All Questions ⁺⁰235602 Questions

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Geometry

Let $ABC$ be a triangle. The side lengths of triangle $ABC$ are $AB = 15,$ $BC = 9,$ and $AC = 10.$ Find the length of the shortest altitude in this triangle.

booboo44 Nov 4, 2024

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+688

Geometry

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.

booboo44 Nov 4, 2024

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+688

Geometry

Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1

booboo44 Nov 4, 2024

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+688

Geometry

In rectangle $EFGH$, let $M$ be the midpoint of $\overline{EF}$, and let $X$ be a point such that $MH = MX$, as shown below. If $\angle EFH = 30^\circ$ and $\angle MHX = 60^\circ,$ then find $\angle XHG,$ in degrees.

booboo44 Nov 4, 2024

Nov 3, 2024

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+755

Geometry

In trapezoid ABCD, \overline{AB} \parallel \overline{CD}. Find the area of the trapezoid.

gnistory Nov 3, 2024

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+755

Geometry

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

gnistory Nov 3, 2024

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+755

Geometry

In parallelogram $EFGH,$ let $M$ be the point on $\overline{EF}$ such that $FM:ME = 1:1,$ and let $N$ be the point on $\overline{EH}$ such that $HN:NE = 1:1.$ Line segments $\overline{FH}$ and $\overline{GM}$ intersect at $P,$ and line segments $\overline{FH}$ read more ..

gnistory Nov 3, 2024

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+637

Algebra

Let $a$ be a real number such that a - \frac{1}{a} = 4 + a + \frac{1}{a} Compute $a^3 - \frac{1}{a^3}$.

siIviajendeukie Nov 3, 2024

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+637

Algebra

Let
f(x) = \sqrt[3]{x + \sqrt[3]{x}}
Find the largest three-digit value of $x$ such that $f(x)$ is an integer.

siIviajendeukie Nov 3, 2024

Nov 2, 2024

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+637

Complex Numbers

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

siIviajendeukie Nov 2, 2024

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+637

Algebra

Compute
1 + \frac{3}{7} + \frac{5}{7^2} + \frac{7}{7^3}

siIviajendeukie Nov 2, 2024

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+637

Algebra

Compute
1 + \frac{3}{4} + \frac{5}{4^2} + \frac{7}{4^3}

siIviajendeukie Nov 2, 2024

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+637

Algebra

A geometric sequence has 400 terms. The first term is 1600 and the common ratio is $-\frac{1}{2}.$ How many terms of this sequence are greater than 1000?

siIviajendeukie Nov 2, 2024

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+637

Algebra

Find all c such that |c + 5| - 3c = 10 - 5|c - 1| + 7|c + 8|.

siIviajendeukie Nov 2, 2024

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+637

Algebra

Assume that $f(3) = 4$. Name a point that must be on the graph of y= f(2/5*x + 4) + f(3x - 1) .

siIviajendeukie Nov 2, 2024

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+637

Algebra

Let a(x) and b(x) be functions. Find (a\circ b)(3) - (b\circ a)(3) if $a(x) = 2x + 5$ and $b(x) = 4 - x^2$.

siIviajendeukie Nov 2, 2024

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+293

Geometry

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 ..

MeIdHunter Nov 2, 2024

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+293

Geometry

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 ..

MeIdHunter Nov 2, 2024

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+293

Geometry

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 ..

MeIdHunter Nov 2, 2024

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+293

Geometry

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 $a + b$.

MeIdHunter Nov 2, 2024

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+1439

Algebra

Fill in the blanks to make the statement true:
The infinite geometric series with first term and common ratio has a sum of .
4
10 read more ..

kittykat Nov 2, 2024

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