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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2003 Switzerland Team Selection Test
2003 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(10)
10
1
Hide problems
strictly monotonous functions f : N \to N , f(f(n)) = 3n
Find all strictly monotonous functions
f
:
N
→
N
f : N \to N
f
:
N
→
N
that satisfy
f
(
f
(
n
)
)
=
3
n
f(f(n)) = 3n
f
(
f
(
n
))
=
3
n
for all
n
∈
N
n \in N
n
∈
N
.
9
1
Hide problems
5050 | a_k +a_l -a_m -a_n when 0 < a_1 < a_2 <... < a_{101} < 5050
Given integers
0
<
a
1
<
a
2
<
.
.
.
<
a
101
<
5050
0 < a_1 < a_2 <... < a_{101} < 5050
0
<
a
1
<
a
2
<
...
<
a
101
<
5050
, prove that one can always choose for different numbers
a
k
,
a
l
,
a
m
,
a
n
a_k,a_l,a_m,a_n
a
k
,
a
l
,
a
m
,
a
n
such that
5050
∣
a
k
+
a
l
−
a
m
−
a
n
5050 | a_k +a_l -a_m -a_n
5050∣
a
k
+
a
l
−
a
m
−
a
n
8
1
Hide problems
7 circles related problem, in pairs externally tangent
Let
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
be a triangle and
ω
1
\omega_1
ω
1
be a circle passing through
A
1
A_1
A
1
and
A
2
A_2
A
2
. Suppose that there are circles
ω
2
,
.
.
.
,
ω
7
\omega_2,...,\omega_7
ω
2
,
...
,
ω
7
such that: (a)
ω
k
\omega_k
ω
k
passes through
A
k
A_k
A
k
and
A
k
+
1
A_{k+1}
A
k
+
1
for
k
=
2
,
3
,
.
.
.
,
7
k = 2,3,...,7
k
=
2
,
3
,
...
,
7
, where
A
i
=
A
i
+
3
A_i = A_{i+3}
A
i
=
A
i
+
3
, (b)
ω
k
\omega_k
ω
k
and
ω
k
+
1
\omega_{k+1}
ω
k
+
1
are externally tangent for
k
=
1
,
2
,
.
.
.
,
6
k = 1,2,...,6
k
=
1
,
2
,
...
,
6
. Prove that
ω
1
=
ω
7
\omega_1 = \omega_7
ω
1
=
ω
7
.
7
1
Hide problems
|Q(p_1)| = |Q(p_2)| = |Q(p_3)| = 11 , integer trinomials
Find all polynomials
Q
(
x
)
=
a
x
2
+
b
x
+
c
Q(x)= ax^2+bx+c
Q
(
x
)
=
a
x
2
+
b
x
+
c
with integer coefficients for which there exist three different prime numbers
p
1
,
p
2
,
p
3
p_1, p_2, p_3
p
1
,
p
2
,
p
3
such that
∣
Q
(
p
1
)
∣
=
∣
Q
(
p
2
)
∣
=
∣
Q
(
p
3
)
∣
=
11
|Q(p_1)| = |Q(p_2)| = |Q(p_3)| = 11
∣
Q
(
p
1
)
∣
=
∣
Q
(
p
2
)
∣
=
∣
Q
(
p
3
)
∣
=
11
.
5
1
Hide problems
n pieces on the squares of a 5 x 9 board, game to the end of the world
There are
n
n
n
pieces on the squares of a
5
×
9
5 \times 9
5
×
9
board, at most one on each square at any time during the game. A move in the game consists of simultaneously moving each piece to a neighboring square by side, under the restriction that a piece having been moved horizontally in the previous move must be moved vertically and vice versa. Find the greatest value of
n
n
n
for which there exists an initial position starting at which the game can be continued until the end of the world.
4
1
Hide problems
max n divides a^{25}-a for all integers a
Find the largest natural number
n
n
n
that divides
a
25
−
a
a^{25} -a
a
25
−
a
for all integers
a
a
a
.
2
1
Hide problems
equal segments, altitudes and perpendicular related
In an acute-angled triangle
A
B
C
,
E
ABC, E
A
BC
,
E
and
F
F
F
are the feet of the altitudes from
B
B
B
and
C
C
C
, and
G
G
G
and
H
H
H
are the projections of
B
B
B
and
C
C
C
on
E
F
EF
EF
, respectively. Prove that
H
E
=
F
G
HE = FG
H
E
=
FG
.
1
1
Hide problems
x+y = x^3 +y^3 = x^5 +y^5 = a
Real numbers
x
,
y
,
a
x,y,a
x
,
y
,
a
satisfy the equations
x
+
y
=
x
3
+
y
3
=
x
5
+
y
5
=
a
x+y = x^3 +y^3 = x^5 +y^5 = a
x
+
y
=
x
3
+
y
3
=
x
5
+
y
5
=
a
Find all possible values of
a
a
a
.
3
1
Hide problems
Infimum and supremum for an expression
Find the largest real number
C
1
C_1
C
1
and the smallest real number
C
2
C_2
C
2
, such that, for all reals
a
,
b
,
c
,
d
,
e
a,b,c,d,e
a
,
b
,
c
,
d
,
e
, we have
C
1
<
a
a
+
b
+
b
b
+
c
+
c
c
+
d
+
d
d
+
e
+
e
e
+
a
<
C
2
C_1 < \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a} < C_2
C
1
<
a
+
b
a
+
b
+
c
b
+
c
+
d
c
+
d
+
e
d
+
e
+
a
e
<
C
2
6
1
Hide problems
Simple inequality with a+b+c=2
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers satisfying
a
+
b
+
c
=
2
a+b+c=2
a
+
b
+
c
=
2
. Prove the inequality
1
1
+
a
b
+
1
1
+
b
c
+
1
1
+
c
a
≥
27
13
\frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca} \ge \frac{27}{13}
1
+
ab
1
+
1
+
b
c
1
+
1
+
c
a
1
≥
13
27