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2013 SDMO (Middle School)
Part of
SDMO (Middle School)
Subcontests
(5)
5
1
Hide problems
Nice numbers by splitting into three equal-sum partitions
We say that a positive integer
n
n
n
is nice if we can split the numbers
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
into three sets, so that the sum of the numbers in each set is the same. For example, the number
12
12
12
is nice because we can divide the numbers
1
,
2
,
…
,
12
1,2,\ldots,12
1
,
2
,
…
,
12
into the sets
{
1
,
2
,
4
,
5
,
6
,
8
}
\left\{1,2,4,5,6,8\right\}
{
1
,
2
,
4
,
5
,
6
,
8
}
,
{
7
,
9
,
10
}
\left\{7,9,10\right\}
{
7
,
9
,
10
}
, and
{
3
,
11
,
12
}
\left\{3,11,12\right\}
{
3
,
11
,
12
}
, and the sum of the numbers in each set is
26
26
26
.Find all nice positive integers.
4
1
Hide problems
Power sum inequalities
Let
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
be positive real numbers such that
a
+
b
=
c
+
d
a+b=c+d
a
+
b
=
c
+
d
and
a
2
+
b
2
>
c
2
+
d
2
a^2+b^2>c^2+d^2
a
2
+
b
2
>
c
2
+
d
2
. Prove that
a
3
+
b
3
>
c
3
+
d
3
a^3+b^3>c^3+d^3
a
3
+
b
3
>
c
3
+
d
3
.
3
1
Hide problems
Square, circle, and areas
Let
A
B
C
D
ABCD
A
BC
D
be a square, and let
Γ
\Gamma
Γ
be the circle that is inscribed in square
A
B
C
D
ABCD
A
BC
D
. Let
E
E
E
and
F
F
F
be points on line segments
A
B
AB
A
B
and
A
D
AD
A
D
, respectively, so that
E
F
EF
EF
is tangent to
Γ
\Gamma
Γ
. Find the ratio of the area of triangle
C
E
F
CEF
CEF
to the area of square
A
B
C
D
ABCD
A
BC
D
.
2
1
Hide problems
Sequence a_i shows number of times i shows up
Find all sequences
(
a
0
,
a
1
,
a
2
,
a
3
)
\left(a_0,a_1,a_2,a_3\right)
(
a
0
,
a
1
,
a
2
,
a
3
)
, where for each
k
k
k
,
0
≤
k
≤
3
0\leq k\leq3
0
≤
k
≤
3
,
a
k
a_k
a
k
is the number of times that the number
k
k
k
appears in the sequence
(
a
0
,
a
1
,
a
2
,
a
3
)
\left(a_0,a_1,a_2,a_3\right)
(
a
0
,
a
1
,
a
2
,
a
3
)
.
1
1
Hide problems
Deleting points to destroy squares
Consider the
4
×
4
4\times4
4
×
4
array of
16
16
16
dots, shown below.[asy] size(2cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((0,3)); dot((1,3)); dot((2,3)); dot((3,3)); [/asy]Counting the number of squares whose vertices are among the
16
16
16
dots and whose sides are parallel to the sides of the grid, we find that there are nine
1
×
1
1\times1
1
×
1
squares, four
2
×
2
2\times2
2
×
2
squares, and one
3
×
3
3\times3
3
×
3
square, for a total of
14
14
14
squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the
14
14
14
squares is missing at least one vertex?