Akhain1

283 Questions
19 Answers

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Geometry

A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.
The line is x = 4, and the equation of the circle is x^2 + y^2 = 25.

Akhain1 Jul 12, 2024

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Geometry

Let $k$ be a positive real number. The line $x + y = 3 + k$ and the circle $x^2 + y^2 = k$ are drawn. Find $k$ so that the line is tangent to the circle.

Akhain1 Jul 12, 2024

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Geometry

Two circles intersect at two points, $P$ and $Q$. The equations of the two circles are $x^2 + (y - 1)^2 = 1$ and $(x - 1)^2 + y^2 = 1$. Find the length PQ.

Akhain1 Jul 12, 2024

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Algebra

Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$

Akhain1 Jun 22, 2024

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Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence. If
\[a_n = a_{n - 1} + a_{n - 2}\]
for all $n \ge 3,$ and $a_{11} = 1$ and $a_{10} = 4,$ then find $a_6.$

Akhain1 Jun 22, 2024

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Algebra

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

Akhain1 Jun 22, 2024

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Algebra

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

Akhain1 Jun 22, 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 Jun 8, 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 Jun 8, 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 Jun 8, 2024

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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 Jun 8, 2024

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Geometry

Points $A$ and $B$ are on side $\overline{YZ}$ of rectangle $WXYZ$ such that $\overline{WA}$ and $\overline{WB}$ trisect $\angle ZWX$. If $WX = 2$ and $XY = 3$, then what is the area of rectangle $WXYZ$?

Akhain1 May 21, 2024