All Questions ⁺⁰236337 Questions

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math problem

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 $a + b$.

nathanl6656 Mar 24, 2024

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43

0

+368

Algebra

Let $x$ and $y$ be complex numbers. If $x + y =2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?

nathanl6656 Mar 24, 2024

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30

1

+1927

help with coordinates

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 ..

tomtom Mar 24, 2024

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38

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+1927

help! coordinates

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 ..

tomtom Mar 24, 2024

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30

1

+1927

help coordinates

The line $y = mx$ bisects the angle between the two lines shown below. Find $m$.

tomtom Mar 24, 2024

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33

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+1927

Coordinates

Let $x$ and $y$ be real numbers. If $x$ and $y$ satisfy
\[x^2 + y^2 = 4x - 8y + 17x - 5y + 25,\]
then find the largest possible value of $x.$ Give your answer in exact form using radicals, simplified as far as possible.

tomtom Mar 24, 2024

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29

1

+823

Pyramid

Let $ABCDE$ be a right square pyramid, with base $ABCD$ and apex $E.$ Find the volume of the pyramid.

booboo44 Mar 24, 2024

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24

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+823

Prism

Let $ABCDEFGH$ be right rectangular prism. The total surface area of the prism $15.$ Also, the sum of all the edges of the prism is $17.$ Find the length of the diagonal joining one corner of the prism to the opposite corner.

booboo44 Mar 24, 2024

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30

1

+359

need help

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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42

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+359

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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36

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+359

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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35

0

+1520

Angles

Let $B,$ $A,$ and $D$ be three consecutive vertices of a regular $20$-gon. A regular heptagon is constructed on $\overline{AB},$ with a vertex $C$ next to $A.$ Find $\angle BAD,$ in degrees.

kittykat Mar 24, 2024

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36

0

+1520

Polygon

Let $IJKLMN$ be a hexagon with side lengths $IJ = LM = 3,$ $JK = MN = 3,$ and $KL = NI = 3$. Also, all the interior angles of the hexagon are equal. Find the area of hexagon $IJKLMN$.

kittykat Mar 24, 2024

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28

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+1520

Angles

The interior angles of a polygon form an arithmetic sequence. The difference between the largest angle and smallest angle is $56^\circ$. If the polygon has $3$ sides, then find the smallest angle, in degrees.

kittykat Mar 24, 2024

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24

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+1560

Triangle

In triangle $ABC,$ $AC = BC$ and $\angle ACB = 90^\circ.$ Points $P$ and $Q$ are on $\overline{AB}$ such that $P$ is between $A$ and $Q$ and $\angle QCP = 45^\circ.$ If cos ACP = 2/3, then find cos BCQ.

blackpanther Mar 24, 2024

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28

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+1560

Triangle

In triangle PQR, M is the midpoint of PQ. Let X be the point on QR such that PX bisects angle QPR, and let the perpendicular bisector of PQ intersect AX at Y. If PQ = 36, PR = 22, QR = 26, and MY = 8, then find the area of triangle PQR

blackpanther Mar 24, 2024

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37

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+1560

geometry problem

In square ABCD, P is on BC such that BP = 4 and PC = 1, and Q is on line CD such that DQ = 2 and QC = 3. Find sin angle PAQ.

blackpanther Mar 24, 2024

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32

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+1577

help with triangle

In triangle $ABC$, $\angle A = 30^\circ$ and $\angle B = 90^\circ$. Point $X$ is on side $\overline{AC}$ such that line segment $\overline{BX}$ bisects $\angle ABC$. If $BC = 12$, then find the area of triangle $BXA$.

parmen Mar 24, 2024

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30

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+1577

Triangle

In triangle ABC, the angle bisector of angle BAC meets BC at D. If angle BAC=60 degrees, angle ABC=60 degrees and AD=24 then find the area of triangle ABC.

parmen Mar 24, 2024

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27

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+1577

Pyramid

A right pyramid has a square base. The area of each triangular face is one-third the area of the square face. If the total surface area of the pyramid is $432$ square units, then what is the volume of the pyramid in cubic units? read more ..

parmen Mar 24, 2024

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38

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+359

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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31

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+359

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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28

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+359

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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32

1

+70

last one for the month

Semicircles are constructed on AB, AC,and BC. A circle is tangent to all three semicircles. Find the radius of the circle.
read more ..

faiafronk Mar 24, 2024

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23

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+70

trianlge ABC

In triangle ABC, point X is on side BC such that AX = 13, BC = 10, CX = 4, and the circumcircles of traingles ABX and ACX have the same radius. Find the area of triangle ABC.
(no diagram)

faiafronk Mar 24, 2024

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