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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1999 Federal Competition For Advanced Students, Part 2
1999 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
3
2
Hide problems
Solve y^2 - [x]^2 = 19.99 and x^2 + [y]^2 = 1999 in R
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of real numbers such that
y
2
−
[
x
]
2
=
19.99
and
x
2
+
[
y
]
2
=
1999
y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999
y
2
−
[
x
]
2
=
19.99
and
x
2
+
[
y
]
2
=
1999
where
f
(
x
)
=
[
x
]
f(x)=[x]
f
(
x
)
=
[
x
]
is the floor function.
If player B plays correctly, then player A cannot win
Two players
A
A
A
and
B
B
B
play the following game. An even number of cells are placed on a circle.
A
A
A
begins and
A
A
A
and
B
B
B
play alternately, where each move consists of choosing a free cell and writing either
O
O
O
or
M
M
M
in it. The player after whose move the word
O
M
O
OMO
OMO
(OMO = Osterreichische Mathematik Olympiade) occurs for the first time in three successive cells wins the game. If no such word occurs, then the game is a draw. Prove that if player
B
B
B
plays correctly, then player
A
A
A
cannot win.
2
2
Hide problems
The lines intersect on the sphere
Let
ϵ
\epsilon
ϵ
be a plane and
k
1
,
k
2
,
k
3
k_1, k_2, k_3
k
1
,
k
2
,
k
3
be spheres on the same side of
ϵ
\epsilon
ϵ
. The spheres
k
1
,
k
2
,
k
3
k_1, k_2, k_3
k
1
,
k
2
,
k
3
touch the plane at points
T
1
,
T
2
,
T
3
T_1, T_2, T_3
T
1
,
T
2
,
T
3
, respectively, and
k
2
k_2
k
2
touches
k
1
k_1
k
1
at
S
1
S_1
S
1
and
k
3
k_3
k
3
at
S
3
S_3
S
3
. Prove that the lines
S
1
T
1
S_1T_1
S
1
T
1
and
S
3
T
3
S_3T_3
S
3
T
3
intersect on the sphere
k
2
k_2
k
2
. Describe the locus of the intersection point.
Find all polynomials P such that P(P(P(x))) = Ax^n +19x+99
Given a real number
A
A
A
and an integer
n
n
n
with
2
≤
n
≤
19
2 \leq n \leq 19
2
≤
n
≤
19
, find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that
P
(
P
(
P
(
x
)
)
)
=
A
x
n
+
19
x
+
99
P(P(P(x))) = Ax^n +19x+99
P
(
P
(
P
(
x
)))
=
A
x
n
+
19
x
+
99
.
1
2
Hide problems
Sum of the numbers of digits of 4^n and of 25^n is odd
Prove that for each positive integer
n
n
n
, the sum of the numbers of digits of
4
n
4^n
4
n
and of
2
5
n
25^n
2
5
n
(in the decimal system) is odd.
Ninety-nine points on one of the diagonals of a unit square
Ninety-nine points are given on one of the diagonals of a unit square. Prove that there is at most one vertex of the square such that the average squared distance from a given point to the vertex is less than or equal to
1
/
2
1/2
1/2
.