MathDB

Cubic Polynomial and Trigonometry

Source:

8/8/2024

The polynomial

P(x)=3x^3-2x^2+ax-b

has roots

\sin^2\theta

,

\cos^2\theta

, and

\sin\theta\cos\theta

for some angle

\theta

. Find

P(1)

.

algebrapolynomialtrigonometry2022

Stack of Apples and Oranges

Source:

8/8/2024

Four mutually externally tangent spherical apples of radius

4

are placed on a horizontal flat table. Then, a spherical orange of radius

3

is placed such that it rests on all the apples. Find the distance from the center of the orange to the table.

3D geometryspheregeometry2022

Bounding Value of a Family of Functions

Source:

8/9/2024

Suppose that

f

is a differentiable function such that

f(0) = 20

and

|f'(x)| \leq 4

for all real numbers

x

. Let

a

and

b

be real numbers such that every such function

f

satisfies

a \leq f(22) \leq b

. Find the smallest possible value of

|a| + |b|

.

calculusderivativefunction2022

Uncertain Subsets

Source:

8/8/2024

Erica intends to construct a subset

T

of

S=\{ I,J,K,L,M,N \}

, but if she is unsure about including an element

x

of

S

in

T

, she will write

x

in bold and include it in

T

. For example,

\{ I,J \},

\{ J,\mathbf{K},L \},

and

\{ \mathbf{I},\mathbf{J},\mathbf{M},\mathbf{N} \}

are valid examples of

T

, while

\{ I,J,\mathbf{J},K \}

is not. Find the total number of such subsets

T

that Erica can construct.

combinatorics2022

Factorial Sum is Multiple of 10

Source:

8/8/2024

Find the number of ordered pairs of positive integers

(m,n)

, where

m,n\leq 10

, such that

m!+n!

is a multiple of

10

.

factorialnumber theory2022

2

Problems(5)