MathDB

SMT 2022 Algebra #9

Source:

3/27/2023

Let

P(x)=8x^3+ax+b+1

for

a,b\in\mathbb{Z}

. It is known that

P

has a root

x_0=p+\sqrt{q}+\sqrt[3]{r}

, where

p,q,r\in\mathbb{Q}

,

q\ge0

; however,

P

has no

<span class='latex-italic'>rational</span>

roots. Find the smallest possible value of

a+b

.

SMT 2022 Calculus #9

Source:

3/29/2023

Let

f(x,y)=(\cos x+y\sin x)^2

. We may express

\text{max}_xf(x,y)

, the maximum value of

f(x,y)

over all values of

x

for a given fixed value of

y

, as a function of

y

, call it

g(y)

. Let the smallest positive value

x

which achieves this maximum value of

f(x,y)

for a given

y

be

h(y)

. Compute

\int_1^{2+\sqrt{3}}\frac{h(y)}{g(y)}\text{d}y.

SMT 2022 Discrete #9

Source:

4/1/2023

For any positive integer

n

, let

f(n)

be the maximum number of groups formed by a total of

n

people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute

\textstyle\sum_{n=1}^{2022}f(n)\text{ mod }1000

.

SMT 2022 Geometry #9

Source:

4/1/2023

The bisector of

\angle BAC

in

\triangle ABC

intersects

BC

in point

L

. The external bisector of

\angle ACB

intersects

\overrightarrow{BA}

in point

K

. If the length of

AK

is equal to the perimeter of

\triangle ACL

,

LB=1

, and

\angle ABC=36^\circ

, find the length of

AC

.

9

Problems(4)