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Contests
International Contests
Czech-Polish-Slovak Junior Match
2018 Czech-Polish-Slovak Junior Match
2018 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
6
1
Hide problems
a^3 + b^3 = 2 => 1/a+ 1/b \ge 2(a^2 - a + 1)(b^2 - b + 1) for a,b>0
Positive real numbers
a
,
b
a, b
a
,
b
are such that
a
3
+
b
3
=
2
a^3 + b^3 = 2
a
3
+
b
3
=
2
. Show that that
1
a
+
1
b
≥
2
(
a
2
−
a
+
1
)
(
b
2
−
b
+
1
)
\frac{1}{a}+\frac{1}{b}\ge 2(a^2 - a + 1)(b^2 - b + 1)
a
1
+
b
1
≥
2
(
a
2
−
a
+
1
)
(
b
2
−
b
+
1
)
.
5
2
Hide problems
perpendicularity for Czech, Polish and Slovak juniors
An acute triangle
A
B
C
ABC
A
BC
is given in which
A
B
<
A
C
AB <AC
A
B
<
A
C
. Point
E
E
E
lies on the
A
C
AC
A
C
side of the triangle, with
A
B
=
A
E
AB = AE
A
B
=
A
E
. The segment
A
D
AD
A
D
is the diameter of the circumcircle of the triangle
A
B
C
ABC
A
BC
, and point
S
S
S
is the center of this arc
B
C
BC
BC
of this circle to which point
A
A
A
does not belong. Point
F
F
F
is symmetric of point
D
D
D
wrt
S
S
S
. Prove that lines
F
E
F E
FE
and
A
C
AC
A
C
are perpendicular.
2n people sit, knowing both neighbors in a round table
There are
2
n
2n
2
n
people (
n
≥
2
n \ge 2
n
≥
2
) sitting around the round table, with each person getting to know both with his neighbors and exactly opposite him sits a person he does not know. Prove that people can rearrange in such a way that everyone knows one of their two neighbors.
4
2
Hide problems
no B created by removing middle digit of no A, divisible by 2018
Determine the smallest positive integer
A
A
A
with an odd number of digits and this property, that both
A
A
A
and the number
B
B
B
created by removing the middle digit of the number
A
A
A
are divisible by
2018
2018
2018
.
circumencter lies on perp. bisector, equilateral and 3 circumcircles related
A line passing through the center
M
M
M
of the equilateral triangle
A
B
C
ABC
A
BC
intersects sides
B
C
BC
BC
and
C
A
CA
C
A
, respectively, in points
D
D
D
and
E
E
E
. Circumcircles of triangle
A
E
M
AEM
A
EM
and
B
D
M
BDM
B
D
M
intersects, besides point
M
M
M
, also at point
P
P
P
. Prove that the center of circumcircle of triangle
D
E
P
DEP
D
EP
lies on the perpendicular bisector of the segment
A
B
AB
A
B
.
3
2
Hide problems
teacher gave each of her 37 students 36 pencils in different colors
The teacher gave each of her
37
37
37
students
36
36
36
pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed.
| ax^2 + bx + c | <= 1 => |cx^2 + bx + a | <= r
Calculate all real numbers
r
r
r
with the following properties: If real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy the inequality
∣
a
x
2
+
b
x
+
c
∣
≤
1
| ax^2 + bx + c | \le 1
∣
a
x
2
+
b
x
+
c
∣
≤
1
for each
x
∈
[
−
1
,
1
]
x \in [ - 1, 1]
x
∈
[
−
1
,
1
]
, then they also satisfy the inequality
∣
c
x
2
+
b
x
+
a
∣
≤
r
| cx^2 + bx + a | \le r
∣
c
x
2
+
b
x
+
a
∣
≤
r
for each
x
∈
[
−
1
,
1
]
x \in [- 1, 1]
x
∈
[
−
1
,
1
]
.
2
2
Hide problems
hexagon, 2 // sides, diagonal divides into quadrilaterals equal perimeters
A convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is given whose sides
A
B
AB
A
B
and
D
E
DE
D
E
are parallel. Each of the diagonals
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
divides this hexagon into two quadrilaterals of equal perimeters. Show that these three diagonals intersect at one point.
ratio of areas wanted, right triangle and 2 symmetric points related
Given a right triangle
A
B
C
ABC
A
BC
with the hypotenuse
A
B
AB
A
B
. Let
K
K
K
be any interior point of triangle
A
B
C
ABC
A
BC
and points
L
,
M
L, M
L
,
M
are symmetric of point
K
K
K
wrt lines
B
C
,
A
C
BC, AC
BC
,
A
C
respectively. Specify all possible values for
S
A
B
L
M
/
S
A
B
C
S_{ABLM} / S_{ABC}
S
A
B
L
M
/
S
A
BC
, where
S
X
Y
.
.
.
Z
S_{XY ... Z}
S
X
Y
...
Z
indicates the area of the polygon
X
Y
.
.
.
Z
XY...Z
X
Y
...
Z
.
1
2
Hide problems
ad + bc = x, ac + bd = y, ab + cd = z, one of a, b,c,d equals to sum of other 3
Are there four real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
for every three positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
with the property
a
d
+
b
c
=
x
ad + bc = x
a
d
+
b
c
=
x
,
a
c
+
b
d
=
y
ac + bd = y
a
c
+
b
d
=
y
,
a
b
+
c
d
=
z
ab + cd = z
ab
+
c
d
=
z
and one of the numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
is equal to the sum of the other three?
(a + b + c) | (a - b)^2 + (b - c)^2 + (c - a)^2
For natural numbers
a
,
b
c
a, b c
a
,
b
c
it holds that
(
a
+
b
+
c
)
2
∣
a
b
(
a
+
b
)
+
b
c
(
b
+
c
)
+
c
a
(
c
+
a
)
+
3
a
b
c
(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc
(
a
+
b
+
c
)
2
∣
ab
(
a
+
b
)
+
b
c
(
b
+
c
)
+
c
a
(
c
+
a
)
+
3
ab
c
. Prove that
(
a
+
b
+
c
)
∣
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2
(
a
+
b
+
c
)
∣
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2