MathDB

p1. Find the number of ordered triples

(a, b, c)

satisfying

\bullet

a, b, c

are are single-digit positive integers, and

\bullet

a \cdot b + c = a + b \cdot c

.
p2. In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form an increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to

94

. How many possible combinations of test scores could they have had? (Test scores at Greendale range between

0

and

100

, inclusive.)
p3. Suppose that

a + \frac{1}{b} = 2

and

b +\frac{1}{a} = 3

. If

\frac{a}{b} + \frac{b}{a}

can be expressed as

\frac{p}{q}

in simplest terms, find

p + q

.
p4. Suppose that

A

and

B

are digits between

1

and

9

such that

0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B_1B_1B_1...}) + 1

Find the sum of all possible values of

10A + B

.
p5. Let

ABC

be an isosceles right triangle with

m\angle ABC = 90^o

. Let

D

and

E

lie on segments

AC

and

BC

, respectively, such that triangles

\vartriangle ADB

and

\vartriangle CDE

are similar and

DE = EB

. If

\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}

with

a, b

positive integers and a squarefree, then find

a + b

.
p6. Five bowling pins

P_1

P_2

,...,

P_5

are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is a b where a and b are relatively prime, find

a + b

. (Pins

P_i

and

P_j

are adjacent if and only if

|i -j| = 1

.)
p7. Let triangle

ABC

have side lengths

AB = 10

BC = 24

, and

AC = 26

. Let

I

be the incenter of

ABC

. If the maximum possible distance between

I

and a point on the circumcircle of

ABC

can be expressed as

a +\sqrt{b}

for integers

a

and

b

with

b

squarefree, find

a + b

. (The incenter of any triangle

XY Z

is the intersection of the angle bisectors of

\angle Y XZ

\angle XZY

, and

\angle ZY X

.)
p8. How many terms in the expansion of

(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})

have coefficients equal to

1011

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

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