All Questions ⁺⁰236046 Questions

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Triangle

In triangle $PQR$, let $M$ be the midpoint of $\overline{PQ}$, let $N$ be the midpoint of $\overline{PR}$, and let $O$ be the intersection of $\overline{QN}$ and $\overline{RM}$, as shown. If $\overline{QN}\perp\overline{PR}$, $QN = 15$, and $PR = 20$, read more ..

booboo44 Jan 16, 2024

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30

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+1348

Triangle

Points $M$, $N$, and $O$ are the midpoints of sides $\overline{KL}$, $\overline{LJ}$, and $\overline{JK}$, respectively, of triangle $JKL$. Points $P$, $Q$, and $R$ are the midpoints of $\overline{NO}$, $\overline{OM}$, and $\overline{MN}$, respectively. read more ..

sandwich Jan 16, 2024

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28

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+1348

Angles

Let $M$, $N$, and $P$ be the midpoints of sides $\overline{TU}$, $\overline{US}$, and $\overline{ST}$ of triangle $STU$, respectively. Let $\overline{UZ}$ be an altitude of the triangle. If $\angle TSU = 60^\circ$ and $\angle STU = 45^\circ$, read more ..

sandwich Jan 16, 2024

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56

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+1348

Geometry

Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$. Find the area of triangle $BET$ if $AB = 20$, $AC = 20$,
<read more ..

sandwich Jan 16, 2024

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30

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+1348

Triangle

Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. What is the area of triangle $ABC$?

sandwich Jan 16, 2024

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26

1

+816

Probability

A member of the junior varsity swim team will be elected as President. Each member is equally likely to be elected. The probability that a sophomore is elected is times the probability that a freshman is elected. Everyone on the junior varsity read more ..

maximum Jan 16, 2024

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31

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+816

Counting

Find the number of positive integers that satisfy both the following conditions:
- Each digit is a 1 or a 2 or a 3
- The sum of the digits is 5

maximum Jan 16, 2024

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28

1

+1768

Operations

Let $x \mathbin{\spadesuit} y = \frac{x^2}{y}$ for all $x$ and $y$ such that $y\neq 0$. Find all values of $a$ such that $a \mathbin{\spadesuit} (a + 1) = 9$. Write your answer as a list separated by commas.

bader Jan 16, 2024

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51

0

+1768

Functions

Suppose $a(x) = 2x + 5 - 3x^2 + x^3$ and $b(x) = 4 - x^2 + 4x^4$. Find $(a\circ b)(3) - (b\circ a)(3).$

bader Jan 16, 2024

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29

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+13

pls help

In triangle $ABC,$ $\angle C = 90^\circ.$ A semicircle is constructed along side $\overline{AC}$ that is tangent to $\overline{BC}$ and $\overline{AB}.$ If the radius of the semicircle is equal to $\frac{BC}{3},$ then find $\frac{AC}{BC}.$

whatsupbro Jan 16, 2024

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45

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+1768

Inverse Functions

Let
\[f(x) = \frac{2x + 5}{x - 4} + \frac{x^2 + 1}{4x - 3}.\]
Find the inverse $f^{-1}(x)$.

bader Jan 16, 2024

+1

50

1

+13

help urgent

The tangent to the circumcircle of triangle $WXY$ at $X$ is drawn, and the line through $W$ that is parallel to this tangent intersects $\overline{XY}$ at $Z.$ If $XY = 15$ and $WX = 10,$ find $YZ.$

whatsupbro Jan 16, 2024

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45

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+1911

cone

The circular sector below is rolled to form a cone. Find the volume of the cone.

tomtom Jan 16, 2024

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34

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+1911

cone and cylinder

Let $A$ be the lateral surface area of the cone below, and let $B$ be the lateral surface area of the cylinder below. Find $\frac{A}{B}.$

tomtom Jan 16, 2024

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45

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+1911

cone

The cone below has a lateral surface area of $16 \sqrt{5} \cdot \pi$. Find the volume of the cone.
read more ..

tomtom Jan 16, 2024

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34

0

+1911

3d geo

Triangle $ABC$ below is rotated completely around side $\overline{BC}$, sweeping out a solid in space. Find the volume of the solid.

tomtom Jan 16, 2024

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20

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+1911

3d geo

A spherical ball fits snugly inside a cylindrical jar, so that the ball touches the top and bottom of the jar, and the sides of the jar. The volume of the cylinder is $144 \pi.$ What is the difference between the surface area of the sphere and read more ..

tomtom Jan 16, 2024

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31

1

+1911

Sphere

A plane and a sphere intersect in a circle. The circle has an area of $14 \pi$. The radius of the sphere is $10$ units. Find the surface area of the sphere.

ElectricPavlov ●

tomtom Jan 16, 2024

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48

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+1911

Cylinder

The rectangle below is wrapped around a cylinder with height $10$. The rectangle perfectly covers the curved surface of the cylinder without overlap. What is the volume of the cylinder?

tomtom Jan 16, 2024

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35

2

+13

pls help

The tangent to the circumcircle of triangle $WXY$ at $X$ is drawn, and the line through $W$ that is parallel to this tangent intersects $\overline{XY}$ at $Z.$ If $XY = 15$ and $WX = 10,$ find $YZ.$

whatsupbro Jan 16, 2024

-1

26

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+478

Expand

Expand (sqrt(t) + t^3)^4*(sqrt(t) - t^3)^4*(t - sqrt(t))^4.

ChiIIBill Jan 16, 2024

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29

1

+478

Expand

Expand (x^2 + 1/x)*(x + x^4)^3.

ChiIIBill Jan 16, 2024

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