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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2010 Irish Math Olympiad
2010 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
Hide problems
Find Polynomials
Find all polynomials
f
(
x
)
=
x
3
+
b
x
2
+
c
x
+
d
f(x)=x^3+bx^2+cx+d
f
(
x
)
=
x
3
+
b
x
2
+
c
x
+
d
, where
b
,
c
,
d
,
b,c,d,
b
,
c
,
d
,
are real numbers, such that
f
(
x
2
−
2
)
=
−
f
(
−
x
)
f
(
x
)
f(x^2-2)=-f(-x)f(x)
f
(
x
2
−
2
)
=
−
f
(
−
x
)
f
(
x
)
.
Triangles with Equal Area
Suppose
a
,
b
,
c
a,b,c
a
,
b
,
c
are the side lengths of a triangle
A
B
C
ABC
A
BC
. Show that
x
=
a
(
b
+
c
−
a
)
,
y
=
b
(
c
+
a
−
b
)
,
z
=
c
(
a
+
b
−
c
)
x=\sqrt{a(b+c-a)}, y=\sqrt{b(c+a-b)}, z=\sqrt{c(a+b-c)}
x
=
a
(
b
+
c
−
a
)
,
y
=
b
(
c
+
a
−
b
)
,
z
=
c
(
a
+
b
−
c
)
are the side lengths of an acute-angled triangle
X
Y
Z
XYZ
X
Y
Z
, with the same area as
A
B
C
ABC
A
BC
, but with a smaller perimeter, unless
A
B
C
ABC
A
BC
is equilateral.
4
2
Hide problems
Coins Arranged in Triangle
The country of Harpland has three types of coins: green, white and orange. The unit of currency in Harpland is the shilling. Any coin is worth a positive integer number of shillings, but coins of the same color may be worth different amounts. A set of coins is stacked in the form of an equilateral triangle of side
n
n
n
coins, as shown below for the case of
n
=
6
n=6
n
=
6
.[asy] size(100); for (int j=0; j<6; ++j) { for (int i=0; i<6-j; ++i) { draw(Circle((i+j/2,0.866j),0.5)); } } [/asy]The stacking has the following properties:(a) no coin touches another coin of the same color;(b) the total worth, in shillings, of the coins lying on any line parallel to one of the sides of the triangle is divisible by by three.Prove that the total worth in shillings of the green coins in the triangle is divisible by three.
Recursive Sequence Divisibility
Let
n
≥
3
n\ge 3
n
≥
3
be an integer and
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots ,a_n
a
1
,
a
2
,
…
,
a
n
be a finite sequence of positive integers, such that, for
k
=
2
,
3
,
…
,
n
k=2,3,\dots ,n
k
=
2
,
3
,
…
,
n
n
(
a
k
+
1
)
−
(
n
−
1
)
a
k
−
1
=
1.
n(a_k+1)-(n-1)a_{k-1}=1.
n
(
a
k
+
1
)
−
(
n
−
1
)
a
k
−
1
=
1.
Prove that
a
n
a_n
a
n
is not divisible by
(
n
−
1
)
2
(n-1)^2
(
n
−
1
)
2
.
3
2
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Inequality
Suppose
x
,
y
,
z
x,y,z
x
,
y
,
z
are positive numbers such that
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
. Prove that (a)
x
y
+
y
z
+
x
z
≥
9
x
y
z
xy+yz+xz\ge 9xyz
x
y
+
yz
+
x
z
≥
9
x
yz
; (b)
x
y
+
y
z
+
x
z
<
1
4
+
3
x
y
z
xy+yz+xz<\frac{1}{4}+3xyz
x
y
+
yz
+
x
z
<
4
1
+
3
x
yz
;
Obtuse Triangle with Altitude
In triangle
A
B
C
ABC
A
BC
we have
∣
A
B
∣
=
1
|AB|=1
∣
A
B
∣
=
1
and
∠
A
B
C
=
12
0
∘
.
\angle ABC=120^\circ.
∠
A
BC
=
12
0
∘
.
The perpendicular line to
A
B
AB
A
B
at
B
B
B
meets
A
C
AC
A
C
at
D
D
D
such that
∣
D
C
∣
=
1
|DC|=1
∣
D
C
∣
=
1
. Find the length of
A
D
AD
A
D
.
2
2
Hide problems
Triangle is Isosceles
Let
A
B
C
ABC
A
BC
be a triangle and let
P
P
P
denote the midpoint of the side
B
C
BC
BC
. Suppose that there exist two points
M
M
M
and
N
N
N
interior to the side
A
B
AB
A
B
and
A
C
AC
A
C
respectively, such that
∣
A
D
∣
=
∣
D
M
∣
=
2
∣
D
N
∣
,
|AD|=|DM|=2|DN|,
∣
A
D
∣
=
∣
D
M
∣
=
2∣
D
N
∣
,
where
D
D
D
is the intersection point of the lines
M
N
MN
MN
and
A
P
AP
A
P
. Show that
∣
A
C
∣
=
∣
B
C
∣
|AC|=|BC|
∣
A
C
∣
=
∣
BC
∣
.
Real Roots of Polynomial
For each odd integer
p
≥
3
p\ge 3
p
≥
3
find the number of real roots of the polynomial
f
p
(
x
)
=
(
x
−
1
)
(
x
−
2
)
⋯
(
x
−
p
+
1
)
+
1.
f_p(x)=(x-1)(x-2)\cdots (x-p+1)+1.
f
p
(
x
)
=
(
x
−
1
)
(
x
−
2
)
⋯
(
x
−
p
+
1
)
+
1.
1
2
Hide problems
Squares Sum to 2010
Find the least
k
k
k
for which the number
2010
2010
2010
can be expressed as the sum of the squares of
k
k
k
integers.
Amusing puzzle
There are
14
14
14
boys in a class. Each boy is asked how many other boys in the class have his first name, and how many have his last name. It turns out that each number from
0
0
0
to
6
6
6
occurs among the answers.Prove that there are two boys in the class with the same first name and the same last name.