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Problems
Contests
International Contests
APMO
2003 APMO
2003 APMO
Part of
APMO
Subcontests
(5)
5
1
Hide problems
2m, 2n pairs of people
Given two positive integers
m
m
m
and
n
n
n
, find the smallest positive integer
k
k
k
such that among any
k
k
k
people, either there are
2
m
2m
2
m
of them who form
m
m
m
pairs of mutually acquainted people or there are
2
n
2n
2
n
of them forming
n
n
n
pairs of mutually unacquainted people.
4
1
Hide problems
In ineqaulity with sides of triangle
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the sides of a triangle, with
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
, and let
n
≥
2
n\ge 2
n
≥
2
be an integer. Show that
a
n
+
b
n
n
+
b
n
+
c
n
n
+
c
n
+
a
n
n
<
1
+
2
n
2
.
\sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<1+\frac{\sqrt[n]{2}}{2}.
n
a
n
+
b
n
+
n
b
n
+
c
n
+
n
c
n
+
a
n
<
1
+
2
n
2
.
3
1
Hide problems
Divisibility
Let
k
≥
14
k\ge 14
k
≥
14
be an integer, and let
p
k
p_k
p
k
be the largest prime number which is strictly less than
k
k
k
. You may assume that
p
k
≥
3
k
/
4
p_k\ge 3k/4
p
k
≥
3
k
/4
. Let
n
n
n
be a composite integer. Prove: (a) if
n
=
2
p
k
n=2p_k
n
=
2
p
k
, then
n
n
n
does not divide
(
n
−
k
)
!
(n-k)!
(
n
−
k
)!
; (b) if
n
>
2
p
k
n>2p_k
n
>
2
p
k
, then
n
n
n
divides
(
n
−
k
)
!
(n-k)!
(
n
−
k
)!
.
2
1
Hide problems
Constant = m_1+m_2
Suppose
A
B
C
D
ABCD
A
BC
D
is a square piece of cardboard with side length
a
a
a
. On a plane are two parallel lines
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
, which are also
a
a
a
units apart. The square
A
B
C
D
ABCD
A
BC
D
is placed on the plane so that sides
A
B
AB
A
B
and
A
D
AD
A
D
intersect
ℓ
1
\ell_1
ℓ
1
at
E
E
E
and
F
F
F
respectively. Also, sides
C
B
CB
CB
and
C
D
CD
C
D
intersect
ℓ
2
\ell_2
ℓ
2
at
G
G
G
and
H
H
H
respectively. Let the perimeters of
△
A
E
F
\triangle AEF
△
A
EF
and
△
C
G
H
\triangle CGH
△
CG
H
be
m
1
m_1
m
1
and
m
2
m_2
m
2
respectively. Prove that no matter how the square was placed,
m
1
+
m
2
m_1+m_2
m
1
+
m
2
remains constant.
1
1
Hide problems
Polynomial
Let
a
,
b
,
c
,
d
,
e
,
f
a,b,c,d,e,f
a
,
b
,
c
,
d
,
e
,
f
be real numbers such that the polynomial
p
(
x
)
=
x
8
−
4
x
7
+
7
x
6
+
a
x
5
+
b
x
4
+
c
x
3
+
d
x
2
+
e
x
+
f
p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f
p
(
x
)
=
x
8
−
4
x
7
+
7
x
6
+
a
x
5
+
b
x
4
+
c
x
3
+
d
x
2
+
e
x
+
f
factorises into eight linear factors
x
−
x
i
x-x_i
x
−
x
i
, with
x
i
>
0
x_i>0
x
i
>
0
for
i
=
1
,
2
,
…
,
8
i=1,2,\ldots,8
i
=
1
,
2
,
…
,
8
. Determine all possible values of
f
f
f
.