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International Contests
IMO Shortlist
2002 IMO Shortlist
2002 IMO Shortlist
Part of
IMO Shortlist
Subcontests
(8)
6
2
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IMO ShortList 2002, algebra problem 6
Let
A
A
A
be a non-empty set of positive integers. Suppose that there are positive integers
b
1
,
…
b
n
b_1,\ldots b_n
b
1
,
…
b
n
and
c
1
,
…
,
c
n
c_1,\ldots,c_n
c
1
,
…
,
c
n
such that - for each
i
i
i
the set
b
i
A
+
c
i
=
{
b
i
a
+
c
i
:
a
∈
A
}
b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}
b
i
A
+
c
i
=
{
b
i
a
+
c
i
:
a
∈
A
}
is a subset of
A
A
A
, and - the sets
b
i
A
+
c
i
b_iA+c_i
b
i
A
+
c
i
and
b
j
A
+
c
j
b_jA+c_j
b
j
A
+
c
j
are disjoint whenever
i
≠
j
i\ne j
i
=
j
Prove that
1
b
1
+
…
+
1
b
n
≤
1.
{1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.
b
1
1
+
…
+
b
n
1
≤
1.
IMO ShortList 2002, combinatorics problem 6
Let
n
n
n
be an even positive integer. Show that there is a permutation
(
x
1
,
x
2
,
…
,
x
n
)
\left(x_{1},x_{2},\ldots,x_{n}\right)
(
x
1
,
x
2
,
…
,
x
n
)
of
(
1
,
2
,
…
,
n
)
\left(1,\,2,\,\ldots,n\right)
(
1
,
2
,
…
,
n
)
such that for every
i
∈
{
1
,
2
,
.
.
.
,
n
}
i\in\left\{1,\ 2,\ ...,\ n\right\}
i
∈
{
1
,
2
,
...
,
n
}
, the number
x
i
+
1
x_{i+1}
x
i
+
1
is one of the numbers
2
x
i
2x_{i}
2
x
i
,
2
x
i
−
1
2x_{i}-1
2
x
i
−
1
,
2
x
i
−
n
2x_{i}-n
2
x
i
−
n
,
2
x
i
−
n
−
1
2x_{i}-n-1
2
x
i
−
n
−
1
. Hereby, we use the cyclic subscript convention, so that
x
n
+
1
x_{n+1}
x
n
+
1
means
x
1
x_{1}
x
1
.
3
3
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IMO ShortList 2002, number theory problem 3
Let
p
1
,
p
2
,
…
,
p
n
p_1,p_2,\ldots,p_n
p
1
,
p
2
,
…
,
p
n
be distinct primes greater than
3
3
3
. Show that
2
p
1
p
2
⋯
p
n
+
1
2^{p_1p_2\cdots p_n}+1
2
p
1
p
2
⋯
p
n
+
1
has at least
4
n
4^n
4
n
divisors.
IMO ShortList 2002, algebra problem 3
Let
P
P
P
be a cubic polynomial given by
P
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
P(x)=ax^3+bx^2+cx+d
P
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
, where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are integers and
a
≠
0
a\ne0
a
=
0
. Suppose that
x
P
(
x
)
=
y
P
(
y
)
xP(x)=yP(y)
x
P
(
x
)
=
y
P
(
y
)
for infinitely many pairs
x
,
y
x,y
x
,
y
of integers with
x
≠
y
x\ne y
x
=
y
. Prove that the equation
P
(
x
)
=
0
P(x)=0
P
(
x
)
=
0
has an integer root.
IMO ShortList 2002, combinatorics problem 3
Let
n
n
n
be a positive integer. A sequence of
n
n
n
positive integers (not necessarily distinct) is called full if it satisfies the following condition: for each positive integer
k
≥
2
k\geq2
k
≥
2
, if the number
k
k
k
appears in the sequence then so does the number
k
−
1
k-1
k
−
1
, and moreover the first occurrence of
k
−
1
k-1
k
−
1
comes before the last occurrence of
k
k
k
. For each
n
n
n
, how many full sequences are there ?
7
2
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IMO ShortList 2002, geometry problem 7
The incircle
Ω
\Omega
Ω
of the acute-angled triangle
A
B
C
ABC
A
BC
is tangent to its side
B
C
BC
BC
at a point
K
K
K
. Let
A
D
AD
A
D
be an altitude of triangle
A
B
C
ABC
A
BC
, and let
M
M
M
be the midpoint of the segment
A
D
AD
A
D
. If
N
N
N
is the common point of the circle
Ω
\Omega
Ω
and the line
K
M
KM
K
M
(distinct from
K
K
K
), then prove that the incircle
Ω
\Omega
Ω
and the circumcircle of triangle
B
C
N
BCN
BCN
are tangent to each other at the point
N
N
N
.
IMO ShortList 2002, combinatorics problem 7
Among a group of 120 people, some pairs are friends. A weak quartet is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?
8
1
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IMO ShortList 2002, geometry problem 8
Let two circles
S
1
S_{1}
S
1
and
S
2
S_{2}
S
2
meet at the points
A
A
A
and
B
B
B
. A line through
A
A
A
meets
S
1
S_{1}
S
1
again at
C
C
C
and
S
2
S_{2}
S
2
again at
D
D
D
. Let
M
M
M
,
N
N
N
,
K
K
K
be three points on the line segments
C
D
CD
C
D
,
B
C
BC
BC
,
B
D
BD
B
D
respectively, with
M
N
MN
MN
parallel to
B
D
BD
B
D
and
M
K
MK
M
K
parallel to
B
C
BC
BC
. Let
E
E
E
and
F
F
F
be points on those arcs
B
C
BC
BC
of
S
1
S_{1}
S
1
and
B
D
BD
B
D
of
S
2
S_{2}
S
2
respectively that do not contain
A
A
A
. Given that
E
N
EN
EN
is perpendicular to
B
C
BC
BC
and
F
K
FK
F
K
is perpendicular to
B
D
BD
B
D
prove that
∠
E
M
F
=
9
0
∘
\angle EMF=90^{\circ}
∠
EMF
=
9
0
∘
.
5
4
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4
3
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IMO ShortList 2002, geometry problem 4
Circles
S
1
S_1
S
1
and
S
2
S_2
S
2
intersect at points
P
P
P
and
Q
Q
Q
. Distinct points
A
1
A_1
A
1
and
B
1
B_1
B
1
(not at
P
P
P
or
Q
Q
Q
) are selected on
S
1
S_1
S
1
. The lines
A
1
P
A_1P
A
1
P
and
B
1
P
B_1P
B
1
P
meet
S
2
S_2
S
2
again at
A
2
A_2
A
2
and
B
2
B_2
B
2
respectively, and the lines
A
1
B
1
A_1B_1
A
1
B
1
and
A
2
B
2
A_2B_2
A
2
B
2
meet at
C
C
C
. Prove that, as
A
1
A_1
A
1
and
B
1
B_1
B
1
vary, the circumcentres of triangles
A
1
A
2
C
A_1A_2C
A
1
A
2
C
all lie on one fixed circle.
IMO ShortList 2002, number theory problem 4
Is there a positive integer
m
m
m
such that the equation
1
a
+
1
b
+
1
c
+
1
a
b
c
=
m
a
+
b
+
c
{1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c}
a
1
+
b
1
+
c
1
+
ab
c
1
=
a
+
b
+
c
m
has infinitely many solutions in positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
?
IMO ShortList 2002, combinatorics problem 4
Let
T
T
T
be the set of ordered triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
, where
x
,
y
,
z
x,y,z
x
,
y
,
z
are integers with
0
≤
x
,
y
,
z
≤
9
0\leq x,y,z\leq9
0
≤
x
,
y
,
z
≤
9
. Players
A
A
A
and
B
B
B
play the following guessing game. Player
A
A
A
chooses a triple
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
in
T
T
T
, and Player
B
B
B
has to discover
A
A
A
's triple in as few moves as possible. A move consists of the following:
B
B
B
gives
A
A
A
a triple
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
in
T
T
T
, and
A
A
A
replies by giving
B
B
B
the number
∣
x
+
y
−
a
−
b
∣
+
∣
y
+
z
−
b
−
c
∣
+
∣
z
+
x
−
c
−
a
∣
\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|
∣
x
+
y
−
a
−
b
∣
+
∣
y
+
z
−
b
−
c
∣
+
∣
z
+
x
−
c
−
a
∣
. Find the minimum number of moves that
B
B
B
needs to be sure of determining
A
A
A
's triple.
2
3
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IMO ShortList 2002, geometry problem 2
Let
A
B
C
ABC
A
BC
be a triangle for which there exists an interior point
F
F
F
such that
∠
A
F
B
=
∠
B
F
C
=
∠
C
F
A
\angle AFB=\angle BFC=\angle CFA
∠
A
FB
=
∠
BFC
=
∠
CF
A
. Let the lines
B
F
BF
BF
and
C
F
CF
CF
meet the sides
A
C
AC
A
C
and
A
B
AB
A
B
at
D
D
D
and
E
E
E
respectively. Prove that
A
B
+
A
C
≥
4
D
E
.
AB+AC\geq4DE.
A
B
+
A
C
≥
4
D
E
.
IMO ShortList 2002, algebra problem 2
Let
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
be an infinite sequence of real numbers, for which there exists a real number
c
c
c
with
0
≤
a
i
≤
c
0\leq a_i\leq c
0
≤
a
i
≤
c
for all
i
i
i
, such that \left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \text{for all }i,\ j \text{ with } i \neq j. Prove that
c
≥
1
c\geq1
c
≥
1
.
IMO ShortList 2002, combinatorics problem 2
For
n
n
n
an odd positive integer, the unit squares of an
n
×
n
n\times n
n
×
n
chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an
L
L
L
-shape formed by three connected unit squares. For which values of
n
n
n
is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?
1
3
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IMO ShortList 2002, geometry problem 1
Let
B
B
B
be a point on a circle
S
1
S_1
S
1
, and let
A
A
A
be a point distinct from
B
B
B
on the tangent at
B
B
B
to
S
1
S_1
S
1
. Let
C
C
C
be a point not on
S
1
S_1
S
1
such that the line segment
A
C
AC
A
C
meets
S
1
S_1
S
1
at two distinct points. Let
S
2
S_2
S
2
be the circle touching
A
C
AC
A
C
at
C
C
C
and touching
S
1
S_1
S
1
at a point
D
D
D
on the opposite side of
A
C
AC
A
C
from
B
B
B
. Prove that the circumcentre of triangle
B
C
D
BCD
BC
D
lies on the circumcircle of triangle
A
B
C
ABC
A
BC
.
IMO ShortList 2002, number theory problem 1
What is the smallest positive integer
t
t
t
such that there exist integers
x
1
,
x
2
,
…
,
x
t
x_1,x_2,\ldots,x_t
x
1
,
x
2
,
…
,
x
t
with
x
1
3
+
x
2
3
+
…
+
x
t
3
=
200
2
2002
?
x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?
x
1
3
+
x
2
3
+
…
+
x
t
3
=
200
2
2002
?
IMO ShortList 2002, algebra problem 1
Find all functions
f
f
f
from the reals to the reals such that
f
(
f
(
x
)
+
y
)
=
2
x
+
f
(
f
(
y
)
−
x
)
f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)
f
(
f
(
x
)
+
y
)
=
2
x
+
f
(
f
(
y
)
−
x
)
for all real
x
,
y
x,y
x
,
y
.