MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2000 Singapore MO Open
2000 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(4)
4
1
Hide problems
one can find 27 people in party who have all shaken hands with each other
In a party of
1000
1000
1000
people, the number of people who have shaken hands with at most
962
962
962
people is less than or equal to
37
37
37
. Show that one can find
27
27
27
people in the party who have all shaken hands with each other.
3
1
Hide problems
first digit is moved to the end of the number, it increases by 60 %
Is there a positive integer with at most four digits whose value is increased by exactly
60
%
60\%
60%
when the first digit is moved to the end of the number? For example, when the first digit of
1234
1234
1234
is moved to the end of the number, the result is the integer
2341
2341
2341
.
2
1
Hide problems
240 divides all numbers of the form p^4 - q^4 when p,q primes >5
Show that
240
240
240
divides all numbers of the form
p
4
−
q
4
p^4 - q^4
p
4
−
q
4
, where p and q are prime numbers strictly greater than
5
5
5
. Show also that
240
240
240
is the greatest common divisor of all numbers of the form
p
4
−
q
4
p^4 - q^4
p
4
−
q
4
, with
p
p
p
and
q
q
q
prime numbers strictly greater than
5
5
5
.
1
1
Hide problems
B,D,E,C concyclic, circumcircle and perpendicular related
Triangle
A
B
C
ABC
A
BC
is inscribed in a circle with center
O
O
O
. Let
D
D
D
and
E
E
E
be points on the respective sides
A
B
AB
A
B
and
A
C
AC
A
C
so that
D
E
DE
D
E
is perpendicular to
A
O
AO
A
O
. Show that the four points
B
,
D
,
E
B,D,E
B
,
D
,
E
and
C
C
C
lie on a circle.