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2023 UMD Math Competition Part I
#12
Real numbers
Real numbers
Source: UMD 2023 I #12
October 26, 2023
UMD
algebra
Problem Statement
Suppose for real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
we know
a
+
1
b
=
3
,
a + \dfrac 1b = 3,
a
+
b
1
=
3
,
and
b
+
3
c
=
1
3
.
b + \dfrac 3c = \dfrac 13.
b
+
c
3
=
3
1
.
What is the value of
c
+
27
a
?
c + \dfrac{27}a?
c
+
a
27
?
a
.
1
b
.
3
c
.
8
d
.
9
e
.
21
\mathrm a. ~ 1\qquad \mathrm b.~3\qquad \mathrm c. ~8 \qquad \mathrm d. ~9 \qquad \mathrm e. ~21
a
.
1
b
.
3
c
.
8
d
.
9
e
.
21
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