Placement Exam For Level B - Summer High School Program, 2015 Page 4
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42015 Summer High School Program
Placement Exam for Level B
NAME: ______________________
Long Answer: Write your solution in the space below each question. Make sure you include sufficient justification.
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17. Prove that the equation ��
= 3 + 7��
has no solutions in integers.
18. Let �� ( �� ) denote how many positive integers with digits 1 and/or 2 have �� be the sum of their digits.
For example, �� ( 4 ) = 5 because there are 5 such integers (1111, 112, 121, 211, and 22)
a. What pattern is made by the values in the sequence �� ( 1 ) , �� ( 2 ) , �� ( 3 ) , �� ( 4 ) , �� ( 5 ) , … ?
b. Prove that the pattern holds forever.
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New York Math Circle
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