MathDB

Part 6
p16. Let

p(x)

and

q(x)

be polynomials with integer coefficients satisfying

p(1) = q(1)

. Find the greatest integer

n

such that

\frac{p(2023)-q(2023)}{n}

is an integer no matter what

p(x)

and

q(x)

are.
p17. Find all ordered pairs of integers

(m,n)

that satisfy

n^3 +m^3 +231 = n^2m^2 +nm.

p18. Ben rolls the frustum-shaped piece of candy (shown below) in such a way that the lateral area is always in contact with the table. He rolls the candy until it returns to its original position and orientation. Given that

AB = 4

and

BD =CD = 3

, find the length of the path traced by

A

.
Part 7
p19. In their science class, Adam, Chris, Eddie and Sam are independently and randomly assigned an integer grade between

70

and

79

inclusive. Given that they each have a distinct grade, what is the expected value of the maximum grade among their four grades?
p20. Let

ABCD

be a regular tetrahedron with side length

2

. Let point

E

be the foot of the perpendicular from

D

to the plane containing

\vartriangle ABC

. There exist two distinct spheres

\omega_1

and

\omega_2

, centered at points

O_1

and

O_2

respectively, such that both

O_1

and

O_2

lie on

\overrightarrow{DE}

and both spheres are tangent to all four of the planes

ABC

BCD

CDA

, and

DAB

. Find the sum of the volumes of

\omega_1

and

\omega_2

.
p21. Evaluate

\sum^{\infty}_{i=0}\sum^{\infty}_{j=0}\sum^{\infty}_{k=0} \frac{1}{(i + j +k +1)2^{i+j+k+1}}.

Part 8
p22. In

\vartriangle ABC

, let

I_A

I_B

, and

I_C

denote the

A

B

, and

C

-excenters, respectively. Given that

AB = 15

BC = 14

and

C A = 13

, find

\frac{[I_A I_B I_C ]}{[ABC]}

.
p23. The polynomial

x +2x^2 +3x^3 +4x^4 +5x^5 +6x^6 +5x^7 +4x^8 +3x^9 +2x^{10} +x^{11}

has distinct complex roots

z_1, z_2, ..., z_n

. Find

\sum^n_{k=1} |R(z^2n))|+|I(z^2n)|,

where

R(z)

and

I(z)

indicate the real and imaginary parts of

z

, respectively. Express your answer in simplest radical form.
p24. Given that

\sin 33^o +2\sin 161^o \cdot \sin 38^o = \sin n^o

, compute the least positive integer value of

n

.
Part 9
p25. Submit a prime between

2

and

2023

, inclusive. If you don’t, or if you submit the same number as another team’s submission, you will receive

0

points. Otherwise, your score will be

\min \left(30, \lfloor 4 \cdot ln(x) \rfloor \right)

, where

x

is the positive difference between your submission and the closest valid submission made by another team.
p26. Sam, Derek, Jacob, andMuztaba are eating a very large pizza with

2023

slices. Due to dietary preferences, Sam will only eat an even number of slices, Derek will only eat a multiple of

3

slices, Jacob will only eat a multiple of

5

slices, andMuztaba will only eat a multiple of

7

slices. How many ways are there for Sam, Derek, Jacob, andMuztaba to eat the pizza, given that all slices are identical and order of slices eaten is irrelevant? If your answer is

A

and the correct answer is

C

, the number of points you receive will be: irrelevant? If your answer is

A

and the correct answer is

C

, the number of points you receive will be:

\max \left( 0, \left\lfloor 30 \left( 1-2\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)

p27. Let

\Omega_(k)

denote the number of perfect square divisors of

k

. Compute

\sum^{10000}_{k=1} \Omega_(k).

If your answer is

A

and the correct answer is

C

, the number of points you recieve will be

\max \left( 0, \left\lfloor 30 \left( 1-4\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)

PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3267911p30056982]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement