All Questions ⁺⁰236027 Questions

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Geometry

The perimeter of a rectangle is $40,$ and the length of one of its diagonals is $10 \sqrt{2}.$ Find the area of the rectangle.

Akhain1 May 20, 2024

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Geometry

Point $B$ is the image of point $A$ under a dilation with center $O$. If $OB = 30$, and the scale factor of the dilation is $5$, then what is $OA$?

Akhain1 May 20, 2024

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Geometry

Let a be a real number such that 0 < a < \frac{\pi}{2}. Show that \left(\sin(a)\right)^7 + \left(\cos ( a)\right)^7< 2*sin(a)^4*cos(a)^4.

Akhain1 May 20, 2024

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Geometry

Let A1 A2 A3 ... A15 be a regular polygon. A rotation centered at $A_4$ with an angle of $\alpha$ takes $A_3$ to $A_5$. Given that $\alpha < 180^\circ$, find $\alpha,$ in degrees.

Akhain1 May 20, 2024

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Geometry

Let $ABCD$ be a square with side length $1.$ A laser is located at vertex $A,$ which fires a laser beam at point $X$ on side $\overline{BC},$ such that $BX = \frac{1}{2}.$ The beam reflects off the sides of the square, until it ends up at another read more ..

Akhain1 May 20, 2024

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Geometry

Let $G$ be the center of equilateral triangle $XYZ.$ A dilation centered at $G$ with scale factor $-\frac{4}{3}$ is applied to triangle $XYZ,$ to obtain triangle $X'Y'Z'.$ Let $A$ be the area of the region that is contained inside both triangles XYZ and read more ..

Akhain1 May 20, 2024

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+120

Hey guys a couple problems

I was going over some notes and I noticed a pretty big dilemma talking about types of tables and graphs. I was wondering what are some defining features about types of graphs that make them the best option for a statistics problem. If you guys could please read more ..

breadstickim01 May 20, 2024

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+976

Geometry

Let $APQRS$ be a pyramid, where the base $PQRS$ is a square. The total surface area of pyramid $APQRS$ (including the base) is $800$. Let $W$, $X$, $Y$, and $Z$ be the midpoints of $\overline{AP}$, $\overline{AQ}$, $\overline{AR}$, and $\overline{AS}$, read more ..

Hi6942O May 20, 2024

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22

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+976

Geometry

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

Hi6942O May 20, 2024

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Geometry

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 ..

Hi6942O May 20, 2024

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Algebra

The area of a square, in square units, is 38 more than 10 times the length of a side of the square, in units.
Find all possible values for the side length of the square.

Hi6942O May 20, 2024

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Algebra

Fill in the blanks to make a quadratic whose roots are 5 and -5.
x^2 + ___ x + ___

Hi6942O May 20, 2024

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+976

Algebra

Consider the quadratic equation ax^2 + bx + c = 0 with a, b, c integers from -4 to 4 and a is not 0. The solutions to this equation are in the form where D is square free . Determine the largest possible value of D.

Hi6942O May 20, 2024

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+976

Algebra

Let a and b be the solutions to 5x^2 - 11x + 4 = 4x^2 - 17x + 6. Find 1/a^3 + 1/b^3.

Hi6942O May 20, 2024

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Algebra

Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 + 11x - 6. Find u/v^2 + v/u^2.

Hi6942O May 20, 2024

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Algebra

Let a and b be the solutions to 7x^2 + x - 5 = -3x^2 - 9x - 4. Compute (a - 4)/(b - 4) + (b - 4)/(a - 4).

Hi6942O May 20, 2024

May 19, 2024

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Algebra

Find all ordered pairs $(x,y)$ of real numbers such that $x + y = 10$ and $x^2 + y^2 = 32 + 2xy$.
For example, to enter the solutions $(2,4)$ and $(-3,9)$, you would enter "(2,4),(-3,9)" (without the quotation marks).

Hi6942O May 19, 2024

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Algebra

The roots of the quadratic equation $z^2 + bz + c = 0$ are $4 + 2i$ and $-5 + 8i$. What is $b+c$?

Hi6942O May 19, 2024

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Algebra

Find all values of $x$ such that
\frac{x}{x - 5} = \frac{4}{2x - 4} + 2x

Hi6942O May 19, 2024

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26

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+976

Algebra

Find all solutions to the equation $x^2 - 11x - 42 = -2x + 10 - x^2 - 5x + 16$.

Hi6942O May 19, 2024

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+976

Algebra

Find all real values of $s$ such that $x^2 + sx + 144 - 63 + x^2 - 25 - 18$ is the square of a binomial.

Hi6942O May 19, 2024

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Algebra

For what real values of $c$ is $x^2 - 8x - 4x + c + x^2 - 20x + x^2$ the square of a binomial?

Hi6942O May 19, 2024

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Algebra

One of the five quadratics below has a repeated root. (The other four have distinct roots.) What is the repeated root?
x^2 - 2x + 1
x^2 - 4x - 4
x^2 + 6x - 9 read more ..

Hi6942O May 19, 2024

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+868

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 read more ..

eramsby1O1O May 19, 2024

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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 ..

eramsby1O1O May 19, 2024

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