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Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Junior Math Olympiad
2006 Singapore Junior Math Olympiad
2006 Singapore Junior Math Olympiad
Part of
Singapore Junior Math Olympiad
Subcontests
(5)
5
1
Hide problems
tiling n triangular tiles to create convex equiangular hexagon
You have a large number of congruent equilateral triangular tiles on a table and you want to fit
n
n
n
of them together to make a convex equiangular hexagon (i.e. one whose interior angles are
12
0
o
120^o
12
0
o
) . Obviously,
n
n
n
cannot be any positive integer. The first three feasible
n
n
n
are
6
,
10
6, 10
6
,
10
and
13
13
13
. Show that
12
12
12
is not feasible but
14
14
14
is.
3
1
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n people, min no of phone calls
Suppose that each of
n
n
n
people knows exactly one piece of information and all
n
n
n
pieces are different. Every time person
A
A
A
phones person
B
B
B
,
A
A
A
tells
B
B
B
everything he knows, while tells
A
A
A
nothing. What is the minimum of phone calls between pairs of people needed for everyone to know everything?
2
1
Hide problems
(p-1)/p cannot be a sum of distinct unit fractions
The fraction
2
3
\frac23
3
2
can be eypressed as a sum of two distinct unit fractions:
1
2
+
1
6
\frac12 + \frac16
2
1
+
6
1
. Show that the fraction
p
−
1
p
\frac{p-1}{p}
p
p
−
1
where
p
≥
5
p\ge 5
p
≥
5
is a prime cannot be expressed as a sum of two distinct unit fractions.
1
1
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x+y=x^2-xy+y^2 diophantine
Find all integers
x
,
y
x,y
x
,
y
that satisfy the equation
x
+
y
=
x
2
−
x
y
+
y
2
x+y=x^2-xy+y^2
x
+
y
=
x
2
−
x
y
+
y
2
4
1
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anglechasing with incenter, 60^o wanted (Singapore Junior 2006)
In
△
A
B
C
\vartriangle ABC
△
A
BC
, the bisector of
∠
B
\angle B
∠
B
meets
A
C
AC
A
C
at
D
D
D
and the bisector of
∠
C
\angle C
∠
C
meets
A
B
AB
A
B
at
E
E
E
. These bisectors intersect at
O
O
O
and
O
D
=
O
E
OD = OE
O
D
=
OE
. If
A
D
≠
A
E
AD \ne AE
A
D
=
A
E
, prove that
∠
A
=
6
0
o
\angle A = 60^o
∠
A
=
6
0
o
.