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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2016 Switzerland Team Selection Test
2016 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(9)
Problem 12
1
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competition and two contestants
In an EGMO exam, there are three exercises, each of which can yield a number of points between
0
0
0
and
7
7
7
. Show that, among the
49
49
49
participants, one can always find two such that the first in each of the three tasks was at least as good as the other.
Problem 10
1
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colinear points
Let
A
B
C
ABC
A
BC
be a non-rectangle triangle with
M
M
M
the middle of
B
C
BC
BC
. Let
D
D
D
be a point on the line
A
B
AB
A
B
such that
C
A
=
C
D
CA=CD
C
A
=
C
D
and let
E
E
E
be a point on the line
B
C
BC
BC
such that
E
B
=
E
D
EB=ED
EB
=
E
D
. The parallel to
E
D
ED
E
D
passing through
A
A
A
intersects the line
M
D
MD
M
D
at the point
I
I
I
and the line
A
M
AM
A
M
intersects the line
E
D
ED
E
D
at the point
J
J
J
. Show that the points
C
,
I
C, I
C
,
I
and
J
J
J
are aligned.
Problem 9
1
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functional equation
Find all functions
f
:
R
↦
R
f : \mathbb{R} \mapsto \mathbb{R}
f
:
R
↦
R
such that
(
f
(
x
)
+
y
)
(
f
(
x
−
y
)
+
1
)
=
f
(
f
(
x
f
(
x
+
1
)
)
−
y
f
(
y
−
1
)
)
\left(f(x)+y\right)\left(f(x-y)+1\right)=f\left(f(xf(x+1))-yf(y-1)\right)
(
f
(
x
)
+
y
)
(
f
(
x
−
y
)
+
1
)
=
f
(
f
(
x
f
(
x
+
1
))
−
y
f
(
y
−
1
)
)
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
Problem 8
1
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relation between sides
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
and let
M
M
M
be the middle of
B
C
BC
BC
. The bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersects the line
B
C
BC
BC
in
Q
Q
Q
. Let
H
H
H
be the foot of
A
A
A
on
B
C
BC
BC
. The perpendicular to
A
Q
AQ
A
Q
passing through
A
A
A
intersects the line
B
C
BC
BC
in
S
S
S
. Show that
M
H
×
Q
S
=
A
B
×
A
C
MH \times QS=AB \times AC
M
H
×
QS
=
A
B
×
A
C
.
Problem 7
1
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sum of square of divisors
Find all positive integers
n
n
n
such that
∑
d
∣
n
,
1
≤
d
<
n
d
2
=
5
(
n
+
1
)
\sum_{d|n, 1\leq d <n}d^2=5(n+1)
d
∣
n
,
1
≤
d
<
n
∑
d
2
=
5
(
n
+
1
)
Problem 4
1
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Inequality for all real numbers
Find all integers
n
≥
1
n \geq 1
n
≥
1
such that for all
x
1
,
.
.
.
,
x
n
∈
R
x_1,...,x_n \in \mathbb{R}
x
1
,
...
,
x
n
∈
R
the following inequality is satisfied
(
x
1
n
+
.
.
.
+
x
n
n
n
−
x
1
.
.
.
.
x
n
)
(
x
1
+
.
.
.
+
x
n
)
≥
0
\left(\frac{x_1^n+...+x_n^n}{n}-x_1....x_n\right)\left(x_1+...+x_n\right) \geq 0
(
n
x
1
n
+
...
+
x
n
n
−
x
1
....
x
n
)
(
x
1
+
...
+
x
n
)
≥
0
Problem 2
1
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Polynomial equation
Find all polynomial functions with real coefficients for which
(
x
−
2
)
P
(
x
+
2
)
+
(
x
+
2
)
P
(
x
−
2
)
=
2
x
P
(
x
)
(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)
(
x
−
2
)
P
(
x
+
2
)
+
(
x
+
2
)
P
(
x
−
2
)
=
2
x
P
(
x
)
for all real
x
x
x
Problem 1
1
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Unsociable numbers
Let
n
n
n
be a natural number. Two numbers are called "unsociable" if their greatest common divisor is
1
1
1
. The numbers
{
1
,
2
,
.
.
.
,
2
n
}
\{1,2,...,2n\}
{
1
,
2
,
...
,
2
n
}
are partitioned into
n
n
n
pairs. What is the minimum number of "unsociable" pairs that are formed?
Problem 6
1
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$7^{7^n}+1$ is the product of at least $2n + 3$ primes
Prove that for every nonnegative integer
n
n
n
, the number
7
7
n
+
1
7^{7^{n}}+1
7
7
n
+
1
is the product of at least
2
n
+
3
2n+3
2
n
+
3
(not necessarily distinct) primes.