MathDB
Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1999 Irish Math Olympiad
1999 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
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find a term of a sequence
The sequence
u
n
u_n
u
n
, n\equal{}0,1,2,... is defined by u_0\equal{}0, u_1\equal{}1 and for each
n
≥
1
n \ge 1
n
≥
1
, u_{n\plus{}1} is the smallest positive integer greater than
u
n
u_n
u
n
such that \{ u_0,u_1,...,u_{n\plus{}1} \} contains no three elements in arithmetic progression. Find
u
100
u_{100}
u
100
.
concurrent lines
A convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
satisfies AB\equal{}BC, CD\equal{}DE, EF\equal{}FA and: \angle ABC\plus{}\angle CDE\plus{}\angle EFA \equal{} 360^{\circ}. Prove that the perpendiculars from
A
,
C
A,C
A
,
C
and
E
E
E
to
F
B
,
B
D
FB,BD
FB
,
B
D
and
D
F
DF
D
F
respectively are concurrent.
4
2
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divisors
Find all positive integers
m
m
m
with the property that the fourth power of the number of (positive) divisors of
m
m
m
equals
m
m
m
.
tiling
A
100
×
100
100 \times 100
100
×
100
square floor consisting of
10000
10000
10000
squares is to be tiled by rectangular
1
×
3
1 \times 3
1
×
3
tiles, fitting exactly over three squares of the floor.
(
a
)
(a)
(
a
)
If a
2
×
2
2 \times 2
2
×
2
square is removed from the center of the floor, prove that the rest of the floor can be tiled with the available tiles.
(
b
)
(b)
(
b
)
If, instead, a
2
×
2
2 \times 2
2
×
2
square is removed from the corner, prove that such a tiling is not possble.
3
2
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prove an identity
If
A
D
AD
A
D
is the altitude,
B
E
BE
BE
the angle bisector, and
C
F
CF
CF
the median of a triangle
A
B
C
ABC
A
BC
, prove that
A
D
,
B
E
,
AD,BE,
A
D
,
BE
,
and
C
F
CF
CF
are concurrent if and only if: a^2(a\minus{}c)\equal{}(b^2\minus{}c^2)(a\plus{}c), where
a
,
b
,
c
a,b,c
a
,
b
,
c
are the lengths of the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively.
easy
The sum of positive real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
is
1
1
1
. Prove that: \frac{a^2}{a\plus{}b}\plus{}\frac{b^2}{b\plus{}c}\plus{}\frac{c^2}{c\plus{}d}\plus{}\frac{d^2}{d\plus{}a} \ge \frac{1}{2}, with equality if and only if a\equal{}b\equal{}c\equal{}d\equal{}\frac{1}{4}.
2
2
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Fibonacci
Show that there is a positive number in the Fibonacci sequence which is divisible by
1000
1000
1000
.
function
A function
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
satisfies:
(
a
)
(a)
(
a
)
f(ab)\equal{}f(a)f(b) whenever
a
a
a
and
b
b
b
are coprime;
(
b
)
(b)
(
b
)
f(p\plus{}q)\equal{}f(p)\plus{}f(q) for all prime numbers
p
p
p
and
q
q
q
. Prove that f(2)\equal{}2,f(3)\equal{}3 and f(1999)\equal{}1999.
1
2
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find all real numbers
Find all real numbers
x
x
x
which satisfy: \frac{x^2}{(x\plus{}1\minus{}\sqrt{x\plus{}1})^2}<\frac{x^2\plus{}3x\plus{}18}{(x\plus{}1)^2}.
system of equations
Solve the system of equations: y^2\equal{}(x\plus{}8)(x^2\plus{}2), y^2\minus{}(8\plus{}4x)y\plus{}(16\plus{}16x\minus{}5x^2)\equal{}0.