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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1992 Irish Math Olympiad
1992 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
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Rational Coordinates
Let
A
B
C
ABC
A
BC
be a triangle such that the coordinates of the points
A
A
A
and
B
B
B
are rational numbers. Prove that the coordinates of
C
C
C
are rational if, and only if,
tan
A
\tan A
tan
A
,
tan
B
\tan B
tan
B
, and
tan
C
\tan C
tan
C
, when defined, are all rational numbers.
nth Root Inequality
If, for
k
=
1
,
2
,
…
,
n
k=1,2,\dots ,n
k
=
1
,
2
,
…
,
n
,
a
k
a_k
a
k
and
b
k
b_k
b
k
are positive real numbers, prove that
a
1
a
2
⋯
a
n
n
+
b
1
b
2
⋯
b
n
n
≤
(
a
1
+
b
1
)
(
a
2
+
b
2
)
⋯
(
a
n
+
b
n
)
n
;
\sqrt[n]{a_1a_2\cdots a_n}+\sqrt[n]{b_1b_2\cdots b_n}\le \sqrt[n]{(a_1+b_1)(a_2+b_2)\cdots (a_n+b_n)};
n
a
1
a
2
⋯
a
n
+
n
b
1
b
2
⋯
b
n
≤
n
(
a
1
+
b
1
)
(
a
2
+
b
2
)
⋯
(
a
n
+
b
n
)
;
and that equality holds if, and only if,
a
1
b
1
=
a
2
b
2
=
⋯
=
a
n
b
n
.
\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}.
b
1
a
1
=
b
2
a
2
=
⋯
=
b
n
a
n
.
4
2
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Diameter of Circumcircle
In a triangle
A
B
C
ABC
A
BC
, the points
A
’
A’
A
’
,
B
’
B’
B
’
and
C
’
C’
C
’
on the sides opposite
A
A
A
,
B
B
B
and
C
C
C
, respectively, are such that the lines
A
A
’
AA’
AA
’
,
B
B
’
BB’
BB
’
and
C
C
’
CC’
CC
’
are concurrent. Prove that the diameter of the circumscribed circle of the triangle
A
B
C
ABC
A
BC
equals the product
∣
A
B
’
∣
⋅
∣
B
C
’
∣
⋅
∣
C
A
’
∣
|AB’|\cdot |BC’|\cdot |CA’|
∣
A
B
’∣
⋅
∣
BC
’∣
⋅
∣
C
A
’∣
divided by the area of the triangle
A
’
B
’
C
’
A’B’C’
A
’
B
’
C
’
.
Area of Pentagon
A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.
2
2
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Number of Solutions
How many ordered triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of real numbers satisfy the system of equations
x
2
+
y
2
+
z
2
=
9
,
x^2+y^2+z^2=9,
x
2
+
y
2
+
z
2
=
9
,
x
4
+
y
4
+
z
4
=
33
,
x^4+y^4+z^4=33,
x
4
+
y
4
+
z
4
=
33
,
x
y
z
=
−
4
?
xyz=-4?
x
yz
=
−
4
?
Digital Root
If
a
1
a_1
a
1
is a positive integer, form the sequence
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\dots
a
1
,
a
2
,
a
3
,
…
by letting
a
2
a_2
a
2
be the product of the digits of
a
1
a_1
a
1
, etc.. If
a
k
a_k
a
k
consists of a single digit, for some
k
≥
1
k\ge 1
k
≥
1
,
a
k
a_k
a
k
is called a digital root of
a
1
a_1
a
1
. It is easy to check that every positive integer has a unique root.
(
(
(
For example, if
a
1
=
24378
a_1=24378
a
1
=
24378
, then
a
2
=
1344
a_2=1344
a
2
=
1344
,
a
3
=
48
a_3=48
a
3
=
48
,
a
4
=
32
a_4=32
a
4
=
32
,
a
5
=
6
a_5=6
a
5
=
6
, and thus
6
6
6
is the digital root of
24378.
)
24378.)
24378.
)
Prove that the digital root of a positive integer
n
n
n
equals
1
1
1
if, and only if, all the digits of
n
n
n
equal
1
1
1
.
1
2
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Describing Set of Points
Describe in geometric terms the set of points
(
x
,
y
)
(x,y)
(
x
,
y
)
in the plane such that
x
x
x
and
y
y
y
satisfy the condition
t
2
+
y
t
+
x
≥
0
t^2+yt+x\ge 0
t
2
+
y
t
+
x
≥
0
for all
t
t
t
with
−
1
≤
t
≤
1
-1\le t\le 1
−
1
≤
t
≤
1
.
argh!!!!!!!!!!!
Let
n
>
2
n > 2
n
>
2
be an integer and let
m
=
∑
k
3
m = \sum k^3
m
=
∑
k
3
, where the sum is taken over all integers
k
k
k
with
1
≤
k
<
n
1 \leq k < n
1
≤
k
<
n
that are relatively prime to
n
n
n
. Prove that
n
n
n
divides
m
m
m
.
3
2
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no of ways of choosing a pair of subsets (B,C) of A ...
Let
A
A
A
be a nonempty set with
n
n
n
elements. Find the number of ways of choosing a pair of subsets
(
B
,
C
)
(B,C)
(
B
,
C
)
of
A
A
A
such that
B
B
B
is a nonempty subset of
C
C
C
.
complex roots
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
be real numbers with
a
≠
0
a \neq 0
a
=
0
. Prove that if all the roots of the cubic equation
a
z
3
+
b
z
2
+
c
z
+
d
=
0
az^{3} +bz^{2} +cz+d=0
a
z
3
+
b
z
2
+
cz
+
d
=
0
lie to the left of the imaginary axis in the complex plane, then
a
b
>
0
,
b
c
−
a
d
>
0
,
a
d
>
0
ab >0, bc-ad >0, ad>0
ab
>
0
,
b
c
−
a
d
>
0
,
a
d
>
0
.