MathDB

SMT 2022 Algebra #1

Source:

3/27/2023

Compute

\frac{5+\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{7+\sqrt{12}}{\sqrt{3}+\sqrt{4}}+\dots+\frac{63+\sqrt{992}}{\sqrt{31}+\sqrt{32}}.

SMT 2022 Calculus #1

Source:

3/29/2023

If

f(x)=x^4+4x^3+7x^2+6x+2022

, compute

f'(3)

.

SMT 2022 Algebra Tiebreaker #1

Source:

3/28/2023

If

x

,

y

, and

z

are real numbers such that

x^2+2y^2+3z^2=96

, what is the maximum possible value of

x+2y+3z

?

inequalitiesalgebra

SMT 2022 Calculus Tiebreaker #1

Source:

3/29/2023

Compute

\int_0^{10}(x-5)+(x-5)^2+(x-3)^2dx.

SMT 2022 Discrete #1

Source:

4/1/2023

An ant starts at the point

(1,1)

. It can travel along the integer lattice, only moving in the positive

x

and

y

directions. What is the number of ways it can reach

(5,5)

without passing through

(3,3)

?

SMT 2022 Discrete Tiebreaker #1

Source:

4/1/2023

How many

4

element subsets of

\{0,1,2,\dots,20\}

contain their sum modulo

21

?

SMT 2022 Geometry #1

Source:

4/1/2023

Points

A

,

B

,

C

, and

D

lie on a circle. Let

AC

and

BD

intersect at point

E

inside the circle. If

[ABE]\cdot[CDE]=36

, what is the value of

[ADE]\cdot[BCE]

? (Given a triangle

\triangle ABC

,

[ABC]

denotes its area.)

SMT 2022 Geometry Tiebreaker #1

Source:

4/1/2023

George is drawing a Christmas tree; he starts with an isosceles triangle

AB_0C_0

with

AB_0=AC_0=41

and

B_0C_0=18

. Then, he draws points

B_i

and

C_i

on sides

AB_0

and

AC_0

, respectively, such that

B_iB_{i+1}=1

and

C_iC_{i+1}=1

(

B_{41}=C_{41}=A

). Finally, he uses a green crayon to color in triangles

B_iC_iC_{i+1}

for

i

from

0

to

40

. What is the total area that he colors in?

1

Problems(8)