MathDB
Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2004 Irish Math Olympiad
2004 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
Hide problems
Inequality
Let
a
,
b
≥
0
a,b\ge 0
a
,
b
≥
0
. Prove that
2
(
a
(
a
+
b
)
3
+
b
a
2
+
b
2
)
≤
3
(
a
2
+
b
2
)
\sqrt{2}\left(\sqrt{a(a+b)^3}+b\sqrt{a^2+b^2}\right)\le 3(a^2+b^2)
2
(
a
(
a
+
b
)
3
+
b
a
2
+
b
2
)
≤
3
(
a
2
+
b
2
)
with equality if and only if
a
=
b
a=b
a
=
b
.
Number of Solutions mod p
Suppose
p
,
q
p,q
p
,
q
are distinct primes and
S
S
S
is a subset of
{
1
,
2
,
…
,
p
−
1
}
\{1,2,\dots ,p-1\}
{
1
,
2
,
…
,
p
−
1
}
. Let
N
(
S
)
N(S)
N
(
S
)
denote the number of solutions to the equation
∑
i
=
1
q
x
i
≡
0
m
o
d
p
\sum_{i=1}^{q}x_i\equiv 0\mod p
i
=
1
∑
q
x
i
≡
0
mod
p
where
x
i
∈
S
x_i\in S
x
i
∈
S
,
i
=
1
,
2
,
…
,
q
i=1,2,\dots ,q
i
=
1
,
2
,
…
,
q
. Prove that
N
(
S
)
N(S)
N
(
S
)
is a multiple of
q
q
q
.
2
2
Hide problems
Tennis Tournament
Each of the players in a tennis tournament played one match against each of the others. If every player won at least one match, show that there is a group A; B; C of three players for which A beat B, B beat C and C beat A.
Angle in Circle
A
A
A
and
B
B
B
are distinct points on a circle
T
T
T
.
C
C
C
is a point distinct from
B
B
B
such that
∣
A
B
∣
=
∣
A
C
∣
|AB|=|AC|
∣
A
B
∣
=
∣
A
C
∣
, and such that
B
C
BC
BC
is tangent to
T
T
T
at
B
B
B
. Suppose that the bisector of
∠
A
B
C
\angle ABC
∠
A
BC
meets
A
C
AC
A
C
at a point
D
D
D
inside
T
T
T
. Show that
∠
A
B
C
>
7
2
∘
\angle ABC>72^\circ
∠
A
BC
>
7
2
∘
.
3
2
Hide problems
Square Inscribed in Sector
A
B
AB
A
B
is a chord of length
6
6
6
of a circle centred at
O
O
O
and of radius
5
5
5
. Let
P
Q
R
S
PQRS
PQRS
denote the square inscribed in the sector
O
A
B
OAB
O
A
B
such that
P
P
P
is on the radius
O
A
OA
O
A
,
S
S
S
is on the radius
O
B
OB
OB
and
Q
Q
Q
and
R
R
R
are points on the arc of the circle between
A
A
A
and
B
B
B
. Find the area of
P
Q
R
S
PQRS
PQRS
.
Find the digit
Suppose
n
n
n
is an integer
≥
2
\geq 2
≥
2
. Determine the first digit after the decimal point in the decimal expansion of the number
n
3
+
2
n
2
+
n
3
\sqrt[3]{n^{3}+2n^{2}+n}
3
n
3
+
2
n
2
+
n
4
2
Hide problems
Only two real roots
Prove that there are only two real numbers
x
x
x
such that
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
(
x
−
4
)
(
x
−
5
)
(
x
−
6
)
=
720
(x-1)(x-2)(x-3)(x-4)(x-5)(x-6) = 720
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
(
x
−
4
)
(
x
−
5
)
(
x
−
6
)
=
720
Functions question
Define the function
m
m
m
of the three real variables
x
x
x
,
y
y
y
,
z
z
z
by
m
m
m
(
x
x
x
,
y
y
y
,
z
z
z
) = max(
x
2
x^2
x
2
,
y
2
y^2
y
2
,
z
2
z^2
z
2
),
x
x
x
,
y
y
y
,
z
z
z
∈
R
R
R
.Determine, with proof, the minimum value of
m
m
m
if
x
x
x
,
y
y
y
,
z
z
z
vary in
R
R
R
subject to the following restrictions:
x
x
x
+
y
y
y
+
z
z
z
= 0,
x
2
x^2
x
2
+
y
2
y^2
y
2
+
z
2
z^2
z
2
= 1.
1
2
Hide problems
Simple Problem (with sum of first n integers)
1. (a) For which positive integers n, does 2n divide the sum of the first n positive integers? (b) Determine, with proof, those positive integers n (if any) which have the property that 2n + 1 divides the sum of the first n positive integers.
Primes and again primes
Determine all pairs of prime numbers
(
p
,
q
)
(p, q)
(
p
,
q
)
, with
2
≤
p
,
q
<
100
2 \leq p, q < 100
2
≤
p
,
q
<
100
, such that
p
+
6
,
p
+
10
,
q
+
4
,
q
+
10
p+6, p+10, q+4, q+10
p
+
6
,
p
+
10
,
q
+
4
,
q
+
10
and
p
+
q
+
1
p+q+1
p
+
q
+
1
are all prime numbers.