MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2022 Switzerland Team Selection Test
2022 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(5)
8
1
Hide problems
Game on coordinate plane
Johann and Nicole are playing a game on the coordinate plane. First, Johann draws any polygon
S
\mathcal{S}
S
and then Nicole can shift
S
\mathcal{S}
S
to wherever she wants. Johann wins if there exists a point with coordinates
(
x
,
y
)
(x, y)
(
x
,
y
)
in the interior of
S
\mathcal{S}
S
, where
x
x
x
and
y
y
y
are coprime integers. Otherwise, Nicole wins. Determine who has a winning strategy.
5
1
Hide problems
Inequalities and lambda
Let
a
,
b
,
c
,
λ
a, b, c, \lambda
a
,
b
,
c
,
λ
be positive real numbers with
λ
≥
1
/
4
\lambda \geq 1/4
λ
≥
1/4
. Show that
a
b
2
+
λ
b
c
+
c
2
+
b
c
2
+
λ
c
a
+
a
2
+
c
a
2
+
λ
a
b
+
b
2
≥
3
λ
+
2
.
\frac{a}{\sqrt{b^2+\lambda bc+c^2}}+\frac{b}{\sqrt{c^2+\lambda ca+a^2}}+\frac{c}{\sqrt{a^2+\lambda ab+b^2}} \geq \frac{3}{\sqrt{\lambda +2}}.
b
2
+
λb
c
+
c
2
a
+
c
2
+
λ
c
a
+
a
2
b
+
a
2
+
λab
+
b
2
c
≥
λ
+
2
3
.
4
1
Hide problems
Graphs and winning strategies
Given a (simple) graph
G
G
G
with
n
≥
2
n \geq 2
n
≥
2
vertices
v
1
,
v
2
,
…
,
v
n
v_1, v_2, \dots, v_n
v
1
,
v
2
,
…
,
v
n
and
m
≥
1
m \geq 1
m
≥
1
edges, Joël and Robert play the following game with
m
m
m
coins:[*]Joël first assigns to each vertex
v
i
v_i
v
i
a non-negative integer
w
i
w_i
w
i
such that
w
1
+
⋯
+
w
n
=
m
w_1+\cdots+w_n=m
w
1
+
⋯
+
w
n
=
m
. [*]Robert then chooses a (possibly empty) subset of edges, and for each edge chosen he places a coin on exactly one of its two endpoints, and then removes that edge from the graph. When he is done, the amount of coins on each vertex
v
i
v_i
v
i
should not be greater than
w
i
w_i
w
i
. [*]Joël then does the same for all the remaining edges. [*]Joël wins if the number of coins on each vertex
v
i
v_i
v
i
is equal to
w
i
w_i
w
i
.Determine all graphs
G
G
G
for which Joël has a winning strategy.
1
1
Hide problems
Bases and divisibility
Let
n
n
n
be a positive integer. Prove that there exists a finite sequence
S
S
S
consisting of only zeros and ones, satisfying the following property: for any positive integer
d
≥
2
d \geq 2
d
≥
2
, when
S
S
S
is interpreted in base
d
d
d
, the resulting number is non-zero and divisible by
n
n
n
. Remark: The sequence
S
=
s
k
s
k
−
1
⋯
s
1
s
0
S=s_ks_{k-1} \cdots s_1s_0
S
=
s
k
s
k
−
1
⋯
s
1
s
0
interpreted in base
d
d
d
is the number
∑
i
=
0
k
s
i
d
i
\sum_{i=0}^{k}s_id^i
∑
i
=
0
k
s
i
d
i
12
1
Hide problems
Nice FE over R+
Let
R
+
\mathbb{R}^+
R
+
denote the set of positive real numbers. Find all functions
f
:
R
+
→
R
+
f:\mathbb{R}^+ \to \mathbb{R}^+
f
:
R
+
→
R
+
such that
x
+
f
(
y
f
(
x
)
+
1
)
=
x
f
(
x
+
y
)
+
y
f
(
y
f
(
x
)
)
x+f(yf(x)+1)=xf(x+y)+yf(yf(x))
x
+
f
(
y
f
(
x
)
+
1
)
=
x
f
(
x
+
y
)
+
y
f
(
y
f
(
x
))
for all
x
,
y
>
0.
x,y>0.
x
,
y
>
0.