maximum

260 Questions
128 Answers

-2

150

0

+834

inequalities

(1) Let $a_1,$ $a_2,$ $a_3$ be real numbers such that
\[|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_1| = 1.\]
What is the largest possible value of $|a_1 - a_2|$?
(2) Let $a_1,$ $a_2,$ $a_3,$ $\dots,$ $a_{10}$ be real numbers mehr ..

maximum 17.08.2023

-2

156

0

+834

floor function

Find the number of positive integers ${}n$ that satisfy $\lfloor \sqrt{n} \rfloor + \lfloor \sqrt{4n} \rfloor = 5.$

maximum 17.08.2023

-2

142

1

+834

help

I need help with algebra
Find constants $A$ and $B$ such that
\[\frac{2x + 11 - x - 13}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1}\]
for all ${}x$ such that $x\neq -1$ and $x\neq 2$.

maximum 17.08.2023

-2

84

1

+834

circle question

Points $A$, $B$, $Q$, $D$, and $C$ lie on the circle shown and the measures of arcs $BQ$ and $QD$ are $30^\circ$ and $30^\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$, in degrees. mehr ..

maximum 17.08.2023

-2

102

1

+834

help pls

Help with circles plz
A hexagon is inscribed in a circle. Find the angle $\alpha$, in degrees.

maximum 17.08.2023

-2

153

0

+834

help

The product $(3x+2y+1)(x+4y+5)(7x-y+2)$ is expanded. What is the sum of the coefficients of the terms which contain a nonzero power of $y$?

maximum 17.08.2023

-2

69

1

+834

help

Help I have trouble with complex numbers.
In the complex plane, the complex numbers $2+i$, $a$, $7-3i$, $b$ form the vertices of a square, in counterclockwise order. Find the following quantities, in rectangular form.
b/a mehr ..

maximum 16.08.2023

-1

132

1

+834

Complex numbers

Determine the complex number $z$ satisfying the equation $3z+4i\bar{z}=1-8i+3iz-5\bar{z}$. Note that $\bar{z}$ denotes the conjugate of $z$.

maximum 16.08.2023

-1

141

1

+834

help with calculus

The function $f(x),$ defined for $x \ge 0,$ has the following properties:
$f(x) \ge \sqrt{x}$ for all $x \ge 0.$
The function $f(x)$ is increasing.
The area between the graph of $y = f(x)$ for $0 \le x \le a$ and the graph of $y = x^2$ mehr ..

maximum 15.08.2023

-1

137

1

+834

domain

Can you help?
Find the domain of the function $\frac{x^4-4x^3+6x^2-4x+1}{\sqrt{x^2 + 4}}$.

maximum 15.08.2023

+834

-1

Do not post AoPS homework.

maximum20.01.2025