falafronk

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Donatello starts with a marble cube. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length. If the side length of the original cube is $6$, then find the volume of the resulting read more ..

falafronk Mar 24, 2024

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help pyramid

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 $ABCD$ is $18.$ Find the volume of pyramid $DEFG$.

falafronk Mar 24, 2024

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Prism

Let $ABCDEFGH$ be right rectangular prism. The total surface area of the prism $1.$ Also, the sum of all the edges of the prism is $2.$ Find the length of the diagonal joining one corner of the prism to the opposite corner.

falafronk Mar 24, 2024

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Circles

Let $\overline{TU}$ and $\overline{VW}$ be chords of a circle, which intersect at $S$, as shown. Find $SW$.

falafronk Mar 24, 2024

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Circles

A circle is centered at $O.$ The tangent to the circle at $P$ is extended to $Q.$ Line segment $\overline{QS}$ intersects the circle at $R.$ Find the radius of the circle.

falafronk Mar 24, 2024

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Angle bisector

Let $\overline{AB}$ be a diameter of a circle, and let $C$ be a point on the circle such that $AC = 4$ and $BC = 4.$ The angle bisector of $\angle ACB$ intersects the circle at point $M.$ Find $CM.$

falafronk Mar 24, 2024

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28

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Algebra

Describe all solutions to zw -3w - 2iw + 4iz = - 8 + 12i where z and w are complex numbers.

falafronk Mar 20, 2024

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Integers

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

falafronk Mar 20, 2024

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46

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Complex Numbers

Fill in the blanks with real numbers
(sqrt(3) + i)*6i = ___ + ___i

falafronk Mar 20, 2024

0

33

1

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Algebra

Select a sign ($+$ or $-$) and fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear factors.
2mn - 21m + 5n - mn + 14m + 6n + ___

falafronk Mar 20, 2024