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Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
1995 South africa National Olympiad
1995 South africa National Olympiad
Part of
South Africa National Olympiad
Subcontests
(4)
4
1
Hide problems
Ratio between areas of triangles defined by tangent circles
Three circles, with radii
p
p
p
,
q
q
q
and
r
r
r
and centres
A
A
A
,
B
B
B
and
C
C
C
respectively, touch one another externally at points
D
D
D
,
E
E
E
and
F
F
F
. Prove that the ratio of the areas of
△
D
E
F
\triangle DEF
△
D
EF
and
△
A
B
C
\triangle ABC
△
A
BC
equals
2
p
q
r
(
p
+
q
)
(
q
+
r
)
(
r
+
p
)
.
\frac{2pqr}{(p+q)(q+r)(r+p)}.
(
p
+
q
)
(
q
+
r
)
(
r
+
p
)
2
pq
r
.
3
2
Hide problems
Sum of squares of difference between permutations is even
Suppose that
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
are the numbers
1
,
2
,
3
,
…
,
n
1,2,3,\dots,n
1
,
2
,
3
,
…
,
n
but written in any order. Prove that
(
a
1
−
1
)
2
+
(
a
2
−
2
)
2
+
⋯
+
(
a
n
−
n
)
2
(a_1-1)^2+(a_2-2)^2+\cdots+(a_n-n)^2
(
a
1
−
1
)
2
+
(
a
2
−
2
)
2
+
⋯
+
(
a
n
−
n
)
2
is always even.
AP BP CP < (1+x)^2 (1-x)
The circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
has radius
1
1
1
and centre
O
O
O
and
P
P
P
is a point inside the triangle such that
O
P
=
x
OP=x
OP
=
x
. Prove that
A
P
⋅
B
P
⋅
C
P
≤
(
1
+
x
)
2
(
1
−
x
)
,
AP\cdot BP\cdot CP\le(1+x)^2(1-x),
A
P
⋅
BP
⋅
CP
≤
(
1
+
x
)
2
(
1
−
x
)
,
with equality if and only if
P
=
O
P=O
P
=
O
.
2
2
Hide problems
Prove that this angle is right
A
B
C
ABC
A
BC
is a triangle with
A
^
<
C
^
\hat{A}<\hat{C}
A
^
<
C
^
, and
D
D
D
is the point on
B
C
BC
BC
such that
B
A
^
D
=
A
C
^
B
B\hat{A}D=A\hat{C}B
B
A
^
D
=
A
C
^
B
. The perpendicular bisectors of
A
D
AD
A
D
and
A
C
AC
A
C
intersect in the point
E
E
E
. Prove that
B
A
^
E
=
9
0
∘
B\hat{A}E=90^\circ
B
A
^
E
=
9
0
∘
.
n divides m^2+1 and m divides n^2+1
Find all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of natural numbers with
m
<
n
m<n
m
<
n
such that
m
2
+
1
m^2+1
m
2
+
1
is a multiple of
n
n
n
and
n
2
+
1
n^2+1
n
2
+
1
is a multiple of
m
m
m
.
1
2
Hide problems
Quadratic Diophantine equation with no solutions
Prove that there are no integers
m
m
m
and
n
n
n
such that
19
m
2
+
95
m
n
+
2000
n
2
=
1995.
19m^2+95mn+2000n^2=1995.
19
m
2
+
95
mn
+
2000
n
2
=
1995.
Find the area of the larger quadrilateral
The convex quadrilateral
A
B
C
D
ABCD
A
BC
D
has area
1
1
1
, and
A
B
AB
A
B
is produced to
E
E
E
,
B
C
BC
BC
to
F
F
F
,
C
D
CD
C
D
to
G
G
G
and
D
A
DA
D
A
to
H
H
H
, such that
A
B
=
B
E
AB=BE
A
B
=
BE
,
B
C
=
C
F
BC=CF
BC
=
CF
,
C
D
=
D
G
CD=DG
C
D
=
D
G
and
D
A
=
A
H
DA=AH
D
A
=
A
H
. Find the area of the quadrilateral
E
F
G
H
EFGH
EFG
H
.