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International Contests
JBMO ShortLists
2001 JBMO ShortLists
2001 JBMO ShortLists
Part of
JBMO ShortLists
Subcontests
(13)
13
1
Hide problems
Mathematcians at a conference - JBMO Shortlist
At a conference there are
n
n
n
mathematicians. Each of them knows exactly
k
k
k
fellow mathematicians. Find the smallest value of
k
k
k
such that there are at least three mathematicians that are acquainted each with the other two.[color=#BF0000]Rewording of the last line for clarification: Find the smallest value of
k
k
k
such that there (always) exists
3
3
3
mathematicians
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
such that
X
X
X
and
Y
Y
Y
know each other,
X
X
X
and
Z
Z
Z
know each other and
Y
Y
Y
and
Z
Z
Z
know each other.
12
1
Hide problems
AKOM is a paralleogram - JBMO Shortlist
Consider the triangle
A
B
C
ABC
A
BC
with
∠
A
=
9
0
∘
\angle A= 90^{\circ}
∠
A
=
9
0
∘
and
∠
B
≠
∠
C
\angle B \not= \angle C
∠
B
=
∠
C
. A circle
C
(
O
,
R
)
\mathcal{C}(O,R)
C
(
O
,
R
)
passes through
B
B
B
and
C
C
C
and intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
, respectively. Let
S
S
S
be the foot of the perpendicular from
A
A
A
to
B
C
BC
BC
and let
K
K
K
be the intersection point of
A
S
AS
A
S
with the segment
D
E
DE
D
E
. If
M
M
M
is the midpoint of
B
C
BC
BC
, prove that
A
K
O
M
AKOM
A
K
OM
is a parallelogram.
11
1
Hide problems
The point E on AB=AC - JBMO Shortlist
Consider a triangle
A
B
C
ABC
A
BC
with
A
B
=
A
C
AB=AC
A
B
=
A
C
, and
D
D
D
the foot of the altitude from the vertex
A
A
A
. The point
E
E
E
lies on the side
A
B
AB
A
B
such that
∠
A
C
E
=
∠
E
C
B
=
1
8
∘
\angle ACE= \angle ECB=18^{\circ}
∠
A
CE
=
∠
ECB
=
1
8
∘
.If
A
D
=
3
AD=3
A
D
=
3
, find the length of the segment
C
E
CE
CE
.
10
1
Hide problems
Ratio of incircle and excircle ? - JBMO Shortlist
A triangle
A
B
C
ABC
A
BC
is inscribed in the circle
C
(
O
,
R
)
\mathcal{C}(O,R)
C
(
O
,
R
)
. Let
α
<
1
\alpha <1
α
<
1
be the ratio of the radii of the circles tangent to
C
\mathcal{C}
C
, and both of the rays
(
A
B
(AB
(
A
B
and
(
A
C
(AC
(
A
C
. The numbers
β
<
1
\beta <1
β
<
1
and
γ
<
1
\gamma <1
γ
<
1
are defined analogously. Prove that
α
+
β
+
γ
=
1
\alpha + \beta + \gamma =1
α
+
β
+
γ
=
1
.
9
1
Hide problems
Equal angles in a convex quadrilateral
Consider a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
B
=
C
D
AB=CD
A
B
=
C
D
and
∠
B
A
C
=
3
0
∘
\angle BAC=30^{\circ}
∠
B
A
C
=
3
0
∘
. If
∠
A
D
C
=
15
0
∘
\angle ADC=150^{\circ}
∠
A
D
C
=
15
0
∘
, prove that
∠
B
C
A
=
∠
A
C
D
\angle BCA= \angle ACD
∠
BC
A
=
∠
A
C
D
.
8
1
Hide problems
Equilateral triangle and lattice points - JBMO Shortlist
Prove that no three points with integer coordinates can be the vertices of an equilateral triangle.
7
1
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5-th powers is a no-go - JBMO Shortlist
Prove that there are are no positive integers
x
x
x
and
y
y
y
such that
x
5
+
y
5
+
1
=
(
x
+
2
)
5
+
(
y
−
3
)
5
x^5+y^5+1=(x+2)^5+(y-3)^5
x
5
+
y
5
+
1
=
(
x
+
2
)
5
+
(
y
−
3
)
5
.[hide="Note"] The restriction
x
,
y
x,y
x
,
y
are positive isn't necessary.
6
1
Hide problems
Diophantine equation with 2001p - JBMO Shortlist
Find all integers
x
x
x
and
y
y
y
such that
x
3
±
y
3
=
2001
p
x^3\pm y^3 =2001p
x
3
±
y
3
=
2001
p
, where
p
p
p
is prime.
5
1
Hide problems
At least one square between T_n-1 and T_n - JBMO Shortlist
Let
x
k
=
k
(
k
+
1
)
2
x_k=\frac{k(k+1)}{2}
x
k
=
2
k
(
k
+
1
)
for all integers
k
≥
1
k\ge 1
k
≥
1
. Prove that for any integer
n
≥
10
n \ge 10
n
≥
10
, between the numbers
A
=
x
1
+
x
2
+
…
+
x
n
−
1
A=x_1+x_2 + \ldots + x_{n-1}
A
=
x
1
+
x
2
+
…
+
x
n
−
1
and
B
=
A
+
x
n
B=A+x_n
B
=
A
+
x
n
there is at least one square.
4
1
Hide problems
Roots of the equation are integers - JBMO Shortlist
The discriminant of the equation
x
2
−
a
x
+
b
=
0
x^2-ax+b=0
x
2
−
a
x
+
b
=
0
is the square of a rational number and
a
a
a
and
b
b
b
are integers. Prove that the roots of the equation are integers.
3
1
Hide problems
abcabc...abc is divisible by 91 - JBMO Shortlist
Find all the three-digit numbers
a
b
c
‾
\overline{abc}
ab
c
such that the
6003
6003
6003
-digit number
a
b
c
a
b
c
…
a
b
c
‾
\overline{abcabc\ldots abc}
ab
c
ab
c
…
ab
c
is divisible by
91
91
91
.
2
1
Hide problems
Squares in P_n - JBMO Shortlist
Let
P
n
(
n
=
3
,
4
,
5
,
6
,
7
)
P_n \ (n=3,4,5,6,7)
P
n
(
n
=
3
,
4
,
5
,
6
,
7
)
be the set of positive integers
n
k
+
n
l
+
n
m
n^k+n^l+n^m
n
k
+
n
l
+
n
m
, where
k
,
l
,
m
k,l,m
k
,
l
,
m
are positive integers. Find
n
n
n
such that:i) In the set
P
n
P_n
P
n
there are infinitely many squares.ii) In the set
P
n
P_n
P
n
there are no squares.
1
1
Hide problems
Perfect cube when n is not divisible by 3 - JBMO Shortlist
Find the positive integers
n
n
n
that are not divisible by
3
3
3
if the number
2
n
2
−
10
+
2133
2^{n^2-10}+2133
2
n
2
−
10
+
2133
is a perfect cube.[hide="Note"] [color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation
x
3
=
2
n
2
−
10
+
2133
x^3=2^{n^2-10}+2133
x
3
=
2
n
2
−
10
+
2133
where
x
,
n
∈
N
x,n \in \mathbb{N}
x
,
n
∈
N
and
3
∤
n
3\nmid n
3
∤
n
.