MathDB

3

Part of 2017 ASDAN Math Tournament

Problems(9)

2017 Algebra #3

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8/20/2022
Let aa and bb be real numbers such that a5b8=12a^5b^8=12 and a8b13=18a^8b^{13}=18. Find abab.
2017Algebra Test
2017 Algebra Tiebreaker #3

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9/10/2022
For some integers bb and cc, neither of the equations below have real solutions: \begin{align*} 2x^2+bx+c&=0\\ 2x^2+cx+b&=0. \end{align*} What is the largest possible value of b+cb+c?
2017Algebra Tiebreaker
2017 Calculus #3

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9/17/2022
Let f(x)=x4+2x+1f(x)=x^4+2x+1. Find the slope of the tangent line to the curve at (0,1)(0,1).
2017Calculus Test
2017 Calculus Tiebreaker #3

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9/17/2022
Compute ∫01x2017−1log⁡xdx.\int_0^1\frac{x^{2017}-1}{\log x}dx.
2017Calculus Tiebreaker
2017 Discrete #3

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10/12/2022
What is the remainder when 210232^{1023} is divided by 10231023?
2017Discrete Math Test
2017 Discrete Tiebreaker #3

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10/12/2022
Alex and Zev are two members of a group of 20172017 friends who all know each other. Alex is trying to send a package to Zev. The delivery process goes as follows: Alex sends the package randomly to one of the people in the group. If this person is Zev, the delivery is done. Otherwise, the person who received the package also randomly sends it to someone in the group who hasn't held the package before and this process repeats until Zev gets the package. What is the expected number of deliveries made?
2017Discrete Math Tiebreaker
2017 Geometry #3

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11/19/2022
Line segment ABAB has length 1010. A circle centered at AA has radius 55, and a circle centered at BB has radius 535\sqrt{3}. What is the area of the intersection of the two circles?
2017Geometry Test
2017 Geometry Tiebreaker #3

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11/19/2022
Triangle ABCABC has AB=4,BC=6,CA=5AB=4,BC=6,CA=5. Let MM be the midpoint of BC‾\overline{BC} and PP the point on the circumcircle of △ABC\triangle ABC such that ∠MPA=90∘\angle MPA=90^\circ. Let points DD and EE lie on AC‾\overline{AC} and AB‾\overline{AB} respectively such that BD‾⊥AC‾\overline{BD}\perp\overline{AC} and CE‾⊥AB‾\overline{CE}\perp\overline{AB}. Find PDPE\tfrac{PD}{PE}.
2017Geometry Tiebreaker
2017 Guts #3

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11/25/2022
Four mathematicians, four physicists, and four programmers gather in a classroom. The 1212 people organize themselves into four teams, with each team having one mathematician, one physicist, and one programmer. How many possible arrangements of teams can exist?
2017Guts Round