MathDB

7

Part of 2022 BMT

Problems(5)

BMT 2022 Fall Algebra P7

Source:

5/5/2023
Let r,s,r, s, and tt be the distinct roots of x3−2022x2+2022x+2022.x^3- 2022x^2 + 2022x + 2022. Compute 11−r2+11−s2+11−t2.\frac{1}{1-r^2} + \frac{1}{1-s^2} + \frac{1}{1-t^2}.
BMT 2022 Fall Discrete P7

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5/6/2023
Luke the frog has a standard deck of 5252 cards shuffled uniformly at random placed face down on a table. The deck contains four aces and four kings (no card is both an ace and a king). He now begins to flip over the cards one by one, leaving a card face up once he has flipped it over. He continues until the set of cards he has flipped over contains at least one ace and at least one king, at which point he stops. What is the expected value of the number of cards he flips over?
BMT 2022 Geometry #7

Source:

8/12/2023
In triangle △ABC\vartriangle ABC with orthocenter HH, the internal angle bisector of ∠BAC\angle BAC intersects BC‾\overline{BC} at YY . Given that AH=4AH = 4, AY=6AY = 6, and the distance from YY to AC‾\overline{AC} is 15\sqrt{15}, compute BCBC.
geometry
BMT 2022 Guts #7

Source:

8/31/2023
A regular hexagon is inscribed in a circle of radius 11, and all diagonals between vertices that have exactly one vertex between them are drawn. Compute the area of the hexagon enclosed by all of the diagonals.
geometry
BMT 2022 General p7

Source:

9/28/2023
Let f(x)=x2+⌊x⌋2−2x⌊x⌋+1f(x) = x^2 + \lfloor x\rfloor ^2 - 2x \lfloor x \rfloor + 1. Compute f(4+56)f\left(4 + \frac56 \right). Here, ⌊m⌋\lfloor m \rfloor is defined as the greatest integer less than or equal to mm. For example, ⌊3⌋=3\lfloor 3 \rfloor = 3 and ⌊−4.25⌋=−5\lfloor - 4.25 \rfloor = -5.
algebra