MathDB

3

Part of 2016 ASDAN Math Tournament

Problems(10)

2016 Algebra Tiebreaker #3

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8/8/2022
Denote the dot product of two sequences ⟨x1,…,xn⟩\langle x_1,\dots,x_n\rangle and ⟨y1,…,yn⟩\langle y_1,\dots,y_n\rangle to be x1y1+x2y2+⋯+xnyn.x_1y_1+x_2y_2+\dots+x_ny_n. Let ⟨a1,…,an⟩\langle a_1,\dots,a_n\rangle and ⟨b1,…,bn⟩\langle b_1,\dots,b_n\rangle be two sequences of consecutive integers (i.e. for 1≤i,i+1≤n1\leq i,i+1\leq n, ai+1=ai+1a_i+1=a_{i+1} and similarly for bib_i). Minnie permutes the two sequences so that their dot product, mm, is minimized. Maximilian permutes the two sequences so that their dot product, MM, is maximized. Given that m=4410m=4410 and M=4865M=4865, compute nn, the number of terms in each sequence.
2016Algebra Tiebreaker
2016 Algebra #3

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8/6/2022
Real numbers x,y,zx,y,z form an arithmetic sequence satisfying \begin{align*} x+y+z&=6\\ xy+yz+zx&=10. \end{align*} What is the absolute value of their common difference?
2016Algebra Test
2016 Calculus #3

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8/8/2022
If f(x)=exg(x)f(x)=e^xg(x), where g(2)=1g(2)=1 and g′(2)=2g'(2)=2, find f′(2)f'(2).
2016Calculus Test
2016 Calculus Tiebreaker #3

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8/8/2022
Compute ∫0π1−sin⁡x1+sin⁡xdx.\int_0^\pi\frac{1-\sin x}{1+\sin x}dx.
2016Calculus Tiebreaker
2016 Discrete #3

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8/10/2022
Julia adds up the numbers from 11 to 20162016 in a calculator. However, every time she inputs a 22, the calculator malfunctions and inputs a 33 instead (for example, when Julia inputs 202202, the calculator inputs 303303 instead). How much larger is the total sum returned by the broken calculator? (No 22s are replaced by 33s in the output, and the calculator only malfunctions while Julia is inputting numbers.)
2016Discrete Math Test
2016 Discrete Tiebreaker #3

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8/10/2022
Find the 20162016th smallest positive integer that is a solution to xx≡x(mod5)x^x\equiv x\pmod{5}.
2016Discrete Math Tiebreaker
2016 Geometry Tiebreaker #3

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8/11/2022
Let HH be the orthocenter of triangle ABCABC, and DD be the foot of AA onto BCBC. Given that DB=3DB=3, DH=2DH=2, and DC=6DC=6, calculate HAHA.
2016Geometry Tiebreaker
2016 Geometry #3

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8/11/2022
Let ABCDABCD be a unit square, and let there be two unit circles centered at CC and DD. Let PP be the point of intersection of the two circles inside the square. Compute ∠APB\angle APB in degrees.
2016Geometry Test
2016 Guts #3

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8/14/2022
A number nn is <spanclass=′latex−italic′>almostprime</span><span class='latex-italic'>almost prime</span> if any of n−2n-2, n−1n-1, nn, n+1n+1, or n+2n+2 is prime. Compute the smallest positive integer that is not <spanclass=′latex−italic′>almostprime</span><span class='latex-italic'>almost prime</span>.
2016Guts Round
2016 Team #3

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8/17/2022
Moor has 20162016 white rabbit candies. He and his nn friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that nn is a positive integer (Moor has at least 11 friend), how many possible values of nn exist?
2016team test