MathDB
Problems
Contests
National and Regional Contests
Cuba Contests
Cuba MO
2011 Cuba MO
2011 Cuba MO
Part of
Cuba MO
Subcontests
(7)
6
1
Hide problems
<O_1DM = <ODO_2 -- 2011 Cuba MO 2.6
Let
A
B
C
ABC
A
BC
be a triangle with circumcenter
O
O
O
. Let
ω
(
O
1
)
\omega (O_1)
ω
(
O
1
)
be the circumference which passes through
A
A
A
and
B
B
B
and is tangent to
B
C
BC
BC
at
B
B
B
.
ω
(
O
2
)
\omega (O_2)
ω
(
O
2
)
the circle that passes through
A
A
A
and
C
C
C
and is tangent to
B
C
BC
BC
at
C
C
C
. Let
M
M
M
the midpoint of
O
1
O
2
O_1O_2
O
1
O
2
and
D
D
D
the symmetric point of
O
O
O
with respect to
A
A
A
. Prove that
∠
O
1
D
M
=
∠
O
D
O
2
\angle O_1DM = \angle ODO_2
∠
O
1
D
M
=
∠
O
D
O
2
.
7
1
Hide problems
max n, such that lcm of n numbers <2011
Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than
2011
2011
2011
.
4
1
Hide problems
x_1 + 2x_2 + 3x_3 +...+ 24x_{24} - 439 <= (x^2_1+... + x^2_{24})/2 + 2011
Let
x
1
,
x
2
,
.
.
.
,
x
24
x_1, x_2, ..., x_{24}
x
1
,
x
2
,
...
,
x
24
be real numbers. prove that
x
1
+
2
x
2
+
3
x
3
+
.
.
.
+
24
x
24
−
439
≤
x
1
2
+
x
2
2
+
.
.
.
+
x
24
2
2
+
2011.
x_1 + 2x_2 + 3x_3 +...+ 24x_{24} - 439 \le \frac{x^2_1+x^2_2+... + x^2_{24}}{2}+ 2011.
x
1
+
2
x
2
+
3
x
3
+
...
+
24
x
24
−
439
≤
2
x
1
2
+
x
2
2
+
...
+
x
24
2
+
2011.
5
1
Hide problems
f(x)f(y) = 2f(x + y) + 9xy
Determine all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
f
(
x
)
f
(
y
)
=
2
f
(
x
+
y
)
+
9
x
y
∀
x
,
y
∈
R
.
f(x)f(y) = 2f(x + y) + 9xy \ \ \forall x, y \in R.
f
(
x
)
f
(
y
)
=
2
f
(
x
+
y
)
+
9
x
y
∀
x
,
y
∈
R
.
3
2
Hide problems
1x2 in 2011x2011
We have a board of
2011
×
2011
2011 \times 2011
2011
×
2011
, divided by lines parallel to the edges into
1
×
1
1 \times 1
1
×
1
squares. Manuel, Reinaldo and Jorge (at that time order) play to form squares with vertices at the vertices of the grid. The one who forms the last possible square wins, so that its sides do not cut the sides of any unit square. Who can be sure that he will win?
n^2 = d_1 + d_2^2+d_3^3 +d_4^4
Let
n
n
n
be a positive integer and let
1
=
d
1
<
d
2
<
d
3
<
d
4
1 = d_1 < d_2 < d_3 < d_4
1
=
d
1
<
d
2
<
d
3
<
d
4
the four smallest divisors of
n
n
n
. Find all
n
n
n
such that
n
2
=
d
1
+
d
2
2
+
d
3
3
+
d
4
4
.
n^2 = d_1 + d_2^2+d_3^3 +d_4^4.
n
2
=
d
1
+
d
2
2
+
d
3
3
+
d
4
4
.
2
2
Hide problems
1x2x2 in 20x20x20
A cube of dimensions
20
×
20
×
20
20 \times 20 \times 20
20
×
20
×
20
is constructed with blocks of
1
×
2
×
2
1 \times 2 \times 2
1
×
2
×
2
. Prove that there is a line that passes through the cube but not any block.
3x^4-2024y+1= 0 diophantine
Determine all the integer solutions of the equation
3
x
4
−
2024
y
+
1
=
0
3x^4-2024y+1= 0
3
x
4
−
2024
y
+
1
=
0
.
1
2
Hide problems
3 ways to move a token in 2010x2001
There is a board with
2010
2010
2010
rows and
2001
2001
2001
columns, on it there is a token located in the upper left box that can perform one of the following operations:(A) Walk 3 steps horizontally or vertically. (B) Walk 2 steps to the right and 3 steps down. (C) Walk 2 steps to the left and 2 steps up.With the condition that immediately after carrying out an operation on (B) or (C) it is mandatory to take a step to the right before perform the following operation. It is possible to exit the board, so count the number of steps necessary, entering through the other end of the row or column from which it exits, as if the board outside circular (example: from the beginning you can walk to the square located in row
1
1
1
and column
1999
1999
1999
). Will it be possible that after
2011
2011
2011
operations allowed the checker to land exactly on the bottom square right?
P(x) = x^3 + (t - 1)x^2 - (t + 3)x + 1
Let
P
(
x
)
=
x
3
+
(
t
−
1
)
x
2
−
(
t
+
3
)
x
+
1
P(x) = x^3 + (t - 1)x^2 - (t + 3)x + 1
P
(
x
)
=
x
3
+
(
t
−
1
)
x
2
−
(
t
+
3
)
x
+
1
. For what values of real
t
t
t
the sum of the squares and the reciprocals of the roots of
P
(
x
)
P(x)
P
(
x
)
is minimum?