let
n
n
n
be any red number. The sum of any
n
n
n
(not necessarily different) red numbers is red. - Let
m
m
m
be any green number. The sum of any
m
m
m
(not necessarily different) green numbers is green. Determine all such colorings.
sum of the three largest divisors of n is 1457
Find all natural numbers
n
n
n
such that the sum of the three largest divisors of
n
n
n
is
1457
1457
1457
.
1
2
Hide problems
\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy} diophantine
Determine all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of integers for which satisfy the equality
x
−
y
+
x
+
y
=
x
y
\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}
x
−
y
+
x
+
y
=
x
y
infinite primes p having a multiple in the form n^2 + n + 1 ?
Decide whether there are infinitely many primes
p
p
p
having a multiple in the form
n
2
+
n
+
1
n^2 + n + 1
n
2
+
n
+
1
for some natural number
n
n
n