MathDB

Powers of 3 and Floors

Source:

8/8/2024

Let

\lfloor x \rfloor

denote the greatest integer less than or equal to

x

. Find the number of positive integers

m

between

1

and

2022

inclusive such that

\left\lfloor \frac{3^m}{11} \right\rfloor

is even.

floor functionnumber theory2022

Parabola Intersecting Triangle

Source:

8/8/2024

Triangle

ABC

has side lengths

AC = 3

,

BC = 4

, and

AB = 5

. Let

I

be the incenter of

\triangle{ABC}

, and let

\mathcal{P}

be the parabola with focus

I

and directrix

\overleftrightarrow{AC}

. Suppose that

\mathcal{P}

intersects

\overline{AB}

and

\overline{BC}

at points

D

and

E

, respectively. Find

DI+EI

.

conicsparabolageometry2022

Infinite Sum Equality

Source:

8/9/2024

There is a unique choice of positive integers

a

,

b

, and

c

such that

c

is not divisible by the square of any prime and the infinite sums \sum_{n=0}^{\infty} \left(\left(\frac{a - b\sqrt{c}}{10}\right)^{n-10}\cdot\prod_{k=0}^{9} (n - k)\right) \text{and} \sum_{n=0}^{\infty} \left((a - b\sqrt{c})^{n+1}\cdot\prod_{k=0}^{9} (n - k)\right) are equal (i.e., converging to the same finite value). Compute

a + b + c

.

calculus2022

Arranging Letters in Johns Hopkins

Source:

8/8/2024

Let

A

be the number of arrangements of the letters in JOHNS HOPKINS such that no two Os are adjacent, no two Hs are adjacent, no two Ns are adjacent, and no two Ss are adjacent. Find

\frac{A}{8!}

.

combinatorics2022

Maximizing Triangle Area

Source:

8/8/2024

For positive real numbers

a

and

b,

let

f(a,b)

denote the real number

x

such that area of the (non-degenerate) triangle with side lengths

a,b,

and

x

is maximized. Find

\sum_{n=2}^{100}f\left(\sqrt{\tbinom{n}{2}},\sqrt{\tbinom{n+1}{2}}\right).

algebrageometryoptimization2022

6

Problems(5)