All Questions ⁺⁰236457 Questions

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+1536

Geometry

A circular table is pushed into a corner of the room, where two walls meet at a right angle. A point P on the edge of the table (as shown below) has a distance of 6 from one wall, and a distance of 6 from the other wall. Find the radius of the read more ..

kittykat 1 hour ago

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+1536

Geometry

Lines $XQ$ and $XR$ are tangent to a circle, as shown below. If $\angle XTU = 47^\circ$ and $\angle QAU = 65^\circ,$ then find $\angle QXR,$ in degrees.

kittykat 1 hour ago

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+1536

Geometry

Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{AB}$ is $4,$ and the circle has a radius of $5.$ Find the length $AB.$ read more ..

kittykat 3 hours ago

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+1536

Geometry

We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude to $\triangle ABC$. If $AC = 8,$ $BK = 2$, and $CK = 3,$ then what is $AB$?

kittykat 3 hours ago

Jan 19, 2025

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+5

Geometry

NM is the median of trapezoid ABCD. If [ABMN] = 8 and [NMCD] = 12, what is CD/AB

Tigergaming2464 Jan 19, 2025, 10:26 PM

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+705

Geometry

In triangle $ABC$, let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. If $BC = 20$ and $\angle C = 15^\circ$, then find the length of $BE$.

cooIcooIcooI17 Jan 19, 2025

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+705

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 $4,$ and the area of triangle $PSR$ is $8.$ Find the area of trapezoid read more ..

cooIcooIcooI17 Jan 19, 2025

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+569

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

crimefightingvigiI Jan 19, 2025

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+569

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.

crimefightingvigiI Jan 19, 2025

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+569

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

crimefightingvigiI Jan 19, 2025

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+569

Geometry

In triangle $ABC$, point $D$ is on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 45^\circ$, $\angle C = 45^\circ$, and $AC = 12$, then find the area of triangle $ABD$.

crimefightingvigiI Jan 19, 2025

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+758

Algebra

Let f(x) = x^2 + bx + 12 - 3x + 10 for all real numbers $x$. Find the greatest integer value of $b$ such that $-4$ is not in the range of $f(x)$.

bingboy Jan 19, 2025

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+758

Algebra

Let f(x) be a polynomial with integer coefficients. There exist distinct integers $p,$ $q,$ $r,$ $s,$ $t$ such that
f(p) = f(q) = f(r) = f(s) = 1
and $f(t) > 1.$ What is the smallest possible value of $f(t)?$

bingboy Jan 19, 2025

Jan 18, 2025

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+758

Algebra

Solve the inequality
\frac{3-z}{z+1} \ge 2(z + 4).
Write your answer in interval notation.

bingboy Jan 18, 2025

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+758

Algebra

Solve the inequality 4t^2 \le -9t + 12 - 4t + 23 + (2t - 1)(2t + 1). Write your answer in interval notation.

bingboy Jan 18, 2025

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+190

Geometry

Triangle $ABC$ has altitudes $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$ If $AD = 12,$ $BE = 20,$ and $CF$ is a positive integer, then find the largest possible value of $CF.$

cwimie Jan 18, 2025

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+190

Geometry

There are four cities: Capital City, Little Village, Mytown, and Yourtown. The distance from Capital City to Little Village is $5$ miles. The distance from Capital City to Mytown is $2$ miles. The distance from Mytown to Yourtown is $3$ read more ..

cwimie Jan 18, 2025

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+190

Geometry

All the sides of a triangle are integers. If the perimeter of the triangle is $3,$ then how many different possible triangles are there? (Assume that the triangle is non-degenerate. Two triangles are considered the same if they are co read more ..

cwimie Jan 18, 2025

Jan 17, 2025

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+984

Algebra

Let a and b be positive real numbers such that a + b = 1. Find the set of all possible values of a/b.

Hi6942O Jan 17, 2025

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+984

Algebra

Find the number of ordered pairs $(a,b)$ of integers such that
\frac{a + 2}{a + 1} = \frac{b}{8}.

Hi6942O Jan 17, 2025

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