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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2017 Irish Math Olympiad
2017 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
Hide problems
a_{n+2} = 2a_{n+1} + 41a_n
The sequence
a
=
(
a
0
,
a
1
,
a
2
,
.
.
.
)
a = (a_0, a_1,a_2,...)
a
=
(
a
0
,
a
1
,
a
2
,
...
)
is defined by
a
0
=
0
,
a
1
=
2
a_0 = 0, a_1 =2
a
0
=
0
,
a
1
=
2
and
a
n
+
2
=
2
a
n
+
1
+
41
a
n
a_{n+2} = 2a_{n+1} + 41a_n
a
n
+
2
=
2
a
n
+
1
+
41
a
n
Prove that
a
2016
a_{2016}
a
2016
is divisible by
2017.
2017.
2017.
(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}
Given a positive integer
m
m
m
, a sequence of real numbers
a
=
(
a
1
,
a
2
,
a
3
,
.
.
.
)
a= (a_1,a_2,a_3,...)
a
=
(
a
1
,
a
2
,
a
3
,
...
)
is called
m
m
m
-powerful if it satisfies
(
∑
k
=
1
n
a
k
)
m
=
∑
k
=
1
n
a
k
m
(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}
(
k
=
1
∑
n
a
k
)
m
=
k
=
1
∑
n
a
k
m
for all positive integers
n
n
n
. (a) Show that a sequence is
30
30
30
-powerful if and only if at most one of its terms is non-zero. (b) Find a sequence none of whose terms are zero but which is
2017
2017
2017
-powerful.
2
2
Hide problems
a + b + c = 0 , a^2 + b^2 + c^2 = 1 , a^3 + b^3 +c^3 = 4abc
Solve the equations :
{
a
+
b
+
c
=
0
a
2
+
b
2
+
c
2
=
1
a
3
+
b
3
+
c
3
=
4
a
b
c
\begin{cases} a + b + c = 0 \\ a^2 + b^2 + c^2 = 1\\a^3 + b^3 +c^3 = 4abc \end{cases}
⎩
⎨
⎧
a
+
b
+
c
=
0
a
2
+
b
2
+
c
2
=
1
a
3
+
b
3
+
c
3
=
4
ab
c
for
a
,
b
,
a,b,
a
,
b
,
and
c
.
c.
c
.
5 teams play in a soccer competition
5
5
5
teams play in a soccer competition where each team plays one match against each of the other four teams. A winning team gains
5
5
5
points and a losing team
0
0
0
points. For a
0
−
0
0-0
0
−
0
draw both teams gain
1
1
1
point, and for other draws (
1
−
1
,
2
−
2
,
3
−
3
,
1-1,2-2,3-3,
1
−
1
,
2
−
2
,
3
−
3
,
etc.) both teams gain 2 points. At the end of the competition, we write down the total points for each team, and we find that they form 5 consecutive integers. What is the minimum number of goals scored?
1
2
Hide problems
min multiple of 99 with digits 1 or 2
Determine, with proof, the smallest positive multiple of
99
99
99
all of whose digits are either
1
1
1
or
2
2
2
.
n+1 is divisible by 5 , 2^n + n and 2^n -1 are coprime
Does there exist an even positive integer
n
n
n
for which
n
+
1
n+1
n
+
1
is divisible by
5
5
5
and the two numbers
2
n
+
n
2^n + n
2
n
+
n
and
2
n
−
1
2^n -1
2
n
−
1
are co-prime?
4
2
Hide problems
subtriangles of equilateral
An equilateral triangle of integer side length
n
≥
1
n \geq 1
n
≥
1
is subdivided into small triangles of unit side length, as illustrated in the figure below for
n
=
5
n = 5
n
=
5
. In this diagram a subtriangle is a triangle of any size which is formed by connecting vertices of the small triangles along the grid lines. https://cdn.artofproblemsolving.com/attachments/e/9/17e83ad4872fcf9e97f0479104b9569bf75ad0.jpg It is desired to color each vertex of the small triangles either red or blue in such a way that there is no subtriangle with all of its vertices having the same color. Let
f
(
n
)
f(n)
f
(
n
)
denote the number of distinct colorings satisfying this condition. Determine, with proof,
f
(
n
)
f(n)
f
(
n
)
for every
n
≥
1
n \geq 1
n
≥
1
1 + a^{2017} + b^{2017} \geq a^{10}b^{7} + a^{7}b^{2000} + a^{2000}b^{10}
Show that for all non-negative numbers
a
,
b
a,b
a
,
b
,
1
+
a
2017
+
b
2017
≥
a
10
b
7
+
a
7
b
2000
+
a
2000
b
10
1 + a^{2017} + b^{2017} \geq a^{10}b^{7} + a^{7}b^{2000} + a^{2000}b^{10}
1
+
a
2017
+
b
2017
≥
a
10
b
7
+
a
7
b
2000
+
a
2000
b
10
When is equality attained?
3
2
Hide problems
A'B'C'D' is similar to ABCD
Four circles are drawn with the sides of quadrilateral
A
B
C
D
ABCD
A
BC
D
as diameters. The two circles passing through
A
A
A
meet again at
A
′
A'
A
′
, two circles through
B
B
B
at
B
′
B'
B
′
, two circles at
C
C
C
at
C
′
C'
C
′
and the two circles at
D
D
D
at
D
′
D'
D
′
. Suppose the points
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
and
D
′
D'
D
′
are distinct. Prove quadrilateral
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
is similar to
A
B
C
D
ABCD
A
BC
D
.
\sum |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2
A line segment
B
0
B
n
B_0B_n
B
0
B
n
is divided into
n
n
n
equal parts at points
B
1
,
B
2
,
.
.
.
,
B
n
−
1
B_1,B_2,...,B_{n-1}
B
1
,
B
2
,
...
,
B
n
−
1
and
A
A
A
is a point such that
∠
B
0
A
B
n
\angle B_0AB_n
∠
B
0
A
B
n
is a right angle. Prove that :
∑
k
=
0
n
∣
A
B
k
∣
2
=
∑
k
=
0
n
∣
B
0
B
k
∣
2
\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2
k
=
0
∑
n
∣
A
B
k
∣
2
=
k
=
0
∑
n
∣
B
0
B
k
∣
2