MathDB

p1. Given any four digit number

X = \underline{ABCD}

, consider the quantity

Y(X) = 2 \cdot \underline{AB}+\underline{CD}

. For example, if

X = 1234

, then

Y(X) = 2 \cdot 12+34 = 58

. Find the sum of all natural numbers

n \le 10000

such that over all four digit numbers

X

, the number

n

divides

X

if and only if it also divides

Y(X)

.
p2. A sink has a red faucet, a blue faucet, and a drain. The two faucets release water into the sink at constant but different rates when turned on, and the drain removes water from the sink at a constant rate when opened. It takes

5

minutes to fill the sink (from empty to full) when the drain is open and only the red faucet is on, it takes

10

minutes to fill the sink when the drain is open and only the blue faucet is on, and it takes

15

seconds to fill the sink when both faucets are on and the drain is closed. Suppose that the sink is currently one-thirds full of water, and the drain is opened. Rounded to the nearest integer, how many seconds will elapse before the sink is emptied (keeping the two faucets closed)?
p3. One of the bases of a right triangular prism is a triangle

XYZ

with side lengths

XY = 13

YZ = 14

ZX = 15

. Suppose that a sphere may be positioned to touch each of the five faces of the prism at exactly one point. A plane parallel to the rectangular face of the prism containing

\overline{YZ}

cuts the prism and the sphere, giving rise to a cross-section of area

A

for the prism and area

15\pi

for the sphere. Find the sum of all possible values of

A

.
p4. Albert, Brian, and Christine are hanging out by a magical tree. This tree gives each of them a stick, each of which have a non-negative real length. Say that Albert gets a branch of length

x

, Brian a branch of length

y

, and Christine a branch of length

z

, and the lengths follow the condition that

x+y+z = 2

. Let

m

and

n

be the minimum and maximum possible values of

xy+yz+xz-xyz

, respectively. What is

m+n

?
p5. Let

S := MATHEMATICSMATHEMATICSMATHE...

be the sequence where

7

copies of the word

MATHEMATICS

are concatenated together. How many ways are there to delete all but five letters of

S

such that the resulting subsequence is

CHMMC

?
p6. Consider two sequences of integers

a_n

and

b_n

such that

a_1 = a_2 = 1

b_1 = b_2 = 1

and that the following recursive relations are satisfied for integers

n > 2

a_n = a_{n-1}a_{n-2}-b_{n-1}b_{n-2},

b_n = b_{n-1}a_{n-2}+a_{n-1}b_{n-2}.

Determine the value of

\sum_{1\le n\le2023,b_n \ne 0} \frac{a_n}{b_n}.

p7. Suppose

ABC

is a triangle with circumcenter

O

. Let

A'

be the reflection of

A

across

\overline{BC}

. If

BC =12

\angle BAC = 60^o

, and the perimeter of

ABC

30

, then find

A'O

.
p8. A class of

10

students wants to determine the class president by drawing slips of paper from a box. One of the students, Bob, puts a slip of paper with his name into the box. Each other student has a

\frac12

probability of putting a slip of paper with their own name into the box and a

\frac12

probability of not doing so. Later, one slip is randomly selected from the box. Given that Bob’s slip is selected, find the expected number of slips of paper in the box before the slip is selected.
p9. Let

a

and

b

be positive integers,

a > b

, such that

6! \cdot 11

divides

x^a -x^b

for all positive integers

x

. What is the minimum possible value of

a+b

?
p10. Find the number of pairs of positive integers

(m,n)

such that

n < m \le 100

and the polynomial

x^m+x^n+1

has a root on the unit circle.
p11. Let

ABC

be a triangle and let

\omega

be the circle passing through

A

B

C

with center

O

. Lines

\ell_A

\ell_B

\ell_C

are drawn tangent to

\omega

A

B

C

respectively. The intersections of these lines form a triangle

XYZ

where

X

is the intersection of

\ell_B

and

\ell_C

Y

is the intersection of

\ell_C

and

\ell_A

, and

Z

is the intersection of

\ell_A

and

\ell_B

. Let

P

be the intersection of lines

\overline{OX}

and

\overline{YZ}

. Given

\angle ACB = \frac32 \angle ABC

and

\frac{AC}{AB} = \frac{15}{16}

, find

\frac{ZP}{YP}

.
p12. Compute the remainder when

\sum_{1\le a,k\le 2021} a^k

is divided by

2022

(in the above summation

a,k

are integers).
p13. Consider a

7\times 2

grid of squares, each of which is equally likely to be colored either red or blue. Madeline would like to visit every square on the grid exactly once, starting on one of the top two squares and ending on one of the bottom two squares. She can move between two squares if they are adjacent or diagonally adjacent. What is the probability that Madeline may visit the squares of the grid in this way such that the sequence of colors she visits is alternating (i.e., red, blue, red,... or blue, red, blue,... )?
p14. Let

ABC

be a triangle with

AB = 8

BC = 10

, and

CA = 12

. Denote by

\Omega_A

the

A

-excircle of

ABC

, and suppose that

\Omega_A

is tangent to

\overline{AB}

and

\overline{AC}

F

and

E

, respectively. Line

\ell \ne \overline{BC}

is tangent to

\Omega_A

and passes through the midpoint of

\overline{BC}

. Let

T

be the intersection of

\overline{EF}

and

\ell

. Compute the area of triangle

ATB

.
p15. For any positive integer

n

, let

D_n

be the set of ordered pairs of positive integers

(m,d)

such that

d

divides

n

and gcd

(m,n) = 1

1 \le m \le n

. For any positive integers

a

b

, let

r(a,b)

be the non-negative remainder when

a

is divided by

b

. Denote by

S_n

the sum

S_n = \sum_{(m,d)\in D_n} r(m,d).

Determine the value of

S_{396}

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

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