MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
1999 Switzerland Team Selection Test
1999 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(10)
8
1
Hide problems
sum a_k = 96, sum a_k^2 = 144, sum a_k^3 = 216
Find all
n
n
n
for which there are real numbers
0
<
a
1
≤
a
2
≤
.
.
.
≤
a
n
0 < a_1 \le a_2 \le ... \le a_n
0
<
a
1
≤
a
2
≤
...
≤
a
n
with
{
∑
k
=
1
n
a
k
=
96
∑
k
=
1
n
a
k
2
=
144
∑
k
=
1
n
a
k
3
=
216
\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}
⎩
⎨
⎧
∑
k
=
1
n
a
k
=
96
∑
k
=
1
n
a
k
2
=
144
∑
k
=
1
n
a
k
3
=
216
7
1
Hide problems
square dissected into rectangles, sum of ratios is at least 1.
A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least
1
1
1
.
6
1
Hide problems
m^2 + n^2 - m is divisible by 2mn, then m is a perfect square.
Prove that if
m
m
m
and
n
n
n
are positive integers such that
m
2
+
n
2
−
m
m^2 + n^2 - m
m
2
+
n
2
−
m
is divisible by
2
m
n
2mn
2
mn
, then
m
m
m
is a perfect square.
4
1
Hide problems
\frac{4x^2}{1+4x^2}= y,\frac{4y^2}{1+4y^2}= z, \frac{4z^2}{1+4z^2}= x
Find all real solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of the system
{
4
x
2
1
+
4
x
2
=
y
4
y
2
1
+
4
y
2
=
z
4
z
2
1
+
4
z
2
=
x
\begin{cases}\dfrac{4x^2}{1+4x^2}= y\\ \\\dfrac{4y^2}{1+4y^2}= z\\ \\ \dfrac{4z^2}{1+4z^2}= x \end{cases}
⎩
⎨
⎧
1
+
4
x
2
4
x
2
=
y
1
+
4
y
2
4
y
2
=
z
1
+
4
z
2
4
z
2
=
x
5
1
Hide problems
MN bisects the <QNP , inside rectangle ABCD
In a rectangle
A
B
C
D
,
M
ABCD, M
A
BC
D
,
M
and
N
N
N
are the midpoints of
A
D
AD
A
D
and
B
C
BC
BC
respectively and
P
P
P
is a point on line
C
D
CD
C
D
. The line
P
M
PM
PM
meets
A
C
AC
A
C
at
Q
Q
Q
. Prove that MN bisects the angle
∠
Q
N
P
\angle QNP
∠
QNP
.
3
1
Hide problems
1/x f(-x)+ f(/x)= x
Find all functions
f
:
R
−
{
0
}
→
R
f : R -\{0\} \to R
f
:
R
−
{
0
}
→
R
that satisfy
1
x
f
(
−
x
)
+
f
(
1
x
)
=
x
\frac{1}{x}f(-x)+ f\left(\frac{1}{x}\right)= x
x
1
f
(
−
x
)
+
f
(
x
1
)
=
x
for all
x
≠
0
x \ne 0
x
=
0
.
2
1
Hide problems
{1,2,...,33} partitioned into 11 three-element sets
Can the set
{
1
,
2
,
.
.
.
,
33
}
\{1,2,...,33\}
{
1
,
2
,
...
,
33
}
be partitioned into
11
11
11
three-element sets, in each of which one element equals the sum of the other two?
1
1
Hide problems
tangent to first circle at A is parallel to BC
Two circles intersect at points
M
M
M
and
N
N
N
. Let
A
A
A
be a point on the first circle, distinct from
M
,
N
M,N
M
,
N
. The lines
A
M
AM
A
M
and
A
N
AN
A
N
meet the second circle again at
B
B
B
and
C
C
C
, respectively. Prove that the tangent to the first circle at
A
A
A
is parallel to
B
C
BC
BC
.
10
1
Hide problems
product of five consecutive positive integers never not a perfect square
Prove that the product of five consecutive positive integers cannot be a perfect square.
9
1
Hide problems
An infinite arithmetic progression not contain value of P
Suppose that
P
(
x
)
P(x)
P
(
x
)
is a polynomial with degree
10
10
10
and integer coefficients. Prove that, there is an infinite arithmetic progression (open to bothside) not contain value of
P
(
k
)
P(k)
P
(
k
)
with
k
∈
Z
k\in\mathbb{Z}
k
∈
Z