MathDB
Problems
Contests
International Contests
APMO
1995 APMO
1995 APMO
Part of
APMO
Subcontests
(5)
5
1
Hide problems
Function
Find the minimum positive integer
k
k
k
such that there exists a function
f
f
f
from the set
Z
\Bbb{Z}
Z
of all integers to
{
1
,
2
,
…
k
}
\{1, 2, \ldots k\}
{
1
,
2
,
…
k
}
with the property that
f
(
x
)
≠
f
(
y
)
f(x) \neq f(y)
f
(
x
)
=
f
(
y
)
whenever
∣
x
−
y
∣
∈
{
5
,
7
,
12
}
|x-y| \in \{5, 7, 12\}
∣
x
−
y
∣
∈
{
5
,
7
,
12
}
.
1
1
Hide problems
Sequence
Determine all sequences of real numbers
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\ldots
…
,
a
1995
a_{1995}
a
1995
which satisfy: 2\sqrt{a_n - (n - 1)} \geq a_{n+1} - (n - 1), \ \mbox{for} \ n = 1, 2, \ldots 1994, and
2
a
1995
−
1994
≥
a
1
+
1.
2\sqrt{a_{1995} - 1994} \geq a_1 + 1.
2
a
1995
−
1994
≥
a
1
+
1.
2
1
Hide problems
Smallest n to ensure a prime
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\ldots
…
,
a
n
a_n
a
n
be a sequence of integers with values between 2 and 1995 such that: (i) Any two of the
a
i
a_i
a
i
's are relatively prime, (ii) Each
a
i
a_i
a
i
is either a prime or a product of primes. Determine the smallest possible values of
n
n
n
to make sure that the sequence will contain a prime number.
4
1
Hide problems
A moves around the circle
Let
C
C
C
be a circle with radius
R
R
R
and centre
O
O
O
, and
S
S
S
a fixed point in the interior of
C
C
C
. Let
A
A
′
AA'
A
A
′
and
B
B
′
BB'
B
B
′
be perpendicular chords through
S
S
S
. Consider the rectangles
S
A
M
B
SAMB
S
A
MB
,
S
B
N
′
A
′
SBN'A'
SB
N
′
A
′
,
S
A
′
M
′
B
′
SA'M'B'
S
A
′
M
′
B
′
, and
S
B
′
N
A
SB'NA
S
B
′
N
A
. Find the set of all points
M
M
M
,
N
′
N'
N
′
,
M
′
M'
M
′
, and
N
N
N
when
A
A
A
moves around the whole circle.
3
1
Hide problems
Determine tangency points
Let
P
Q
R
S
PQRS
PQRS
be a cyclic quadrilateral such that the segments
P
Q
PQ
PQ
and
R
S
RS
RS
are not parallel. Consider the set of circles through
P
P
P
and
Q
Q
Q
, and the set of circles through
R
R
R
and
S
S
S
. Determine the set
A
A
A
of points of tangency of circles in these two sets.