MathDB

9

Part of 2016 ASDAN Math Tournament

Problems(6)

2016 Algebra #9

Source:

8/6/2022
Let P(x)P(x) be a monic cubic polynomial. The line y=0y=0 and y=my=m intersect P(x)P(x) at points A,C,EA,C,E and B,D<FB,D<F from left to right for a positive real number mm. If AB=7AB=\sqrt{7}, CD=15CD=\sqrt{15}, and EF=10EF=\sqrt{10}, what is the value of mm?
2016Algebra Test
2016 Calculus #9

Source:

8/8/2022
Compute ∫0∞ln⁡(1+x111+x3)(1+x2)ln⁡xdx.\int_0^\infty\frac{\ln\left(\frac{1+x^{11}}{1+x^3}\right)}{(1+x^2)\ln x}dx.
2016Calculus Test
2016 Discrete #9

Source:

8/10/2022
Define ϕn(x)\phi_n(x) to be the number of integers yy less than or equal to nn such that gcd⁡(x,y)=1\gcd(x,y)=1. Also, define m=lcm(2016,6102)m=\text{lcm}(2016,6102). Compute ϕmm(2016)ϕmm(6102).\frac{\phi_{m^m}(2016)}{\phi_{m^m}(6102)}.
2016Discrete Math Test
2016 Geometry #9

Source:

8/11/2022
A cyclic quadrilateral ABCDABCD has side lengths AB=14AB=14, BC=19BC=19, CD=26CD=26, and DA=29DA=29. Compute the sine of the smaller angle between diagonals ACAC and BDBD.
2016Geometry Test
2016 Guts #9

Source:

8/14/2022
An equilateral triangle △ABC\triangle ABC with side length 33 has center OO. A circle is drawn centered at OO with radius 11. Find the area of the region contained inside both the triangle and circle.
2016Guts Round
2016 Team #9

Source:

8/17/2022
A cake in the shape of a rectangular prism has dimensions 6 cm×14 cm×21 cm6\text{ cm}\times14\text{ cm}\times21\text{ cm}. It is cut into 17641764 equally sized cubes such that each cube is 1 cm31\text{ cm}^3. Andy the ant starts at one corner of the cake and eats through the cake in a straight line to the opposite corner of the cake. How many of the 1 cm31\text{ cm}^3 cubes does Andy bite through?
2016team test