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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2015 Irish Math Olympiad
2015 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(10)
7
1
Hide problems
sum- free subsets of set of all positive integers not larger than 2n
Let
n
>
1
n > 1
n
>
1
be an integer and
Ω
=
{
1
,
2
,
.
.
.
,
2
n
−
1
,
2
n
}
\Omega=\{1,2,...,2n-1,2n\}
Ω
=
{
1
,
2
,
...
,
2
n
−
1
,
2
n
}
the set of all positive integers that are not larger than
2
n
2n
2
n
. A nonempty subset
S
S
S
of
Ω
\Omega
Ω
is called sum-free if, for all elements
x
,
y
x, y
x
,
y
belonging to
S
,
x
+
y
S, x + y
S
,
x
+
y
does not belong to
S
S
S
. We allow
x
=
y
x = y
x
=
y
in this condition. Prove that
Ω
\Omega
Ω
has more than
2
n
2^n
2
n
distinct sum-free subsets.
9
1
Hide problems
integer polynomials, p(x)q(x) - 2015 has at least 33 different integer roots
Let
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
be non-constant polynomial functions with integer coeffcients. It is known that the polynomial
p
(
x
)
q
(
x
)
−
2015
p(x)q(x) - 2015
p
(
x
)
q
(
x
)
−
2015
has at least
33
33
33
different integer roots. Prove that neither
p
(
x
)
p(x)
p
(
x
)
nor
q
(
x
)
q(x)
q
(
x
)
can be a polynomial of degree less than three.
10
1
Hide problems
binomial inequality, with pairs of nonnegative integers
Prove that, for all pairs of nonnegative integers,
j
,
n
j,n
j
,
n
,
∑
K
=
0
n
k
j
(
n
k
)
≥
2
n
−
j
n
j
\sum_{K=0}^{n}k^j\binom n k \ge 2^{n-j} n^j
K
=
0
∑
n
k
j
(
k
n
)
≥
2
n
−
j
n
j
8
1
Hide problems
if |DE| =|BC|, find angle <BAC, perpendicular - circumcircle
In triangle
△
A
B
C
\triangle ABC
△
A
BC
, the angle
∠
B
A
C
\angle BAC
∠
B
A
C
is less than
9
0
o
90^o
9
0
o
. The perpendiculars from
C
C
C
on
A
B
AB
A
B
and from
B
B
B
on
A
C
AC
A
C
intersect the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
again at
D
D
D
and
E
E
E
respectively. If
∣
D
E
∣
=
∣
B
C
∣
|DE| =|BC|
∣
D
E
∣
=
∣
BC
∣
, find the measure of the angle
∠
B
A
C
\angle BAC
∠
B
A
C
.
4
1
Hide problems
starting with a circle with diameter the centres of 2 externally tangent circles
Two circles
C
1
C_1
C
1
and
C
2
C_2
C
2
, with centres at
D
D
D
and
E
E
E
respectively, touch at
B
B
B
. The circle having
D
E
DE
D
E
as diameter intersects the circle
C
1
C_1
C
1
at
H
H
H
and the circle
C
2
C_2
C
2
at
K
K
K
. The points
H
H
H
and
K
K
K
both lie on the same side of the line
D
E
DE
D
E
.
H
K
HK
HK
extended in both directions meets the circle
C
1
C_1
C
1
at
L
L
L
and meets the circle
C
2
C_2
C
2
at
M
M
M
. Prove that (a)
∣
L
H
∣
=
∣
K
M
∣
|LH| = |KM|
∣
L
H
∣
=
∣
K
M
∣
(b) the line through
B
B
B
perpendicular to
D
E
DE
D
E
bisects
H
K
HK
HK
.
2
1
Hide problems
coloring vertices of regular polygon (red, green or blue)
A regular polygon with
n
≥
3
n \ge 3
n
≥
3
sides is given. Each vertex is coloured either red, green or blue, and no two adjacent vertices of the polygon are the same colour. There is at least one vertex of each colour. Prove that it is possible to draw certain diagonals of the polygon in such a way that they intersect only at the vertices of the polygon and they divide the polygon into triangles so that each such triangle has vertices of three different colours.
1
1
Hide problems
altitude from A to BC equals 1, D midpoint AC, BD=?
In the triangle
A
B
C
ABC
A
BC
, the length of the altitude from
A
A
A
to
B
C
BC
BC
is equal to
1
1
1
.
D
D
D
is the midpoint of
A
C
AC
A
C
. What are the possible lengths of
B
D
BD
B
D
?
6
1
Hide problems
x,y>0 and x + y <= 1 prove 8xy <= 5x(1 - x) + 5y(1 - y)
Suppose
x
,
y
x,y
x
,
y
are nonnegative real numbers such that
x
+
y
≤
1
x + y \le 1
x
+
y
≤
1
. Prove that
8
x
y
≤
5
x
(
1
−
x
)
+
5
y
(
1
−
y
)
8xy \le 5x(1 - x) + 5y(1 - y)
8
x
y
≤
5
x
(
1
−
x
)
+
5
y
(
1
−
y
)
and determine the cases of equality.
3
1
Hide problems
find n so that (837 + n) and (837 − n) are cubes of positive integers
Find all positive integers
n
n
n
for which both
837
+
n
837 + n
837
+
n
and
837
−
n
837 - n
837
−
n
are cubes of positive integers.
5
1
Hide problems
Bounded sequence is constant
Suppose a doubly infinite sequence of real numbers
.
.
.
,
a
−
2
,
a
−
1
,
a
0
,
a
1
,
a
2
,
.
.
.
. . . , a_{-2}, a_{-1}, a_0, a_1, a_2, . . .
...
,
a
−
2
,
a
−
1
,
a
0
,
a
1
,
a
2
,
...
has the property that
a
n
+
3
=
a
n
+
a
n
+
1
+
a
n
+
2
3
,
a_{n+3} =\frac{a_n + a_{n+1} + a_{n+2}}{3},
a
n
+
3
=
3
a
n
+
a
n
+
1
+
a
n
+
2
,
for all integers
n
.
n .
n
.
Show that if this sequence is bounded (i.e., if there exists a number
R
R
R
such that
∣
a
n
∣
≤
R
|a_n| \leq R
∣
a
n
∣
≤
R
for all
n
n
n
), then
a
n
a_n
a
n
has the same value for all
n
.
n.
n
.