MathDB

7

Part of 2016 ASDAN Math Tournament

Problems(6)

2016 Algebra #7

Source:

8/6/2022
Let xx, yy, and zz be real numbers satisfying the equations \begin{align*} 4x+2yz-6z+9xz^2&=4\\ xyz&=1. \end{align*} Find all possible values of x+y+zx+y+z.
2016Algebra Test
2016 Discrete #7

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8/10/2022
Heesu, Xingyou, and Bill are in a class with 99 other children. The teacher randomly arranges the children in a circle for story time. However, Heesu, Xingyou, and Bill want to sit near each other. Compute the probability that all 33 children are seated within a consecutive group of 55 seats.
2016Discrete Math Test
2016 Calculus #7

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8/8/2022
Compute ∫0π2ex(sin⁡x+cos⁡x−2)(cos⁡x−2)2dx.\int_0^{\frac{\pi}{2}}\frac{e^x(\sin x+\cos x-2)}{(\cos x-2)^2}dx.
2016Calculus Test
2016 Guts #7

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8/14/2022
What is ∑n=19962016⌊n⌋?\sum_{n=1996}^{2016}\lfloor\sqrt{n}\rfloor?
2016Guts Round
2016 Geometry #7

Source:

8/11/2022
The side lengths of triangle ABCABC are 1313, 1414, and 1515. Let II be the incenter of the triangle. Compute the product AI⋅BI⋅CIAI\cdot BI\cdot CI.
2016Geometry Test
2016 Team #7

Source:

8/17/2022
Eddy and Moor play a game with the following rules:
[*] The game begins with a pile of NN stones, where NN is a positive integer. [/*] [*] The 22 players alternate taking turns (e.g. Eddy moves, then Moor moves, then Eddy moves, and so on). [/*] [*] During a player's turn, given aa stones remaining in the pile, the player may remove bb stones from the pile, where gcd⁡(a,b)=1\gcd(a,b)=1 and b≤ab\leq a. [/*] [*] If a player cannot make a move, they lose. [/*]
For example, if Eddy goes first and N=4N=4, then Eddy can remove 33 stones from the pile (since 3≤43\leq4 and gcd⁡(3,4)=1\gcd(3,4)=1), leaving 11 stone in the pile. Moor can then remove 11 stone from the pile (since 1≤11\leq1 and gcd⁡(1,1)=1\gcd(1,1)=1), leaving 00 stones in the pile. Since Eddy cannot remove stones from an empty pile, he cannot make a move, and therefore loses.
Both Eddy and Moor want to win, so they will both always make the best possible move. If Eddy moves first, for how many values of N<2016N<2016 can Eddy win no matter what moves Moor chooses?
2016team test