MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2009 Switzerland - Final Round
2009 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(8)
2
1
Hide problems
11... 11 x 11... 11 is palindrome
A palindrome is a natural number that works in the decimal system forwards and backwards read is the same size (e.g.
1129211
1129211
1129211
or
7337
7337
7337
). Determine all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of natural numbers, such that
(
11...11
⏟
m
)
⋅
(
11...11
⏟
n
)
(\underbrace{11... 11}_{m}) \cdot (\underbrace{11... 11}_{n})
(
m
11...11
)
⋅
(
n
11...11
)
is a palindrome.
9
1
Hide problems
f(f(n)) <= 1/2 (f(n) + n)
Find all injective functions
f
:
N
→
N
f : N\to N
f
:
N
→
N
such that holds for all natural numbers
n
n
n
:
f
(
f
(
n
)
)
≤
f
(
n
)
+
n
2
f(f(n)) \le \frac{f(n) + n}{2}
f
(
f
(
n
))
≤
2
f
(
n
)
+
n
10
1
Hide problems
4^n + 1 has a prime divisor > 20
Let
n
>
3
n > 3
n
>
3
be a natural number. Prove that
4
n
+
1
4^n + 1
4
n
+
1
has a prime divisor
>
20
> 20
>
20
.
8
1
Hide problems
1x2 and T-tetrominos to cover n x n floor, colors related
Given is a floor plan composed of
n
n
n
unit squares. Albert and Berta want to cover this floor with tiles, with all tiles having the shape of a
1
×
2
1\times 2
1
×
2
domino or a
T
T
T
-tetromino. Albert only has tiles from one color, while Berta has two-color dominoes and tetrominoes available in four colors. Albert can use this floor plan in
a
a
a
ways to cover tiles, Berta in
b
b
b
ways. Assuming that
a
≠
0
a \ne 0
a
=
0
, determine the ratio
b
/
a
b/a
b
/
a
.
6
1
Hide problems
f(x - y + z) = f(x) + f(y) + f(z) - xy - yz + xz
Find all functions
f
:
R
>
0
→
R
>
0
f : R_{>0} \to R_{>0}
f
:
R
>
0
→
R
>
0
, which for all
x
>
y
>
z
>
0
x > y > z > 0
x
>
y
>
z
>
0
is the following equation holds
f
(
x
−
y
+
z
)
=
f
(
x
)
+
f
(
y
)
+
f
(
z
)
−
x
y
−
y
z
+
x
z
.
f(x - y + z) = f(x) + f(y) + f(z) - xy - yz + xz.
f
(
x
−
y
+
z
)
=
f
(
x
)
+
f
(
y
)
+
f
(
z
)
−
x
y
−
yz
+
x
z
.
4
1
Hide problems
1 of n symbols in n x n square
Let
n
n
n
be a natural number. Each cell of a
n
×
n
n \times n
n
×
n
square contains one of
n
n
n
different symbols, such that each of the symbols is in exactly
n
n
n
cells. Show that a row or a column exists that contains at least \sqrt{n} different symbols.
3
1
Hide problems
sum (a-b)(b+c) >=0
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be positive real numbers. Prove the following inequality and determine all cases in which the equality holds :
a
−
b
b
+
c
+
b
−
c
c
+
d
+
c
−
d
d
+
a
+
d
−
a
a
+
b
≥
0.
\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a}+\frac{d - a}{a + b} \ge 0.
b
+
c
a
−
b
+
c
+
d
b
−
c
+
d
+
a
c
−
d
+
a
+
b
d
−
a
≥
0.
1
1
Hide problems
locus related to distances from vertices of regular hexagon
Let
P
P
P
be a regular hexagon. For a point
A
A
A
, let
d
1
≤
d
2
≤
.
.
.
≤
d
6
d_1\le d_2\le ...\le d_6
d
1
≤
d
2
≤
...
≤
d
6
the distances from
A
A
A
to the six vertices of
P
P
P
, ordered by magnitude. Find the locus of all points
A
A
A
in the interior or on the boundary of
P
P
P
such that: (a)
d
3
d_3
d
3
takes the smallest possible value. (b)
d
4
d_4
d
4
takes the smallest possible value.