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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2018 Irish Math Olympiad
2018 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(10)
10
1
Hide problems
2 players game of Greed, moving stones between piles, winning strategy
The game of Greed starts with an initial configuration of one or more piles of stones. Player
1
1
1
and Player
2
2
2
take turns to remove stones, beginning with Player
1
1
1
. At each turn, a player has two choices: • take one stone from any one of the piles (a simple move); • take one stone from each of the remaining piles (a greedy move). The player who takes the last stone wins. Consider the following two initial configurations: (a) There are
2018
2018
2018
piles, with either
20
20
20
or
18
18
18
stones in each pile. (b) There are four piles, with
17
,
18
,
19
17, 18, 19
17
,
18
,
19
, and
20
20
20
stones, respectively. In each case, find an appropriate strategy that guarantees victory to one of the players.
9
1
Hide problems
a_{n+1} = a^2_{n} + 2018 for n>=1, exists at most one perfect cube
The sequence of positive integers
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
satisfies
a
n
+
1
=
a
n
2
+
2018
a_{n+1} = a^2_{n} + 2018
a
n
+
1
=
a
n
2
+
2018
for
n
≥
1
n \ge 1
n
≥
1
. Prove that there exists at most one
n
n
n
for which
a
n
a_n
a
n
is the cube of an integer.
6
1
Hide problems
find real f so that f(2x + f(y)) + f(f(y)) = 4x + 8y
Find all real-valued functions
f
f
f
satisfying
f
(
2
x
+
f
(
y
)
)
+
f
(
f
(
y
)
)
=
4
x
+
8
y
f(2x + f(y)) + f(f(y)) = 4x + 8y
f
(
2
x
+
f
(
y
))
+
f
(
f
(
y
))
=
4
x
+
8
y
for all real numbers
x
x
x
and
y
y
y
.
7
1
Hide problems
(a^3+b^3+c^3)<(a+b+c)(a^2+b^2+c^2)<=3(a^3+b^3+c^3) where a,b,c sidelengths
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the side lengths of a triangle. Prove that
2
(
a
3
+
b
3
+
c
3
)
<
(
a
+
b
+
c
)
(
a
2
+
b
2
+
c
2
)
≤
3
(
a
3
+
b
3
+
c
3
)
2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)
2
(
a
3
+
b
3
+
c
3
)
<
(
a
+
b
+
c
)
(
a
2
+
b
2
+
c
2
)
≤
3
(
a
3
+
b
3
+
c
3
)
8
1
Hide problems
equal angles, starting with an equailateral triangle
Let
M
M
M
be the midpoint of side
B
C
BC
BC
of an equilateral triangle
A
B
C
ABC
A
BC
. The point
D
D
D
is on
C
A
CA
C
A
extended such that
A
A
A
is between
D
D
D
and
C
C
C
. The point
E
E
E
is on
A
B
AB
A
B
extended such that
B
B
B
is between
A
A
A
and
E
E
E
, and
∣
M
D
∣
=
∣
M
E
∣
|MD| = |ME|
∣
M
D
∣
=
∣
ME
∣
. The point
F
F
F
is the intersection of
M
D
MD
M
D
and
A
B
AB
A
B
. Prove that
∠
B
F
M
=
∠
B
M
E
\angle BFM = \angle BME
∠
BFM
=
∠
BME
.
5
1
Hide problems
chord of a circle tangent to another circle
Points
A
,
B
A, B
A
,
B
and
P
P
P
lie on the circumference of a circle
Ω
1
\Omega_1
Ω
1
such that
∠
A
P
B
\angle APB
∠
A
PB
is an obtuse angle. Let
Q
Q
Q
be the foot of the perpendicular from
P
P
P
on
A
B
AB
A
B
. A second circle
Ω
2
\Omega_2
Ω
2
is drawn with centre
P
P
P
and radius
P
Q
PQ
PQ
. The tangents from
A
A
A
and
B
B
B
to
Ω
2
\Omega_2
Ω
2
intersect
Ω
1
\Omega_1
Ω
1
at
F
F
F
and
H
H
H
respectively. Prove that
F
H
FH
F
H
is tangent to
Ω
2
\Omega_2
Ω
2
.
2
1
Hide problems
angle chasing in a right triangle
The triangle
A
B
C
ABC
A
BC
is right-angled at
A
A
A
. Its incentre is
I
I
I
, and
H
H
H
is the foot of the perpendicular from
I
I
I
on
A
B
AB
A
B
. The perpendicular from
H
H
H
on
B
C
BC
BC
meets
B
C
BC
BC
at
E
E
E
, and it meets the bisector of
∠
A
B
C
\angle ABC
∠
A
BC
at
D
D
D
. The perpendicular from
A
A
A
on
B
C
BC
BC
meets
B
C
BC
BC
at
F
F
F
. Prove that
∠
E
F
D
=
4
5
o
\angle EFD = 45^o
∠
EF
D
=
4
5
o
4
1
Hide problems
a rectangle fits inside another rectangle, 2019 integer sidelength rectangles
We say that a rectangle with side lengths
a
a
a
and
b
b
b
fits inside a rectangle with side lengths
c
c
c
and
d
d
d
if either (
a
≤
c
a \le c
a
≤
c
and
b
≤
d
b \le d
b
≤
d
) or (
a
≤
d
a \le d
a
≤
d
and
b
≤
c
b \le c
b
≤
c
). For instance, a rectangle with side lengths
1
1
1
and
5
5
5
fits inside another rectangle with side lengths
1
1
1
and
5
5
5
, and also fits inside a rectangle with side lengths
6
6
6
and
2
2
2
. Suppose
S
S
S
is a set of
2019
2019
2019
rectangles, all with integer side lengths between
1
1
1
and
2018
2018
2018
inclusive. Show that there are three rectangles
A
A
A
,
B
B
B
, and
C
C
C
in
S
S
S
such that
A
A
A
fits inside
B
B
B
, and
B
B
B
fits inside
C
C
C
.
3
1
Hide problems
find f(x)=ax^2+bx+c, if a \ne 0 and f(f(1)) = f(f(0)) = f(f(-1))
Find all functions
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x) = ax^2 + bx + c
f
(
x
)
=
a
x
2
+
b
x
+
c
, with
a
≠
0
a \ne 0
a
=
0
, such that
f
(
f
(
1
)
)
=
f
(
f
(
0
)
)
=
f
(
f
(
−
1
)
)
f(f(1)) = f(f(0)) = f(f(-1))
f
(
f
(
1
))
=
f
(
f
(
0
))
=
f
(
f
(
−
1
))
.
1
1
Hide problems
Mary and Pat play a number game, smallest initial integer for Pat not winning
Mary and Pat play the following number game. Mary picks an initial integer greater than
2017
2017
2017
. She then multiplies this number by
2017
2017
2017
and adds
2
2
2
to the result. Pat will add
2019
2019
2019
to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by
2017
2017
2017
and add
2
2
2
to the result when it is her turn. Pat will always add
2019
2019
2019
to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by
2018
2018
2018
. Mary wants to prevent Pat from winning the game. Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.