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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1998 Czech And Slovak Olympiad IIIA
1998 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
2
1
Hide problems
1/a_1+1/a_2+...+1/a_k -(1/b_1+1/b_2+...+1/b_k)<0.001
Given any set of
14
14
14
(different) natural numbers, prove that for some
k
k
k
(
1
≤
k
≤
7
1 \le k \le 7
1
≤
k
≤
7
) there exist two disjoint
k
k
k
-element subsets
{
a
1
,
.
.
.
,
a
k
}
\{a_1,...,a_k\}
{
a
1
,
...
,
a
k
}
and
{
b
1
,
.
.
.
,
b
k
}
\{b_1,...,b_k\}
{
b
1
,
...
,
b
k
}
such that
A
=
1
a
1
+
1
a
2
+
.
.
.
+
1
a
k
A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}
A
=
a
1
1
+
a
2
1
+
...
+
a
k
1
and
B
=
1
b
1
+
1
b
2
+
.
.
.
+
1
b
k
B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}
B
=
b
1
1
+
b
2
1
+
...
+
b
k
1
differ by less than
0.001
0.001
0.001
, i.e.
∣
A
−
B
∣
<
0.001
|A-B| < 0.001
∣
A
−
B
∣
<
0.001
6
1
Hide problems
y/z+z/y=a/x,z/x+x/z=b/y, x/y+y/x=c/z => exists triangle with a,b,c sides
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive numbers. Prove that a triangle with sides
a
,
b
,
c
a,b,c
a
,
b
,
c
exists if and only if the system of equations
{
y
z
+
z
y
=
a
x
z
x
+
x
z
=
b
y
x
y
+
y
x
=
c
z
\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}
⎩
⎨
⎧
z
y
+
y
z
=
x
a
x
z
+
z
x
=
y
b
y
x
+
x
y
=
z
c
has a real solution.
4
1
Hide problems
greatest power of 3 that divides day^{month} −1998
For each date of year
1998
1998
1998
, we calculate day
m
o
n
t
h
^{month}
m
o
n
t
h
−year and determine the greatest power of
3
3
3
that divides it. For example, for April
21
21
21
we get
2
1
4
−
1998
=
192483
=
3
3
⋅
7129
21^4 - 1998 =192483 = 3^3 \cdot 7129
2
1
4
−
1998
=
192483
=
3
3
⋅
7129
, which is divisible by
3
3
3^3
3
3
and not by
3
4
3^4
3
4
. Find all dates for which this power of
3
3
3
is the greatest.
3
1
Hide problems
sum of edges of 4 tetrahedra equals sum of edges of another tetrahedron
A sphere is inscribed in a tetrahedron
A
B
C
D
ABCD
A
BC
D
. The tangent planes to the sphere parallel to the faces of the tetrahedron cut off four smaller tetrahedra. Prove that sum of all the
24
24
24
edges of the smaller tetrahedra equals twice the sum of edges of the tetrahedron
A
B
C
D
ABCD
A
BC
D
.
5
1
Hide problems
all trapezoids have the same intersection of diagonals
A circle
k
k
k
and a point
A
A
A
outside it are given in the plane. Prove that all trapezoids, whose non-parallel sides meet at
A
A
A
, have the same intersection of diagonals.
1
1
Hide problems
x\cdot [x \cdot [x \cdot [x]]] = 88
Solve the equation
x
⋅
[
x
⋅
[
x
⋅
[
x
]
]
]
=
88
x\cdot [x\cdot [x \cdot [x]]] = 88
x
⋅
[
x
⋅
[
x
⋅
[
x
]]]
=
88
in the set of real numbers.