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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2001 Irish Math Olympiad
2001 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
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find the cases of equality
Prove that for all real numbers
a
,
b
a,b
a
,
b
with
a
b
>
0
ab>0
ab
>
0
we have: \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}10ab\plus{}b^2}{12} and find the cases of equality. Hence, or otherwise, prove that for all real numbers
a
,
b
a,b
a
,
b
\sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}ab\plus{}b^2}{3} and find the cases of equality.
function (maybe posted before)
Determine all functions
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
which satisfy: f(x\plus{}f(y))\equal{}f(x)\plus{}y for all
x
,
y
∈
N
x,y \in \mathbb{N}
x
,
y
∈
N
.
4
2
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nice inequality
Prove that for all positive integers
n
n
n
: \frac{2n}{3n\plus{}1} \le \displaystyle\sum_{k\equal{}n\plus{}1}^{2n}\frac{1}{k} \le \frac{3n\plus{}1}{4(n\plus{}1)}.
nonnegative real numbers
Find all nonnegative real numbers
x
x
x
for which \sqrt[3]{13\plus{}\sqrt{x}}\plus{}\sqrt[3]{13\minus{}\sqrt{x}} is an integer.
3
2
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identity
Show that if an odd prime number
p
p
p
can be expressed in the form x^5\minus{}y^5 for some integers
x
,
y
,
x,y,
x
,
y
,
then: \sqrt{\frac{4p\plus{}1}{5}}\equal{}\frac{v^2\plus{}1}{2} for some odd integer
v
v
v
.
easy exercise
In an acute-angled triangle
A
B
C
ABC
A
BC
,
D
D
D
is the foot of the altitude from
A
A
A
, and
P
P
P
a point on segment
A
D
AD
A
D
. The lines
B
P
BP
BP
and
C
P
CP
CP
meet
A
C
AC
A
C
and
A
B
AB
A
B
at
E
E
E
and
F
F
F
respectively. Prove that
A
D
AD
A
D
bisects the angle
E
D
F
EDF
E
D
F
.
2
2
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perpendicular medians (old exercise)
Let
A
B
C
ABC
A
BC
be a triangle with sides BC\equal{}a, CA\equal{}b,AB\equal{}c and let
D
D
D
and
E
E
E
be the midpoints of
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. Prove that the medians
B
D
BD
B
D
and
C
E
CE
CE
are perpendicular to each other if and only if b^2\plus{}c^2\equal{}5a^2.
hoops
Three hoops are arranged concentrically as in the diagram. Each hoop is threaded with
20
20
20
beads,
10
10
10
of which are black and
10
10
10
are white. On each hoop the positions of the beads are labelled
1
1
1
through
20
20
20
as shown. We say there is a match at position
i
i
i
if all three beads at position
i
i
i
have the same color. We are free to slide beads around a hoop, not breaking the hoop. Show that it is always possible to move them into a configuration involving no less than
5
5
5
matches.
1
2
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equation
Find all positive integer solutions
(
a
,
b
,
c
,
n
)
(a,b,c,n)
(
a
,
b
,
c
,
n
)
of the equation: 2^n\equal{}a!\plus{}b!\plus{}c!.
find a number
Find the least positive integer
a
a
a
such that
2001
2001
2001
divides 55^n\plus{}a \cdot 32^n for some odd
n
n
n
.