MathDB
Problems
Contests
International Contests
Austrian-Polish
2003 Austrian-Polish Competition
2003 Austrian-Polish Competition
Part of
Austrian-Polish
Subcontests
(10)
7
1
Hide problems
n!^{f(n)} divides (n^n)! when f(n) = (n^n - 1)/(n - 1)
Put
f
(
n
)
=
n
n
−
1
n
−
1
f(n) = \frac{n^n - 1}{n - 1}
f
(
n
)
=
n
−
1
n
n
−
1
. Show that
n
!
f
(
n
)
n!^{f(n)}
n
!
f
(
n
)
divides
(
n
n
)
!
(n^n)!
(
n
n
)!
. Find as many positive integers as possible for which
n
!
f
(
n
)
+
1
n!^{f(n)+1}
n
!
f
(
n
)
+
1
does not divide
(
n
n
)
!
(n^n)!
(
n
n
)!
.
8
1
Hide problems
sum (-1)^{k+1} x_{k}^n >= (sum (-1)^{i} x_{k})^n for k=1 to 2003
Given reals
x
1
≥
x
2
≥
.
.
.
≥
x
2003
≥
0
x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0
x
1
≥
x
2
≥
...
≥
x
2003
≥
0
, show that
x
1
n
−
x
2
n
+
x
2
n
−
.
.
.
−
x
2002
n
+
x
2003
n
≥
(
x
1
−
x
2
+
x
3
−
x
4
+
.
.
.
−
x
2002
+
x
2003
)
n
x_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n
x
1
n
−
x
2
n
+
x
2
n
−
...
−
x
2002
n
+
x
2003
n
≥
(
x
1
−
x
2
+
x
3
−
x
4
+
...
−
x
2002
+
x
2003
)
n
for any positive integer
n
n
n
.
10
1
Hide problems
smallest no of 5x1 tiles which must be placed on a 31x5 rectangle
What is the smallest number of
5
×
1
5\times 1
5
×
1
tiles which must be placed on a
31
×
5
31\times 5
31
×
5
rectangle (each covering exactly
5
5
5
unit squares) so that no further tiles can be placed? How many different ways are there of placing the minimal number (so that further tiles are blocked)? What are the answers for a
52
×
5
52\times 5
52
×
5
rectangle?
4
1
Hide problems
product of alpine numbers is alpine, when m divides 2^{2n+1} + 1
A positive integer
m
m
m
is alpine if
m
m
m
divides
2
2
n
+
1
+
1
2^{2n+1} + 1
2
2
n
+
1
+
1
for some positive integer
n
n
n
. Show that the product of two alpine numbers is alpine.
1
1
Hide problems
p(x-1)p(x+1)= p(x^2-1) polynomial
Find all real polynomials
p
(
x
)
p(x)
p
(
x
)
such that
p
(
x
−
1
)
p
(
x
+
1
)
=
p
(
x
2
−
1
)
p(x-1)p(x+1)= p(x^2-1)
p
(
x
−
1
)
p
(
x
+
1
)
=
p
(
x
2
−
1
)
.
3
1
Hide problems
trisecting sides + constructing similar triangles - Austrian-Polish 2003
A
B
C
ABC
A
BC
is a triangle. Take
a
=
B
C
a = BC
a
=
BC
etc as usual. Take points
T
1
,
T
2
T_1, T_2
T
1
,
T
2
on the side
A
B
AB
A
B
so that
A
T
1
=
T
1
T
2
=
T
2
B
AT_1 = T_1T_2 = T_2B
A
T
1
=
T
1
T
2
=
T
2
B
. Similarly, take points
T
3
,
T
4
T_3, T_4
T
3
,
T
4
on the side BC so that
B
T
3
=
T
3
T
4
=
T
4
C
BT_3 = T_3T_4 = T_4C
B
T
3
=
T
3
T
4
=
T
4
C
, and points
T
5
,
T
6
T_5, T_6
T
5
,
T
6
on the side
C
A
CA
C
A
so that
C
T
5
=
T
5
T
6
=
T
6
A
CT_5 = T_5T_6 = T_6A
C
T
5
=
T
5
T
6
=
T
6
A
. Show that if
a
′
=
B
T
5
,
b
′
=
C
T
1
,
c
′
=
A
T
3
a' = BT_5, b' = CT_1, c'=AT_3
a
′
=
B
T
5
,
b
′
=
C
T
1
,
c
′
=
A
T
3
, then there is a triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
with sides
a
′
,
b
′
,
c
′
a', b', c'
a
′
,
b
′
,
c
′
(
a
′
=
B
′
C
a' = B'C
a
′
=
B
′
C
' etc). In the same way we take points
T
i
′
T_i'
T
i
′
on the sides of
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
and put
a
′
′
=
B
′
T
6
′
,
b
′
′
=
C
′
T
2
′
,
c
′
′
=
A
′
T
4
′
a'' = B'T_6', b'' = C'T_2', c'' = A'T_4'
a
′′
=
B
′
T
6
′
,
b
′′
=
C
′
T
2
′
,
c
′′
=
A
′
T
4
′
. Show that there is a triangle
A
′
′
B
′
′
C
′
′
A'' B'' C''
A
′′
B
′′
C
′′
with sides
a
′
′
b
′
′
,
c
′
′
a'' b'' , c''
a
′′
b
′′
,
c
′′
and that it is similar to
A
B
C
ABC
A
BC
. Find
a
′
′
/
a
a'' /a
a
′′
/
a
.
6
1
Hide problems
sphere intersecting planes of tetrahedron in circles - Austrian-Polish 2003
A
B
C
D
ABCD
A
BC
D
is a tetrahedron such that we can find a sphere
k
(
A
,
B
,
C
)
k(A,B,C)
k
(
A
,
B
,
C
)
through
A
,
B
,
C
A, B, C
A
,
B
,
C
which meets the plane
B
C
D
BCD
BC
D
in the circle diameter
B
C
BC
BC
, meets the plane
A
C
D
ACD
A
C
D
in the circle diameter
A
C
AC
A
C
, and meets the plane
A
B
D
ABD
A
B
D
in the circle diameter
A
B
AB
A
B
. Show that there exist spheres
k
(
A
,
B
,
D
)
k(A,B,D)
k
(
A
,
B
,
D
)
,
k
(
B
,
C
,
D
)
k(B,C,D)
k
(
B
,
C
,
D
)
and
k
(
C
,
A
,
D
)
k(C,A,D)
k
(
C
,
A
,
D
)
with analogous properties.
9
1
Hide problems
Product of numbers from the set
Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the
26
26
26
whose product is a square. I think that the upper limit for such subset is 37.
2
1
Hide problems
Beautiful sequence (help)
The sequence
a
0
,
a
1
,
a
2
,
.
.
a_0, a_1, a_2, ..
a
0
,
a
1
,
a
2
,
..
is defined by
a
0
=
a
,
a
n
+
1
=
a
n
+
L
(
a
n
)
a_0 = a, a_{n+1} = a_n + L(a_n)
a
0
=
a
,
a
n
+
1
=
a
n
+
L
(
a
n
)
, where
L
(
m
)
L(m)
L
(
m
)
is the last digit of
m
m
m
(eg
L
(
14
)
=
4
L(14) = 4
L
(
14
)
=
4
). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by
d
=
3
d = 3
d
=
3
. For what other d is this true?
5
1
Hide problems
(x+y+z)^2 <= (a^2+b^2+c^2)/2 + 2S.sqrt(3), in a triangle?
A triangle with sides a, b, c has area S. The distances of its centroid from the vertices are x, y, z. Show that: if (x + y + z)^2 ≤ (a^2 + b^2 + c^2)/2 + 2S√3, then the triangle is equilateral.