MathDB
Problems
Contests
International Contests
JBMO ShortLists
2010 JBMO Shortlist
2010 JBMO Shortlist
Part of
JBMO ShortLists
Subcontests
(5)
3
2
Hide problems
JBMO 2010 Shortlist A3
Find all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of real numbers such that
∣
x
∣
+
∣
y
∣
=
1340
|x|+ |y|=1340
∣
x
∣
+
∣
y
∣
=
1340
and
x
3
+
y
3
+
2010
x
y
=
67
0
3
x^{3}+y^{3}+2010xy= 670^{3}
x
3
+
y
3
+
2010
x
y
=
67
0
3
.
2010 JBMO Shortlist G2
Consider a triangle
A
B
C
{ABC}
A
BC
and let
M
{M}
M
be the midpoint of the side
B
C
.
{BC.}
BC
.
Suppose
∠
M
A
C
=
∠
A
B
C
{\angle MAC=\angle ABC}
∠
M
A
C
=
∠
A
BC
and
∠
B
A
M
=
10
5
∘
.
{\angle BAM=105^{\circ}.}
∠
B
A
M
=
10
5
∘
.
Find the measure of
∠
A
B
C
{\angle ABC}
∠
A
BC
.
4
1
Hide problems
Inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real positive numbers such that
a
b
c
(
a
+
b
+
c
)
=
3
abc(a+b+c)=3
ab
c
(
a
+
b
+
c
)
=
3
Prove that
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≥
8
(a+b)(b+c)(c+a) \geq 8
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≥
8
5
1
Hide problems
xyz <= 36
Let
x
,
y
,
z
>
0
x, y, z > 0
x
,
y
,
z
>
0
with
x
≤
2
,
y
≤
3
x \leq 2, \;y \leq 3 \;
x
≤
2
,
y
≤
3
and
x
+
y
+
z
=
11
x+y+z = 11
x
+
y
+
z
=
11
Prove that
x
y
z
≤
36
xyz \leq 36
x
yz
≤
36
1
2
Hide problems
Strategical Game with 2010 coins in a pile
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
P
r
o
b
l
e
m
C
.
1
<
/
s
p
a
n
>
<span class='latex-bold'>Problem C.1</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
P
ro
b
l
e
m
C
.1
<
/
s
p
an
>
There are two piles of coins, each containing
2010
2010
2010
pieces. Two players
A
A
A
and
B
B
B
play a game taking turns (
A
A
A
plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
In a right ∆ABC prove that A,Q,R are collinear & AP=AC
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
P
r
o
b
l
e
m
G
1
<
/
s
p
a
n
>
<span class='latex-bold'>Problem G1</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
P
ro
b
l
e
m
G
1
<
/
s
p
an
>
Consider a triangle
A
B
C
ABC
A
BC
with
∠
A
C
B
=
9
0
∘
\angle ACB=90^{\circ}
∠
A
CB
=
9
0
∘
. Let
F
F
F
be the foot of the altitude from
C
C
C
. Circle
ω
\omega
ω
touches the line segment
F
B
FB
FB
at point
P
P
P
, the altitude
C
F
CF
CF
at point
Q
Q
Q
and the circumcircle of
A
B
C
ABC
A
BC
at point
R
R
R
. Prove that points
A
,
Q
,
R
A, Q, R
A
,
Q
,
R
are collinear and
A
P
=
A
C
AP = AC
A
P
=
A
C
.
2
3
Hide problems
JBMO 2010 Shortlist A2
Determine all four digit numbers
a
ˉ
b
ˉ
c
ˉ
d
ˉ
\bar{a}\bar{b}\bar{c}\bar{d}
a
ˉ
b
ˉ
c
ˉ
d
ˉ
such that
a
(
a
+
b
+
c
+
d
)
(
a
2
+
b
2
+
c
2
+
d
2
)
(
a
6
+
2
b
6
+
3
c
6
+
4
d
6
)
=
a
ˉ
b
ˉ
c
ˉ
d
ˉ
a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\bar{a}\bar{b}\bar{c}\bar{d}
a
(
a
+
b
+
c
+
d
)
(
a
2
+
b
2
+
c
2
+
d
2
)
(
a
6
+
2
b
6
+
3
c
6
+
4
d
6
)
=
a
ˉ
b
ˉ
c
ˉ
d
ˉ
Geometry
Let
A
B
C
ABC
A
BC
be acute-angled triangle . A circle
ω
1
(
O
1
,
R
1
)
\omega_1(O_1,R_1)
ω
1
(
O
1
,
R
1
)
passes through points
B
B
B
and
C
C
C
and meets the sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
D
D
D
and
E
E
E
,respectively . Let
ω
2
(
O
2
,
R
2
)
\omega_2(O_2,R_2)
ω
2
(
O
2
,
R
2
)
be the circumcircle of triangle
A
D
E
ADE
A
D
E
. Prove that
O
1
O
2
O_1O_2
O
1
O
2
is equal to the circumradius of triangle
A
B
C
ABC
A
BC
.
Number theory
Find n such that
3
6
n
−
6
36^n-6
3
6
n
−
6
is the product of three consecutive natural numbers