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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
1998 Switzerland Team Selection Test
1998 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(10)
1
1
Hide problems
f(x)- f(y) = f(x)f(1/y)- f(y)f(1/x), periodic
A function
f
:
R
−
{
0
}
→
R
f : R -\{0\} \to R
f
:
R
−
{
0
}
→
R
has the following properties: (i)
f
(
x
)
−
f
(
y
)
=
f
(
x
)
f
(
1
y
)
−
f
(
y
)
f
(
1
x
)
f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)
f
(
x
)
−
f
(
y
)
=
f
(
x
)
f
(
y
1
)
−
f
(
y
)
f
(
x
1
)
for all
x
,
y
≠
0
x,y \ne 0
x
,
y
=
0
, (ii)
f
f
f
takes the value
1
2
\frac12
2
1
at least once. Determine
f
(
−
1
)
f(-1)
f
(
−
1
)
. Prove that
f
f
f
is a periodic function
10
1
Hide problems
f(x+3/42)+ f(x) = f(x+1/6)+f\left(x+1/7), periodic
5. Let
f
:
R
→
R
f : R \to R
f
:
R
→
R
be a function that satisfies for all
x
∈
R
x \in R
x
∈
R
(i)
∣
f
(
x
)
∣
≤
1
| f(x)| \le 1
∣
f
(
x
)
∣
≤
1
, and (ii)
f
(
x
+
13
42
)
+
f
(
x
)
=
f
(
x
+
1
6
)
+
f
(
x
+
1
7
)
f\left(x+\frac{13}{42}\right)+ f(x) = f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right)
f
(
x
+
42
13
)
+
f
(
x
)
=
f
(
x
+
6
1
)
+
f
(
x
+
7
1
)
Prove that
f
f
f
is a periodic function
9
1
Hide problems
frac{x}{x^4 +y^2 }+\frac{y}{x^2 +y^4} \le \frac{1}{xy}
If
x
x
x
and
y
y
y
are positive numbers, prove the inequality
x
x
4
+
y
2
+
y
x
2
+
y
4
≤
1
x
y
\frac{x}{x^4 +y^2 }+\frac{y}{x^2 +y^4} \le \frac{1}{xy}
x
4
+
y
2
x
+
x
2
+
y
4
y
≤
x
y
1
.
8
1
Hide problems
XY \cdot YZ\cdot ZX \ge XB\cdot YC\cdot ZA, equilateral
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an equilateral triangle and let
P
P
P
be a point in its interior. Let the lines
A
P
,
B
P
,
C
P
AP,BP,CP
A
P
,
BP
,
CP
meet the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
in the points
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
respectively. Prove that
X
Y
⋅
Y
Z
⋅
Z
X
≥
X
B
⋅
Y
C
⋅
Z
A
XY \cdot YZ\cdot ZX \ge XB\cdot YC\cdot ZA
X
Y
⋅
Y
Z
⋅
ZX
≥
XB
⋅
Y
C
⋅
Z
A
.
7
1
Hide problems
entry at intersection of i-th row and the j-th column equals i+ j -1
Consider an
n
×
n
n\times n
n
×
n
matrix whose entry at the intersection of the
i
i
i
-th row and the
j
−
j-
j
−
th column equals
i
+
j
−
1
i+ j -1
i
+
j
−
1
. What is the largest possible value of the product of
n
n
n
entries of the matrix, no two of which are in the same row or column?
6
1
Hide problems
p^2 +11 has exactly six positive divisors, where p prime
Find all prime numbers
p
p
p
for which
p
2
+
11
p^2 +11
p
2
+
11
has exactly six positive divisors.
5
1
Hide problems
line AB bisects the segment PQ, circle related
Points
A
A
A
and
B
B
B
are chosen on a circle
k
k
k
. Let AP and
B
Q
BQ
BQ
be segments of the same length tangent to
k
k
k
, drawn on different sides of line
A
B
AB
A
B
. Prove that the line
A
B
AB
A
B
bisects the segment
P
Q
PQ
PQ
.
4
1
Hide problems
cut a square into n smaller squares
Find all numbers
n
n
n
for which it is possible to cut a square into
n
n
n
smaller squares.
3
1
Hide problems
f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2} , min
Given positive numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
, find the minimum of the function
f
(
x
)
=
a
2
+
x
2
+
(
b
−
x
)
2
+
c
2
f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2}
f
(
x
)
=
a
2
+
x
2
+
(
b
−
x
)
2
+
c
2
.
2
1
Hide problems
1/(x+2)+1/(y+2)=1/2 +1/(z+2)
Find all nonnegative integer solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of the equation
1
x
+
2
+
1
y
+
2
=
1
2
+
1
z
+
2
\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}
x
+
2
1
+
y
+
2
1
=
2
1
+
z
+
2
1