MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
1996 Argentina National Olympiad
1996 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
10 players in a tennis tournament
In a tennis tournament of
10
10
10
players, everyone played against everyone once. In this tournament, if player
i
i
i
won the match against player
j
j
j
, then the total number of matches
i
i
i
lost plus the total number of matches
j
j
j
won is greater than or equal to
8
8
8
. We will say that three players
i
i
i
,
j
j
j
,
k
k
k
form an atypical trio if
i
i
i
beat
j
j
j
,
j
j
j
beat
k
k
k
and
k
k
k
beat
i
i
i
. Prove that in the tournament there were exactly
40
40
40
atypical trios.
5
1
Hide problems
[x]+[1996x ]=1996
Determine all positive real numbers
x
x
x
for which
[
x
]
+
[
1996
x
]
=
1996
\left [x\right ]+\left [\sqrt{1996x}\right ]=1996
[
x
]
+
[
1996
x
]
=
1996
is verifiedClarification:The brackets indicate the integer part of the number they enclose.
2
1
Hide problems
10-digit number existence questioned, rearranging digits
Decide if there exists any number of
10
10
10
digits such that rearranging
10
,
000
10,000
10
,
000
times its digits results in
10
,
000
10,000
10
,
000
different numbers that are multiples of
7
7
7
.
1
1
Hide problems
100 numbers written around a circle.
100
100
100
numbers were written around a circle. The sum of the
100
100
100
numbers is equal to
100
100
100
and the sum of six consecutive numbers is always less than or equal to
6
6
6
. The first number is
6
6
6
. Find all the numbers.
4
1
Hide problems
A_1D _|_ B_1C , <A_1B_1C = 1/2<ABC , A_1B_1 // AB, ABCD parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram with center
O
O
O
such that
∠
B
A
D
<
9
0
o
\angle BAD <90^o
∠
B
A
D
<
9
0
o
and
∠
A
O
B
>
9
0
o
\angle AOB> 90^o
∠
A
OB
>
9
0
o
. Consider points
A
1
A_1
A
1
and
B
1
B_1
B
1
on the rays
O
A
OA
O
A
and
O
B
OB
OB
respectively, such that
A
1
B
1
A_1B_1
A
1
B
1
is parallel to
A
B
AB
A
B
and
∠
A
1
B
1
C
=
1
2
∠
A
B
C
\angle A_1B_1C = \frac12 \angle ABC
∠
A
1
B
1
C
=
2
1
∠
A
BC
. Prove that
A
1
D
A_1D
A
1
D
is perpendicular to
B
1
C
B_1C
B
1
C
.
3
1
Hide problems
orthocenter of MNK is the circumcenter of the inscribed hexagon
The non-regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is inscribed on a circle of center
O
O
O
and
A
B
=
C
D
=
E
F
AB = CD = EF
A
B
=
C
D
=
EF
. If diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
M
M
M
, diagonals
C
E
CE
CE
and
D
F
DF
D
F
intersect at
N
N
N
, and diagonals
A
E
AE
A
E
and
B
F
BF
BF
intersect at
K
K
K
, show that the heights of triangle
M
N
K
MNK
MN
K
intersect at
O
O
O
.