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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1995 Irish Math Olympiad
1995 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
1
Hide problems
divisors
For each integer
n
n
n
of the form n\equal{}p_1 p_2 p_3 p_4, where
p
1
,
p
2
,
p
3
,
p
4
p_1,p_2,p_3,p_4
p
1
,
p
2
,
p
3
,
p
4
are distinct primes, let 1\equal{}d_1
n
n
n
. Prove that if
n
<
1995
n<1995
n
<
1995
, then d_9\minus{}d_8 \not\equal{} 22.
4
2
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disk
Consider the following one-person game played on the real line. During the game disks are piled at some of the integer points on the line. To perform a move in the game, the player chooses a point
j
j
j
at which at least two disks are piled and then takes two disks from the point
j
j
j
and places one of them at j\minus{}1 and one at j\plus{}1. Initially, 2n\plus{}1 disks are placed at point
0
0
0
. The player proceeds to perform moves as long as possible. Prove that after \frac{1}{6} n(n\plus{}1)(2n\plus{}1) moves no further moves will be possible and that at this stage, one disks remains at each of the positions \minus{}n,\minus{}n\plus{}1,...,0,...n.
easy geometry
Points
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
are given in the plane. It is known that there is a triangle
A
B
C
ABC
A
BC
such that
P
P
P
is the midpoint of
B
C
BC
BC
,
Q
Q
Q
the point on side
C
A
CA
C
A
with \frac{CQ}{QA}\equal{}2, and
R
R
R
the point on side
A
B
AB
A
B
with \frac{AR}{RB}\equal{}2. Determine with proof how the triangle
A
B
C
ABC
A
BC
may be reconstructed from
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
.
3
2
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inequality
Points
A
,
X
,
D
A,X,D
A
,
X
,
D
lie on a line in this order, point
B
B
B
is on the plane such that
∠
A
B
X
>
12
0
∘
\angle ABX>120^{\circ}
∠
A
BX
>
12
0
∘
, and point
C
C
C
is on the segment
B
X
BX
BX
. Prove the inequality: 2AD \ge \sqrt{3} (AB\plus{}BC\plus{}CD).
Common Points are Above Curve
Let
S
S
S
be the square consisting of all pints
(
x
,
y
)
(x,y)
(
x
,
y
)
in the plane with
0
≤
x
,
y
≤
1
0\le x,y\le 1
0
≤
x
,
y
≤
1
. For each real number
t
t
t
with
0
<
t
<
1
0<t<1
0
<
t
<
1
, let
C
t
C_t
C
t
denote the set of all points
(
x
,
y
)
∈
S
(x,y)\in S
(
x
,
y
)
∈
S
such that
(
x
,
y
)
(x,y)
(
x
,
y
)
is on or above the line joining
(
t
,
0
)
(t,0)
(
t
,
0
)
to
(
0
,
1
−
t
)
(0,1-t)
(
0
,
1
−
t
)
. Prove that the points common to all
C
t
C_t
C
t
are those points in
S
S
S
that are on or above the curve
x
+
y
=
1
\sqrt{x}+\sqrt{y}=1
x
+
y
=
1
.
2
2
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infinitely many solutions
Determine all integers
a
a
a
for which the equation x^2\plus{}axy\plus{}y^2\equal{}1 has infinitely many distinct integer solutions
x
,
y
x,y
x
,
y
.
complex numbers
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be complex numbers. Prove that if all the roots of the equation x^3\plus{}ax^2\plus{}bx\plus{}c\equal{}0 are of module
1
1
1
, then so are the roots of the equation x^3\plus{}|a|x^2\plus{}|b|x\plus{}|c|\equal{}0.
1
2
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students
There are
n
2
n^2
n
2
students in a class. Each week all the students participate in a table quiz. Their teacher arranges them into
n
n
n
teams of
n
n
n
players each. For as many weeks as possible, this arrangement is done in such a way that any pair of students who were members of the same team one week are not in the same team in subsequent weeks. Prove that after at most n\plus{}2 weeks, it is necessary for some pair of students to have been members of the same team in at least two different weeks.
inequality in integers
Prove that for every positive integer
n
n
n
, n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.