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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2019 Switzerland Team Selection Test
2019 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(9)
11
1
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Inequality with All Means Except the QM
Let
n
n
n
be a positive integer. Determine whether there exists a positive real number
ϵ
>
0
\epsilon >0
ϵ
>
0
(depending on
n
n
n
) such that for all positive real numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots ,x_n
x
1
,
x
2
,
…
,
x
n
, the inequality
x
1
x
2
…
x
n
n
≤
(
1
−
ϵ
)
x
1
+
x
2
+
⋯
+
x
n
n
+
ϵ
n
1
x
1
+
1
x
2
+
⋯
+
1
x
n
,
\sqrt[n]{x_1x_2\dots x_n}\leq (1-\epsilon)\frac{x_1+x_2+\dots+x_n}{n}+\epsilon \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\dots +\frac{1}{x_n}},
n
x
1
x
2
…
x
n
≤
(
1
−
ϵ
)
n
x
1
+
x
2
+
⋯
+
x
n
+
ϵ
x
1
1
+
x
2
1
+
⋯
+
x
n
1
n
,
holds.
10
1
Hide problems
Different Colors of Balls
Let
n
≥
5
n \geq 5
n
≥
5
be an integer. A shop sells balls in
n
n
n
different colors. Each of
n
+
1
n + 1
n
+
1
children bought three balls with different colors, but no two children bought exactly the same color combination. Show that there are at least two children who bought exactly one ball of the same color.
9
1
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Perpendicular to Median
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
.
E
,
F
E,F
E
,
F
are foots of the altitudes drawn from
B
,
C
B,C
B
,
C
respectively. Let
M
M
M
be the midpoint of segment
B
C
BC
BC
. The tangent at
A
A
A
to the circumcircle of
A
B
C
ABC
A
BC
cuts
B
C
BC
BC
in
P
P
P
and
E
F
EF
EF
cuts the parallel to
B
C
BC
BC
from
A
A
A
at
Q
Q
Q
. Prove that
P
Q
PQ
PQ
is perpendicular to
A
M
AM
A
M
.
8
1
Hide problems
Ordering a Closet of Socks
Let
k
,
n
,
r
k,n,r
k
,
n
,
r
be positive integers and
r
<
n
r<n
r
<
n
. Quirin owns
k
n
+
r
kn+r
kn
+
r
black and
k
n
+
r
kn+r
kn
+
r
white socks. He want to clean his cloths closet such there does not exist
2
n
2n
2
n
consecutive socks
n
n
n
of which black and the other
n
n
n
white. Prove that he can clean his closet in the desired manner if and only if
r
≥
k
r\geq k
r
≥
k
and
n
>
k
+
r
n>k+r
n
>
k
+
r
.
6
1
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The Sequence Formed by Wining Strategy of a Nim-Type Game
Let
(
a
,
b
)
(a,b)
(
a
,
b
)
be a pair of natural numbers. Henning and Paul play the following game. At the beginning there are two piles of
a
a
a
and
b
b
b
coins respectively. We say that
(
a
,
b
)
(a,b)
(
a
,
b
)
is the starting position of the game. Henning and Paul play with the following rules:
∙
\bullet
∙
They take turns alternatively where Henning begins.
∙
\bullet
∙
In every step each player either takes a positive integer number of coins from one of the two piles or takes same natural number of coins from both piles.
∙
\bullet
∙
The player how take the last coin wins. Let
A
A
A
be the set of all positive integers like
a
a
a
for which there exists a positive integer
b
<
a
b<a
b
<
a
such that Paul has a wining strategy for the starting position
(
a
,
b
)
(a,b)
(
a
,
b
)
. Order the elements of
A
A
A
to construct a sequence
a
1
<
a
2
<
a
3
<
…
a_1<a_2<a_3<\dots
a
1
<
a
2
<
a
3
<
…
(
a
)
(a)
(
a
)
Prove that
A
A
A
has infinity many elements.
(
b
)
(b)
(
b
)
Prove that the sequence defined by
m
k
:
=
a
k
+
1
−
a
k
m_k:=a_{k+1}-a_{k}
m
k
:=
a
k
+
1
−
a
k
will never become periodic. (This means the sequence
m
k
0
+
k
m_{k_0+k}
m
k
0
+
k
will not be periodic for any choice of
k
0
k_0
k
0
)
7
1
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Solution to Divisibility
Prove that for all positive integers
n
n
n
there are positive integers
a
,
b
a,b
a
,
b
such that
n
∣
4
a
2
+
9
b
2
−
1.
n\mid 4a^2+9b^2-1.
n
∣
4
a
2
+
9
b
2
−
1.
4
1
Hide problems
Another Sequence with P^n(x)
Let
p
p
p
be a prime number. Find all polynomials
P
P
P
with integer coefficients with the following properties:
(
a
)
(a)
(
a
)
P
(
x
)
>
x
P(x)>x
P
(
x
)
>
x
for all positive integers
x
x
x
.
(
b
)
(b)
(
b
)
The sequence defined by
p
0
:
=
p
p_0:=p
p
0
:=
p
,
p
n
+
1
:
=
P
(
p
n
)
p_{n+1}:=P(p_n)
p
n
+
1
:=
P
(
p
n
)
for all positive integers
n
n
n
, satisfies the property that for all positive integers
m
m
m
there exists some
l
≥
0
l\geq 0
l
≥
0
such that
m
∣
p
l
m\mid p_l
m
∣
p
l
.
2
1
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Prime Expressible in Root of a Rational
Find the largest prime
p
p
p
such that there exist positive integers
a
,
b
a,b
a
,
b
satisfying
p
=
b
2
a
−
b
a
+
b
.
p=\frac{b}{2}\sqrt{\frac{a-b}{a+b}}.
p
=
2
b
a
+
b
a
−
b
.
1
1
Hide problems
Classical Orthocenter Figure
Let
A
B
C
ABC
A
BC
be a triangle and
D
,
E
,
F
D, E, F
D
,
E
,
F
be the foots of altitudes drawn from
A
,
B
,
C
A,B,C
A
,
B
,
C
respectively. Let
H
H
H
be the orthocenter of
A
B
C
ABC
A
BC
. Lines
E
F
EF
EF
and
A
D
AD
A
D
intersect at
G
G
G
. Let
K
K
K
the point on circumcircle of
A
B
C
ABC
A
BC
such that
A
K
AK
A
K
is a diameter of this circle.
A
K
AK
A
K
cuts
B
C
BC
BC
in
M
M
M
. Prove that
G
M
GM
GM
and
H
K
HK
HK
are parallel.