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Problems
Contests
International Contests
Czech-Polish-Slovak Match
2012 Czech-Polish-Slovak Match
2012 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(3)
1
2
Hide problems
One of the three is the AM of other two
Given a positive integer
n
n
n
, let
τ
(
n
)
\tau(n)
τ
(
n
)
denote the number of positive divisors of
n
n
n
and
φ
(
n
)
\varphi(n)
φ
(
n
)
denote the number of positive integers not exceeding
n
n
n
that are relatively prime to
n
n
n
. Find all
n
n
n
for which one of the three numbers
n
,
τ
(
n
)
,
φ
(
n
)
n,\tau(n), \varphi(n)
n
,
τ
(
n
)
,
φ
(
n
)
is the arithmetic mean of the other two.
Ratio of area is independent of the position of point
Let
A
B
C
ABC
A
BC
be a right angled triangle with hypotenuse
A
B
AB
A
B
and
P
P
P
be a point on the shorter arc
A
C
AC
A
C
of the circumcircle of triangle
A
B
C
ABC
A
BC
. The line, perpendicuar to
C
P
CP
CP
and passing through
C
C
C
, intersects
A
P
AP
A
P
,
B
P
BP
BP
at points
K
K
K
and
L
L
L
respectively. Prove that the ratio of area of triangles
B
K
L
BKL
B
K
L
and
A
C
P
ACP
A
CP
is independent of the position of point
P
P
P
.
3
2
Hide problems
ABCD is cyclic implies CFIJ is cyclic
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with circumcircle
ω
\omega
ω
. Let
I
,
J
I, J
I
,
J
and
K
K
K
be the incentres of the triangles
A
B
C
,
A
C
D
ABC, ACD
A
BC
,
A
C
D
and
A
B
D
ABD
A
B
D
respectively. Let
E
E
E
be the midpoint of the arc
D
B
DB
D
B
of circle
ω
\omega
ω
containing the point
A
A
A
. The line
E
K
EK
E
K
intersects again the circle
ω
\omega
ω
at point
F
F
F
(
F
≠
E
)
(F \neq E)
(
F
=
E
)
. Prove that the points
C
,
F
,
I
,
J
C, F, I, J
C
,
F
,
I
,
J
lie on a circle.
abcd=4, a^2+b^2+c^2+d^2=10, maximize (a+c)(b+d)
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive real numbers such that
a
b
c
d
=
4
abcd=4
ab
c
d
=
4
and
a
2
+
b
2
+
c
2
+
d
2
=
10.
a^2+b^2+c^2+d^2=10.
a
2
+
b
2
+
c
2
+
d
2
=
10.
Find the maximum possible value of
a
b
+
b
c
+
c
d
+
d
a
ab+bc+cd+da
ab
+
b
c
+
c
d
+
d
a
.
2
2
Hide problems
f(x + f(y)) - f(x) = (x + f(y))^4 - x^4
Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
satisfying
f
(
x
+
f
(
y
)
)
−
f
(
x
)
=
(
x
+
f
(
y
)
)
4
−
x
4
f(x+f(y))-f(x)=(x+f(y))^4-x^4
f
(
x
+
f
(
y
))
−
f
(
x
)
=
(
x
+
f
(
y
)
)
4
−
x
4
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
.
Streets of Mar del Plata
City of Mar del Plata is a square shaped
W
S
E
N
WSEN
W
SEN
land with
2
(
n
+
1
)
2(n + 1)
2
(
n
+
1
)
streets that divides it into
n
×
n
n \times n
n
×
n
blocks, where
n
n
n
is an even number (the leading streets form the perimeter of the square). Each block has a dimension of
100
×
100
100 \times 100
100
×
100
meters. All streets in Mar del Plata are one-way. The streets which are parallel and adjacent to each other are directed in opposite direction. Street
W
S
WS
W
S
is driven in the direction from
W
W
W
to
S
S
S
and the street
W
N
WN
W
N
travels from
W
W
W
to
N
N
N
. A street cleaning car starts from point
W
W
W
. The driver wants to go to the point
E
E
E
and in doing so, he must cross as much as possible roads. What is the length of the longest route he can go, if any
100
100
100
-meter stretch cannot be crossed more than once? (The figure shows a plan of the city for
n
=
6
n=6
n
=
6
and one of the possible - but not the longest - routes of the street cleaning car. See http://goo.gl/maps/JAzD too.) http://s14.postimg.org/avfg7ygb5/CPS_2012_P5.jpg