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Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia BMO TST
2013 Saudi Arabia BMO TST
2013 Saudi Arabia BMO TST
Part of
Saudi Arabia BMO TST
Subcontests
(8)
6
2
Hide problems
AS = AT wanted, midpoints, incenter related
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
,
I,
I
,
and let
D
,
E
,
F
D,E,F
D
,
E
,
F
be the midpoints of sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively. Lines
B
I
BI
B
I
and
D
E
DE
D
E
meet at
P
P
P
and lines
C
I
CI
C
I
and
D
F
DF
D
F
meet at
Q
Q
Q
. Line
P
Q
PQ
PQ
meets sides
A
B
AB
A
B
and
A
C
AC
A
C
at
T
T
T
and
S
S
S
, respectively. Prove that
A
S
=
A
T
AS = AT
A
S
=
A
T
a\sqrt{b^2+c^2+bc}+b\sqrt{c^2+a^2+ca}+c\sqrt{a^2+b^2+ab}>= \sqrt3
Let
a
,
b
,
c
a, b,c
a
,
b
,
c
be positive real numbers such that
a
b
+
b
c
+
c
a
=
1
ab + bc + ca = 1
ab
+
b
c
+
c
a
=
1
. Prove that
a
b
2
+
c
2
+
b
c
+
b
c
2
+
a
2
+
c
a
+
c
a
2
+
b
2
+
a
b
≥
3
a\sqrt{b^2 + c^2 + bc} + b\sqrt{c^2 + a^2 + ca} + c\sqrt{a^2 + b^2 + ab} \ge \sqrt3
a
b
2
+
c
2
+
b
c
+
b
c
2
+
a
2
+
c
a
+
c
a
2
+
b
2
+
ab
≥
3
8
2
Hide problems
1^1 + 3^3 + 5^5 + ... + (2^{2013} - 1)^{(2^{2013} - 1)} / 2^{2013} is odd
Prove that the ratio
1
1
+
3
3
+
5
5
+
.
.
.
+
(
2
2013
−
1
)
(
2
2013
−
1
)
2
2013
\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}
2
2013
1
1
+
3
3
+
5
5
+
...
+
(
2
2013
−
1
)
(
2
2013
−
1
)
is an odd integer.
social club has 101 members, each is fluent in same 50 languages
A social club has
101
101
101
members, each of whom is fluent in the same
50
50
50
languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let
A
A
A
be the number of three-member subsets such that the three distinct pairs among them use different languages. Find the maximum possible value of
A
A
A
.
7
2
Hide problems
olor the cells of a 50 x 50 chessboard into black and white
Ayman wants to color the cells of a
50
×
50
50 \times 50
50
×
50
chessboard into black and white so that each
2
×
3
2 \times 3
2
×
3
or
3
×
2
3 \times 2
3
×
2
rectangle contains an even number of white cells. Determine the number of ways Ayman can color the chessboard.
line tangent to excircle wanted
The excircle
ω
B
\omega_B
ω
B
of triangle
A
B
C
ABC
A
BC
opposite
B
B
B
touches side
A
C
AC
A
C
, rays
B
A
BA
B
A
and
B
C
BC
BC
at
B
1
,
C
1
B_1, C_1
B
1
,
C
1
and
A
1
A_1
A
1
, respectively. Point
D
D
D
lies on major arc
A
1
C
1
A_1C_1
A
1
C
1
of
ω
B
\omega_B
ω
B
. Rays
D
A
1
DA_1
D
A
1
and
C
1
B
1
C_1B_1
C
1
B
1
meet at
E
E
E
. Lines
A
B
1
AB_1
A
B
1
and
B
E
BE
BE
meet at
F
F
F
. Prove that line
F
D
FD
F
D
is tangent to
ω
B
\omega_B
ω
B
(at
D
D
D
).
5
2
Hide problems
X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0
Let
k
k
k
be a real number such that the product of real roots of the equation
X
4
+
2
X
3
+
(
2
+
2
k
)
X
2
+
(
1
+
2
k
)
X
+
2
k
=
0
X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0
X
4
+
2
X
3
+
(
2
+
2
k
)
X
2
+
(
1
+
2
k
)
X
+
2
k
=
0
is
−
2013
-2013
−
2013
. Find the sum of the squares of these real roots.
sum of the squares of its digits is a perfect square
We call a positive integer good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example,
122
122
122
and
34
34
34
are good and
304
304
304
and
12
12
12
are not not good. Prove that there exists a
n
n
n
-digit good number for every positive integer
n
n
n
.
2
4
Show problems
4
4
Show problems
3
4
Show problems
1
4
Show problems