BRAlNBOLT

169 Questions
15 Answers

-1

2

0

+195

Counting

Find the number of $7$-digit positive integers, where the sum of the digits is divisible by $3.$

BRAlNBOLT 23.10.2024

0

1

1

+195

Algebra

Expand (sqrt(x) + 3x)*4 + (sqrt(x) - 3x)*4.

BRAlNBOLT 23.10.2024

-1

1

1

+195

Algebra

Fill in the blanks with positive integers:
(3 + sqrt(6))*3 = ___ + ___ * sqrt(6)

BRAlNBOLT 23.10.2024

0

3

0

+195

Number Theory

A positive integer is called terrific if it has exactly $10$ positive divisors. What is the smallest number of primes that could divide a terrific positive integer?

BRAlNBOLT 14.10.2024

0

4

0

+195

Number Theory

A positive integer is called terrific if it has exactly $10$ positive divisors. What is the largest number of primes that could divide a terrific positive integer?

BRAlNBOLT 14.10.2024

0

3

0

+195

Number Theory

A positive integer is called terrific if it has exactly $10$ positive divisors. What is the smallest terrific positive integer?

BRAlNBOLT 14.10.2024

0

5

0

+195

Algebra

Let $f(x)$ be a polynomial with integer coefficients. There exist distinct integers $p,$ $q,$ $r,$ $s,$ $t$ such that
f(p) = f(q) = f(r) = f(s) = 1
and $f(t) > 1.$ What is the smallest possible value of $f(t)?$

BRAlNBOLT 01.10.2024

0

8

0

+195

Algebra

Find the number of ordered pairs $(a,b)$ of integers such that
\frac{a + 2}{a + 1} = \frac{b}{8}.

BRAlNBOLT 11.09.2024

0

2

0

+195

Algebra

Suppose $f(x)$ is a function defined for all real $x$, and suppose $f$ is invertible (that is, $f^{-1}(x)$ exists for all $x$ in the range of $f$). If the graphs of $y=f(x)$ and $y=f(1/x)$ are drawn, at how many points do they intersect?

BRAlNBOLT 07.09.2024

-1

3

1

+195

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

BRAlNBOLT 31.08.2024

0

6

1

+195

Coefficient

What's the coefficient of the x^2 y^4 term in the expansion of (x + y)*6*(3x - 2y)*(5x + y)^4?

BRAlNBOLT 10.08.2024

0

1

1

+195

Coefficient

What's the coefficient of xy in the expansion of (3x + 3y + 3)^3?

BRAlNBOLT 10.08.2024

-1

9

0

+195

Algebra

Let $f(x) = px + q$, where $p$ and $q$ are real numbers. Find $p+q$ if $f(f(f(x))) = 64x - 105 - 30x + 70 + 32x - 12$.

BRAlNBOLT 10.08.2024

-1

7

1

+195

Counting

We call a number cozy if every digit in the number is either even or next to an even digit. For example, a number is cozy if every digit in the number is either a 2 or next to a 2. For example, the number 222, 72, 45, and 810 and mehr ..

BRAlNBOLT 08.08.2024

-1

4

0

+195

Number Theory

Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in base $12$, it has the same three digits, in reverse order. What is $N$? (Express your answer in decim mehr ..

BRAlNBOLT 19.07.2024

-1

3

1

+195

Number Theory

A positive integer is called terrific if it has exactly $10$ positive divisors. What is the smallest terrific positive integer?

BRAlNBOLT 19.07.2024

-1

6

1

+195

Algebra

Find the minimum value of \frac{x^2}{x - 1 + x^3} for x > 1

BRAlNBOLT 18.07.2024

-1

12

1

+195

Algebra

Find all values of t such that floor(t) = 3t + 4 - t^2. If you find more than one value, then list the values you find in increasing order, separated by commas.

BRAlNBOLT 18.07.2024

-1

3

1

+195

Algebra

Let
f(x) = \left\lfloor\frac{2 - 3x}{3x + 1}\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.)

BRAlNBOLT 18.07.2024