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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2000 Switzerland Team Selection Test
2000 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(15)
15
1
Hide problems
x_1^2 +x_2^2 +...+x_{2000}^2<9, points interior in circle of radius 1
Let
S
=
{
P
1
,
P
2
,
.
.
.
,
P
2000
}
S = \{P_1,P_2,...,P_{2000}\}
S
=
{
P
1
,
P
2
,
...
,
P
2000
}
be a set of
2000
2000
2000
points in the interior of a circle of radius
1
1
1
, one of which at its center. For
i
=
1
,
2
,
.
.
.
,
2000
i = 1,2,...,2000
i
=
1
,
2
,
...
,
2000
denote by
x
i
x_i
x
i
the distance from
P
i
P_i
P
i
to the closest point
P
j
≠
P
i
P_j \ne P_i
P
j
=
P
i
. Prove that
x
1
2
+
x
2
2
+
.
.
.
+
x
2000
2
<
9
x_1^2 +x_2^2 +...+x_{2000}^2<9
x
1
2
+
x
2
2
+
...
+
x
2000
2
<
9
.
14
1
Hide problems
P(k) = \frac{k}{k +1} , find P(n+1), polynomial
The polynomial
P
P
P
of degree
n
n
n
satisfies
P
(
k
)
=
k
k
+
1
P(k) = \frac{k}{k +1}
P
(
k
)
=
k
+
1
k
for
k
=
0
,
1
,
2
,
.
.
.
,
n
k = 0,1,2,...,n
k
=
0
,
1
,
2
,
...
,
n
. Find
P
(
n
+
1
)
P(n+1)
P
(
n
+
1
)
.
13
1
Hide problems
concyclic wanted, incircle related
The incircle of a triangle
A
B
C
ABC
A
BC
touches the sides
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
at points
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Let
P
P
P
be an internal point of triangle
A
B
C
ABC
A
BC
such that the incircle of triangle
A
B
P
ABP
A
BP
touches
A
B
AB
A
B
at
D
D
D
and the sides
A
P
AP
A
P
and
B
P
BP
BP
at
Q
Q
Q
and
R
R
R
. Prove that the points
E
,
F
,
R
,
Q
E,F,R,Q
E
,
F
,
R
,
Q
lie on a circle.
12
1
Hide problems
f(f(x)+y) = f(x^2 -y)+4y f(x)
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that for all real
x
,
y
x,y
x
,
y
,
f
(
f
(
x
)
+
y
)
=
f
(
x
2
−
y
)
+
4
y
f
(
x
)
f(f(x)+y) = f(x^2 -y)+4y f(x)
f
(
f
(
x
)
+
y
)
=
f
(
x
2
−
y
)
+
4
y
f
(
x
)
11
1
Hide problems
regular 2n-gon are labelled with the numbers 1,2,...,2n
The vertices of a regular
2
n
2n
2
n
-gon (
n
≥
3
n \ge 3
n
≥
3
) are labelled with the numbers
1
,
2
,
.
.
.
,
2
n
1,2,...,2n
1
,
2
,
...
,
2
n
so that the sum of the numbers at any two adjacent vertices equals the sum of the numbers at the vertices diametrically opposite to them. Show that this is only possible if
n
n
n
is odd.
10
1
Hide problems
n cars around a circular race course
At
n
n
n
distinct points of a circular race course there are
n
n
n
cars ready to start. Each car moves at a constant speed and covers the circle in an hour. On hearing the initial signal, each of them selects a direction and starts moving immediately. If two cars meet, both of them change directions and go on without loss of speed. Show that at a certain moment each car will be at its starting point.
9
1
Hide problems
a segment construction, passing through intersections of two circles
Two given circles
k
1
k_1
k
1
and
k
2
k_2
k
2
intersect at points
P
P
P
and
Q
Q
Q
. Construct a segment
A
B
AB
A
B
through
P
P
P
with the endpoints at
k
1
k_1
k
1
and
k
2
k_2
k
2
for which
A
P
⋅
P
B
AP \cdot PB
A
P
⋅
PB
is maximal.
8
1
Hide problems
sum f (k/1921) when f(x) = 4^x/ (4^x+2)
Let
f
(
x
)
=
4
x
4
x
+
2
f(x) = \frac{4^x}{4^x+2}
f
(
x
)
=
4
x
+
2
4
x
for
x
>
0
x > 0
x
>
0
. Evaluate
∑
k
=
1
1920
f
(
k
1921
)
\sum_{k=1}^{1920}f\left(\frac{k}{1921}\right)
∑
k
=
1
1920
f
(
1921
k
)
7
1
Hide problems
14x^2 +15y^2 = 7^{2000}
Show that the equation
14
x
2
+
15
y
2
=
7
2000
14x^2 +15y^2 = 7^{2000}
14
x
2
+
15
y
2
=
7
2000
has no integer solutions.
6
1
Hide problems
\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7 when x,y,z have sum 1
Positive real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
have the sum
1
1
1
. Prove that
7
x
+
3
+
7
y
+
3
+
7
z
+
3
≤
7
\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7
7
x
+
3
+
7
y
+
3
+
7
z
+
3
≤
7
. Can number
7
7
7
on the right hand side be replaced with a smaller constant?
5
1
Hide problems
all words of length n consisting of letters I,O,M with no 2 consecutive M
Consider all words of length
n
n
n
consisting of the letters
I
,
O
,
M
I,O,M
I
,
O
,
M
. How many such words are there, which contain no two consecutive
M
M
M
’s?
4
1
Hide problems
q(q(q(2000^{2000}))) , where q(n) is sum of digits of n
Let
q
(
n
)
q(n)
q
(
n
)
denote the sum of the digits of a natural number
n
n
n
. Determine
q
(
q
(
q
(
200
0
2000
)
)
)
q(q(q(2000^{2000})))
q
(
q
(
q
(
200
0
2000
)))
.
3
1
Hide problems
equilateral of side 1 is covered by 5 congruent equilateral of side s<1
An equilateral triangle of side
1
1
1
is covered by five congruent equilateral triangles of side
s
<
1
s < 1
s
<
1
with sides parallel to those of the larger triangle. Show that some four of these smaller triangles also cover the large triangle.
2
1
Hide problems
\sum a_i = 100 and \sum a_i^2 = 1000 , max a_i =?
Real numbers
a
1
,
a
2
,
.
.
.
,
a
16
a_1,a_2,...,a_{16}
a
1
,
a
2
,
...
,
a
16
satisfy the conditions
∑
i
=
1
16
a
i
=
100
\sum_{i=1}^{16}a_i = 100
∑
i
=
1
16
a
i
=
100
and
∑
i
=
1
16
a
i
2
=
1000
\sum_{i=1}^{16}a_i^2 = 1000
∑
i
=
1
16
a
i
2
=
1000
. What is the greatest possible value of
a
1
6
a_16
a
1
6
?
1
1
Hide problems
segments joining arc midpoints are perpendicular
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle. Show that the line connecting the midpoints of the arcs
A
B
AB
A
B
and
C
D
CD
C
D
and the line connecting the midpoints of the arcs
B
C
BC
BC
and
D
A
DA
D
A
are perpendicular.