All Questions ⁺⁰234777 Questions

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Counting

Consider a house with a finite number of rooms. Some of the rooms have an even number of doors, and some of the rooms have an odd number of doors. In the house below, how many doors are there from the house to the outside world?

cooIcooIcooI17 Sep 26, 2024

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+1344

Geometry

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

blackpanther Sep 26, 2024

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+1344

Geometry

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

blackpanther Sep 26, 2024

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+1344

Geometry

Triangle ABC is rotated completely around side BC, sweeping out a solid in space. Find the volume of the solid.
AB = 8, AC = 12, CB = 8

blackpanther Sep 26, 2024

Sep 25, 2024

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+228

Geometry

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

nathanl6656 Sep 25, 2024

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+228

Geometry

In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $CE =3$ and $DF = 3$, then what is $BD$?

nathanl6656 Sep 25, 2024

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+228

Geometry

In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

nathanl6656 Sep 25, 2024

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+872

Geometry

In quadrilateral $PQRS$, $PQ = QR = RS = SP$, $PR = \sqrt{2}$, and $QS = \sqrt{2}$. Find the perimeter of $PQRS$.

Rangcr897 Sep 25, 2024

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+687

Algebra

At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15. Gary orders a bagel and a muffin, which comes out to $3.50. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15. How many cents does read more ..

magenta Sep 25, 2024

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+543

Counting

Consider a house with a finite number of rooms. Some of the rooms have an even number of doors, and some of the rooms have an odd number of doors. In the house below, how many doors are there from the house to the outside world?

gnistory Sep 25, 2024

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+543

Counting

Twelve points are given in the plane, so that no three points lie on the same line.
Five distinct segments joining pairs of these points are chosen at random, with all \binom{12}{2} = 66 such segments equally likely to be chosen. What read more ..

gnistory Sep 25, 2024

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+543

Counting

A standard 8 times 8 chessboard has 64 unit squares.
Eight rooks are randomly placed on different squares of a chessboard. A rook is said to attack all of the squares in its row and its column.
Compute the probability read more ..

gnistory Sep 25, 2024

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+70

Geometry

In a rhombus, all the side lengths are $12,$ and one of the angles is equal to $90^\circ.$ Find the area of the rhombus.

AUnVerifedTaxPayer Sep 25, 2024

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+70

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.

AUnVerifedTaxPayer Sep 25, 2024

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+70

Geometry

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

AUnVerifedTaxPayer Sep 25, 2024

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+70

Geometry

A rectangle contains a strip of width $1,$ as shown below. Find the area of the strip.

AUnVerifedTaxPayer Sep 25, 2024

Sep 24, 2024

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+471

Algebra

For some real number a and some positive integer n, the first few terms in the expansion of (1 + ax)^n are 1 - 12x + 54x^2 + cx^3 + ... Find c.

Pythagorearn Sep 24, 2024

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+471

Algebra

Find the sum of all values of $k$ for which $x^2 + kx - 9x + 25 - 9$ is the square of a binomial.

Pythagorearn Sep 24, 2024

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+543

Algebra

Find the largest real number $c$ such that $1$ is in the range of $f(x)=x^2-5x+c-3x+8$.

gnistory Sep 24, 2024

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