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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2017 Singapore Senior Math Olympiad
2017 Singapore Senior Math Olympiad
Part of
Singapore Senior Math Olympiad
Subcontests
(4)
4
1
Hide problems
f(k + 1) >f(f(k)) for k > 1 when f : Z^+ \to Z^+
Find all functions
f
:
Z
+
→
Z
+
f : Z^+ \to Z^+
f
:
Z
+
→
Z
+
such that
f
(
k
+
1
)
>
f
(
f
(
k
)
)
f(k + 1) >f(f(k))
f
(
k
+
1
)
>
f
(
f
(
k
))
for
k
>
1
k > 1
k
>
1
, where
Z
+
Z^+
Z
+
is the set of positive integers.
5
1
Hide problems
arithmetic progression of a, a + d, a + 2d,.. with 3,4,7 of given numbers
Given
7
7
7
distinct positive integers, prove that there is an infinite arithmetic progression of positive integers
a
,
a
+
d
,
a
+
2
d
,
.
.
a, a + d, a + 2d,..
a
,
a
+
d
,
a
+
2
d
,
..
with
a
<
d
a < d
a
<
d
, that contains exactly
3
3
3
or
4
4
4
of the
7
7
7
given integers.
3
1
Hide problems
2017 distinct points, min distinct midpoints of all pairs
There are
2017
2017
2017
distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?
2
1
Hide problems
right angle leads to right angle, intersections of sides of cyclic ABCD related
In the cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
, the sides
A
B
,
D
C
AB, DC
A
B
,
D
C
meet at
Q
Q
Q
, the sides
A
D
,
B
C
AD,BC
A
D
,
BC
meet at
P
,
M
P, M
P
,
M
is the midpoint of
B
D
BD
B
D
, If
∠
A
P
Q
=
9
0
o
\angle APQ=90^o
∠
A
PQ
=
9
0
o
, prove that
P
M
PM
PM
is perpendicular to
A
B
AB
A
B
.