Akhain1

260 Questions
16 Answers

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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 lire plus ..

Akhain1 21 mins ago

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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 lire plus ..

Akhain1 37 mins ago

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Quadratic Functions

What real value of $t$ produces the smallest value of the quadratic $t^2 -9t - 36 + 13t - 60$?

Akhain1 14 mai 2024

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Quadratic Function

If s is a real number, then what is the smallest possible value of 2s^2 - 8s + 19 - 7s^2 - 8s + 20?

Akhain1 14 mai 2024

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Roots of Quadratic

Let a and b be the solutions to 7x^2 + x - 5 = 6x^2 - 5x + 12. Compute 1/(a^2 - 4) + 1/(b^2 - 4).

Akhain1 14 mai 2024

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Quadratic Functions

If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4 +5t^2 - 4t + 18$?

Akhain1 14 mai 2024

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Algebra

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

Akhain1 14 mai 2024

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Algebra

Let a and b be complex numbers. If a + b = 4 and a^2 + b^2 = 6 + ab, then what is a^3 + b^3?

Akhain1 14 mai 2024

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+1075

Algebra

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

Akhain1 14 mai 2024

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Algebra

Compute
\frac{1}{1 \times 4} + \frac{1}{7 \times 10}

Akhain1 14 mai 2024