2 assignment2.pdf - Adobe Acrobat Reader DCFile Edit View Sign Window HelpHomeToolsassignment2.pdfa2-Q.pdfa2sol.pdfDiscreteStructuresf...Sign In2 / 4Question 3:• Let a1, a2, ..., a7 be real numbers that are all contained in the interval [-1/2, 7/2].Use the Pigeonhole Principle to prove that there are two distinct indices i and j suchthat 0 < a; – aj < T/6.1• Let a1, a2, ..., a7 be real numbers such that a;a; -1 for all i + j.Prove that there are two distinct indices i and j such thata; – aj11+ a¿ajV3Hint: For each i, let p; be the point with coordinates (1, a;), and consider the anglebetween the x-axis and the vector from the origin to p;. You learned in high schoolthattan a – tan 3tan(a – B)1+ tan a tan 311:21 PM10/8/2020

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1. The pigeonhole principle states that if n items are put into k boxes with n>k, then at least…

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Let a1, a2,...,a, be real numbers that are all contained in the interval [-7 /2, 7 /2]. Use the Pigeonhole Principle to prove that there are two distinct indices i and j such that 0

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Chapter2: Second-order Linear Odes

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Question 3:
• Let a1, a2, ..., a7 be real numbers that are all contained in the interval [-1/2, 7/2].
Use the Pigeonhole Principle to prove that there are two distinct indices i and j such
that 0 < a; – aj < T/6.
1
• Let a1, a2, ..., a7 be real numbers such that a;a; -1 for all i + j.
Prove that there are two distinct indices i and j such that
a; – aj
1
1+ a¿aj
V3
Hint: For each i, let p; be the point with coordinates (1, a;), and consider the angle
between the x-axis and the vector from the origin to p;. You learned in high school
that
tan a – tan 3
tan(a – B)
1+ tan a tan 3
11:21 PM
10/8/2020

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