All Questions ⁺⁰236031 Questions

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20

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+868

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots,$ $a_{10},$ $a_{11},$ $a_{12}$ be an arithmetic sequence. If $a_1 + a_3 + a_5 + a_7 + a_9 + a_{11} = 0$ and $a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12} = 0$, then find $a_1$.

eramsby1O1O May 24, 2024

0

16

1

+868

Algebra

When the same constant is added to the numbers $60,$ $120,$ and $160,$ a three-term geometric sequence arises. What is the common ratio of the resulting sequence?

eramsby1O1O May 24, 2024

0

9

2

+868

Algebra

What real value of $t$ produces the smallest value of the quadratic $t^2 -9t - 36 + 13t - 60$?

eramsby1O1O May 24, 2024

0

15

1

+868

Algebra

If $s$ is a real number, then what is the smallest possible value of $2s^2 - 8s + 19 - 7s^2 - 8s + 20$?

eramsby1O1O May 24, 2024

0

19

1

+868

Algebra

Find the sum of all positive integers less than $1000$ ending in $3$ or $4$ or $5$.

eramsby1O1O May 24, 2024

0

14

1

+868

Algebra

If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4 +5t^2 - 4t + 18$?

NotThatSmart ●

eramsby1O1O May 24, 2024

0

21

1

+868

Algebra

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

eramsby1O1O May 24, 2024

0

16

1

+868

Algebra

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

eramsby1O1O May 24, 2024

-1

18

1

+219

Algebra

Five workers have been hired to complete a job. If one additional worker is hired, they could complete the job $4$ days earlier. If the job needs to be completed $20$ days earlier, how many additional workers should be hired?

BRAlNBOLT May 24, 2024

-1

12

1

+219

Algebra

Three runners, Dirk, Edith, and Foley all start at the same time for a $24$ km race, and each of them runs at a constant speed. When Dirk finishes the race, Edith is $10$ km behind, and Foley is $15$ km behind. When Edith finishes the race, how far read more ..

NotThatSmart ●

BRAlNBOLT May 24, 2024

May 23, 2024

-1

16

1

+219

Geometry

Point D is the midpoint of median AM of triangle ABC. Point E is the midpoint of AB, and point T is the intersection of BD and ME. Find the area of triangle BEC if [ABC] = 14.

BRAlNBOLT May 23, 2024

-1

20

1

+219

Geometry

In triangle $ABC$, $\angle ABC = 90^\circ$, and $D$ is on side $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. If $AB = 4,$ $BC = 3$, and $AC = 5,$ then find the area of $\triangle ADC$. Round your answer to the nearest integer.

BRAlNBOLT May 23, 2024

-1

14

1

+219

Geometry

In triangle $ABC$, let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. If $BC = 20$ and $\angle C = 15^\circ$, then find the length of $BE$.

BRAlNBOLT May 23, 2024

0

16

1

+2653

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 ..

LiIIiam0216 May 23, 2024

0

17

1

+2653

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 ..

LiIIiam0216 May 23, 2024

0

25

1

+2653

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 of the freshmen are standing next to each other?

NotThatSmart ●

LiIIiam0216 May 23, 2024

0

21

2

+2653

Counting

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 psychology textbooks are together?

LiIIiam0216 May 23, 2024

0

27

1

+2653

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 together? (Two seatings are considered the same if one can be rotated to form the other.)

LiIIiam0216 May 23, 2024

0

10

2

+355

Math

Let $n$ be a positive integer.
(a) There are $n^2$ ordered pairs $(a,b)$ of positive integers, where $1 < a,$ $b < n.$ Using a counting argument, show that this number is also equal to \[n + 2 \binom{n}{2}.\]
(b) There are $n^3$ read more ..

Lilliam0216 May 23, 2024

-1

16

1

+219

Geometry

Find the area of parallelogram ABCD.

BRAlNBOLT May 23, 2024

-1

24

1

+219

Geometry

The perimeter of a rectangle is $40,$ and the length of one of its diagonals is $10 \sqrt{2}.$ Find the area of the rectangle.

BRAlNBOLT May 23, 2024

+2

11

1

+280

Help!

Let $ABCD$ be a regular tetrahedron. Let $E$, $F$, $G,$ $H$ be the centers of faces $BCD$, $ACD$, $ABD$, $ABC$, respectively. The volume of pyramid $DEFG$ is $18.$ Find the volume of pyramid $EFGH$

BRAINBOLT May 23, 2024

+2

15

1

+280

Probability

what is the probability a number between 1 and 1000 is divible by 5?

BRAINBOLT May 23, 2024

0

26

0

+976

Dilation

Let $G$ be the center of equilateral triangle $XYZ.$ A dilation centered at $G$ with scale factor $-1$ is applied to triangle $XYZ,$ to obtain triangle $X'Y'Z'.$ Let $A$ be the area of the region that is contained in both triangles $XYZ$ and $X'Y'Z'.$ Find read more ..

Hi6942O May 23, 2024

0

17

1

+976

Rotations

Let A1 A2 A3 ... A15 be a regular polygon. A rotation centered at $A_4$ with an angle of $\alpha$ takes $A_3$ to $A_5$. Given that $\alpha < 180^\circ$, find $\alpha,$ in degrees.

Hi6942O May 23, 2024

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