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JBMO ShortLists
2000 JBMO ShortLists
2000 JBMO ShortLists
Part of
JBMO ShortLists
Subcontests
(23)
5
1
Hide problems
A,B,C have a common divisor greater than 1 - JBMO Shortlist
Find all pairs of integers
(
m
,
n
)
(m,n)
(
m
,
n
)
such that the numbers
A
=
n
2
+
2
m
n
+
3
m
2
+
2
A=n^2+2mn+3m^2+2
A
=
n
2
+
2
mn
+
3
m
2
+
2
,
B
=
2
n
2
+
3
m
n
+
m
2
+
2
B=2n^2+3mn+m^2+2
B
=
2
n
2
+
3
mn
+
m
2
+
2
,
C
=
3
n
2
+
m
n
+
2
m
2
+
1
C=3n^2+mn+2m^2+1
C
=
3
n
2
+
mn
+
2
m
2
+
1
have a common divisor greater than
1
1
1
.
6
1
Hide problems
Sum of prime factors = Sum of exponents - JBMO Shortlist
Find all four-digit numbers such that when decomposed into prime factors, each number has the sum of its prime factors equal to the sum of the exponents.
23
1
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P is inside an equailateral triangle - JBMO Shortlist
The point
P
P
P
is inside of an equilateral triangle with side length
10
10
10
so that the distance from
P
P
P
to two of the sides are
1
1
1
and
3
3
3
. Find the distance from
P
P
P
to the third side.
22
1
Hide problems
Area of quadrilateral ABCD - JBMO Shortlist
Consider a quadrilateral with
∠
D
A
B
=
6
0
∘
\angle DAB=60^{\circ}
∠
D
A
B
=
6
0
∘
,
∠
A
B
C
=
9
0
∘
\angle ABC=90^{\circ}
∠
A
BC
=
9
0
∘
and
∠
B
C
D
=
12
0
∘
\angle BCD=120^{\circ}
∠
BC
D
=
12
0
∘
. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
M
M
M
. If
M
B
=
1
MB=1
MB
=
1
and
M
D
=
2
MD=2
M
D
=
2
, find the area of the quadrilateral
A
B
C
D
ABCD
A
BC
D
.
21
1
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All angles in a hexagon are equal - JBMO Shortlist
All the angles of the hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
are equal. Prove that
A
B
−
D
E
=
E
F
−
B
C
=
C
D
−
F
A
AB-DE=EF-BC=CD-FA
A
B
−
D
E
=
EF
−
BC
=
C
D
−
F
A
20
1
Hide problems
Inequality of angles - JBMO Shortlist
Let
A
B
C
ABC
A
BC
be a triangle and let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the lengths of the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
respectively. Consider a triangle
D
E
F
DEF
D
EF
with the side lengths
E
F
=
a
u
EF=\sqrt{au}
EF
=
a
u
,
F
D
=
b
u
FD=\sqrt{bu}
F
D
=
b
u
,
D
E
=
c
u
DE=\sqrt{cu}
D
E
=
c
u
. Prove that
∠
A
>
∠
B
>
∠
C
\angle A >\angle B >\angle C
∠
A
>
∠
B
>
∠
C
implies
∠
A
>
∠
D
>
∠
E
>
∠
F
>
∠
C
\angle A >\angle D >\angle E >\angle F >\angle C
∠
A
>
∠
D
>
∠
E
>
∠
F
>
∠
C
.
19
1
Hide problems
7 regions of equal area - JBMO Shortlist
Let
A
B
C
ABC
A
BC
be a triangle. Find all the triangles
X
Y
Z
XYZ
X
Y
Z
with vertices inside triangle
A
B
C
ABC
A
BC
such that
X
Y
,
Y
Z
,
Z
X
XY,YZ,ZX
X
Y
,
Y
Z
,
ZX
and six non-intersecting segments from the following
A
X
,
A
Y
,
A
Z
,
B
X
,
B
Y
,
B
Z
,
C
X
,
C
Y
,
C
Z
AX, AY, AZ, BX, BY, BZ, CX, CY, CZ
A
X
,
A
Y
,
A
Z
,
BX
,
B
Y
,
BZ
,
CX
,
C
Y
,
CZ
divide the triangle
A
B
C
ABC
A
BC
into seven regions with equal areas.
18
1
Hide problems
5 regions of equal area - JBMO Shortlist
A triangle
A
B
C
ABC
A
BC
is given. Find all the segments
X
Y
XY
X
Y
that lie inside the triangle such that
X
Y
XY
X
Y
and five of the segments
X
A
,
X
B
,
X
C
,
Y
A
,
Y
B
,
Y
C
XA,XB, XC, YA,YB,YC
X
A
,
XB
,
XC
,
Y
A
,
Y
B
,
Y
C
divide the triangle
A
B
C
ABC
A
BC
into
5
5
5
regions with equal areas. Furthermore, prove that all the segments
X
Y
XY
X
Y
have a common point.
17
1
Hide problems
Four segments divide ABC into equal areas - JBMO Shortlist
A triangle
A
B
C
ABC
A
BC
is given. Find all the pairs of points
X
,
Y
X,Y
X
,
Y
so that
X
X
X
is on the sides of the triangle,
Y
Y
Y
is inside the triangle, and four non-intersecting segments from the set
{
X
Y
,
A
X
,
A
Y
,
B
X
,
B
Y
,
C
X
,
C
Y
}
\{XY, AX, AY, BX,BY, CX, CY\}
{
X
Y
,
A
X
,
A
Y
,
BX
,
B
Y
,
CX
,
C
Y
}
divide the triangle
A
B
C
ABC
A
BC
into four triangles with equal areas.
16
1
Hide problems
Find all x,y,z - JBMO Shortlist
Find all the triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of real numbers such that
2
x
y
−
1
+
2
y
z
−
1
+
2
z
x
−
1
≥
x
y
+
y
z
+
z
x
2x\sqrt{y-1}+2y\sqrt{z-1}+2z\sqrt{x-1} \ge xy+yz+zx
2
x
y
−
1
+
2
y
z
−
1
+
2
z
x
−
1
≥
x
y
+
yz
+
z
x
15
1
Hide problems
Fraction is greater than 1 - JBMO Shortlist
Let
x
,
y
,
a
,
b
x,y,a,b
x
,
y
,
a
,
b
be positive real numbers such that
x
≠
y
x\not= y
x
=
y
,
x
≠
2
y
x\not= 2y
x
=
2
y
,
y
≠
2
x
y\not= 2x
y
=
2
x
,
a
≠
3
b
a\not=3b
a
=
3
b
and
2
x
−
y
2
y
−
x
=
a
+
3
b
a
−
3
b
\frac{2x-y}{2y-x}=\frac{a+3b}{a-3b}
2
y
−
x
2
x
−
y
=
a
−
3
b
a
+
3
b
. Prove that
x
2
+
y
2
x
2
−
y
2
≥
1
\frac{x^2+y^2}{x^2-y^2}\ge 1
x
2
−
y
2
x
2
+
y
2
≥
1
.
14
1
Hide problems
Smallest value of k - JBMO Shortlist
Let
m
m
m
and
n
n
n
be positive integers with
m
≤
2000
m\le 2000
m
≤
2000
and
k
=
3
−
m
n
k=3-\frac{m}{n}
k
=
3
−
n
m
. Find the smallest positive value of
k
k
k
.
13
1
Hide problems
Sum of k-th powers - JBMO Shortlist
Prove that
(
1
k
+
2
k
)
(
1
k
+
2
k
+
3
k
)
…
(
1
k
+
2
k
+
…
+
n
k
)
\sqrt{(1^k+2^k)(1^k+2^k+3^k)\ldots (1^k+2^k+\ldots +n^k)}
(
1
k
+
2
k
)
(
1
k
+
2
k
+
3
k
)
…
(
1
k
+
2
k
+
…
+
n
k
)
≥
1
k
+
2
k
+
…
+
n
k
−
2
k
−
1
+
2
⋅
3
k
−
1
+
…
+
(
n
−
1
)
⋅
n
k
−
1
n
\ge 1^k+2^k+\ldots +n^k-\frac{2^{k-1}+2\cdot 3^{k-1}+\ldots + (n-1)\cdot n^{k-1}}{n}
≥
1
k
+
2
k
+
…
+
n
k
−
n
2
k
−
1
+
2
⋅
3
k
−
1
+
…
+
(
n
−
1
)
⋅
n
k
−
1
for all integers
n
,
k
≥
2
n,k \ge 2
n
,
k
≥
2
.
12
1
Hide problems
Two sequences imply a third - JBMO Shortlist
Consider a sequence of positive integers
x
n
x_n
x
n
such that:
(
A
)
x
2
n
+
1
=
4
x
n
+
2
n
+
2
(\text{A})\ x_{2n+1}=4x_n+2n+2
(
A
)
x
2
n
+
1
=
4
x
n
+
2
n
+
2
(\text{B})\ x_{3n+\color[rgb]{0.9529,0.0980,0.0118}2}=3x_{n+1}+6x_n for all
n
≥
0
n\ge 0
n
≥
0
. Prove that
(
C
)
x
3
n
−
1
=
x
n
+
2
−
2
x
n
+
1
+
10
x
n
(\text{C})\ x_{3n-1}=x_{n+2}-2x_{n+1}+10x_n
(
C
)
x
3
n
−
1
=
x
n
+
2
−
2
x
n
+
1
+
10
x
n
for all
n
≥
0
n\ge 0
n
≥
0
.
11
1
Hide problems
For any integer n - JBMO Shortlist
Prove that for any integer
n
n
n
one can find integers
a
a
a
and
b
b
b
such that
n
=
[
a
2
]
+
[
b
3
]
n=\left[ a\sqrt{2}\right]+\left[ b\sqrt{3}\right]
n
=
[
a
2
]
+
[
b
3
]
10
1
Hide problems
No integers satisfying the equation - JBMO Shortlist
Prove that there are no integers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
x
4
+
y
4
+
z
4
−
2
x
2
y
2
−
2
y
2
z
2
−
2
z
2
x
2
=
2000
x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000
x
4
+
y
4
+
z
4
−
2
x
2
y
2
−
2
y
2
z
2
−
2
z
2
x
2
=
2000
9
1
Hide problems
xy+yz+zx-xyz=2 - JBMO Shortlist
Find all the triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of positive integers such that
x
y
+
y
z
+
z
x
−
x
y
z
=
2
xy+yz+zx-xyz=2
x
y
+
yz
+
z
x
−
x
yz
=
2
.
8
1
Hide problems
Do you know the Sophie-Germain Identity? - JBMO Shortlist
Find all positive integers
a
,
b
a,b
a
,
b
for which
a
4
+
4
b
4
a^4+4b^4
a
4
+
4
b
4
is a prime number.
7
1
Hide problems
A,B,C are consecutive - JBMO Shortlist
Find all the pairs of positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
such that the numbers
A
=
n
2
+
2
m
n
+
3
m
2
+
3
n
A=n^2+2mn+3m^2+3n
A
=
n
2
+
2
mn
+
3
m
2
+
3
n
,
B
=
2
n
2
+
3
m
n
+
m
2
B=2n^2+3mn+m^2
B
=
2
n
2
+
3
mn
+
m
2
,
C
=
3
n
2
+
m
n
+
2
m
2
C=3n^2+mn+2m^2
C
=
3
n
2
+
mn
+
2
m
2
are consecutive in some order.
4
1
Hide problems
abcd to dcba - JBMO Shortlist
Find all the integers written as
a
b
c
d
‾
\overline{abcd}
ab
c
d
in decimal representation and
d
c
b
a
‾
\overline{dcba}
d
c
ba
in base
7
7
7
.
3
1
Hide problems
Exponent of 23 in 2000! - JBMO Shortlist
Find the greatest positive integer
x
x
x
such that
2
3
6
+
x
23^{6+x}
2
3
6
+
x
divides
2000
!
2000!
2000
!
2
1
Hide problems
Perfect cubes after erasing three digits - JBMO Shortlist
Find all the positive perfect cubes that are not divisible by
10
10
10
such that the number obtained by erasing the last three digits is also a perfect cube.
1
1
Hide problems
Composite numbers with 2006 digits - JBMO Shortlist
Prove that there are at least
666
666
666
positive composite numbers with
2006
2006
2006
digits, having a digit equal to
7
7
7
and all the rest equal to
1
1
1
.