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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2002 Irish Math Olympiad
2002 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
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interesting
Let
0
<
a
,
b
,
c
<
1
0<a,b,c<1
0
<
a
,
b
,
c
<
1
. Prove the inequality: \frac{a}{1\minus{}a}\plus{}\frac{b}{1\minus{}b}\plus{}\frac{c}{1\minus{}c} \ge \frac {3 \sqrt[3]{abc}}{1\minus{} \sqrt[3]{abc}}. Determine the cases of equality.
equality case
Let
A
B
C
ABC
A
BC
be a triangle with integer side lengths, and let its incircle touch
B
C
BC
BC
at
D
D
D
and
A
C
AC
A
C
at
E
E
E
. If |AD^2\minus{}BE^2| \le 2, show that AC\equal{}BC.
4
2
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positive integer
The sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by a_1\equal{}a_2\equal{}a_3\equal{}1 and a_{n\plus{}1}a_{n\minus{}2}\minus{}a_n a_{n\minus{}1}\equal{}2 for all
n
≥
3.
n \ge 3.
n
≥
3.
Prove that
a
n
a_n
a
n
is a positive integer for all
n
≥
1
n \ge 1
n
≥
1
.
identity
Let \alpha\equal{}2\plus{}\sqrt{3}. Prove that \alpha^n\minus{}[\alpha^n]\equal{}1\minus{}\alpha^{\minus{}n} for all
n
∈
N
0
n \in \mathbb{N}_0
n
∈
N
0
.
3
2
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triples
Find all triples of positive integers
(
p
,
q
,
n
)
(p,q,n)
(
p
,
q
,
n
)
, with
p
p
p
and
q
q
q
primes, satisfying: p(p\plus{}3)\plus{}q(q\plus{}3)\equal{}n(n\plus{}3).
find all functions
Find all functions
f
:
Q
→
Q
f: \mathbb{Q} \rightarrow \mathbb{Q}
f
:
Q
→
Q
such that: f(x\plus{}f(y))\equal{}y\plus{}f(x) for all
x
,
y
∈
Q
x,y \in \mathbb{Q}
x
,
y
∈
Q
.
2
2
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acquaintances
(
a
)
(a)
(
a
)
A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present?
(
b
)
(b)
(
b
)
If, in addition, the group contains three mutual acquaintances, what is the maximum possible number of people?
determine n
Suppose
n
n
n
is a product of four distinct primes
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that:
(
i
)
(i)
(
i
)
a\plus{}c\equal{}d;
(
i
i
)
(ii)
(
ii
)
a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);
(
i
i
i
)
(iii)
(
iii
)
1\plus{}bc\plus{}d\equal{}bd. Determine
n
n
n
.
1
2
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grid
A
3
×
n
3 \times n
3
×
n
grid is filled as follows. The first row consists of the numbers from
1
1
1
to
n
n
n
arranged in ascending order. The second row is a cyclic shift of the top row: i,i\plus{}1,...,n,1,2,...,i\minus{}1 for some
i
i
i
. The third row has the numbers
1
1
1
to
n
n
n
in some order so that in each of the
n
n
n
columns, the sum of the three numbers is the same. For which values of
n
n
n
is it possible to fill the grid in this way? For all such
n
n
n
, determine the number of different ways of filling the grid.
calculate an angle
In a triangle
A
B
C
ABC
A
BC
with AB\equal{}20, AC\equal{}21 and BC\equal{}29, points
D
D
D
and
E
E
E
are taken on the segment
B
C
BC
BC
such that BD\equal{}8 and EC\equal{}9. Calculate the angle
∠
D
A
E
\angle DAE
∠
D
A
E
.