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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2014 Irish Math Olympiad
2014 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(10)
8
1
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if Q(i),Q(i+1),Q(i+2),Q(i+3) are integers for 1 integer i, then Q(n) is integer
(a) Let
a
0
,
a
1
,
a
2
a_0, a_1,a_2
a
0
,
a
1
,
a
2
be real numbers and consider the polynomial
P
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
P(x) = a_0 + a_1x + a_2x^2
P
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
. Assume that
P
(
−
1
)
,
P
(
0
)
P(-1), P(0)
P
(
−
1
)
,
P
(
0
)
and
P
(
1
)
P(1)
P
(
1
)
are integers. Prove that
P
(
n
)
P(n)
P
(
n
)
is an integer for all integers
n
n
n
. (b) Let
a
0
,
a
1
,
a
2
,
a
3
a_0,a_1, a_2, a_3
a
0
,
a
1
,
a
2
,
a
3
be real numbers and consider the polynomial
Q
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
Q(x) = a0 + a_1x + a_2x^2 + a_3x^3
Q
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
. Assume that there exists an integer
i
i
i
such that
Q
(
i
)
,
Q
(
i
+
1
)
,
Q
(
i
+
2
)
Q(i),Q(i+1),Q(i+2)
Q
(
i
)
,
Q
(
i
+
1
)
,
Q
(
i
+
2
)
and
Q
(
i
+
3
)
Q(i+3)
Q
(
i
+
3
)
are integers. Prove that
Q
(
n
)
Q(n)
Q
(
n
)
is an integer for all integers
n
n
n
.
9
1
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f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}
Let
n
n
n
be a positive integer and
a
1
,
.
.
.
,
a
n
a_1,...,a_n
a
1
,
...
,
a
n
be positive real numbers. Let
g
(
x
)
g(x)
g
(
x
)
denote the product
(
x
+
a
1
)
⋅
.
.
.
⋅
(
x
+
a
n
)
(x + a_1)\cdot ... \cdot (x + a_n)
(
x
+
a
1
)
⋅
...
⋅
(
x
+
a
n
)
. Let
a
0
a_0
a
0
be a real number and let
f
(
x
)
=
(
x
−
a
0
)
g
(
x
)
=
x
n
+
1
+
b
1
x
n
+
b
2
x
n
−
1
+
.
.
.
+
b
n
x
+
b
n
+
1
f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}
f
(
x
)
=
(
x
−
a
0
)
g
(
x
)
=
x
n
+
1
+
b
1
x
n
+
b
2
x
n
−
1
+
...
+
b
n
x
+
b
n
+
1
. Prove that all the coeffcients
b
1
,
b
2
,
.
.
.
,
b
n
+
1
b_1,b_2,..., b_{n+1}
b
1
,
b
2
,
...
,
b
n
+
1
of the polynomial
f
(
x
)
f(x)
f
(
x
)
are negative if and only if
a
0
>
a
1
+
a
2
+
.
.
.
+
a
n
a_0 > a_1 + a_2 +...+ a_n
a
0
>
a
1
+
a
2
+
...
+
a
n
.
10
1
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2014 babies born over a period of k consecutive days, at least 1 each day
Over a period of
k
k
k
consecutive days, a total of
2014
2014
2014
babies were born in a certain city, with at least one baby being born each day. Show that: (a) If
1014
<
k
≤
2014
1014 < k \le 2014
1014
<
k
≤
2014
, there must be a period of consecutive days during which exactly
100
100
100
babies were born. (b) By contrast, if
k
=
1014
k = 1014
k
=
1014
, such a period might not exist.
7
1
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Geometry Irish MO
The square
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle with center
O
O
O
. Let
E
E
E
be the midpoint of
A
D
AD
A
D
. The line
C
E
CE
CE
meets the circle again at
F
F
F
. The lines
F
B
FB
FB
and
A
D
AD
A
D
meet at
H
H
H
. Prove
H
D
=
2
A
H
HD = 2AH
HD
=
2
A
H
2
1
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Prove divisibility by a given expression
Prove that for
N
>
1
N>1
N
>
1
that
(
N
2
)
2014
−
(
N
11
)
106
(N^{2})^{2014} - (N^{11})^{106}
(
N
2
)
2014
−
(
N
11
)
106
is divisible by
N
6
+
N
3
+
1
N^6 + N^3 +1
N
6
+
N
3
+
1
Is this just a proof by induction or is there a more elegant method? I don't think calculating
N
=
2
N = 2
N
=
2
was expected.
3
1
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Find the angle
In the triangle ABC, D is the foot of the altitude from A to BC, and M is the midpoint of the line segment BC. The three angles ∠BAD, ∠DAM and ∠MAC are all equal. Find the angles of the triangle ABC.
1
1
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A chess board
Given an
8
×
8
8\times 8
8
×
8
chess board, in how many ways can we select
56
56
56
squares on the board while satisfying both of the following requirements: (a) All black squares are selected. (b) Exactly seven squares are selected in each column and in each row.
6
1
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An easy problem from Irish Mathematical Olympiad
Each of the four positive integers
N
,
N
+
1
,
N
+
2
,
N
+
3
N,N +1,N +2,N +3
N
,
N
+
1
,
N
+
2
,
N
+
3
has exactly six positive divisors. There are exactly
20
20
20
di erent positive numbers which are exact divisors of at least one of the numbers. One of these is
27
27
27
. Find all possible values of
N
N
N
.(Both
1
1
1
and
m
m
m
are counted as divisors of the number
m
m
m
.)
5
1
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Irish Mathematical Olympiad 2014 (2)
Suppose
a
1
,
a
2
,
…
,
a
n
>
0
a_1,a_2,\ldots,a_n>0
a
1
,
a
2
,
…
,
a
n
>
0
, where
n
>
1
n>1
n
>
1
and
∑
i
=
1
n
a
i
=
1
\sum_{i=1}^{n}a_i=1
∑
i
=
1
n
a
i
=
1
. For each
i
=
1
,
2
,
…
,
n
i=1,2,\ldots,n
i
=
1
,
2
,
…
,
n
, let
b
i
=
a
i
2
∑
j
=
1
n
a
j
2
b_i=\frac{a^2_i}{\sum\limits_{j=1}^{n}a^2_j}
b
i
=
j
=
1
∑
n
a
j
2
a
i
2
. Prove that
∑
i
=
1
n
a
i
1
−
a
i
≤
∑
i
=
1
n
b
i
1
−
b
i
.
\sum_{i=1}^{n}\frac{a_i}{1-a_i}\le \sum_{i=1}^{n}\frac{b_i}{1-b_i} .
i
=
1
∑
n
1
−
a
i
a
i
≤
i
=
1
∑
n
1
−
b
i
b
i
.
When does equality occur ?
4
1
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Irish Mathematical Olympiad 2014 (1)
Three different non-zero real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy the equations
a
+
2
b
=
b
+
2
c
=
c
+
2
a
=
p
a+\frac{2}{b}=b+\frac{2}{c}=c+\frac{2}{a}=p
a
+
b
2
=
b
+
c
2
=
c
+
a
2
=
p
, where
p
p
p
is a real number. Prove that
a
b
c
+
2
p
=
0.
abc+2p=0.
ab
c
+
2
p
=
0.