MathDB

2021 Algebra/NT #9: Cubic applied to itself solutions

Source:

5/30/2021

Let

f

be a monic cubic polynomial satisfying

f(x) + f(-x) = 0

for all real numbers

x

. For all real numbers

y

, define

g(y)

to be the number of distinct real solutions

x

to the equation

f(f(x)) = y

. Suppose that the set of possible values of

g(y)

over all real numbers

y

is exactly

\{1, 5, 9\}

. Compute the sum of all possible values of

f(10)

.

algebra

2021 Combo #9: Region path counting

Source:

5/30/2021

An up-right path between two lattice points

P

and

Q

is a path from

P

to

Q

that takes steps of length

1

unit either up or to the right.
How many up-right paths from

(0, 0)

to

(7, 7),

when drawn in the plane with the line

y = x - 2.021

, enclose exactly one bounded region below that line?

Combo

2021 Geo #9: Another trapezoid

Source:

5/30/2021

Let

ABCD

be a trapezoid with

AB \parallel CD

and

AD = BD

. Let

M

be the midpoint of

AB,

and let

P \neq C

be the second intersection of the circumcircle of

\triangle BCD

and the diagonal

AC.

Suppose that

BC = 27, CD = 25,

and

AP = 10.

If

MP = \tfrac {a}{b}

for relatively prime positive integers

a

and

b,

compute

100a + b

.

geometrytrapezoid

2021 Team #9

Source:

6/27/2021

Let scalene triangle

ABC

have circumcenter

O

and incenter

I

. Its incircle

\omega

is tangent to sides

BC,CA,

and

AB

at

D,E,

and

F

, respectively. Let

P

be the foot of the altitude from

D

to

EF

, and let line

DP

intersect

\omega

again at

Q \ne D

. The line

OI

intersects the altitude from

A

to

BC

at

T

. Given that

OI \|BC,

show that

PQ=PT

.

geometrycircumcircleincenter

9

Problems(4)