MathDB

8

Part of 2015 ASDAN Math Tournament

Problems(5)

2015 Advanced #8

Source:

7/8/2022
You have 88 friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different paths can you take?
2015Advanced Topics Test
2015 Algebra #8

Source:

7/1/2022
Let {x}\{x\} denote the fractional part of xx, which means the unique real 0≤{x}<10\leq\{x\}<1 such that x−{x}x-\{x\} is an integer. Let fa,b(x)={x+a}+2{x+b}f_{a,b}(x)=\{x+a\}+2\{x+b\} and let its range be [ma,b,Ma,b)[m_{a,b},M_{a,b}). Find the minimum value of Ma,bM_{a,b} as aa and bb range along all real numbers.
2015Algebra Test
2015 Guts #8

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7/28/2022
Lynnelle and Moor love toy cars, and together, they have 2727 red cars, 2727 purple cars, and 2727 green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has 1717 more cars of any color than Moor has of any color. How many purple cars does Lynnelle have?
2015Guts Test
2015 Geometry #8

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7/1/2022
In triangle ABCABC, point DD is on side BCBC such that ADAD is the angle bisector of ∠BAC\angle BAC. If AB=12AB=12, AD=9AD=9, and AC=15AC=15, compute cos⁡∠BAC2\cos\tfrac{\angle BAC}{2}.
2015Geometry Test
2015 Team #8

Source:

8/3/2022
Let f(x)=x+ax+bf(x)=\tfrac{x+a}{x+b} for real numbers xx such that x≠−bx\neq -b. Compute all pairs of real numbers (a,b)(a,b) such that f(f(x))=−1xf(f(x))=-\tfrac{1}{x} for x≠0x\neq0.
2015team test