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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2008 Irish Math Olympiad
2008 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
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Obtuse Triangle Ratio Condition
A triangle
A
B
C
ABC
A
BC
has an obtuse angle at
B
B
B
. The perpindicular at
B
B
B
to
A
B
AB
A
B
meets
A
C
AC
A
C
at
D
D
D
, and |CD| \equal{} |AB|. Prove that |AD|^2 \equal{} |AB|.|BC| if and only if \angle CBD \equal{} 30^\circ.
2 Inequalities When xyz>1
Suppose that
x
,
y
x, y
x
,
y
and
z
z
z
are positive real numbers such that
x
y
z
≥
1
xyz \ge 1
x
yz
≥
1
. (a) Prove that 27 \le (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2, with equality if and only if x \equal{} y \equal{} z \equal{} 1. (b) Prove that (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2 \le 3(x \plus{} y \plus{} z)^2, with equality if and only if x \equal{} y \equal{} z \equal{} 1.
4
2
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How Many Such Sequences Exist?
How many sequences
a
1
,
a
2
,
.
.
.
,
a
2
0
0
8
a_1,a_2,...,a{}_2{}_0{}_0{}_8
a
1
,
a
2
,
...
,
a
2
0
0
8
are there such that each of the numbers
1
,
2
,
.
.
.
,
2008
1,2,...,2008
1
,
2
,
...
,
2008
occurs once in the sequence, and
i
∈
(
a
1
,
a
2
,
.
.
.
,
a
i
)
i \in (a_1,a_2,...,a_i)
i
∈
(
a
1
,
a
2
,
...
,
a
i
)
for each
i
i
i
such that
2
≤
i
≤
2008
2\le i \le2008
2
≤
i
≤
2008
?
Sequences Obeying Certain Conditions Modulo 4
Given
k
∈
[
0
,
1
,
2
,
3
]
k \in [0,1,2,3]
k
∈
[
0
,
1
,
2
,
3
]
and a positive integer
n
n
n
, let
f
k
(
n
)
f_k(n)
f
k
(
n
)
be the number of sequences
x
1
,
.
.
.
,
x
n
,
x_1,...,x_n,
x
1
,
...
,
x
n
,
where x_i \in [\minus{}1,0,1] for i\equal{}1,...,n, and x_1\plus{}...\plus{}x_n \equiv k mod 4 a) Prove that f_1(n) \equal{} f_3(n) for all positive integers
n
n
n
. (b) Prove that f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4 for all positive integers
n
n
n
.
3
2
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When Is Given Product A Square?
Determine, with proof, all integers
x
x
x
for which x(x\plus{}1)(x\plus{}7)(x\plus{}8) is a perfect square.
Show the Given Sequence is Unique
Find
a
3
,
a
4
,
.
.
.
,
a
2
0
0
8
a_3,a_4,...,a{}_2{}_0{}_0{}_8
a
3
,
a
4
,
...
,
a
2
0
0
8
, such that
a
i
=
±
1
a_i =\pm1
a
i
=
±
1
for
i
=
3
,
.
.
.
,
2008
i=3,...,2008
i
=
3
,
...
,
2008
and
∑
i
=
3
2008
a
i
2
i
=
2008
\sum\limits_{i=3}^{2008} a_i2^i = 2008
i
=
3
∑
2008
a
i
2
i
=
2008
and show that the numbers
a
3
,
a
4
,
.
.
.
,
a
2008
a_3,a_4,...,a_{2008}
a
3
,
a
4
,
...
,
a
2008
are uniquely determined by these conditions.
2
2
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Positive Real Numbers Such That (a^2 + b^2 + c^2 + d^2) = 1
For positive real numbers
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
such that a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2 \equal{} 1 prove that a^2b^2cd \plus{} \plus{}ab^2c^2d \plus{} abc^2d^2 \plus{} a^2bcd^2 \plus{} a^2bc^2d \plus{} ab^2cd^2 \le 3/32, and determine the cases of equality.
Equal Lines in Two Circles
Circles
S
S
S
and
T
T
T
intersect at
P
P
P
and
Q
Q
Q
, with
S
S
S
passing through the centre of
T
T
T
. Distinct points
A
A
A
and
B
B
B
lie on
S
S
S
, inside
T
T
T
, and are equidistant from the centre of
T
T
T
. The line
P
A
PA
P
A
meets
T
T
T
again at
D
D
D
. Prove that |AD| \equal{} |PB|.
1
2
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Find All Primes Satisfying Given Equations
Let
p
1
,
p
2
,
p
3
p_1, p_2, p_3
p
1
,
p
2
,
p
3
and
p
4
p_4
p
4
be four different prime numbers satisying the equations 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162 Find all possible values of the product
p
1
p
2
p
3
p
4
p_1p_2p_3p_4
p
1
p
2
p
3
p
4
Find All Triples (a,b,c) of Integers
Find, with proof, all triples of integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that
a
,
b
a, b
a
,
b
and
c
c
c
are the lengths of the sides of a right angled triangle whose area is a \plus{} b \plus{} c