MathDB

10

Part of 2017 ASDAN Math Tournament

Problems(5)

2017 Algebra #10

Source:

8/20/2022
Let ζ=e2πi/36\zeta=e^{2\pi i/36}. Compute ∏gcd⁡(a,36)=1a=135(ζa−2).\prod_{\stackrel{a=1}{\gcd(a,36)=1}}^{35}(\zeta^a-2).
2017Algebra Test
2017 Calculus #10

Source:

9/17/2022
Compute lim⁡k→∞(20171/kk+1+20172/kk+12+⋯+2017k/kk+1k).\lim_{k\rightarrow\infty}\left(\frac{2017^{1/k}}{k+1}+\frac{2017^{2/k}}{k+\frac{1}{2}}+\dots+\frac{2017^{k/k}}{k+\frac{1}{k}}\right).
2017Calculus Test
2017 Discrete #10

Source:

10/12/2022
Alice lives on a continent with 66 countries labeled 11 through 66. Each country randomly chooses one other country to allow entry from. Alice can travel to any country that allows entry from the country she is currently in, and can travel along a path through multiple countries in this manner. If Alice starts in county 11, what is the expected number of countries that she can reach (including country 11)?
2017Discrete Math Test
2017 Geometry #10

Source:

11/19/2022
Triangle ABCABC is inscribed in circle γ1\gamma_1 with radius r1r_1. Let γ2\gamma_2 (with radius r2r_2) be the circle internally tangent to γ1\gamma_1 at AA and tangent to BCBC at DD. Let II be the incenter of ABCABC, and PP and QQ be the intersection of γ2\gamma_2 with ABAB and ACAC respectively. Given that PP, II, and QQ are collinear, AI=25AI=25, and the circumradius of triangle BICBIC is 2424, compute the ratio of the radii r2r1\tfrac{r_2}{r_1}.
2017Geometry Test
2017 Guts #10

Source:

11/25/2022
The perimeter of an isosceles trapezoid is 2424. If each of the legs is two times the length of the shorter base and is two-thirds the length of the longer base, what is the area of the trapezoid?
2017Guts Round