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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1990 Irish Math Olympiad
1990 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(6)
6
1
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Equation with 1s and -1s
Let
n
n
n
be a natural number, and suppose that the equation
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+
x
4
x
5
+
⋯
+
x
n
−
1
x
n
+
x
n
x
1
=
0
x_1x_2+x_2x_3+x_3x_4+x_4x_5+\dots +x_{n-1}x_n+x_nx_1=0
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+
x
4
x
5
+
⋯
+
x
n
−
1
x
n
+
x
n
x
1
=
0
has a solution with all the
x
i
x_i
x
i
s equal to
±
1
\pm 1
±
1
. Prove that
n
n
n
is divisible by
4
4
4
.
5
1
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Right Triangle
Let
A
B
C
ABC
A
BC
be a right-angled triangle with right-angle at
A
A
A
. Let
X
X
X
be the foot of the perpendicular from
A
A
A
to
B
C
BC
BC
, and
Y
Y
Y
the mid-point of
X
C
XC
XC
. Let
A
B
AB
A
B
be extended to
D
D
D
so that
∣
A
B
∣
=
∣
B
D
∣
|AB|=|BD|
∣
A
B
∣
=
∣
B
D
∣
. Prove that
D
X
DX
D
X
is perpendicular to
A
Y
AY
A
Y
.
4
2
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2^k Inequality
The real number
x
x
x
satisfies all the inequalities
2
k
<
x
k
+
x
k
+
1
<
2
k
+
1
2^k<x^k+x^{k+1}<2^{k+1}
2
k
<
x
k
+
x
k
+
1
<
2
k
+
1
for
k
=
1
,
2
,
…
,
n
k=1,2,\dots ,n
k
=
1
,
2
,
…
,
n
. What is the greatest possible value of
n
n
n
?
Comparing n-tuples
Let
n
=
2
k
−
1
n=2k-1
n
=
2
k
−
1
, where
k
≥
6
k\ge 6
k
≥
6
is an integer. Let
T
T
T
be the set of all
n
n
n
-tuples
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
a
n
>
=
(
x
1
,
x
2
,
…
,
x
n
)
,
where, for
i
=
1
,
2
,
…
,
n
,
x
i
is
0
or
1.
<span class='latex-bold'>x</span>=(x_1,x_2,\dots ,x_n), \text{ where, for } i=1,2,\dots ,n, \text{ } x_i \text{ is } 0 \text{ or } 1.
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
an
>=
(
x
1
,
x
2
,
…
,
x
n
)
,
where, for
i
=
1
,
2
,
…
,
n
,
x
i
is
0
or
1.
For
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
a
n
>
=
(
x
1
,
x
2
,
…
,
x
n
)
<span class='latex-bold'>x</span>=(x_1,x_2,\dots ,x_n)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
an
>=
(
x
1
,
x
2
,
…
,
x
n
)
and
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
y
<
/
s
p
a
n
>
=
(
y
1
,
y
2
,
…
,
y
n
)
<span class='latex-bold'>y</span>=(y_1,y_2,\dots ,y_n)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
y
<
/
s
p
an
>=
(
y
1
,
y
2
,
…
,
y
n
)
in
T
T
T
, let
d
(
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
a
n
>
,
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
y
<
/
s
p
a
n
>
)
d(<span class='latex-bold'>x</span>,<span class='latex-bold'>y</span>)
d
(
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
an
>
,
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
y
<
/
s
p
an
>
)
denote the number of integers
j
j
j
with
1
≤
j
≤
n
1\le j\le n
1
≤
j
≤
n
such that
x
j
≠
x
y
x_j\neq x_y
x
j
=
x
y
.
(
(
(
In particular,
d
(
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
a
n
>
,
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
a
n
>
)
=
0
)
d(<span class='latex-bold'>x</span>,<span class='latex-bold'>x</span>)=0)
d
(
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
an
>
,
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
an
>
)
=
0
)
.Suppose that there exists a subset
S
S
S
of
T
T
T
with
2
k
2^k
2
k
elements which has the following property: given any element
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
a
n
>
<span class='latex-bold'>x</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
an
>
in
T
T
T
, there is a unique
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
y
<
/
s
p
a
n
>
<span class='latex-bold'>y</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
y
<
/
s
p
an
>
in
S
S
S
with
d
(
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
a
n
>
,
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
y
<
/
s
p
a
n
>
)
≤
3
d(<span class='latex-bold'>x</span>,<span class='latex-bold'>y</span>)\le 3
d
(
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
x
<
/
s
p
an
>
,
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
y
<
/
s
p
an
>
)
≤
3
.Prove that
n
=
23
n=23
n
=
23
.
1
2
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Number of Rectangles in the Plane
Given a natural number
n
n
n
, calculate the number of rectangles in the plane, the coordinates of whose vertices are integers in the range
0
0
0
to
n
n
n
, and whose sides are parallel to the axes.
Irish 1990 paper 2 #1
Let
n
>
3
n>3
n
>
3
be a natural number . Prove that
1
3
3
+
1
4
3
+
⋯
+
1
n
3
<
1
12
.
\frac{1}{3^3}+\frac{1}{4^3}+\cdots+\frac{1}{n^3}<\frac{1}{12}.
3
3
1
+
4
3
1
+
⋯
+
n
3
1
<
12
1
.
2
2
Hide problems
Greatest Prime divisor problem
A sequence of primes
a
n
a_n
a
n
is defined as follows:
a
1
=
2
a_1 = 2
a
1
=
2
, and, for all n \geq 2,
a
n
a_n
a
n
is the largest prime divisor of
a
1
a
2
.
.
.
a
n
−
1
+
1
a_1a_2...a_{n-1} + 1
a
1
a
2
...
a
n
−
1
+
1
. Prove that
a
n
≠
5
a_n \neq 5
a
n
=
5
for all n. I'm presuming it must involve proving it's never equal to 0 mod 5, but I don't know what to do. Thanks
15 Prime Numbers in Arithmetic Progression
Suppose that
p
1
<
p
2
<
⋯
<
p
15
p_1<p_2<\dots <p_{15}
p
1
<
p
2
<
⋯
<
p
15
are prime numbers in arithmetic progression, with common difference
d
d
d
. Prove that
d
d
d
is divisible by
2
,
3
,
5
,
7
,
11
2,3,5,7,11
2
,
3
,
5
,
7
,
11
and
13
13
13
.
3
2
Hide problems
Determine if f(x) exists
Determine whether there exists a function
f
:
N
⟶
N
f: \mathbb{N}\longrightarrow \mathbb{N}
f
:
N
⟶
N
such that f(n)\equal{}f(f(n\minus{}1))\plus{}f(f(n\plus{}1)) for all natural numbers
n
≥
2
n\ge 2
n
≥
2
.
Trigonometric Sequence
Let
t
t
t
be a real number, and let a_n=2\cos \left(\frac{t}{2^n}\right)-1, n=1,2,3,\dots Let
b
n
b_n
b
n
be the product
a
1
a
2
a
3
⋯
a
n
a_1a_2a_3\cdots a_n
a
1
a
2
a
3
⋯
a
n
. Find a formula for
b
n
b_n
b
n
that does not involve a product of
n
n
n
terms, and deduce that
lim
n
→
∞
b
n
=
2
cos
t
+
1
3
\lim_{n\to \infty}b_n=\frac{2\cos t+1}{3}
n
→
∞
lim
b
n
=
3
2
cos
t
+
1