MathDB

10

Part of 2015 ASDAN Math Tournament

Problems(5)

2015 Advanced #10

Source:

7/8/2022
Let σ(n)\sigma(n) be the sum of all the positive divisors of nn. Let aa be the smallest positive integer greater than or equal to 20152015 for which there exists some positive integer nn satisfying σ(n)=a\sigma(n)=a. Finally, let bb be the largest such value of nn. Compute a+ba+b.
2015Advanced Topics Test
2015 Algebra #10

Source:

7/1/2022
The polynomial f(x)=x3−43x2+13x−23f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3} has three real roots, aa, bb, and cc. Find max⁡{a+b−c,a−b+c,−a+b+c}.\max\{a+b-c,a-b+c,-a+b+c\}.
2015Algebra Test
2015 Guts #10

Source:

7/28/2022
Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other 22 players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once?
2015Guts Test
2015 Geometry #10

Source:

7/1/2022
Triangle ABCABC has ∠BAC=90∘\angle BAC=90^\circ. A semicircle with diameter XYXY is inscribed inside △ABC\triangle ABC such that it is tangent to a point DD on side BCBC, with XX on ABAB and YY on ACAC. Let OO be the midpoint of XYXY. Given that AB=3AB=3, AC=4AC=4, and AX=94AX=\tfrac{9}{4}, compute the length of AOAO.
2015Geometry Test
2015 Team #10

Source:

8/3/2022
An ant is walking on the edges of an icosahedron of side length 11. Compute the length of the longest path that the ant can take if it never travels over the same edge twice, but is allowed to revisit vertices.
2015team test