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Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2013 Irish Math Olympiad
2013 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(10)
7
1
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numbers of squares with any possible area chosen from 12 labeled points
Consider the collection of different squares which may be formed by sets of four points chosen from the
12
12
12
labelled points in the diagram on the right. For each possible area such a square may have, determine the number of squares which have this area. Make sure to explain why your list is complete. https://cdn.artofproblemsolving.com/attachments/b/a/faf00c2faa7b949ab2894942f8bd99505543e8.png
5
1
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circles & tangents, two lines intersect on circumcircle, irmo geometry 2013
A
,
B
A, B
A
,
B
and
C
C
C
are points on the circumference of a circle with centre
O
O
O
. Tangents are drawn to the circumcircles of triangles
O
A
B
OAB
O
A
B
and
O
A
C
OAC
O
A
C
at
P
P
P
and
Q
Q
Q
respectively, where
P
P
P
and
Q
Q
Q
are diametrically opposite
O
O
O
. The two tangents intersect at
K
K
K
. The line
C
A
CA
C
A
meets the circumcircle of
△
O
A
B
\triangle OAB
△
O
A
B
at
A
A
A
and
X
X
X
. Prove that
X
X
X
lies on the line
K
O
KO
K
O
.
3
1
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altitudes of a triangle form a second triangle, altitudes of second form a third
The altitudes of a triangle
△
A
B
C
\triangle ABC
△
A
BC
are used to form the sides of a second triangle
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
. The altitudes of
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
are then used to form the sides of a third triangle
△
A
2
B
2
C
2
\triangle A_2B_2C_2
△
A
2
B
2
C
2
. Prove that
△
A
2
B
2
C
2
\triangle A_2B_2C_2
△
A
2
B
2
C
2
is similar to
△
A
B
C
\triangle ABC
△
A
BC
.
4
1
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coloring Red, Yellow or Blue, each square of a 6x6 table
Each of the
36
36
36
squares of a
6
×
6
6 \times 6
6
×
6
table is to be coloured either Red, Yellow or Blue. (a) No row or column is contain more than two squares of the same colour. (b) In any four squares obtained by intersecting two rows with two columns, no colour is to occur exactly three times. In how many di erent ways can the table be coloured if both of these rules are to be respected?
8
1
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smallest N so that (x^2 -1)(y^2 -1)=N has at least two pairs (x,y) of solutions
Find the smallest positive integer
N
N
N
for which the equation
(
x
2
−
1
)
(
y
2
−
1
)
=
N
(x^2 -1)(y^2 -1)=N
(
x
2
−
1
)
(
y
2
−
1
)
=
N
is satis ed by at least two pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
with
1
<
x
≤
y
1 < x \le y
1
<
x
≤
y
.
1
1
Hide problems
Exponents of integer question
Find the smallest positive integer
m
m
m
such that
5
m
5m
5
m
is an exact 5th power,
6
m
6m
6
m
is an exact 6th power, and
7
m
7m
7
m
is an exact 7th power.
9
1
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A sequence
We say that a doubly infinite sequence
.
.
.
,
s
−
2
,
s
−
1
,
s
0
,
s
1
,
s
2
,
.
.
.
. . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . .
...
,
s
−
2
,
s
−
1
,
s
0
,
s
1
,
s
2
,
...
is subaveraging if
s
n
=
(
s
n
−
1
+
s
n
+
1
)
/
4
s_n = (s_{n−1} + s_{n+1})/4
s
n
=
(
s
n
−
1
+
s
n
+
1
)
/4
for all integers n. (a) Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct. (b) Show that if
(
s
n
)
(s_n)
(
s
n
)
is a subaveraging sequence such that there exist distinct integers m, n such that
s
m
=
s
n
s_m = s_n
s
m
=
s
n
, then there are infinitely many pairs of distinct integers i, j with
s
i
=
s
j
s_i = s_j
s
i
=
s
j
.
2
1
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Irish Mathematical Olympiad 2013 (1)
Prove that
1
−
1
2012
(
1
2
+
1
3
+
⋯
+
1
2013
)
≥
1
2013
2012
.
1-\frac{1}{2012}\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2013}\right)\ge \frac{1}{\sqrt[2012]{2013}}.
1
−
2012
1
(
2
1
+
3
1
+
⋯
+
2013
1
)
≥
2012
2013
1
.
10
1
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Irish Mathematical Olympiad 2013 (2)
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers and let
x
=
a
+
b
+
c
,
y
=
a
2
+
b
2
+
c
2
,
z
=
a
3
+
b
3
+
c
3
x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3
x
=
a
+
b
+
c
,
y
=
a
2
+
b
2
+
c
2
,
z
=
a
3
+
b
3
+
c
3
and
S
=
2
x
3
−
9
x
y
+
9
z
.
S=2x^3-9xy+9z .
S
=
2
x
3
−
9
x
y
+
9
z
.
(
a
)
(a)
(
a
)
Prove that
S
S
S
is unchanged when
a
,
b
,
c
a,b,c
a
,
b
,
c
are replaced by
a
+
t
,
b
+
t
,
c
+
t
a+t,b+t,c+t
a
+
t
,
b
+
t
,
c
+
t
, respectively , for any real number
t
t
t
.
(
b
)
(b)
(
b
)
Prove that
(
3
y
−
x
2
)
3
≥
S
2
.
(3y-x^2)^3\ge S^2 .
(
3
y
−
x
2
)
3
≥
S
2
.
6
1
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distinct points
The three distinct points
B
,
C
,
D
B, C, D
B
,
C
,
D
are collinear with C between B and D. Another point A not on the line BD is such that
∣
A
B
∣
=
∣
A
C
∣
=
∣
C
D
∣
.
|AB| = |AC| = |CD|.
∣
A
B
∣
=
∣
A
C
∣
=
∣
C
D
∣.
Prove that ∠
B
A
C
=
36
BAC = 36
B
A
C
=
36
if and only if
1
/
∣
C
D
∣
−
1
/
∣
B
D
∣
=
1
/
(
∣
C
D
∣
+
∣
B
D
∣
)
1/|CD|-1/|BD|=1/(|CD| + |BD|)
1/∣
C
D
∣
−
1/∣
B
D
∣
=
1/
(
∣
C
D
∣
+
∣
B
D
∣
)
.