MathDB
Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1996 Irish Math Olympiad
1996 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
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dissection
Show how to dissect a square into at most five pieces in such a way that the pieces can be reassembled to form three squares of (pairwise) distinct areas.
game
The following game is played on a rectangular chessboard
5
×
9
5 \times 9
5
×
9
(with five rows and nine columns). Initially, a number of discs are randomly placed on some of the squares of the chessboard, with at most one disc on each square. A complete move consists of the moving every disc subject to the following rules:
(
1
)
(1)
(
1
)
Each disc may be moved one square up, down, left or right;
(
2
)
(2)
(
2
)
If a particular disc is moved up or down as part of a complete move, then it must be moved left or right in the next complete move;
(
3
)
(3)
(
3
)
If a particular disc is moved left or right as part of a complete move, then it must be moved up or down in the next complete move;
(
4
)
(4)
(
4
)
At the end of a complete move, no two discs can be on the same square. The game stops if it becomes impossible to perform a complete move. Prove that if initially
33
33
33
discs are placed on the board then the game must eventually stop. Prove also that it is possible to place
32
32
32
discs on the boards in such a way that the game could go on forever.
4
2
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easy geometry
Let
F
F
F
be the midpoint of the side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
. Isosceles right-angled triangles
A
B
D
ABD
A
B
D
and
A
C
E
ACE
A
CE
are constructed externally on
A
B
AB
A
B
and
A
C
AC
A
C
with the right angles at
D
D
D
and
E
E
E
. Prove that the triangle
D
E
F
DEF
D
EF
is right-angled and isosceles.
concurrent lines
In an acute-angled triangle
A
B
C
ABC
A
BC
,
D
,
E
,
F
D,E,F
D
,
E
,
F
are the feet of the altitudes from
A
,
B
,
C
A,B,C
A
,
B
,
C
, respectively, and
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
are the feet of the perpendiculars from
A
,
B
,
C
A,B,C
A
,
B
,
C
onto
E
F
,
F
D
,
D
E
EF,FD,DE
EF
,
F
D
,
D
E
, respectively. Prove that the lines
A
P
,
B
Q
,
C
R
AP,BQ,CR
A
P
,
BQ
,
CR
are concurrent.
3
2
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function
A function
f
f
f
from
[
0
,
1
]
[0,1]
[
0
,
1
]
to
R
\mathbb{R}
R
has the following properties:
(
i
)
(i)
(
i
)
f(1)\equal{}1;
(
i
i
)
(ii)
(
ii
)
f
(
x
)
≥
0
f(x) \ge 0
f
(
x
)
≥
0
for all
x
∈
[
0
,
1
]
x \in [0,1]
x
∈
[
0
,
1
]
;
(
i
i
i
)
(iii)
(
iii
)
If x,y,x\plus{}y \in [0,1], then f(x\plus{}y) \ge f(x)\plus{}f(y). Prove that
f
(
x
)
≤
2
x
f(x) \le 2x
f
(
x
)
≤
2
x
for all
x
∈
[
0
,
1
]
x \in [0,1]
x
∈
[
0
,
1
]
.
prime number
Suppose that
p
p
p
is a prime number and
a
a
a
and
n
n
n
positive integers such that: 2^p\plus{}3^p\equal{}a^n. Prove that n\equal{}1.
2
2
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sum of digits
Let
S
(
n
)
S(n)
S
(
n
)
denote the sum of the digits of a natural number
n
n
n
(in base
10
10
10
). Prove that for every
n
n
n
,
S
(
2
n
)
≤
2
S
(
n
)
≤
10
S
(
2
n
)
S(2n) \le 2S(n) \le 10S(2n)
S
(
2
n
)
≤
2
S
(
n
)
≤
10
S
(
2
n
)
. Prove also that there is a positive integer
n
n
n
with S(n)\equal{}1996S(3n).
inequality with integers
Show that for every positive integer
n
n
n
,
2
1
2
⋅
4
1
4
⋅
8
1
8
⋅
.
.
.
⋅
(
2
n
)
1
2
n
<
4
2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot ... \cdot (2^n)^{\frac{1}{2^n}}<4
2
2
1
⋅
4
4
1
⋅
8
8
1
⋅
...
⋅
(
2
n
)
2
n
1
<
4
.
1
2
Hide problems
compute f(n)
For each positive integer
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
denote the greatest common divisor of n!\plus{}1 and (n\plus{}1)!. Find, without proof, a formula for
f
(
n
)
f(n)
f
(
n
)
.
Fibonacci
The Fibonacci sequence is defined by F_0\equal{}0, F_1\equal{}1 and F_{n\plus{}2}\equal{}F_n\plus{}F_{n\plus{}1} for
n
≥
0
n \ge 0
n
≥
0
. Prove that:
(
a
)
(a)
(
a
)
The statement "F_{n\plus{}k}\minus{}F_n is divisible by
10
10
10
for all
n
∈
N
"
n \in \mathbb{N}"
n
∈
N
"
is true if k\equal{}60 but false for any positive integer
k
<
60
k<60
k
<
60
.
(
b
)
(b)
(
b
)
The statement "F_{n\plus{}t}\minus{}F_n is divisible by
100
100
100
for all
n
∈
N
"
n \in \mathbb{N}"
n
∈
N
"
is true if t\equal{}300 but false for any positive integer
t
<
300
t<300
t
<
300
.