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Contests
International Contests
Czech-Polish-Slovak Junior Match
2016 Czech-Polish-Slovak Junior Match
2016 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
6
1
Hide problems
diophantine system parameter, a + b + c = 3k + 1, ab + bc + ca = 3k^2 + 2k
Let
k
k
k
be a given positive integer. Find all triples of positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
, such that
a
+
b
+
c
=
3
k
+
1
a + b + c = 3k + 1
a
+
b
+
c
=
3
k
+
1
,
a
b
+
b
c
+
c
a
=
3
k
2
+
2
k
ab + bc + ca = 3k^2 + 2k
ab
+
b
c
+
c
a
=
3
k
2
+
2
k
.Slovakia
5
2
Hide problems
fill the fields of the table 10 x10 with no 1-100, minimum
Determine the smallest integer
j
j
j
such that it is possible to fill the fields of the table
10
×
10
10\times 10
10
×
10
with numbers from
1
1
1
to
100
100
100
so that every
10
10
10
consecutive numbers lie in some of the
j
×
j
j\times j
j
×
j
squares of the table.Czech Republic
circumcenter of ABN is midpoint of incenter's segment of ABC and ABM
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
:
A
C
:
B
C
=
5
:
5
:
6
AB : AC : BC =5:5:6
A
B
:
A
C
:
BC
=
5
:
5
:
6
. Denote by
M
M
M
the midpoint of
B
C
BC
BC
and by
N
N
N
the point on the segment
B
C
BC
BC
such that
B
N
=
5
⋅
C
N
BN = 5 \cdot CN
BN
=
5
⋅
CN
. Prove that the circumcenter of triangle
A
B
N
ABN
A
BN
is the midpoint of the segment connecting the incenters of triangles
A
B
C
ABC
A
BC
and
A
B
M
ABM
A
BM
.Slovakia
4
2
Hide problems
AK + KM = BL + LM wanted, AB = CK = CL given, perp. bisectors related
We are given an acute-angled triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
<
B
C
AB < AC < BC
A
B
<
A
C
<
BC
. Points
K
K
K
and
L
L
L
are chosen on segments
A
C
AC
A
C
and
B
C
BC
BC
, respectively, so that
A
B
=
C
K
=
C
L
AB = CK = CL
A
B
=
C
K
=
C
L
. Perpendicular bisectors of segments
A
K
AK
A
K
and
B
L
BL
B
L
intersect the line
A
B
AB
A
B
at points
P
P
P
and
Q
Q
Q
, respectively. Segments
K
P
KP
K
P
and
L
Q
LQ
L
Q
intersect at point
M
M
M
. Prove that
A
K
+
K
M
=
B
L
+
L
M
AK + KM = BL + LM
A
K
+
K
M
=
B
L
+
L
M
.Poland
6 square tile in a 11x11 square table
Several tiles congruent to the one shown in the picture below are to be fit inside a
11
×
11
11 \times 11
11
×
11
square table, with each tile covering
6
6
6
whole unit squares, no sticking out the square and no overlapping. (a) Determine the greatest number of tiles which can be placed this way. (b) Find, with a proof, all unit squares which have to be covered in any tiling with the maximal number of tiles. https://cdn.artofproblemsolving.com/attachments/c/d/23d93e9d05eab94925fc54006fe05123f0dba9.png Poland
3
2
Hide problems
assign pairwise different positive integers to vertices of n-gonal prism
Find all integers
n
≥
3
n \ge 3
n
≥
3
with the following property: it is possible to assign pairwise different positive integers to the vertices of an
n
n
n
-gonal prism in such a way that vertices with labels
a
a
a
and
b
b
b
are connected by an edge if and only if
a
∣
b
a | b
a
∣
b
or
b
∣
a
b | a
b
∣
a
.Poland
each lines of a set in a plane intersects exactly 15 other lines
On a plane several straight lines are drawn in such a way that each of them intersects exactly
15
15
15
other lines. How many lines are drawn on the plane? Find all possibilities and justify your answer.Poland
2
2
Hide problems
if x^2 + y^2 - 1 < xy then x + y - |x - y| < 2
Let
x
x
x
and
y
y
y
be real numbers such that
x
2
+
y
2
−
1
<
x
y
x^2 + y^2 - 1 < xy
x
2
+
y
2
−
1
<
x
y
. Prove that
x
+
y
−
∣
x
−
y
∣
<
2
x + y - |x - y| < 2
x
+
y
−
∣
x
−
y
∣
<
2
.Slovakia
max d, divides all 3 of abc, bca, cab where a,b,c non zero diff. digits
Find the largest integer
d
d
d
divides all three numbers
a
b
c
,
b
c
a
abc, bca
ab
c
,
b
c
a
and
c
a
b
cab
c
ab
with
a
,
b
a, b
a
,
b
and
c
c
c
being some nonzero and mutually different digits.Czech Republic
1
2
Hide problems
largest possible area of the quadrilateral MKCL
Let
A
B
AB
A
B
be a given segment and
M
M
M
be its midpoint. We consider the set of right-angled triangles
A
B
C
ABC
A
BC
with hypotenuses
A
B
AB
A
B
. Denote by
D
D
D
the foot of the altitude from
C
C
C
. Let
K
K
K
and
L
L
L
be feet of perpendiculars from
D
D
D
to the legs
B
C
BC
BC
and
A
C
AC
A
C
, respectively. Determine the largest possible area of the quadrilateral
M
K
C
L
MKCL
M
K
C
L
.Czech Republic
angle chasing candidate, <AP B + < QCR = 180^o , 3 midpoints related
Let
A
B
C
ABC
A
BC
be a right-angled triangle with hypotenuse
A
B
AB
A
B
. Denote by
D
D
D
the foot of the altitude from
C
C
C
. Let
Q
,
R
Q, R
Q
,
R
, and
P
P
P
be the midpoints of the segments
A
D
,
B
D
AD, BD
A
D
,
B
D
, and
C
D
CD
C
D
, respectively. Prove that
∠
A
P
B
+
∠
Q
C
R
=
18
0
o
\angle AP B + \angle QCR = 180^o
∠
A
PB
+
∠
QCR
=
18
0
o
.Czech Republic