MathDB

10

Part of 2016 ASDAN Math Tournament

Problems(6)

2016 Algebra #10

Source:

8/6/2022
Let a1,a2,…a_1,a_2,\dots be a sequence of real numbers satisfying an+1an−an+2an−an+1an+2an2=nan+2an+1an.\frac{a_{n+1}}{a_n}-\frac{a_{n+2}}{a_n}-\frac{a_{n+1}a_{n+2}}{a_n^2}=\frac{na_{n+2}a_{n+1}}{a_n}. Given that a1=−1a_1=-1 and a2=−12a_2=-\tfrac{1}{2}, find the value of a9a20\tfrac{a_9}{a_{20}}.
2016Algebra Test
2016 Calculus #10

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8/8/2022
Using the fact that ∑n=1∞1n2=π26,\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}, compute ∫01(ln⁡x)ln⁡(1−x)dx.\int_0^1(\ln x)\ln(1-x)dx.
2016Calculus Test
2016 Discrete #10

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8/10/2022
Let S\mathcal{S} be the set of all possible 99-digit numbers that use 1,2,3,…,91,2,3,\dots,9 each exactly once as a digit. What is the probability that a randomly selected number nn from S\mathcal{S} is divisible by 2727?
2016Discrete Math Test
2016 Geometry #10

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8/11/2022
Compute the radius of the sphere inscribed in the tetrahedron with coordinates (2,0,0)(2,0,0), (4,0,0)(4,0,0), (0,1,0)(0,1,0), and (0,0,3)(0,0,3).
2016Geometry Test
2016 Guts #10

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8/14/2022
A point PP and a segment ABAB with length 2020 are randomly drawn on a plane. Suppose that the probability that a randomly selected line passing through PP intersects segment ABAB is 12\tfrac{1}{2}. Next, randomly choose point QQ on segment ABAB. What is the probability with respect to choosing QQ that a circle centered at QQ passing through PP contains both AA and BB in its interior?
2016Guts Round
2016 Team #10

Source:

8/17/2022
Compute the smallest positive integer xx which satisfies x2−8x+1≡0(mod22)x^2-8x+1\equiv0\pmod{22} and x2−22x+1≡0(mod8)x^2-22x+1\equiv0\pmod{8}.
2016team test