MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2018 Switzerland - Final Round
2018 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(7)
3
1
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n=(a x lcm(b, c) + b x lcm(c, a) + cxt lcm(a, b)/ lcm(a, b, c)
Determine all natural integers
n
n
n
for which there is no triplet
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of natural numbers such that:
n
=
a
⋅
l
c
m
(
b
,
c
)
+
b
⋅
l
c
m
(
c
,
a
)
+
c
⋅
l
c
m
(
a
,
b
)
l
c
m
(
a
,
b
,
c
)
n = \frac{a \cdot \,\,lcm(b, c) + b \cdot lcm \,\,(c, a) + c \cdot lcm \,\, (a, b)}{lcm \,\,(a, b, c)}
n
=
l
c
m
(
a
,
b
,
c
)
a
⋅
l
c
m
(
b
,
c
)
+
b
⋅
l
c
m
(
c
,
a
)
+
c
⋅
l
c
m
(
a
,
b
)
2
1
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min sum a / gcd(a + b, a - c)
Let
a
,
b
a, b
a
,
b
and
c
c
c
be natural numbers. Determine the smallest value that the following expression can take:
a
g
c
d
(
a
+
b
,
a
−
c
)
+
b
g
c
d
(
b
+
c
,
b
−
a
)
+
c
g
c
d
(
c
+
a
,
c
−
b
)
.
\frac{a}{gcd\,\,(a + b, a - c)} + \frac{b}{gcd\,\,(b + c, b - a)} + \frac{c}{gcd\,\,(c + a, c - b)}.
g
c
d
(
a
+
b
,
a
−
c
)
a
+
g
c
d
(
b
+
c
,
b
−
a
)
b
+
g
c
d
(
c
+
a
,
c
−
b
)
c
.
. Remark:
g
c
d
(
6
,
0
)
=
6
gcd \,\, (6, 0) = 6
g
c
d
(
6
,
0
)
=
6
and
g
c
d
(
3
,
−
6
)
=
3
gcd\,\,(3, -6) = 3
g
c
d
(
3
,
−
6
)
=
3
.
7
1
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n as sum of one or more consecutive natural integers
Let
n
n
n
be a natural integer and let
k
k
k
be the number of ways to write
n
n
n
as the sum of one or more consecutive natural integers. Prove that
k
k
k
is equal to the number of odd positive divisors of
n
n
n
.Example:
9
9
9
has three positive odd divisors and
9
=
9
9 = 9
9
=
9
,
9
=
4
+
5
9 = 4 + 5
9
=
4
+
5
,
9
=
2
+
3
+
4
9 = 2 + 3 + 4
9
=
2
+
3
+
4
.
1
1
Hide problems
white cells in 8x8 chessboard
The cells of an
8
×
8
8\times 8
8
×
8
chessboard are all coloured in white. A move consists in inverting the colours of a rectangle
1
×
3
1 \times 3
1
×
3
horizontal or vertical (the white cells become black and conversely). Is it possible to colour all the cells of the chessboard in black in a finite number of moves ?
9
1
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Parallelogram-free subsets [Swiss 2018]
Let
n
n
n
be a positive integer and let
G
G
G
be the set of points
(
x
,
y
)
(x, y)
(
x
,
y
)
in the plane such that
x
x
x
and
y
y
y
are integers with
1
≤
x
,
y
≤
n
1 \leq x, y \leq n
1
≤
x
,
y
≤
n
. A subset of
G
G
G
is called parallelogram-free if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of
G
G
G
can have?
8
1
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max (ab+bc+cd+de)/(2a^2+b^2+2c^2+d^2+2e^2) [Swiss 2018]
Let
a
,
b
,
c
,
d
,
e
a,b,c,d,e
a
,
b
,
c
,
d
,
e
be positive real numbers. Find the largest possible value for the expression
a
b
+
b
c
+
c
d
+
d
e
2
a
2
+
b
2
+
2
c
2
+
d
2
+
2
e
2
.
\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.
2
a
2
+
b
2
+
2
c
2
+
d
2
+
2
e
2
ab
+
b
c
+
c
d
+
d
e
.
5
1
Hide problems
Functions f:R+ to R satisfying f(xf(x)+yf(y))=xy
Does there exist any function
f
:
R
+
→
R
f: \mathbb{R}^+ \to \mathbb{R}
f
:
R
+
→
R
such that for every positive real number
x
,
y
x,y
x
,
y
the following is true :
f
(
x
f
(
x
)
+
y
f
(
y
)
)
=
x
y
f(xf(x)+yf(y)) = xy
f
(
x
f
(
x
)
+
y
f
(
y
))
=
x
y