2023 AMC 10B Problems/Problem 5
Problem
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds to each number in the list and finds that the sum of her new numbers is
. Lara multiplies each number in the list by
and finds that the sum of her new numbers is also
. How many numbers are written on the blackboard?
Solution
Let there be numbers in the list of numbers, and let their sum be
. Then we have the following
From the second equation, , so
. Solving, we find
~Mintylemon66
Solution 2
Let where
represents the
th number written on the board. Lara's multiplied each number by
, so her sum will be
. This is the same as
. We are given this quantity is equal to
, so the original numbers add to
. Maddy adds
to each of the
terms which yields,
. This is the same as the sum of the original series plus
. Setting this equal to
,
~vsinghminhas