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In an archery contest, each arrow fired by a contestant earns exactly 0, 3, or 6 points: 3 points if it hits the outer ring, 6 points if it hits the bullseye, and 0 points if it misses the target completely. Alexandra scored a total of T points in the contest. Is T an even number?
(1) Exactly five of Alexandra’s arrows hit the bullseye.
(2) Exactly three of Alexandra’s arrows hit the outer ring.
(1) Exactly five of Alexandra’s arrows hit the bullseye.
(2) Exactly three of Alexandra’s arrows hit the outer ring.
02 May 2025, 02:29
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Official Solution:
In an archery contest, each arrow fired by a contestant earns exactly 0, 3, or 6 points: 3 points if it hits the outer ring, 6 points if it hits the bullseye, and 0 points if it misses the target completely. Alexandra scored a total of T points in the contest. Is T an even number?
Assuming Alexandra hit the outer ring with \(m\) arrows and the bullseye with \(n\) arrows, her total score is \(T = 3m + 6n\). Since \(6n\) is always even, whether \(T\) is even or odd depends only on \(3m\). If \(m\) is even, then \(T\) is even; if \(m\) is odd, then \(T\) is odd.
(1) Exactly five of Alexandra’s arrows hit the bullseye.
This gives \(n = 5\). As established, the value of \(n\) has no effect on whether \(T\) is even or odd. Not sufficient.
(2) Exactly three of Alexandra’s arrows hit the outer ring.
This gives \(m = 3\). Since \(m\) is odd, \(T\) is odd. Thus, the answer to the question is NO. Sufficient.
Answer: B
In an archery contest, each arrow fired by a contestant earns exactly 0, 3, or 6 points: 3 points if it hits the outer ring, 6 points if it hits the bullseye, and 0 points if it misses the target completely. Alexandra scored a total of T points in the contest. Is T an even number?
Assuming Alexandra hit the outer ring with \(m\) arrows and the bullseye with \(n\) arrows, her total score is \(T = 3m + 6n\). Since \(6n\) is always even, whether \(T\) is even or odd depends only on \(3m\). If \(m\) is even, then \(T\) is even; if \(m\) is odd, then \(T\) is odd.
(1) Exactly five of Alexandra’s arrows hit the bullseye.
This gives \(n = 5\). As established, the value of \(n\) has no effect on whether \(T\) is even or odd. Not sufficient.
(2) Exactly three of Alexandra’s arrows hit the outer ring.
This gives \(m = 3\). Since \(m\) is odd, \(T\) is odd. Thus, the answer to the question is NO. Sufficient.
Answer: B
