BRAlNBOLT

178 Questions
15 Answers

0

7

1

+216

Counting

A standard six-sided die is rolled $7$ times. You are told that among the rolls, there was one $1,$ one $2$, one $3$, one $4$, one $5$, and two $6$s. How many possible sequences of rolls could there have been? (For example, 2, 3, read more ..

BRAlNBOLT Jan 31, 2025

0

10

0

+216

Counting

At a meeting, four scientists, two mathematicians, and a journalist are to be seated around a circular table. How many different arrangements are possible if the mathematicians must sit next to each other? (Two seatings are considered equivalent if read more ..

BRAlNBOLT Jan 31, 2025

0

10

0

+216

Counting

Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if all the children are next to each other, and all the adults are next to each other? (Two seatings are considered the same if read more ..

BRAlNBOLT Jan 31, 2025

-1

16

0

+216

Counting

In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven numbers (1, 2, 3, 4, 5, 6, and 7). Among all the cards of each color, there is exactly one card labeled with each number. read more ..

BRAlNBOLT Dec 8, 2024

-1

18

0

+216

Counting

Catherine rolls a standard $6$-sided die six times. If the product of her rolls is $300,$ then how many different sequences of rolls could there have been? (The order of the rolls matters.)

BRAlNBOLT Dec 8, 2024

-1

15

0

+216

Counting

Joanna has six beads that she wants to assemble into a bracelet. Four of the beads have the same color, and the other two all have different colors. In how many different ways can Joanna assemble her bracelet? (Two bracelets are considered identical read more ..

BRAlNBOLT Dec 8, 2024

-1

27

0

+216

Geometry

In triangle ABO, M is the midpoint of \overline{AB}. Points C, D, and N lie on line segments $\overline{AO},$ $\overline{BO},$ and $\overline{MO}$ extended, respectively, so that $DO = BO,$ $NO = MO,$ and $CO = AO.$ If $K$ is the area of triangle read more ..

BRAlNBOLT Oct 30, 2024

-1

28

0

+216

Geometry

In the diagram below, ABCD is a square, DE = EF = FG = 2, GH = 1, HB = 5, and \angle DEF = \angle EFG = \angle FGH = \angle GHB = 60^\circ. Find the area of the shaded region.

BRAlNBOLT Oct 30, 2024

-1

23

0

+216

Geometry

Unit square ABCD is divided into two triangles, which are congruent to each other, and two trapezoids, which are also congruent to each other. These pieces are then re-assembled into rectangle $PQRS,$ as shown below. Find the perimeter of rectangle read more ..

BRAlNBOLT Oct 30, 2024

-2

14

0

+216

Counting

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

BRAlNBOLT Oct 23, 2024

-1

23

1

+216

Algebra

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

BRAlNBOLT Oct 23, 2024

-2

11

1

+216

Algebra

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

BRAlNBOLT Oct 23, 2024

-1

16

0

+216

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 Oct 14, 2024

-1

20

0

+216

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 Oct 14, 2024

-1

25

0

+216

Number Theory

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

BRAlNBOLT Oct 14, 2024

-1

27

0

+216

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 Oct 1, 2024

-1

36

0

+216

Algebra

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

BRAlNBOLT Sep 11, 2024

-1

19

0

+216

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 Sep 7, 2024