MathDB
Problems
Contests
International Contests
APMO
1994 APMO
1994 APMO
Part of
APMO
Subcontests
(5)
5
1
Hide problems
Different bases
You are given three lists
A
A
A
,
B
B
B
, and
C
C
C
. List
A
A
A
contains the numbers of the form
1
0
k
10^k
1
0
k
in base
10
10
10
, with
k
k
k
any integer greater than or equal to
1
1
1
. Lists
B
B
B
and
C
C
C
contain the same numbers translated into base
2
2
2
and
5
5
5
respectively:
A
B
C
10
1010
20
100
1100100
400
1000
1111101000
13000
⋮
⋮
⋮
\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}
A
10
100
1000
⋮
B
1010
1100100
1111101000
⋮
C
20
400
13000
⋮
Prove that for every integer
n
>
1
n > 1
n
>
1
, there is exactly one number in exactly one of the lists
B
B
B
or
C
C
C
that has exactly
n
n
n
digits.
3
1
Hide problems
N=a^2+b^2
Let
n
n
n
be an integer of the form
a
2
+
b
2
a^2 + b^2
a
2
+
b
2
, where
a
a
a
and
b
b
b
are relatively prime integers and such that if
p
p
p
is a prime,
p
≤
n
p \leq \sqrt{n}
p
≤
n
, then
p
p
p
divides
a
b
ab
ab
. Determine all such
n
n
n
.
1
1
Hide problems
Functional equation
Let
f
:
R
→
R
f: \Bbb{R} \rightarrow \Bbb{R}
f
:
R
→
R
be a function such that (i) For all
x
,
y
∈
R
x,y \in \Bbb{R}
x
,
y
∈
R
,
f
(
x
)
+
f
(
y
)
+
1
≥
f
(
x
+
y
)
≥
f
(
x
)
+
f
(
y
)
f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y)
f
(
x
)
+
f
(
y
)
+
1
≥
f
(
x
+
y
)
≥
f
(
x
)
+
f
(
y
)
(ii) For all
x
∈
[
0
,
1
)
x \in [0,1)
x
∈
[
0
,
1
)
,
f
(
0
)
≥
f
(
x
)
f(0) \geq f(x)
f
(
0
)
≥
f
(
x
)
, (iii)
−
f
(
−
1
)
=
f
(
1
)
=
1
-f(-1) = f(1) = 1
−
f
(
−
1
)
=
f
(
1
)
=
1
. Find all such functions
f
f
f
.
4
1
Hide problems
Set of points
Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?
2
1
Hide problems
Oh < 3r
Given a nondegenerate triangle
A
B
C
ABC
A
BC
, with circumcentre
O
O
O
, orthocentre
H
H
H
, and circumradius
R
R
R
, prove that
∣
O
H
∣
<
3
R
|OH| < 3R
∣
O
H
∣
<
3
R
.