document.write( "Question 935840: You play the following game with your worst enemy, who wishes to keep your score down. A
\n" ); document.write( "referee gets to pick a positive integer n. Then your enemy gets to pick n different real numbers.
\n" ); document.write( "You get to multiply different pairs of these numbers together, as many different pairs as you
\n" ); document.write( "want. You get one point for each pair whose product is nonnegative. So for instance, if n = 3
\n" ); document.write( "and your enemy picks 0, −8 and −π, you would pick all possible pairs, {0, −8}, {0, −π} and
\n" ); document.write( "{−8, −π}, because all 3 products (0, 0, 8π) are nonnegative. You would get 3 points.
\n" ); document.write( "Determine with proof the highest score you can be sure to get against a smart enemy. Your
\n" ); document.write( "answer will depend on n. \n" ); document.write( "

Algebra.Com's Answer #569204 by KMST(5328) $\"\"$ $\"About$

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The product of a pair of numbers with the same sign (both positive or both negative) is positive.
\n" ); document.write( "The product of zero and any number is zero, which is non-negative.
\n" ); document.write( "A smart enemy will not pick zero, because that will give you a lot of non-negative products.
\n" ); document.write( "He/she will have to pick positive and negative numbers.
\n" ); document.write( "If $\"n\"$ is an even number, $\"n=2k\"$ ,
\n" ); document.write( "a smart enemy would pick $\"k\"$ negative numbers and $\"k\"$ positive numbers would give you $\"k%28k-1%29=highlight%28k%5E2-k%29\"$ positive products,
\n" ); document.write( "from $\"k%28k-1%29%2F2\"$ pairs of positive numbers,
\n" ); document.write( "and $\"k%28k-1%29%2F2\"$ pairs of positive numbers.
\n" ); document.write( "Splitting the negative and positive choices unevenly,
\n" ); document.write( "let's say $\"k%2Ba\"$ positive numbers and $\"k-a\"$ negative numbers, with $\"a%3C=k\"$ ,
\n" ); document.write( "would give you $\"%28k%2Ba%29%28k%2Ba-1%29%2F2\"$ positive products,
\n" ); document.write( "and $\"%28k-a%29%28k-a-1%29%2F2\"$ negative products,
\n" ); document.write( "for a total of $\"k%5E2-k%2Ba%5E2\"$ ,
\n" ); document.write( "so a smart enemy would not make such a choice.
\n" ); document.write( "If one more number could be picked, $\"n=2k%2B1\"$ ,
\n" ); document.write( "a smart enemy would choose $\"k\"$ positive and $\"k\"$ negative numbers,
\n" ); document.write( "plus an additional number.
\n" ); document.write( "The additional number could be positive or negative,
\n" ); document.write( "and paired with the $\"k\"$ numbers of the same sign,
\n" ); document.write( "would give you $\"k\"$ additional positive products,
\n" ); document.write( "for a total of $\"k%5E2-k%2Bk=highlight%28k%5E2%29\"$ points.
\n" ); document.write( "The formulas work even in the case of
\n" ); document.write( " $\"n=1=2%2A0%2B1\"$ , where $\"k=0\"$ and your score would be $\"k%5E2=0%5E2=0\"$ ,
\n" ); document.write( "and $\"n=2=2%2A1\"$ , where $\"k=1\"$ and your score would be $\"k%5E2k=1%5E2-1=1-1=0\"$ . \n" ); document.write( "

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