document.write( "Question 1204042: There are 18 questions in a math competition. The scores of each question are allotted in the following ways: 1 mark will be given for a correct answer, 1 mark will be deducted from a wrong answer and 0 mark will be given for a blank answer. find the minimum number of candidate(s) to ensure that 3 candidates will have the same scores in the competition \n" ); document.write( "

Algebra.Com's Answer #840050 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "There are 18 questions in a math competition. The scores of each question are allotted
\n" ); document.write( "in the following ways: 1 mark will be given for a correct answer, 1 mark will be deducted
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\n" ); document.write( "find the minimum number of candidate(s) to ensure that 3 candidates will have the same scores in the competition
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document.write( "First, the set of possible scores is the set of all integer numbers from -18 to 18\r\n" );
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document.write( "       { -18, -17, -16, . . . , 0, 1, 2, 3, . . . 16, 17, 18 }.\r\n" );
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document.write( "It is so because for all 18 correct answer the score is 18;\r\n" );
document.write( "                 for 17 correct and 1 blank the score is 17;\r\n" );
document.write( "                 for 16 correct and 2 blank the score is 16, same as for 17 correct and 1 incorrect;\r\n" );
document.write( "                 for 15 correct and 3 blank the score is 15;\r\n" );
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document.write( "                 and so on till 18 incorrect, which gives the score -18.\r\n" );
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document.write( "Thus we have 18 + 18 + 1 = 37 possible different scores.\r\n" );
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document.write( "In order to ensure that 3 candidates will have the same scores in the competition,\r\n" );
document.write( "2*37 + 1 = 75 candidates should participate (according to the pigeon principle).\r\n" );
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document.write( "ANSWER.  In order to ensure that 3 candidates will have the same scores in the competition, \r\n" );
document.write( "         minimum 75 candidates should participate.\r\n" );
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