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Contests
International Contests
Czech-Polish-Slovak Junior Match
2012 Czech-Polish-Slovak Junior Match
2012 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
6
1
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tiling a 8x8 board with 1x1 - 1x1 - 1x1 shapes
The
8
×
8
8 \times 8
8
×
8
board is covered with the same shape as in the picture to the right (each of the shapes can be rotated
9
0
o
90^o
9
0
o
) so that any two do not overlap or extend beyond the edge of the chessboard. Determine the largest possible number of fields of this chessboard can be covered as described above. https://cdn.artofproblemsolving.com/attachments/e/5/d7f44f37857eb115edad5ea26400cdca04e107.png
3
2
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projections of point on circumcirle on radii form equilateral of fixed area
Different points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
lie on a circle with a center at the point
O
O
O
at such way that
∠
A
O
B
\angle AOB
∠
A
OB
=
∠
B
O
C
=
= \angle BOC =
=
∠
BOC
=
∠
C
O
D
=
\angle COD =
∠
CO
D
=
6
0
o
60^o
6
0
o
. Point
P
P
P
lies on the shorter arc
B
C
BC
BC
of this circle. Points
K
,
L
,
M
K, L, M
K
,
L
,
M
are projections of
P
P
P
on lines
A
O
,
B
O
,
C
O
AO, BO, CO
A
O
,
BO
,
CO
respectively . Show that (a) the triangle
K
L
M
KLM
K
L
M
is equilateral, (b) the area of triangle
K
L
M
KLM
K
L
M
does not depend on the choice of the position of point
P
P
P
on the shorter arc
B
C
BC
BC
2(n^2 +1)-n not a perfect square
Prove that if
n
n
n
is a positive integer then
2
(
n
2
+
1
)
−
n
2 (n^2 + 1) - n
2
(
n
2
+
1
)
−
n
is not a square of an integer.
4
2
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3 of 51 vertices of 101-regular polygon are vertices of an isosceles
Prove that among any
51
51
51
vertices of the
101
101
101
-regular polygon there are three that are the vertices of an isosceles triangle.
computational inside a rhombus with an angle 60^o
A rhombus
A
B
C
D
ABCD
A
BC
D
is given with
∠
B
A
D
=
6
0
o
\angle BAD = 60^o
∠
B
A
D
=
6
0
o
. Point
P
P
P
lies inside the rhombus such that
B
P
=
1
BP = 1
BP
=
1
,
D
P
=
2
DP = 2
D
P
=
2
,
C
P
=
3
CP = 3
CP
=
3
. Determine the length of the segment
A
P
AP
A
P
.
5
2
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1/2 (c - a) (c - b) is perfect square for positive integers such a^2 + b^2 = c^2
Positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying the equality
a
2
+
b
2
=
c
2
a^2 + b^2 = c^2
a
2
+
b
2
=
c
2
. Show that the number
1
2
(
c
−
a
)
(
c
−
b
)
\frac12(c - a) (c - b)
2
1
(
c
−
a
)
(
c
−
b
)
is square of an integer.
k + a^k = m + 2a^m, diophantine in Z+
Find all triplets
(
a
,
k
,
m
)
(a, k, m)
(
a
,
k
,
m
)
of positive integers that satisfy the equation
k
+
a
k
=
m
+
2
a
m
k + a^k = m + 2a^m
k
+
a
k
=
m
+
2
a
m
.
2
2
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a^2+ab+b^2=c^2+3 prime diophantine
Determine all three primes
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
that satisfied the equality
a
2
+
a
b
+
b
2
=
c
2
+
3
a^2+ab+b^2=c^2+3
a
2
+
ab
+
b
2
=
c
2
+
3
.
parallel and locus of midpoints wanted, as a point moves along a long arc
On the circle
k
k
k
, the points
A
,
B
A,B
A
,
B
are given, while
A
B
AB
A
B
is not the diameter of the circle
k
k
k
. Point
C
C
C
moves along the long arc
A
B
AB
A
B
of circle
k
k
k
so that the triangle
A
B
C
ABC
A
BC
is acute. Let
D
,
E
D,E
D
,
E
be the feet of the altitudes from
A
,
B
A, B
A
,
B
respectively. Let
F
F
F
be the projection of point
D
D
D
on line
A
C
AC
A
C
and
G
G
G
be the projection of point
E
E
E
on line
B
C
BC
BC
. (a) Prove that the lines
A
B
AB
A
B
and
F
G
FG
FG
are parallel. (b) Determine the set of midpoints
S
S
S
of segment
F
G
FG
FG
while along all allowable positions of point
C
C
C
.
1
2
Hide problems
concurrency starting with the symmetric points of an interior wrt midpoints
Point
P
P
P
lies inside the triangle
A
B
C
ABC
A
BC
. Points
K
,
L
,
M
K, L, M
K
,
L
,
M
are symmetrics of point
P
P
P
wrt the midpoints of the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
. Prove that the straight
A
K
,
B
L
,
C
M
AK, BL, CM
A
K
,
B
L
,
CM
intersect at one point.
max no of real numbers in a board such for any 2 their product is included
There are a lot of different real numbers written on the board. It turned out that for each two numbers written, their product was also written. What is the largest possible number of numbers written on the board?