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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2019 Irish Math Olympiad
2019 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(10)
10
1
Hide problems
Island Hopping Holidays offer short holidays to 64 islands
Island Hopping Holidays offer short holidays to
64
64
64
islands, labeled Island
i
,
1
≤
i
≤
64
i, 1 \le i \le 64
i
,
1
≤
i
≤
64
. A guest chooses any Island
a
a
a
for the fi rst night of the holiday, moves to Island
b
b
b
for the second night, and finally moves to Island
c
c
c
for the third night. Due to the limited number of boats, we must have
b
∈
T
a
b \in T_a
b
∈
T
a
and
c
∈
T
b
c \in T_b
c
∈
T
b
, where the sets
T
i
T_i
T
i
are chosen so that (a) each
T
i
T_i
T
i
is non-empty, and
i
∉
T
i
i \notin T_i
i
∈
/
T
i
, (b)
∑
i
=
1
64
∣
T
i
∣
=
128
\sum^{64}_{i=1} |T_i| = 128
∑
i
=
1
64
∣
T
i
∣
=
128
, where
∣
T
i
∣
|T_i|
∣
T
i
∣
is the number of elements of
T
i
T_i
T
i
. Exhibit a choice of sets
T
i
T_i
T
i
giving at least
63
⋅
64
+
6
=
4038
63\cdot 64 + 6 = 4038
63
⋅
64
+
6
=
4038
possible holidays. Note that c = a is allowed, and holiday choices
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
and
(
a
′
,
b
′
,
c
′
)
(a',b',c')
(
a
′
,
b
′
,
c
′
)
are considered distinct if
a
≠
a
′
a \ne a'
a
=
a
′
or
b
≠
b
′
b \ne b'
b
=
b
′
or
c
≠
c
′
c \ne c'
c
=
c
′
.
6
1
Hide problems
2019 is smallest no, sum of fourth powers of 5 int, sum of 6 consecutive
The number
2019
2019
2019
has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact,
2019
=
1
4
+
2
4
+
3
4
+
5
4
+
6
4
2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4
2019
=
1
4
+
2
4
+
3
4
+
5
4
+
6
4
(1)
2019
=
334
+
335
+
336
+
337
+
338
+
339
2019 = 334 + 335 + 336 + 337 + 338 + 339
2019
=
334
+
335
+
336
+
337
+
338
+
339
(2) Prove that
2019
2019
2019
is the smallest number that satis es both (a) and (b). (You may assume that (1) and (2) are correct!)
7
1
Hide problems
ab + bc + ca =? if a + b + c = 0 and a^4 + b^4 + c^4 = 128
Three non-zero real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy
a
+
b
+
c
=
0
a + b + c = 0
a
+
b
+
c
=
0
and
a
4
+
b
4
+
c
4
=
128
a^4 + b^4 + c^4 = 128
a
4
+
b
4
+
c
4
=
128
. Determine all possible values of
a
b
+
b
c
+
c
a
ab + bc + ca
ab
+
b
c
+
c
a
.
9
1
Hide problems
solve 8xyz <= 1 and 8xyz = 1 if x^2 + y^2 + z^2 + 2xyz = 1$
Suppose
x
,
y
,
z
x, y, z
x
,
y
,
z
are real numbers such that
x
2
+
y
2
+
z
2
+
2
x
y
z
=
1
x^2 + y^2 + z^2 + 2xyz = 1
x
2
+
y
2
+
z
2
+
2
x
yz
=
1
. Prove that
8
x
y
z
≤
1
8xyz \le 1
8
x
yz
≤
1
, with equality if and only if
(
x
,
y
,
z
)
(x, y,z)
(
x
,
y
,
z
)
is one of the following:
(
1
2
,
1
2
,
1
2
)
,
(
−
1
2
,
−
1
2
,
1
2
)
,
(
−
1
2
,
1
2
,
−
1
2
)
,
(
1
2
,
−
1
2
,
−
1
2
)
\left( \frac12, \frac12, \frac12 \right) , \left( -\frac12, -\frac12, \frac12 \right), \left(- \frac12, \frac12, -\frac12 \right), \left( \frac12,- \frac12, - \frac12 \right)
(
2
1
,
2
1
,
2
1
)
,
(
−
2
1
,
−
2
1
,
2
1
)
,
(
−
2
1
,
2
1
,
−
2
1
)
,
(
2
1
,
−
2
1
,
−
2
1
)
8
1
Hide problems
intersection of circumcircles lies on diagonal of a parallelogram
Consider a point
G
G
G
in the interior of a parallelogram
A
B
C
D
ABCD
A
BC
D
. A circle
Γ
\Gamma
Γ
through
A
A
A
and
G
G
G
intersects the sides
A
B
AB
A
B
and
A
D
AD
A
D
for the second time at the points
E
E
E
and
F
F
F
respectively. The line
F
G
FG
FG
extended intersects the side
B
C
BC
BC
at
H
H
H
and the line
E
G
EG
EG
extended intersects the side
C
D
CD
C
D
at
I
I
I
. The circumcircle of triangle
H
G
I
HGI
H
G
I
intersects the circle
Γ
\Gamma
Γ
for the second time at
M
≠
G
M \ne G
M
=
G
. Prove that
M
M
M
lies on the diagonal
A
C
AC
A
C
.
5
1
Hide problems
concyclic wanted, circumcenters, reflection wrt line related
Let
M
M
M
be a point on the side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
and let
P
P
P
and
Q
Q
Q
denote the circumcentres of triangles
A
B
M
ABM
A
BM
and
A
C
M
ACM
A
CM
respectively. Let
L
L
L
denote the point of intersection of the extended lines
B
P
BP
BP
and
C
Q
CQ
CQ
and let
K
K
K
denote the reflection of
L
L
L
through the line
P
Q
PQ
PQ
. Prove that
M
,
P
,
Q
M, P, Q
M
,
P
,
Q
and
K
K
K
all lie on the same circle.
4
1
Hide problems
1 +1/x + 2(x + 1)/xy + 3(x + 1)(y + 2)/xyz+4(x + 1)(y + 2)(z + 3)/xyz=0
Find the set of all quadruplets
(
x
,
y
,
z
,
w
)
(x,y, z,w)
(
x
,
y
,
z
,
w
)
of non-zero real numbers which satisfy
1
+
1
x
+
2
(
x
+
1
)
x
y
+
3
(
x
+
1
)
(
y
+
2
)
x
y
z
+
4
(
x
+
1
)
(
y
+
2
)
(
z
+
3
)
x
y
z
w
=
0
1 +\frac{1}{x}+\frac{2(x + 1)}{xy}+\frac{3(x + 1)(y + 2)}{xyz}+\frac{4(x + 1)(y + 2)(z + 3)}{xyzw}= 0
1
+
x
1
+
x
y
2
(
x
+
1
)
+
x
yz
3
(
x
+
1
)
(
y
+
2
)
+
x
yz
w
4
(
x
+
1
)
(
y
+
2
)
(
z
+
3
)
=
0
3
1
Hide problems
angle bisectors intersect at right angles on AD, AB// DC, BC =AB+ CD
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is such that the sides
A
B
AB
A
B
and
D
C
DC
D
C
are parallel, and
∣
B
C
∣
=
∣
A
B
∣
+
∣
C
D
∣
|BC| =|AB| + |CD|
∣
BC
∣
=
∣
A
B
∣
+
∣
C
D
∣
. Prove that the angle bisectors of the angles
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
B
C
D
\angle BCD
∠
BC
D
intersect at right angles on the side
A
D
AD
A
D
.
2
1
Hide problems
Jenny is going to attend a sports camp for 7 days, 1 of 3 sports each day
Jenny is going to attend a sports camp for
7
7
7
days. Each day, she will play exactly one of three sports: hockey, tennis or camogie. The only restriction is that in any period of
4
4
4
consecutive days, she must play all three sports. Find, with proof, the number of possible sports schedules for Jennys week.
1
1
Hide problems
is 1000 a quasi-prime ?
De fine the quasi-primes as follows.
∙
\bullet
∙
The first quasi-prime is
q
1
=
2
q_1 = 2
q
1
=
2
∙
\bullet
∙
For
n
≥
2
n \ge 2
n
≥
2
, the
n
t
h
n^{th}
n
t
h
quasi-prime
q
n
q_n
q
n
is the smallest integer greater than
q
n
1
q_{n_1}
q
n
1
and not of the form
q
i
q
j
q_iq_j
q
i
q
j
for some
1
≤
i
≤
j
≤
n
−
1
1 \le i \le j \le n - 1
1
≤
i
≤
j
≤
n
−
1
. Determine, with proof, whether or not
1000
1000
1000
is a quasi-prime.