Difference between revisions of "2010 AMC 10B Problems/Problem 24"
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Then the quarterly scores for the Raiders are <math>5, 10, 20, 40</math>, and those for the Wildcats are <math>5, 14, 23, 32</math>. Also <math>S_R = 75 = S_W + 1</math>. The total number of points scored by the two teams in the first half is <math>5+10+5+14=\boxed{\textbf{(E)}\ 34}</math>. | Then the quarterly scores for the Raiders are <math>5, 10, 20, 40</math>, and those for the Wildcats are <math>5, 14, 23, 32</math>. Also <math>S_R = 75 = S_W + 1</math>. The total number of points scored by the two teams in the first half is <math>5+10+5+14=\boxed{\textbf{(E)}\ 34}</math>. | ||
− | NOTE: When checking to make sure that r is an integer, we must also check <math>m = 4 </math>n = 3 for it to be complete, as those two solutions also fit the conditions | + | NOTE: When checking to make sure that r is an integer, we must also check <math>m = 4 </math>n = 3 for it to be complete, as those two solutions also fit the conditions |
== Solution 2 == | == Solution 2 == |
Revision as of 04:16, 27 January 2020
Problem
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than points. What was the total number of points scored by the two teams in the first half?
Solution 1
Let be the quarterly scores for the Raiders. We know
because the sequence is said to be increasing. We also know that each of
is an integer. We start by showing that
must also be an integer.
Suppose not, and say where
, and
. Then
must all divide
so
for some integer
. Then
and we see that even if
and
, we get
, which means that the only option for
is
. A quick check shows that even this doesn't work. Thus
must be an integer.
Let be the quarterly scores for the Wildcats. Let
. Let
. Then
implies that
, so
. The Raiders win by one point, so
- If
we get
which means
, which is absurd.
- If
we get
which means
, which is also absurd.
- If
we get
which means
. Reducing modulo 6 we get
. Since
we get
. Thus
. It then follows that
.
Then the quarterly scores for the Raiders are , and those for the Wildcats are
. Also
. The total number of points scored by the two teams in the first half is
.
NOTE: When checking to make sure that r is an integer, we must also check n = 3 for it to be complete, as those two solutions also fit the conditions
Solution 2
Represent the teams' scores as: and
We have
Factoring out the
from the left side of the equation, we can get
, or
Since both are increasing sequences, . We can check cases up to
because when
, we get
. When
Checking each of these cases individually back into the equation , we see that only when
and
, we get an integer value for
, which is
. The original question asks for the first half scores summed, so we must find
- Note: This solution is incomplete because it doesn't consider fractional values of
.
Solution 3
- Note: This solution is incomplete because it doesn't consider fractional values of
.
As above, represent the teams' scores as: and
.
Note that the Wildcat's sum, , is even. Therefore, since the Wildcat's sum is one less than the Raiders, the Raider's team's score should be odd. But if all of
are of the same parity, the sum will be even. If
is even, then the rest of the scores will be even, so clearly
is odd. Then,
is even.
But if , only
satisfies the requirement that the total score of each team is less than
. We can test this out and see it doesn't work.
Therefore, . If we try
, we quickly see only
satisfies all of the conditions. Therefore, our team's scores are
and
, and the answer is
.
Discussion
Couldn't we also say Raiders:1,2,4,8 and Wildcats:1,3,5,7 so the Raiders have won by one point? [s]Is this false because in actual basketball you can only score 2 or more points?[/s] ---> Raiders would have a total of 15 points and the WildCats would have a total of 16 points.
- Note: In example above, the Raiders score a total of 15 points while the Wildcats score 16. So it is the Wildcats who win by one point, not the Raiders.
- Note 2: It is possible in a game of basketball to score 1 point. A person is fouled and makes the free throw. That is 1 point.
See also
2010 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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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