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Chile Junior Math Olympiad
2021 Chile Junior Math Olympiad
2021 Chile Junior Math Olympiad
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Chile Junior Math Olympiad
Subcontests
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2021 Chile NMO Juniors XXXIII
p1. Determine which of the following numbers is greater :
99
!
99!
99
!
or
5
0
99
50^{99}
5
0
99
. p2. Let
x
,
y
x , y
x
,
y
be positive real numbers with
x
>
y
x > y
x
>
y
that satisfy the relation
x
2
+
y
2
=
a
x
y
x^2 +y^2 = axy
x
2
+
y
2
=
a
x
y
where
a
a
a
is a real number greater than
2
2
2
. Find all possible values of
a
a
a
that make
x
+
y
x
−
y
\frac{x + y}{x - y}
x
−
y
x
+
y
an integer. [url=https://artofproblemsolving.com/community/c1068820h2917784p26063850]p3. In the figure below
A
B
C
D
ABCD
A
BC
D
is a square. https://cdn.artofproblemsolving.com/attachments/f/3/999a7e4eb004edcbe11951fc2adab1ebaa2243.png Segment
A
E
AE
A
E
is
3
3
3
units and segment
E
F
EF
EF
is
1
1
1
unit. The angles
∠
A
E
D
\angle AED
∠
A
E
D
and
∠
B
F
A
\angle BFA
∠
BF
A
are right. Calculate the length of the segments
F
C
FC
FC
[url=https://artofproblemsolving.com/community/c6h2945857p26372014]p4. A design
X
X
X
is an array of the digits
1
,
2
,
.
.
.
,
9
1,2,..., 9
1
,
2
,
...
,
9
in the shape of an
X
X
X
, for example, https://cdn.artofproblemsolving.com/attachments/8/e/d371a2cd442cb7a8784e1cc7635344df722e20.png We will say that a design
X
X
X
is balanced if the sum of the numbers of each of the diagonals match. Determine the number of designs
X
X
X
that are balanced.