MathDB
Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2005 Irish Math Olympiad
2005 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
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inequality in a,b,c
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be nonnegative real numbers. Prove that: \frac{1}{3}((a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2) \le a^2\plus{}b^2\plus{}c^2\minus{}3 \sqrt[3]{a^2 b^2 c^2 } \le (a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2.
perfect square
Suppose that
m
m
m
and
n
n
n
are odd integers such that m^2\minus{}n^2\plus{}1 divides n^2\minus{}1. Prove that m^2\minus{}n^2\plus{}1 is a perfect square.
4
2
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arrangements
Determine the number of arrangements
a
1
,
a
2
,
.
.
.
,
a
10
a_1,a_2,...,a_{10}
a
1
,
a
2
,
...
,
a
10
of the numbers
1
,
2
,
.
.
.
,
10
1,2,...,10
1
,
2
,
...
,
10
such that
a
i
>
a
2
i
a_i>a_{2i}
a
i
>
a
2
i
for
1
≤
i
≤
5
1 \le i \le 5
1
≤
i
≤
5
and a_i>a_{2i\plus{}1} for
1
≤
i
≤
4
1 \le i \le 4
1
≤
i
≤
4
.
find two digits
Find the first digit to the left and the first digit to the right of the decimal point in the expansion of (\sqrt{2}\plus{}\sqrt{5})^{2000}.
3
2
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probably known
Prove that the sum of the lengths of the medians of a triangle is at least three quarters of its perimeter.
divisibility
Let
x
x
x
be an integer and
y
,
z
,
w
y,z,w
y
,
z
,
w
be odd positive integers. Prove that
17
17
17
divides x^{y^{z^w}}\minus{}x^{y^z}.
2
2
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centroid
Let
D
,
E
D,E
D
,
E
and
F
F
F
be points on the sides
B
C
,
C
A
BC,CA
BC
,
C
A
and
A
B
AB
A
B
respectively of a triangle
A
B
C
ABC
A
BC
, distinct from the vertices, such that
A
D
,
B
E
AD,BE
A
D
,
BE
and
C
F
CF
CF
meet at a point
G
G
G
. Prove that if the angles
A
G
E
,
C
G
D
,
B
G
F
AGE,CGD,BGF
A
GE
,
CG
D
,
BGF
have equal area, then
G
G
G
is the centroid of
△
A
B
C
\triangle ABC
△
A
BC
.
game
Using the digits:
1
,
2
,
3
,
4
,
5
,
1,2,3,4,5,
1
,
2
,
3
,
4
,
5
,
players
A
A
A
and
B
B
B
compose a
2005
2005
2005
-digit number
N
N
N
by selecting one digit at a time:
A
A
A
selects the first digit,
B
B
B
the second,
A
A
A
the third and so on. Player
A
A
A
wins if and only if
N
N
N
is divisible by
9
9
9
. Who will win if both players play as well as possible?
1
2
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easy
Show that
200
5
2005
2005^{2005}
200
5
2005
is a sum of two perfect squares, but not a sum of two perfect cubes.
inequality
Let
X
X
X
be a point on the side
A
B
AB
A
B
of a triangle
A
B
C
ABC
A
BC
, different from
A
A
A
and
B
B
B
. Let
P
P
P
and
Q
Q
Q
be the incenters of the triangles
A
C
X
ACX
A
CX
and
B
C
X
BCX
BCX
respectively, and let
M
M
M
be the midpoint of
P
Q
PQ
PQ
. Prove that:
M
C
>
M
X
MC>MX
MC
>
MX
.