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Problems
Contests
International Contests
International Zhautykov Olympiad
2012 International Zhautykov Olympiad
2012 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
3
2
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P(Q(x)) + P(R(x))=constant => P(x) or Q(x)+R(x) is constant
Let
P
,
Q
,
R
P, Q,R
P
,
Q
,
R
be three polynomials with real coefficients such that
P
(
Q
(
x
)
)
+
P
(
R
(
x
)
)
=
constant
P(Q(x)) + P(R(x))=\text{constant}
P
(
Q
(
x
))
+
P
(
R
(
x
))
=
constant
for all
x
x
x
. Prove that
P
(
x
)
=
constant
P(x)=\text{constant}
P
(
x
)
=
constant
or
Q
(
x
)
+
R
(
x
)
=
constant
Q(x)+R(x)=\text{constant}
Q
(
x
)
+
R
(
x
)
=
constant
for all
x
x
x
.
All integer solutions of the equation 2x^2 - y^{14} = 1
Find all integer solutions of the equation the equation
2
x
2
−
y
14
=
1
2x^2-y^{14}=1
2
x
2
−
y
14
=
1
.
2
2
Hide problems
Convenient set
A set of (unit) squares of a
n
×
n
n\times n
n
×
n
table is called convenient if each row and each column of the table contains at least two squares belonging to the set. For each
n
≥
5
n\geq 5
n
≥
5
determine the maximum
m
m
m
for which there exists a convenient set made of
m
m
m
squares, which becomes inconvenient when any of its squares is removed.
Find <BAD + CDA if B'C' = AB+CD
Equilateral triangles
A
C
B
′
ACB'
A
C
B
′
and
B
D
C
′
BDC'
B
D
C
′
are drawn on the diagonals of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
so that
B
B
B
and
B
′
B'
B
′
are on the same side of
A
C
AC
A
C
, and
C
C
C
and
C
′
C'
C
′
are on the same sides of
B
D
BD
B
D
. Find
∠
B
A
D
+
∠
C
D
A
\angle BAD + \angle CDA
∠
B
A
D
+
∠
C
D
A
if
B
′
C
′
=
A
B
+
C
D
B'C' = AB+CD
B
′
C
′
=
A
B
+
C
D
.
1
2
Hide problems
The area does not depend on the position of point
An acute triangle
A
B
C
ABC
A
BC
is given. Let
D
D
D
be an arbitrary inner point of the side
A
B
AB
A
B
. Let
M
M
M
and
N
N
N
be the feet of the perpendiculars from
D
D
D
to
B
C
BC
BC
and
A
C
AC
A
C
, respectively. Let
H
1
H_1
H
1
and
H
2
H_2
H
2
be the orthocentres of triangles
M
N
C
MNC
MNC
and
M
N
D
MND
MN
D
, respectively. Prove that the area of the quadrilateral
A
H
1
B
H
2
AH_1BH_2
A
H
1
B
H
2
does not depend on the position of
D
D
D
on
A
B
AB
A
B
.
Do there exist integers m, n and a function f?
Do there exist integers
m
,
n
m, n
m
,
n
and a function
f
:
R
→
R
f\colon \mathbb R \to \mathbb R
f
:
R
→
R
satisfying simultaneously the following two conditions?
∙
\bullet
∙
i)
f
(
f
(
x
)
)
=
2
f
(
x
)
−
x
−
2
f(f(x))=2f(x)-x-2
f
(
f
(
x
))
=
2
f
(
x
)
−
x
−
2
for any
x
∈
R
x \in \mathbb R
x
∈
R
;
∙
\bullet
∙
ii)
m
≤
n
m \leq n
m
≤
n
and
f
(
m
)
=
n
f(m)=n
f
(
m
)
=
n
.