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Contests
International Contests
Czech-Polish-Slovak Junior Match
2015 Czech-Polish-Slovak Junior Match
2015 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
6
1
Hide problems
1 to 8 at each vertex of a cube, product of vertices at each edge, max sum
The vertices of the cube are assigned
1
,
2
,
3...
,
8
1, 2, 3..., 8
1
,
2
,
3...
,
8
and then each edge we assign the product of the numbers assigned to its two extreme points. Determine the greatest possible the value of the sum of the numbers assigned to all twelve edges of the cube.
5
2
Hide problems
For each divisor d> 1 of n, then (d - 1) is a divisor of (n - 1|)
Determine all natural numbers
n
>
1
n> 1
n
>
1
with the property: For each divisor
d
>
1
d> 1
d
>
1
of number
n
n
n
, then
d
−
1
d - 1
d
−
1
is a divisor of
n
−
1
n - 1
n
−
1
.
\sqrt{ab}- 2ab/(a + b) \le p ( (a + b)/2} -\sqrt{ab} ) , in p for any a,b>0
Find the smallest real constant
p
p
p
for which the inequality holds
a
b
−
2
a
b
a
+
b
≤
p
(
a
+
b
2
−
a
b
)
\sqrt{ab}- \frac{2ab}{a + b} \le p \left( \frac{a + b}{2} -\sqrt{ab}\right)
ab
−
a
+
b
2
ab
≤
p
(
2
a
+
b
−
ab
)
with any positive real numbers
a
,
b
a, b
a
,
b
.
4
2
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P is incenter of the CDE wanted, starting with a right triangle
Let
A
B
C
ABC
A
BC
ne a right triangle with
∠
A
C
B
=
9
0
o
\angle ACB=90^o
∠
A
CB
=
9
0
o
. Let
E
,
F
E, F
E
,
F
be respecitvely the midpoints of the
B
C
,
A
C
BC, AC
BC
,
A
C
and
C
D
CD
C
D
be it's altitude. Next, let
P
P
P
be the intersection of the internal angle bisector from
A
A
A
and the line
E
F
EF
EF
. Prove that
P
P
P
is the center of the circle inscribed in the triangle
C
D
E
CDE
C
D
E
.
a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)
Determine all such pairs pf positive integers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that
a
+
b
+
(
g
c
d
(
a
,
b
)
)
2
=
l
c
m
(
a
,
b
)
=
2
⋅
l
c
m
(
a
−
1
,
b
)
a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)
a
+
b
+
(
g
c
d
(
a
,
b
)
)
2
=
l
c
m
(
a
,
b
)
=
2
⋅
l
c
m
(
a
−
1
,
b
)
, where
l
c
m
(
a
,
b
)
lcm (a, b)
l
c
m
(
a
,
b
)
denotes the smallest common multiple, and
g
c
d
(
a
,
b
)
gcd (a, b)
g
c
d
(
a
,
b
)
denotes the greatest common divisor of numbers
a
,
b
a, b
a
,
b
.
3
2
Hide problems
x^2 + y^2 <= 2 => xy + 3 >= 2x + 2y
Real numbers
x
,
y
x, y
x
,
y
satisfy the inequality
x
2
+
y
2
≤
2
x^2 + y^2 \le 2
x
2
+
y
2
≤
2
. Orove that
x
y
+
3
≥
2
x
+
2
y
xy + 3 \ge 2x + 2y
x
y
+
3
≥
2
x
+
2
y
|BE - CE| < AD \sqrt3 when AB = BC= CD and AD \perp BC
Different points
A
A
A
and
D
D
D
are on the same side of the line
B
C
BC
BC
, with
∣
A
B
∣
=
∣
B
C
∣
=
∣
C
D
∣
|AB| = | BC|= |CD|
∣
A
B
∣
=
∣
BC
∣
=
∣
C
D
∣
and lines
A
D
AD
A
D
and
B
C
BC
BC
are perpendicular. Let
E
E
E
be the intersection point of lines
A
D
AD
A
D
and
B
C
BC
BC
. Prove that
∣
∣
B
E
∣
−
∣
C
E
∣
∣
<
∣
A
D
∣
3
||BE| - |CE|| < |AD| \sqrt3
∣∣
BE
∣
−
∣
CE
∣∣
<
∣
A
D
∣
3
2
2
Hide problems
sum of pairs of 1, 2,...., 30 a perfect square?
Decide if the vertices of a regular
30
30
30
-gon can be numbered by numbers
1
,
2
,
.
.
,
30
1, 2,.., 30
1
,
2
,
..
,
30
in such a way that the sum of the numbers of every two neighboring to be a square of a certain natural number.
checkers on a 8x8 board minus the middle 2x2 square
We removed the middle square of
2
×
2
2 \times 2
2
×
2
from the
8
×
8
8 \times 8
8
×
8
board. a) How many checkers can be placed on the remaining
60
60
60
boxes so that there are no two not jeopardize? b) How many at least checkers can be placed on the board so that they are at risk all
60
60
60
squares? (A lady is threatening the box she stands on, as well as any box she can get to in one move without going over any of the four removed boxes.)
1
2
Hide problems
one side of triangle is twice as another, AI=MI, incenter related
Let
I
I
I
be the center of the circle of the inscribed triangle
A
B
C
ABC
A
BC
and
M
M
M
be the center of its side
B
C
BC
BC
. If
∣
A
I
∣
=
∣
M
I
∣
|AI| = |MI|
∣
A
I
∣
=
∣
M
I
∣
, prove that there are two of the sides of triangle
A
B
C
ABC
A
BC
, of which one is twice of the other.
find angle such that a triangle has the largest area, inside a right triangle
In the right triangle
A
B
C
ABC
A
BC
with shorter side
A
C
AC
A
C
the hypotenuse
A
B
AB
A
B
has length
12
12
12
. Denote
T
T
T
its centroid and
D
D
D
the feet of altitude from the vertex
C
C
C
. Determine the size of its inner angle at the vertex
B
B
B
for which the triangle
D
T
C
DTC
D
TC
has the greatest possible area.