2004 AMC 10B Problems/Problem 21
Revision as of 23:04, 23 January 2020 by Scrabbler94 (talk | contribs) (→Solution: solution 2 isn't really correct as the positions of the terms which appear in both sequences change after shifting.)
Problem
Let ;
;
and
;
;
be two arithmetic progressions. The set
is the union of the first
terms of each sequence. How many distinct numbers are in
?
Solution
The two sets of terms are and
.
Now . We can compute
. We will now find
.
Consider the numbers in . We want to find out how many of them lie in
. In other words, we need to find out the number of valid values of
for which
.
The fact "" can be rewritten as "
, and
".
The first condition gives , the second one gives
.
Thus the good values of are
, and their count is
.
Therefore , and thus
.
See also
2004 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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