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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2003 Irish Math Olympiad
2003 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
3
2
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Difference of Floors
For each positive integer
k
k
k
, let
a
k
a_k
a
k
be the greatest integer not exceeding
k
\sqrt{k}
k
and let
b
k
b_k
b
k
be the greatest integer not exceeding
k
3
\sqrt[3]{k}
3
k
. Calculate
∑
k
=
1
2003
(
a
k
−
b
k
)
.
\sum_{k=1}^{2003} (a_k-b_k).
k
=
1
∑
2003
(
a
k
−
b
k
)
.
Easy algebra
Find all the (x,y) integer ,if
y
2
+
2
y
=
x
4
+
20
x
3
+
104
x
2
+
40
x
+
2003
y^2+2y=x^4+20x^3+104x^2+40x+2003
y
2
+
2
y
=
x
4
+
20
x
3
+
104
x
2
+
40
x
+
2003
2
2
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Geometry question
P
P
P
,
Q
Q
Q
,
R
R
R
and
S
S
S
are (distinct) points on a circle.
P
S
PS
PS
is a diameter and
Q
R
QR
QR
is parallel to the diameter
P
S
PS
PS
.
P
R
PR
PR
and
Q
S
QS
QS
meet at
A
A
A
. Let
O
O
O
be the centre of the circle and let
B
B
B
be chosen so that the quadrilateral
P
O
A
B
POAB
PO
A
B
is a parallelogram. Prove that
B
Q
BQ
BQ
=
B
P
BP
BP
.
Quadrilateral- feet of the perpendicular
A
B
C
D
\ ABCD
A
BC
D
is a quadrilateral. the feet of the perpendicular from
D
\ D
D
to
A
B
,
B
C
\ AB, BC
A
B
,
BC
are
P
,
Q
\ P,Q
P
,
Q
respectively, and the feet of the perpendicular from
B
\ B
B
to
A
D
,
C
D
\ AD,CD
A
D
,
C
D
are
R
,
S
\ R,S
R
,
S
respectively. Show that if
∠
P
S
R
=
∠
S
P
Q
\angle PSR= \angle SPQ
∠
PSR
=
∠
SPQ
, then
P
R
=
Q
S
\ PR=QS
PR
=
QS
.
1
2
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Easy number theory
find all solutions, not necessarily positive integers for
(
m
2
+
n
)
(
m
+
n
2
)
=
(
m
+
n
)
3
(m^2+ n)(m+ n^2)= (m+ n)^3
(
m
2
+
n
)
(
m
+
n
2
)
=
(
m
+
n
)
3
Algebraic ineq in triangle
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are the sides of a triangle whose perimeter is equal to 2 then prove that: a)
a
b
c
+
28
27
≥
a
b
+
b
c
+
a
c
abc+\frac{28}{27}\geq ab+bc+ac
ab
c
+
27
28
≥
ab
+
b
c
+
a
c
; b)
a
b
c
+
1
<
a
b
+
b
c
+
a
c
abc+1<ab+bc+ac
ab
c
+
1
<
ab
+
b
c
+
a
c
See also http://www.mathlinks.ro/Forum/viewtopic.php?t=47939&view=next (problem 1) :)
5
2
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Find all
show that thee is no function f definedonthe positive real numbes such that :
f
(
y
)
>
(
y
−
x
)
f
(
x
)
2
f(y) > (y-x)f(x)^2
f
(
y
)
>
(
y
−
x
)
f
(
x
)
2
Combinatorics (from Irish MO)
(a) In how many ways can
1003
1003
1003
distinct integers be chosen from the set
{
1
,
2
,
.
.
.
,
2003
}
\{1, 2, ... , 2003\}
{
1
,
2
,
...
,
2003
}
so that no two of the chosen integers di ffer by
10
?
10?
10
?
(b) Show that there are
(
3
(
5151
)
+
7
(
1700
)
)
10
1
7
(3(5151) + 7(1700)) 101^7
(
3
(
5151
)
+
7
(
1700
))
10
1
7
ways to choose
1002
1002
1002
distinct integers from the set
{
1
,
2
,
.
.
.
,
2003
}
\{1, 2, ... , 2003\}
{
1
,
2
,
...
,
2003
}
so that no two of the chosen integers diff er by
10.
10.
10.
4
2
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Chess Tournament
Eight players, Ann, Bob, Con, Dot, Eve, Fay, Guy and Hal compete in a chess tournament. No pair plays together more than once and there is no group of five people in which each one plays against all of the other four.(a) Write down an arrangement for a tournament of
24
24
24
games satisfying these conditions.(b) Show that it is impossible to have a tournament of more than
24
24
24
games satisfying these conditions.
Inequality
Given real positive a,b , find the larget real c such that
c
≤
m
a
x
(
a
x
+
1
a
x
,
b
x
+
1
b
x
)
c\leq max(ax+\frac{1}{ax},bx+\frac{1}{bx})
c
≤
ma
x
(
a
x
+
a
x
1
,
b
x
+
b
x
1
)
for all positive ral x. There is a solution here,,,, http://www.kalva.demon.co.uk/irish/soln/sol039.html but im wondering if there is a better one . Thank you.