MathDB

BMT 2022 Fall Algebra P3

Source:

5/5/2023

Suppose we have four real numbers

a,b,c,d

such that

a

is nonzero,

a,b,c

form a geometric sequence, in that order, and

b,c,d

form an arithmetic sequence, in that order. Compute the smallest possible value of

\frac{d}{a}.

(A geometric sequence is one where every succeeding term can be written as the previous term multiplied by a constant, and an arithmetic sequence is one where every succeeeding term can be written as the previous term added to a constant.)

algebraSequencegeometric sequence

BMT 2022 Fall Discrete P3

Source:

5/6/2023

Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth

4

and tails is worth

2.

What is the probability Katie’s score is strictly greater than Allie’s?

number theory

BMT 2022 Geometry #3 / General #18

Source:

8/12/2023

Let equilateral triangle

\vartriangle ABC

be inscribed in a circle

\omega_1

with radius

4

. Consider another circle

\omega_2

with radius

2

internally tangent to

\omega_1

at

A

. Let

\omega_2

intersect sides

\overline{AB}

and

\overline{AC}

at

D

and

E

, respectively, as shown in the diagram. Compute the area of the shaded region. https://cdn.artofproblemsolving.com/attachments/a/5/8255cdb8b041719d735607da8139aa4016375d.png

geometry

BMT 2022 Guts #3

Source:

8/31/2023

The polynomial

ax^2 + bx + c

crosses the

x

-axis at

x = 10

and

x = -6

and crosses the

y

-axis at

y = 10

. Compute

a + b + c

.

algebra

3

Problems(4)