MathDB

2004 Algebra #4

Source:

12/26/2011

Evaluate the sum

\dfrac {1}{2 \lfloor \sqrt {1} \rfloor + 1} + \dfrac {1}{2 \lfloor \sqrt {2} \rfloor + 1} + \dfrac {1}{2 \lfloor \sqrt {3} \rfloor + 1} + \cdots + \dfrac {1}{2 \lfloor \sqrt {100} \rfloor + 1}

floor function

2004 Calculus #4

Source:

11/29/2011

Let

f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))

, and suppose that the number

a

satisfies the equation

a=\cos a

. Express

f'(a)

as a polynomial in

a

.

calculustrigonometryalgebrapolynomialinduction

2004 Combinatorics #4

Source:

12/30/2011

Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence

\texttt{HH}

) or flips tails followed by heads (the sequence

\texttt{TH}

). What is the probability that she will stop after flipping

\texttt{HH}

?

probability

2004 HMMT Geometry # 4

Source:

3/3/2024

P

is inside rectangle

ABCD

.

PA = 2

,

PB = 3

, and

PC = 10

. Find

PD

.

geometry

2004 General, part 1 #4

Source:

3/8/2024

How many ways can you mark

8

squares of an

8\times 8

chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)

combinatorics

2004 General, part 2 #4

Source:

3/8/2024

A horse stands at the corner of a chessboard, a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally (like a knight moves). The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice? https://cdn.artofproblemsolving.com/attachments/e/c/c817d92ead6cfb3868f9cb526fb4e1fd7ffe4d.png

combinatorics

4

Problems(6)