2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2
Problem
Suppose , and
all denote distinct digits from
to
. If
, what are
, and
?
Solution
and
Another way of stating the question is that
=
.
This means that the ending digit of
or
has to be the ending digit of a square number, or {
}. Keep in mind that
is a 5-digit number. We know that
cannot be 0 because
would become a 2-digit number. It cannot be 4 or over because
would have 6 digits instead of 5. This only leaves
= 1. Because the last digit of
is
, we can conclude that
must be
or
. However, since all the digits are unique,
=
.
Now the expression becomes:
=
Because the first two digits of are
and
, A$$ (Error compiling LaTeX. Unknown error_msg) cannot take on too many values because if it becomes 5,
will become 2. It cannot be four either because even then,
would be too small. Now, if A is tested from the set {
}:
:
=
(Does not work)
:
=
(Does not work)
:
=
(Does not work)
:
=
This works as it fits the template:
See also
2017 UNM-PNM Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNM-PNM Problems and Solutions |