All Questions ⁺⁰234655 Questions

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+1055

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 $ABD$.

learnmgcat Sep 17, 2024

0

1

1

+1475

Algebra

A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to ten times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right read more ..

parmen Sep 17, 2024

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1

+1475

Algebra

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

parmen Sep 17, 2024

Sep 16, 2024

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1

1

+310

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 ..

cooIcooIcooI17 Sep 16, 2024

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2

1

+310

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 ..

cooIcooIcooI17 Sep 16, 2024

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1

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+310

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 $CF.$

cooIcooIcooI17 Sep 16, 2024

Sep 15, 2024

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1

1

+116

Algebra

Compute the sum
\frac{1}{\sqrt{36} + \sqrt{39}} + \frac{1}{\sqrt{36} + \sqrt{45}} + \frac{1}{\sqrt{39} +\sqrt{96}}

MEMEG0D Sep 15, 2024

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2

+116

Geometry

Let x and y be complex numbers. If $x + y =2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?

MEMEG0D Sep 15, 2024

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1

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+116

Geometry

The circles $x^2 + y^2 = 4$ and $(x - 5)^2 + (y - 8)^2 = 60$ intersect in two points $A$ and $B.$ Find the distance $AB$.

MEMEG0D Sep 15, 2024

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1

0

+116

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 $f(t)?$

MEMEG0D Sep 15, 2024

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1

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+116

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 $f(x)$.

MEMEG0D Sep 15, 2024

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1

0

+116

Algebra

Let S be the set of all real numbers of the form
a1/3 + a2/3^2 + a3/3^3 + ....
where a_i \in {0, 1, 2} for all i.
(a) Is the number 1/7 in the set S?
(b) Is the number 1/4 in the set S? read more ..

MEMEG0D Sep 15, 2024

Sep 14, 2024

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2

1

+673

Algebra

Compute
1 + \frac{3}{6} + \frac{5}{9} + \frac{7}{12} + \dotsb.

magenta Sep 14, 2024

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1

0

+673

Algebra

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

magenta Sep 14, 2024

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1

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+2651

Algebra

The system of equations
{xy}/{x + y} = 1,{xz}/{x + z} = 1, {yz}/{y + z} = 4
has exactly one solution. What is z in this solution?

LiIIiam0216 Sep 14, 2024

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1

0

+2651

Algebra

Let x, y, and z be positive real numbers. Find the largest possible value of
sqrt((3x + 4y)/(6x + 5y + 4z)) + sqrt((y + 4z)/(2x + 3y + 8z)) + sqrt((5z + 7x)/(4x + 3y + 5z))

LiIIiam0216 Sep 14, 2024

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2

+70

help

On the xy-plane, the origin is labeled with an M. The points (1,0), (-1,0), (0,1), and (0,-1) are labeled with A's. The points (2,0), (1,1), (0,2), (-1, 1), (-2, 0), (-1, -1), (0, -2), and (1, -1) are labeled with T's. The points (3,0), (2,1), (1,2), (0, 3), read more ..

Hamburger Sep 14, 2024

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1

1

+116

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

MEMEG0D Sep 14, 2024

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2

+116

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.

MEMEG0D Sep 14, 2024

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1

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+116

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.

MEMEG0D Sep 14, 2024

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1

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+116

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 ..

MEMEG0D Sep 14, 2024

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