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International Contests
Czech-Polish-Slovak Junior Match
2017 Czech-Polish-Slovak Junior Match
2017 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
6
1
Hide problems
a^2 + bc is an integer, x/y is rational
On the board are written
100
100
100
mutually different positive real numbers, such that for any three different numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
is
a
2
+
b
c
a^2 + bc
a
2
+
b
c
is an integer. Prove that for any two numbers
x
,
y
x, y
x
,
y
from the board , number
x
y
\frac{x}{y}
y
x
is rational.
4
2
Hide problems
Czech-Polish-Slovak Junior geometry - excircle related
Given is a right triangle
A
B
C
ABC
A
BC
with perimeter
2
2
2
, with
∠
B
=
9
0
o
\angle B=90^o
∠
B
=
9
0
o
. Point
S
S
S
is the center of the excircle to the side
A
B
AB
A
B
of the triangle and
H
H
H
is the intersection of the heights of the triangle
A
B
S
ABS
A
BS
. Determine the smallest possible length of the segment
H
S
HS
H
S
.
reconstruct a trapezoid given two segments
Bolek draw a trapezoid
A
B
C
D
ABCD
A
BC
D
trapezoid (
A
B
/
/
C
D
AB // CD
A
B
//
C
D
) on the board, with its midsegment line
E
F
EF
EF
in it. Point intersection of his diagonal
A
C
,
B
D
AC, BD
A
C
,
B
D
denote by
P
,
P,
P
,
and his rectangular projection on line
A
B
AB
A
B
denote by
Q
Q
Q
. Lolek, wanting to tease Bolek, blotted from the board everything except segments
E
F
EF
EF
and
P
Q
PQ
PQ
. When Bolek saw it, wanted to complete the drawing and draw the original trapezoid, but did not know how to do it. Can you help Bolek?
5
2
Hide problems
real numbers from [0, 1] into (mn + 1) x (mn + 1) table
Each field of the table
(
m
n
+
1
)
×
(
m
n
+
1
)
(mn + 1) \times (mn + 1)
(
mn
+
1
)
×
(
mn
+
1
)
contains a real number from the interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
. The sum the numbers in each square section of the table with dimensions
n
x
n
n x n
n
x
n
is equal to
n
n
n
. Determine how big it can be sum of all numbers in the table.
1,2,3 in 100x100 square table , balanced 2x2 subtables
In each square of the
100
×
100
100\times 100
100
×
100
square table, type
1
,
2
1, 2
1
,
2
, or
3
3
3
. Consider all subtables
m
×
n
m \times n
m
×
n
, where
m
=
2
m = 2
m
=
2
and
n
=
2
n = 2
n
=
2
. A subtable will be called balanced if it has in its corner boxes of four identical numbers boxes . For as large a number
k
k
k
prove, that we can always find
k
k
k
balanced subtables, of which no two overlap, i.e. do not have a common box.
3
2
Hide problems
(x^2 + 1)(y^2 + 1) >= 2(xy - 1)(x + y)
Prove that for all real numbers
x
,
y
x, y
x
,
y
holds
(
x
2
+
1
)
(
y
2
+
1
)
≥
2
(
x
y
−
1
)
(
x
+
y
)
(x^2 + 1)(y^2 + 1) \ge 2(xy - 1)(x + y)
(
x
2
+
1
)
(
y
2
+
1
)
≥
2
(
x
y
−
1
)
(
x
+
y
)
. For which integers
x
,
y
x, y
x
,
y
does equality occur?
number of 8-digit numbers *2*0*1*7, divisible by 7
How many
8
8
8
-digit numbers are
∗
2
∗
0
∗
1
∗
7
*2*0*1*7
∗
2
∗
0
∗
1
∗
7
, where four unknown numbers are replaced by stars, which are divisible by
7
7
7
?
2
2
Hide problems
concurrency wanted , starting with AB + AC = 3 BC
Given is the triangle
A
B
C
ABC
A
BC
, with
∣
A
B
∣
+
∣
A
C
∣
=
3
⋅
∣
B
C
∣
| AB | + | AC | = 3 \cdot | BC |
∣
A
B
∣
+
∣
A
C
∣
=
3
⋅
∣
BC
∣
. Let's denote
D
,
E
D, E
D
,
E
also points that
B
C
D
A
BCDA
BC
D
A
and
C
B
E
A
CBEA
CBE
A
are parallelograms. On the sides
A
C
AC
A
C
and
A
B
AB
A
B
sides,
F
F
F
and
G
G
G
are selected respectively so that
∣
A
F
∣
=
∣
A
G
∣
=
∣
B
C
∣
| AF | = | AG | = | BC |
∣
A
F
∣
=
∣
A
G
∣
=
∣
BC
∣
. Prove that the lines
D
F
DF
D
F
and
E
G
EG
EG
intersect at the line segment
B
C
BC
BC
convex hexagon with sides >1 and 9 diagonals >2
Decide if exists a convex hexagon with all sides longer than
1
1
1
and all nine of its diagonals are less than
2
2
2
in length.
1
2
Hide problems
max n, there is a_1a_2a_3...a_n divisible by a_2a_3...a_n}
Find the largest integer
n
≥
3
n \ge 3
n
≥
3
for which there is a
n
n
n
-digit number
a
1
a
2
a
3
.
.
.
a
n
‾
\overline{a_1a_2a_3...a_n}
a
1
a
2
a
3
...
a
n
with non-zero digits
a
1
,
a
2
a_1, a_2
a
1
,
a
2
and
a
n
a_n
a
n
, which is divisible by
a
2
a
3
.
.
.
a
n
‾
\overline{a_2a_3...a_n}
a
2
a
3
...
a
n
.
(p^2 + p) (q^2 + q) (r^2 + r) perfect square for p,q,r primes
Decide if there are primes
p
,
q
,
r
p, q, r
p
,
q
,
r
such that
(
p
2
+
p
)
(
q
2
+
q
)
(
r
2
+
r
)
(p^2 + p) (q^2 + q) (r^2 + r)
(
p
2
+
p
)
(
q
2
+
q
)
(
r
2
+
r
)
is a square of an integer.