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Let a, b, c, d, and e be positive integers with a + b + c + d + e = 2010 and let M be the largest of the sum a + b, b + c, c + d and d + e. What is the smallest possible value of M?
A. 670
B. 671
C. 802
D. 803
E. 804
A. 670
B. 671
C. 802
D. 803
E. 804
21 Apr 2019, 21:54
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To make the largest sum as small as possible, we'll want to make all of the sums equal (or as close to equal as we're allowed to make them), because if, say, d+e were the largest sum, and if it were larger than b+c, we'd be able to reduce d or e, and increase b or c by a corresponding amount, and thereby make d+e, the largest sum, smaller. So we wouldn't have found the smallest possible largest sum in this case.
And if all of the sums are equal, so a+b = b+c, b+c = c+d and so on, we find that a=c=e and b=d. If we permit zero, we'd find this arrangement makes M smallest:
670, 0, 670, 0, 670
but since zero is not allowed (the letters must be positive integers), the answer will be 671, which we can get with an arrangement like this:
669, 2, 668, 2, 669
And if all of the sums are equal, so a+b = b+c, b+c = c+d and so on, we find that a=c=e and b=d. If we permit zero, we'd find this arrangement makes M smallest:
670, 0, 670, 0, 670
but since zero is not allowed (the letters must be positive integers), the answer will be 671, which we can get with an arrangement like this:
669, 2, 668, 2, 669
General Discussion
11 Oct 2019, 11:26
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IanStewart
To make the largest sum as small as possible, we'll want to make all of the sums equal (or as close to equal as we're allowed to make them), because if, say, d+e were the largest sum, and if it were larger than b+c, we'd be able to reduce d or e, and increase b or c by a corresponding amount, and thereby make d+e, the largest sum, smaller. So we wouldn't have found the smallest possible largest sum in this case.
And if all of the sums are equal, so a+b = b+c, b+c = c+d and so on, we find that a=c=e and b=d. If we permit zero, we'd find this arrangement makes M smallest:
670, 0, 670, 0, 670
but since zero is not allowed (the letters must be positive integers), the answer will be 671, which we can get with an arrangement like this:
669, 2, 668, 2, 669
And if all of the sums are equal, so a+b = b+c, b+c = c+d and so on, we find that a=c=e and b=d. If we permit zero, we'd find this arrangement makes M smallest:
670, 0, 670, 0, 670
but since zero is not allowed (the letters must be positive integers), the answer will be 671, which we can get with an arrangement like this:
669, 2, 668, 2, 669
I have a doubt. Why can't it be 669,1,669,1,669?
And why the answer can't be 670?
