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Problems
Contests
International Contests
Tuymaada Olympiad
1994 Tuymaada Olympiad
1994 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(8)
2
1
Hide problems
not unique factorization inside a subset of the set 4k-3 , k in N
The set of numbers
M
=
{
4
k
−
3
∣
k
∈
N
}
M=\{4k-3 | k\in N\}
M
=
{
4
k
−
3∣
k
∈
N
}
is considered. A number of of this set is called “simple” if it is impossible to put in the form of a product of numbers from
M
M
M
other than
1
1
1
. Show that in this set, the decomposition of numbers in the product of "simple" factors is ambiguous.
7
1
Hide problems
infinitely relative prime solutions to a+b+c=u+v and a^2+b^2+c^2=u^2+v^2
Prove that there are infinitely many natural numbers
a
,
b
,
c
,
u
a,b,c,u
a
,
b
,
c
,
u
and
v
v
v
with greatest common divisor
1
1
1
satisfying the system of equations:
a
+
b
+
c
=
u
+
v
a+b+c=u+v
a
+
b
+
c
=
u
+
v
and
a
2
+
b
2
+
c
2
=
u
2
+
v
2
a^2+b^2+c^2=u^2+v^2
a
2
+
b
2
+
c
2
=
u
2
+
v
2
6
1
Hide problems
point on a plane so that 3 moving beetles reach 3 points at min time
In three houses
A
,
B
A,B
A
,
B
and
C
C
C
, forming a right triangle with the legs
A
C
=
30
AC=30
A
C
=
30
and
C
B
=
40
CB=40
CB
=
40
, live three beetles
a
,
b
a,b
a
,
b
and
c
c
c
, capable of moving at speeds of
2
,
3
2, 3
2
,
3
and
4
4
4
, respectively. Suppose that you simultaneously release these bugs from point
M
M
M
and mark the time after which beetles reach their homes. Find on the plane such a point
M
M
M
, where is the last time to reach the house a bug would be minimal.
4
1
Hide problems
arranging a ball inside of a convex polyhedron
Let a convex polyhedron be given with volume
V
V
V
and full surface
S
S
S
. Prove that inside a polyhedron it is possible to arrange a ball of radius
V
S
\frac{V}{S}
S
V
.
1
1
Hide problems
new point system in World Cup in America (1994)
World Cup in America introduced a new point system. For a victory
3
3
3
points are given, for a draw
1
1
1
point and for defeat
0
0
0
points. In the preliminary games, the teams are divided into groups of
4
4
4
teams. In groups, teams play with each other, once, then in accordance with the points scored
a
,
b
,
c
a,b,c
a
,
b
,
c
and
d
d
d
(
a
>
b
>
c
>
d
a>b>c>d
a
>
b
>
c
>
d
) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points
a
,
b
,
c
a,b,c
a
,
b
,
c
and
d
d
d
8
1
Hide problems
sphere containing exactly 1994 points with integer coordinates
Prove that in space there is a sphere containing exactly
1994
1994
1994
points with integer coordinates.
5
1
Hide problems
smallest n so that sin ( 1/(n+1934) )< 1/1994
Find the smallest natural number
n
n
n
for which
s
i
n
(
1
n
+
1934
)
<
1
1994
sin \Big(\frac{1}{n+1934}\Big)<\frac{1}{1994}
s
in
(
n
+
1934
1
)
<
1994
1
.
3
1
Hide problems
M,N interior points of triangle ABC, prove that AN>AM or BN>BM or CN>CM
Point
M
M
M
lies inside triangle
A
B
C
ABC
A
BC
. Prove that for any other point
N
N
N
lying inside the triangle
A
B
C
ABC
A
BC
, at least one of the following three inequalities is fulfilled:
A
N
>
A
M
,
B
N
>
B
M
,
C
N
>
C
M
AN>AM, BN>BM, CN>CM
A
N
>
A
M
,
BN
>
BM
,
CN
>
CM
.