All Questions ⁺⁰234662 Questions

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Function

Find the greatest integer value of b for which the expression (9x^3 + 4x^2 + 11x + 7)/(x^2 + bx + 18) has a domain of all real numbers.

bingboy Feb 15, 2024

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19

1

+743

Algebra problem

Find constants A and B such that
(x + 17)/(x^2 - x - 2) = A/(x - 2) + B/(x + 1)
for all x such that $x \neq -1$ and $x \neq 2$. Give your answer as the ordered pair (A,B).

bingboy Feb 15, 2024

Feb 14, 2024

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2

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+1323

textbooks

I have $4$ different mathematics textbooks and $3$ different psychology textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row, if all the mathematics textbooks must be together?

blackpanther Feb 14, 2024

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13

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Counting sequences

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 ..

blackpanther Feb 14, 2024

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27

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Probability

Raina places the following balls into a bag. She draws three balls out of the bag, one at a time, without replacement. What is the probability that the colors of the balls alternate?
There are 6 orange balls and 6 purple read more ..

blackpanther Feb 14, 2024

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38

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+1323

help 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 ..

blackpanther Feb 14, 2024

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30

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+1323

Bracelet

Joanna has seven beads that she wants to assemble into a bracelet. Five 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 ..

blackpanther Feb 14, 2024

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42

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+1323

Probability

The following cards are split into three piles, so that every pile contains the same number of cards. What is the probability that every pile has an Ace?

blackpanther Feb 14, 2024

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33

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+1323

Probability

Two points on a circle of radius $1$ are chosen at random. Find the probability that the distance between the two points is at most $1.5.$

blackpanther Feb 14, 2024

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26

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+1323

Probability question

Let PQR be an equilateral triangle, centered at O. A point X is chosen at random inside the triangle. Find the probability that X is closer to O than to any of the sides. (In other words, find the probability that XO is read more ..

blackpanther Feb 14, 2024

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22

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+1323

Expected Value

The grid below is made up of line segments, like the line segment in red. There are a number of paths that go from A to B in the grid, where every step goes to the right or up. If we choose a line segment at random, then what is the expected number of squares read more ..

blackpanther Feb 14, 2024

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Paths

Find the number of paths from $A$ to $B,$ if each step must go towards the right. One example path is shown below.

blackpanther Feb 14, 2024

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16

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+1756

Counting

Find the number of ways of choosing three circles below, so that no two circles are next to each other.

bader Feb 14, 2024

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15

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+1323

Help with counting

A speech class has $4$ freshmen and $3$ sophomores. (Everyone is distinguishable.) In how many ways can they stand in line, so that all the sophomores are standing together?

blackpanther Feb 14, 2024

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25

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+1323

Counting

North High School has a math club and a science club. There are a students in the math club and b students who are members of both clubs. The total number of students is c. Find an expression for the total number of students read more ..

blackpanther Feb 14, 2024

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22

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Counting

North High School has a math club and a science club. There are a students in the math club and b students who are members of both clubs. The total number of students is c. Find an expression for the total number of students read more ..

blackpanther Feb 14, 2024

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3

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+1756

Angle bisectors

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 12$, $AY = 12$, and $CY = 6$, find $BC$.

bader Feb 14, 2024

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3

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+1756

Angle bisectors

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 12$, $AY = 12$, and $CY = 6$, find $BZ$.

bader Feb 14, 2024

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23

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+400

Quadratic Minimum

The temperature of a point (x,y) in the plane is given by the expression x^2 + y^2 - 4x + 2y - 12x + 14y + 26. What is the temperature of the coldest point in the plane?

omgitsrne Feb 14, 2024

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15

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+400

Minimum

Let x and y be nonnegative real numbers. If xy = \frac{2}{5}, then find the minimum value of 6x + \frac{3}{5y}.

omgitsrne Feb 14, 2024

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