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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1993 Irish Math Olympiad
1993 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
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complex numbers
For a complex number z\equal{}x\plus{}iy we denote by
P
(
z
)
P(z)
P
(
z
)
the corresponding point
(
x
,
y
)
(x,y)
(
x
,
y
)
in the plane. Suppose
z
1
,
z
2
,
z
3
,
z
4
,
z
5
,
α
z_1,z_2,z_3,z_4,z_5,\alpha
z
1
,
z
2
,
z
3
,
z
4
,
z
5
,
α
are nonzero complex numbers such that:
(
i
)
(i)
(
i
)
P
(
z
1
)
,
.
.
.
,
P
(
z
5
)
P(z_1),...,P(z_5)
P
(
z
1
)
,
...
,
P
(
z
5
)
are vertices of a complex pentagon
Q
Q
Q
containing the origin
O
O
O
in its interior, and
(
i
i
)
(ii)
(
ii
)
P
(
α
z
1
)
,
.
.
.
,
P
(
α
z
5
)
P(\alpha z_1),...,P(\alpha z_5)
P
(
α
z
1
)
,
...
,
P
(
α
z
5
)
are all inside
Q
Q
Q
. If \alpha\equal{}p\plus{}iq
(
p
,
q
∈
R
)
(p,q \in \mathbb{R})
(
p
,
q
∈
R
)
, prove that p^2\plus{}q^2 \le 1 and p\plus{}q \tan \frac{\pi}{5} \le 1.
unit squares
(
a
)
(a)
(
a
)
The rectangle
P
Q
R
S
PQRS
PQRS
with PQ\equal{}l and QR\equal{}m
(
l
,
m
∈
N
)
(l,m \in \mathbb{N})
(
l
,
m
∈
N
)
is divided into
l
m
lm
l
m
unit squares. Prove that the diagonal
P
R
PR
PR
intersects exactly l\plus{}m\minus{}d of these squares, where d\equal{}(l,m).
(
b
)
(b)
(
b
)
A box with edge lengths
l
,
m
,
n
∈
N
l,m,n \in \mathbb{N}
l
,
m
,
n
∈
N
is divided into
l
m
n
lmn
l
mn
unit cubes. How many of the cubes does a main diagonal of the box intersect?
4
2
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polynomial
Let f(x)\equal{}x^n\plus{}a_{n\minus{}1} x^{n\minus{}1}\plus{}...\plus{}a_0
(
n
≥
1
)
(n \ge 1)
(
n
≥
1
)
be a polynomial with real coefficients such that |f(0)|\equal{}f(1) and each root
α
\alpha
α
of
f
f
f
is real and lies in the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
. Prove that the product of the roots does not exceed
1
2
n
\frac{1}{2^n}
2
n
1
.
Irish 1993 paper 2 #4
Let
x
x
x
be a real number with
0
<
x
<
π
0<x<\pi
0
<
x
<
π
.Prove that, for all natural number
n
n
n
,
s
i
n
x
+
s
i
n
3
x
3
+
s
i
n
5
x
5
+
⋯
+
s
i
n
(
2
n
−
1
)
x
2
n
−
1
>
0.
sinx+\frac{sin3x}{3}+\frac{sin5x}{5}+\cdots+\frac{sin(2n-1)x}{2n-1}>0.
s
in
x
+
3
s
in
3
x
+
5
s
in
5
x
+
⋯
+
2
n
−
1
s
in
(
2
n
−
1
)
x
>
0.
3
2
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locus
A line
l
l
l
is tangent to a circle
S
S
S
at
A
A
A
. For any points
B
,
C
B,C
B
,
C
on
l
l
l
on opposite sides of
A
A
A
, let the other tangents from
B
B
B
and
C
C
C
to
S
S
S
intersect at a point
P
P
P
. If
B
,
C
B,C
B
,
C
vary on
l
l
l
so that the product
A
B
⋅
A
C
AB \cdot AC
A
B
⋅
A
C
is constant, find the locus of
P
P
P
.
identity
If
1
≤
r
≤
n
1 \le r \le n
1
≤
r
≤
n
are integers, prove the identity: \displaystyle\sum_{d\equal{}1}^{\infty}\binom {n\minus{}r\plus{}1}{d} \binom {r\minus{}1} {d\minus{}1}\equal{}\binom {n}{r}.
2
2
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good numbers
A positive integer
n
n
n
is called
g
o
o
d
good
g
oo
d
if it can be uniquely written simultaneously as a_1\plus{}a_2\plus{}...\plus{}a_k and as
a
1
a
2
.
.
.
a
k
a_1 a_2...a_k
a
1
a
2
...
a
k
, where
a
i
a_i
a
i
are positive integers and
k
≥
2
k \ge 2
k
≥
2
. (For example,
10
10
10
is good because 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1 is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.
real numbers
Let
a
i
,
b
i
a_i,b_i
a
i
,
b
i
(i\equal{}1,2,...,n) be real numbers such that the
a
i
a_i
a
i
are distinct, and suppose that there is a real number
α
\alpha
α
such that the product (a_i\plus{}b_1)(a_i\plus{}b_2)...(a_i\plus{}b_n) is equal to
α
\alpha
α
for each
i
i
i
. Prove that there is a real number
β
\beta
β
such that (a_1\plus{}b_j)(a_2\plus{}b_j)...(a_n\plus{}b_j) is equal to
β
\beta
β
for each
j
j
j
.
1
2
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System of eq. of 3 degree polynomials
The following is known about the reals
α
\alpha
α
and
β
\beta
β
α
3
−
3
α
2
+
5
α
−
17
=
0
\alpha^{3}-3\alpha^{2}+5\alpha-17=0
α
3
−
3
α
2
+
5
α
−
17
=
0
and
β
3
−
3
β
2
+
5
β
+
11
=
0
\beta^{3}-3\beta^{2}+5\beta+11=0
β
3
−
3
β
2
+
5
β
+
11
=
0
Determine
α
+
β
\alpha+\beta
α
+
β
integer coordinates
Show that among any five points
P
1
,
.
.
.
,
P
5
P_1,...,P_5
P
1
,
...
,
P
5
with integer coordinates in the plane, there exists at least one pair
(
P
i
,
P
j
)
(P_i,P_j)
(
P
i
,
P
j
)
, with i \not\equal{} j such that the segment
P
i
P
j
P_i P_j
P
i
P
j
contains a point
Q
Q
Q
with integer coordinates other than
P
i
,
P
j
P_i, P_j
P
i
,
P
j
.