MathDB

p1. For what value of

x > 0

does

f(x) = (x -3)^2(x + 4)

achieve the smallest value?
p2. There are exactly

29

possible values that can be made using one or more of the

5

distinct coins with values

1

3

5

7

, and

X

. What is the smallest positive integral value for

X

?
p3. Define

\star

x \star y = x -\frac{1}{xy}

. What is the sum of all complex

x

such that

x \star (x \star 2x) = 2x

?
p4. Let

\lfloor x \rfloor

be the greatest integer less than or equal to

x

and let

\lceil x \rceil

be the least integer greater than or equal to

x

. Compute the smallest positive value of

a

for which

\lfloor a \rfloor

\lceil a \rceil

\lfloor a^2 \rfloor

is a nonconstant arithmetic sequence.
p5. A right triangle is bounded in a coordinate plane by the lines

x = 0

y = 0

x = x_{100}

, and

y = f(x)

, where

f

is a linear function with a negative slope and

f(x_{100}) = 0

. The lines

x = x_1

x = x_2

...

x = x_{99}

(

x_1 < x_2 <... < x_{100}

) subdivide the triangle into

100

regions of equal area. Compute

\frac{x_{100}}{x_1}

.
p6. There are

10

children in a line to get candy. The pieces of candy are indistinguishable, while the children are not. If there are a total of

390

pieces of candy, how many ways are there to distribute the candy so that the

n^{th}

child in line receives at least

n^2

pieces of candy?
p7. Compute

\frac{ {54 \choose 23}+ 6 {54 \choose 24}+ 15{54 \choose 25}+ 15{54 \choose 27}+ 6{54 \choose 28}+ {54 \choose 29} - {60 \choose 29}}{{54 \choose 26}}

p8. Point

A

lies on the circle centered at

O

\overline{AB}

is tangent to

O

, and

C

is located on the circle so that

m\angle AOC = 120^o

and oriented so that

\angle BAC

is obtuse.

\overline{BC}

intersects the circle at

D

. If

AB = 6

and

BD = 3

, then compute the radius of the circle.
p9. The center of each face of a regular octahedron (a solid figure with

8

equilateral triangles as faces) with side length one unit is marked, and those points are the vertices of some cube. The center of each face of the cube is marked, and these points are the vertices of an even smaller regular octahedron. What is the volume of the smaller octahedron?
p10. Compute the greatest positive integer

n

such that there exists an odd integer

a

, for which

\frac{a^{2^n}-1}{4^{4^4}}

is not an integer.
p11. Three identical balls are painted white and black, so that half of each sphere is a white hemisphere, and the other half is a black one. The three balls are placed on a plane surface, each with a random orientation, so that each ball has a point of contact with the other two. What is the probability that at at least one point of contact between two of the balls, both balls are the same color?
p12. Define an operation

\Phi

whose input is a real-valued function and output is a real number so that it has the following properties:

\bullet

For any two real-valued functions

f(x)

and

g(x)

, and any real numbers

a

and

b

, then

\Phi (af(x) + bg(x)) = a \Phi (f(x)) + b\Phi (g(x))

\bullet

For any real-valued function

h(x)

, there is a polynomial function

p(x)

such that

\Phi (p(x) \cdot h(x)) = \Phi ((h(x))^2)

\bullet

If some function

m(x)

is always non-negative, and

\Phi (m(x)) = 0

, then

m(x)

is always

0

. Let

r(x)

be a real-valued function with

r(5) = 3

. Let

S

be the set of all real-valued functions

s(x)

that satisfy that

\Phi(r(x) \cdot x^n) = \Phi(s(x) \cdot x^{n+1})

. For each

s

S

, give the value of

s(5)

.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement