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Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2007 Irish Math Olympiad
2007 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
2
2
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right-angled triangle
Prove that the triangle ABC is right-angled if it holds:
sin
2
A
+
sin
2
B
+
sin
2
C
=
2
\sin^2 A+\sin^2 B+\sin^2 C = 2
sin
2
A
+
sin
2
B
+
sin
2
C
=
2
nice inequality
Suppose that
a
,
b
,
a,b,
a
,
b
,
and
c
c
c
are positive real numbers. Prove that: \frac{a\plus{}b\plus{}c}{3} \le \sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{3}} \le \frac {\frac{ab}{c}\plus{}\frac{bc}{a}\plus{}\frac{ca}{b}}{3}. For each of the inequalities, find the conditions on
a
,
b
,
a,b,
a
,
b
,
and
c
c
c
such that equality holds.
5
2
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weird inequality
Let
r
r
r
and
n
n
n
be nonnegative integers such that
r
≤
n
r \le n
r
≤
n
.
(
a
)
(a)
(
a
)
Prove that: \frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r} is an integer.
(
b
)
(b)
(
b
)
Prove that: \displaystyle\sum_{r\equal{}0}^{[n/2]}\frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}<2^{n\minus{}2} for all
n
≥
9
n \ge 9
n
≥
9
.
quadratic polynomial
Suppose that
a
a
a
and
b
b
b
are real numbers such that the quadratic polynomial f(x)\equal{}x^2\plus{}ax\plus{}b has no nonnegative real roots. Prove that there exist two polynomials
g
,
h
g,h
g
,
h
whose coefficients are nonnegative real numbers such that: f(x)\equal{}\frac{g(x)}{h(x)} for all real numbers
x
x
x
.
4
2
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flights
Air Michael and Air Patrick operate direct flights connecting Belfast, Cork, Dublin, Galway, Limerick, and Waterord. For each pair of cities exactly one of the airlines operates the route (in both directions) connecting the cities. Prove that there are four cities for which one of the airlines operates a round trip. (Note that a round trip of four cities
P
,
Q
,
R
,
P,Q,R,
P
,
Q
,
R
,
and
S
S
S
, is a journey that follows the path
P
→
Q
→
R
→
S
→
P
P \rightarrow Q \rightarrow R \rightarrow S \rightarrow P
P
→
Q
→
R
→
S
→
P
.)
digits
Find the number of zeros in which the decimal expansion of
2007
!
2007!
2007
!
ends. Also find its last non-zero digit.
3
2
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fixed point
The point
P
P
P
is a fixed point on a circle and
Q
Q
Q
is a fixed point on a line. The point
R
R
R
is a variable point on the circle such that
P
,
Q
,
P,Q,
P
,
Q
,
and
R
R
R
are not collinear. The circle through
P
,
Q
,
P,Q,
P
,
Q
,
and
R
R
R
meets the line again at
V
V
V
. Show that the line
V
R
VR
V
R
passes through a fixed point.
identity
Let
A
B
C
ABC
A
BC
be a triangle the lengths of whose sides
B
C
,
C
A
,
A
B
,
BC,CA,AB,
BC
,
C
A
,
A
B
,
respectively, are denoted by
a
,
b
,
a,b,
a
,
b
,
and
c
c
c
. Let the internal bisectors of the angles
∠
B
A
C
,
∠
A
B
C
,
∠
B
C
A
,
\angle BAC, \angle ABC, \angle BCA,
∠
B
A
C
,
∠
A
BC
,
∠
BC
A
,
respectively, meet the sides
B
C
,
C
A
,
BC,CA,
BC
,
C
A
,
and
A
B
AB
A
B
at
D
,
E
,
D,E,
D
,
E
,
and
F
F
F
. Denote the lengths of the line segments
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
by
d
,
e
,
d,e,
d
,
e
,
and
f
f
f
, respectively. Prove that: def\equal{}\frac{4abc(a\plus{}b\plus{}c) \Delta}{(a\plus{}b)(b\plus{}c)(c\plus{}a)} where
Δ
\Delta
Δ
stands for the area of the triangle
A
B
C
ABC
A
BC
.
1
2
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prime numbers
Find all prime numbers
p
p
p
and
q
q
q
such that
p
p
p
divides q\plus{}6 and
q
q
q
divides p\plus{}6.
polynomial
Let
r
,
s
,
r,s,
r
,
s
,
and
t
t
t
be the roots of the cubic polynomial: p(x)\equal{}x^3\minus{}2007x\plus{}2002. Determine the value of: \frac{r\minus{}1}{r\plus{}1}\plus{}\frac{s\minus{}1}{s\plus{}1}\plus{}\frac{t\minus{}1}{t\plus{}1}.