All Questions ⁺⁰234655 Questions

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Algebra

Evaluate $a^3 - \dfrac{1}{a^3}$ if $a - \dfrac{1}{a} = 0$.

ABJeIIy Aug 3, 2024

Aug 2, 2024

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8

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+1698

Algebra

Let
f(x) = \sqrt{x - \sqrt{x}}.
Find the largest three-digit value of $x$ such that $f(x)$ is an integer.

ABJeIIy Aug 2, 2024

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3

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+779

Number Theory

The number $N$ is a multiple of $7$. The base $2$ representation of $N$ is
10011010011ABC110_2.
Compute the ordered triple of digits $(A,B,C)$.

eramsby1O1O Aug 2, 2024

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6

+42

Number Theory Question

How many digits are in the whole-number representation of The answer is not 101.

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Umeko1.0 Aug 2, 2024

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3

+2651

Number Theory

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.

LiIIiam0216 Aug 2, 2024

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2

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+2651

Number Theory

Find the number of $7$-digit numbers, where the sum of the digits is divisible by $11.$

LiIIiam0216 Aug 2, 2024

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+355

Physics

In the previous problem, we imagined a bathtub draining. Water would leave the tub with a speed of $2.9 \;\mathrm{m/s}$ when the tub is full. Suppose the cross sectional area of the drain is $8.0 \;\mathrm{cm}^2.$ You may assume that all water leaving read more ..

Lilliam0216 Aug 2, 2024

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+355

When the bathtub is full, water leaves the bath tub at $2.9 \;\mathrm{m/s}$ through the drain. At what speed does water leave the bathtub w

When the bathtub is full, water leaves the bath tub at $2.9 \;\mathrm{m/s}$ through the drain. At what speed does water leave the bathtub when it is half full?

Lilliam0216 Aug 2, 2024

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+820

Counting

In how many ways can the numbers 1, 2, 3, 4, 5, 6 be arranged in a row, so that the product of any two adjacent numbers is at least 5?

Hi6942O Aug 2, 2024

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5

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+296

Geometry

In triangle $ABC$, point $D$ is on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 45^\circ$, $\angle C = 45^\circ$, and $AC = 12$, then find the area of triangle $ABD$.

Stanry Aug 2, 2024

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7

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+772

Geometry

In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $BC = 15,$ and $CX = 5$. What is $BX$?

RedDragonl Aug 2, 2024

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8

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+868

Geometry

In triangle $ABC,$ $AB = 15,$ $BC = 9,$ and $AC = 10.$ Find the length of the shortest altitude in this triangle.

Rangcr897 Aug 2, 2024

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+4

help pls

Given R(-6, 19) and S(15, -5), what are the coordinates of the point on segment RS two-thirds of the distance from R to S?

Roteuw Aug 2, 2024

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+1698

Geometry

In triangle $ABC,$ the angle bisector of $\angle BAC$ meets $\overline{BC}$ at $D.$ If $\angle BAC = 60^\circ,$ $\angle CAD = 45^\circ,$ and $AD = 24,$ then find the area of triangle $ABC.$

ABJeIIy Aug 2, 2024

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4

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+1698

Geometry

Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$. The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has slope $2$. Find $a + b$.

ABJeIIy Aug 2, 2024

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+1698

Geometry

Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $4,$ and the slope of line segment $\overline{OQ}$ is $5.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: read more ..

ABJeIIy Aug 2, 2024

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+1698

Geometry

Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If $A = (7,-11),$ $B = (10,13),$ and $C = (18,-22)$, then find the constant read more ..

ABJeIIy Aug 2, 2024

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+1698

Geometry

A circular table is pushed into a corner of the room, where two walls meet at a right angle. A point P on the edge of the table (as shown below) has a distance of 6 from one wall, and a distance of 6 from the other wall. Find the radius of the read more ..

ABJeIIy Aug 2, 2024

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+1698

Geometry

Lines $XQ$ and $XR$ are tangent to a circle, as shown below. If $\angle XTU = 47^\circ$ and $\angle QAU = 65^\circ,$ then find $\angle QXR,$ in degrees.

ABJeIIy Aug 2, 2024

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6

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+1698

Geometry

Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{AB}$ is $4,$ and the circle has a radius of $5.$ Find the length $AB.$ read more ..

ABJeIIy Aug 2, 2024

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+1698

Geometry

We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude to $\triangle ABC$. If $AC = 8,$ $BK = 2$, and $CK = 3,$ then what is $AB$?

ABJeIIy Aug 2, 2024

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+1698

Geometry

Altitudes AD and BE of acute triangle ABC intersect at point H. If angle ABH = 30 and angle DAC = 45, then what is angle HCA in degrees?

ABJeIIy Aug 2, 2024

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+1698

Geometry

In trapezoid EFGH, \overline{EF} \parallel \overline{GH}, and P is the point on \overline{EH} such that EP:PH = 1:2. If the area of triangle PEF is 6, and the area of triangle PGH is 6, then find the area of trapezoid EFGH.

ABJeIIy Aug 2, 2024

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