All Questions ⁺⁰233017 Questions

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Geometry

Let ABCD be a trapezoid with bases AB and CD. Let P be a point on side CD, and let X, Y be the feet of the altitudes from P to AD, BC respectively. Prove that if AD = 5, BC = 7, AB = 6, CD = 12, and CP/PD = 1, then PX = 12*sqrt(6)/5 and PY = read more ..

NotThatSmart ●●

RedDragonl Jun 3, 2024

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+697

Algebra

Find the minimum value of \frac{x^2}{x - 1 + x^3} for x > 1

eramsby1O1O Jun 3, 2024

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+697

Algebra

Find all values of t such that floor(t) = 3t + 4 - t^2. If you find more than one value, then list the values you find in increasing order, separated by commas.

eramsby1O1O Jun 3, 2024

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+697

Algebra

Let
f(x) = \left\lfloor\frac{2 - 3x}{3x + 1}\right\rfloor.
Evaluate f(1)+f(2) + f(3) + \dots + f(999)+f(1000). (This sum has 1000 terms, one for the result when we input each integer from 1 to 1000 into f.)

NotThatSmart ●

eramsby1O1O Jun 3, 2024

Jun 2, 2024

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Algebra

The function f(x) is defined for 1 \le x \le 5 as follows:
f(x) = 2x + 8 if 1 \le x \le 2
f(x) = 13 - 5x if 2 < x \le 3
f(x) = 20 - 14x if 3 < x \le 4 read more ..

eramsby1O1O Jun 2, 2024

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+697

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? read more ..

NotThatSmart ●

eramsby1O1O Jun 2, 2024

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Help plz

Let x,y,z and be nonzero real numbers. Find all possible values of

Thank you

Mememe Jun 2, 2024

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help with algebra

In physics, Ohm's law says that current through a wire, $I$, is directly proportional to voltage, $V$, and inversely proportional to resistance, $R$:
I = V/R
It's also true that resistance is directly proportional to the length of the wire. read more ..

learnmgcat Jun 2, 2024

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Algebra

A plane flies from Penthaven to Jackson and then back to Penthaven. When there is no wind, the round trip takes 3 hours and 36 minutes, but when there is a wind blowing from Penthaven to Jackson at 30 miles per hour, the trip takes 3 hours and 45 read more ..

learnmgcat Jun 2, 2024

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Algebra

When Helen jogs, every $20$ minutes she checks the total distance she traversed. One day, the first time she checked, Helen had jogged a distance of $2.5$ miles, the second time she checked, she had jogged a total distance of $3.8$ miles, and the third read more ..

learnmgcat Jun 2, 2024

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Counting

How many solutions are there to the equation
u + v + w + x + y + z = 7
where u, v, w, x, y, and z are nonnegative integers, and x is at most 2 or y is at least 3?

Rangcr897 Jun 2, 2024

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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.

BRAlNBOLT Jun 2, 2024

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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 ..

BRAlNBOLT Jun 2, 2024

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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.

BRAlNBOLT Jun 2, 2024

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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 ..

BRAlNBOLT Jun 2, 2024

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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 $AB$?

BRAlNBOLT Jun 2, 2024

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Geometry

In triangle $ABC$, $AB = 10$ and $AC = 15$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 1:3$, then find $AD$.

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BRAlNBOLT Jun 2, 2024

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