MathDB
Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1998 Irish Math Olympiad
1998 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
Hide problems
prove that x is an integer
If
x
x
x
is a real number such that x^2\minus{}x and x^n\minus{}x are integers for some
n
≥
3
n \ge 3
n
≥
3
, prove that
x
x
x
is an integer.
minimum possible perimeter
A triangle
A
B
C
ABC
A
BC
has integer sides, \angle A\equal{}2 \angle B and
∠
C
>
9
0
∘
\angle C>90^{\circ}
∠
C
>
9
0
∘
. Find the minimum possible perimeter of this triangle.
4
2
Hide problems
disks
Show that a disk of radius
2
2
2
can be covered by seven (possibly overlapping) disks of radius
1
1
1
.
find a term of a sequence
A sequence
(
x
n
)
(x_n)
(
x
n
)
is given as follows:
x
0
,
x
1
x_0,x_1
x
0
,
x
1
are arbitrary positive real numbers, and x_{n\plus{}2}\equal{}\frac{1\plus{}x_{n\plus{}1}}{x_n} for
n
≥
0
n \ge 0
n
≥
0
. Find
x
1998
x_{1998}
x
1998
.
3
2
Hide problems
base b
Show that no integer of the form
x
y
x
y
xyxy
x
y
x
y
in base
10
10
10
can be a perfect cube. Find the smallest base
b
>
1
b>1
b
>
1
for which there is a perfect cube of the form
x
y
x
y
xyxy
x
y
x
y
in base
b
b
b
.
sets
(
a
)
(a)
(
a
)
Prove that
N
\mathbb{N}
N
can be partitioned into three (mutually disjoint) sets such that, if
m
,
n
∈
N
m,n \in \mathbb{N}
m
,
n
∈
N
and |m\minus{}n| is
2
2
2
or
5
5
5
, then
m
m
m
and
n
n
n
are in different sets.
(
b
)
(b)
(
b
)
Prove that
N
\mathbb{N}
N
can be partitioned into four sets such that, if
m
,
n
∈
N
m,n \in \mathbb{N}
m
,
n
∈
N
and |m\minus{}n| is
2
,
3
,
2,3,
2
,
3
,
or
5
5
5
, then
m
m
m
and
n
n
n
are in different sets. Show, however, that
N
\mathbb{N}
N
cannot be partitioned into three sets with this property.
2
2
Hide problems
nice exercise (maybe posted before)
The distances from a point
P
P
P
inside an equilateral triangle to the vertices of the triangle are
3
,
4
3,4
3
,
4
, and
5
5
5
. Find the area of the triangle.
nice inequality
Prove that if
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers, then: \frac{9}{a\plus{}b\plus{}c} \le 2 \left( \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \right) \le \frac{1}{a}\plus{}\frac{1}{b}\plus{}\frac{1}{c}.
1
2
Hide problems
easy inequality
Prove that if x \not\equal{} 0 is a real number, then: x^8\minus{}x^5\minus{}\frac{1}{x}\plus{}\frac{1}{x^4} \ge 0.
find n
Find all positive integers
n
n
n
having exactly
16
16
16
divisors 1\equal{}d_1