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SDMO (Middle School)
2009 SDMO (Middle School)
2009 SDMO (Middle School)
Part of
SDMO (Middle School)
Subcontests
(5)
5
1
Hide problems
Digits and squares
Let
A
=
33
⋯
3
A=33\cdots3
A
=
33
⋯
3
, where
A
A
A
contains
2009
2009
2009
3
3
3
s. Let
B
=
11
⋯
1088
⋯
89
B=11\cdots1088\cdots89
B
=
11
⋯
1088
⋯
89
, where
B
B
B
contains
2008
2008
2008
1
1
1
s and
2008
2008
2008
8
8
8
s. Prove that
A
2
=
B
A^2=B
A
2
=
B
.
4
1
Hide problems
Minima and maxima of a subset
Sally randomly chooses three different numbers from the set
{
1
,
2
,
…
,
14
}
\left\{1,2,\ldots,14\right\}
{
1
,
2
,
…
,
14
}
. What is the probability that the sum of her smallest number and her biggest number is at least
15
15
15
?
3
1
Hide problems
Fractional Diophantine
Find all ordered pairs of positive integers
(
a
,
b
)
\left(a,b\right)
(
a
,
b
)
such that
1
a
+
a
b
+
1
a
b
=
1.
\frac{1}{a}+\frac{a}{b}+\frac{1}{ab}=1.
a
1
+
b
a
+
ab
1
=
1.
2
1
Hide problems
Square geometry
Let
A
B
C
D
ABCD
A
BC
D
be a square, and let
E
E
E
and
F
F
F
be points on sides
A
B
‾
\overline{AB}
A
B
and
C
D
‾
\overline{CD}
C
D
, respectively, such that
A
E
:
E
B
=
A
F
:
F
D
=
2
:
1
AE:EB=AF:FD=2:1
A
E
:
EB
=
A
F
:
F
D
=
2
:
1
. Let
G
G
G
be the intersection of
A
F
‾
\overline{AF}
A
F
and
D
E
‾
\overline{DE}
D
E
, and let
H
H
H
be the intersection of
B
F
‾
\overline{BF}
BF
and
C
E
‾
\overline{CE}
CE
. Find the ratio of the area of quadrilateral
E
G
F
H
EGFH
EGF
H
to the area of square
A
B
C
D
ABCD
A
BC
D
.[asy] size(5cm,0); draw((0,0)--(3,0)); draw((3,0)--(3,3)); draw((3,3)--(0,3)); draw((0,3)--(0,0)); draw((0,0)--(2,3)); draw((1,0)--(3,3)); draw((0,3)--(1,0)); draw((2,3)--(3,0)); label("
A
A
A
",(0,3),NW); label("
B
B
B
",(3,3),NE); label("
C
C
C
",(3,0),SE); label("
D
D
D
",(0,0),SW); label("
E
E
E
",(2,3),N); label("
F
F
F
",(1,0),S); label("
G
G
G
",(0.66666667,1),E); label("
H
H
H
",(2.33333333,2),W); [/asy]
1
1
Hide problems
A farmer and his trees
A farmer buys a batch of trees, which he wishes to plant in a square grid. For example, if he had
25
25
25
trees, then he could plant them as shown below.[asy] size(3cm,0); dot((0,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((1,0)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((2,0)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((3,0)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((4,0)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); [/asy]However, the farmer finds that he cannot plant his trees in a square grid. If he had
20
20
20
more trees, or if he had
39
39
39
fewer trees, then he could plant his trees in a square grid. How many trees did the farmer buy?