blackpanther

124 Questions
1 Answers

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floor

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

blackpanther 1 hour ago

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hard sum

Let
\[f(x) = \left\lfloor\frac{2 - 3x}{2x + 8}\right\rfloor.\]
Evaluate $f(1)+f(2) + f(3) + \dots + f(999)+f(1000).$ (This sum has $1000$ terms, one for the result when we input each integer from $1$ to $1000$ into ${}f$.)

blackpanther 1 hour ago

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help trig

In right triangle $BCD$ with $\angle D = 90^\circ$, we have $BC = 10$ and $BD = 8$. Find $\sin B$.

blackpanther 1 hour ago

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coordinates

Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

blackpanther Dec 9, 2023

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Counting

Find the number of positive integers that satisfy both the following conditions:
- Each digit is a 1 or a 3 or a 5
- The sum of the digits is 6

blackpanther Dec 9, 2023

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need help

Let $x \mathbin{\spadesuit} y = \frac{x^2}{y}$ for all $x$ and $y$ such that $y\neq 0$. Find all values of $a$ such that $a \mathbin{\spadesuit} (a + 1) = 9$. Write your answer as a list separated by commas.

blackpanther Dec 9, 2023

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help plz

Let
\[f(x) = \frac{2x + 5}{x - 4} + \frac{x^2 + 1}{4x - 3}.\]
Find the inverse $f^{-1}(x)$.

blackpanther Dec 9, 2023

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Functions

Let $f(x) = 3x - 8 + 4x^2$ and $g(x) = 15x + c - 3x^2$. Find $c$ if $(f \circ g)(x) = (g \circ f)(x)$ for all $x$.

blackpanther Dec 9, 2023

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coeffcient

What is the coefficient of $x$ in $(x^3 + x^2 + x + 1)(x^4 - 8x^3 + 17x^2 - 23x + 14)$?

blackpanther Dec 8, 2023

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polynomials

There exists a polynomial $f(x)$ and a constant $k$ such that
(x^2 - 2x - 5) f(x) = 2x^4 + 19x^3 + kx^2 - 15x - 1.
What is $k?$

blackpanther Dec 8, 2023

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polynomial

Find $t$ if the expansion of the product of $x^3$ and $x^2 + tx$ has no $x^2$ term.

blackpanther Dec 8, 2023

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Absolute value

Find all solutions of the equation
|x^2 - 30x - 1| + |x^2 - 30x + 29| = 30

blackpanther Dec 8, 2023

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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 such that read more ..

blackpanther Dec 8, 2023

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rectangle

Rectangle $ABCD$ contains a point $X$ such that $AX = 3$, $BX = 3$, and $CX = 3$. Find $DX$.

blackpanther Dec 3, 2023

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Thanks!

blackpantherNov 29, 2023