All Questions ⁺⁰230619 Questions

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211

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+6

Let $a,$ $b,$ $c,$ $x,$ $y,$ $z$ be nonzero complex numbers such that

Let $a,$ $b,$ $c,$ $x,$ $y,$ $z$ be nonzero complex numbers such that
\[a = \frac{b + c}{x - 2}, \quad b = \frac{a + c}{y - 2}, \quad c = \frac{a + b}{z - 2},\]and $xy + xz + yz = 5$ and $x + y + z = 3,$ find $xyz.$

Quadro Jul 4, 2022

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135

0

quadratic

Find all ordered pairs of real numbers (x,y) such that x^2 + 4x + y^2 + 6y + 13 = 0

Guest Jul 4, 2022

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140

0

help number theory

If a, b, c are positive integers less than 13 such that
2ab + bc + ca = 0 (mod 13)
ab + 2bc + ca = 8abc (mod 13)
ab + bc + 2ca = 6abc (mod 13) read more ..

Guest Jul 4, 2022

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131

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probability

Jeff will pick a card at random from ten cards numbered 1 through 10. The number on this card will indicate his starting point on the number line shown below. He will then spin the fair spinner shown below (which has three congruent sectors) and follow read more ..

Guest Jul 4, 2022

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138

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simplify

Simplify $\frac{2\sqrt[3]{9}}{1+\sqrt[3]{3}}$

Guest Jul 4, 2022

Jul 3, 2022

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128

0

coordinates

Find the equation of the line that passes through the points (-2, 1) and (3, 8). Write your answer in the form y = mx + b

Guest Jul 3, 2022

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199

1

Sequence

The increasing sequence 2, 3, 5, 6, 7, 10, ... consists of all positive integers which are not perfect powers. What is the sum of the squares of the digits of the 10th number in this sequence?

Guest Jul 3, 2022

0

191

1

Probability

If three, standard, 6-faced dice are rolled, what is the probability that the sum of the face up integers is 17?

Guest Jul 3, 2022

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206

1

help probability

If three, standard, 6-faced dice are rolled, what is the probability that the sum of the face up integers is 17?

Guest Jul 3, 2022

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191

1

help probability

Two distinct numbers are selected simultaneously and at random from the set {1,2,3,4,5,6,7,8} . What is the probability that the smaller one divides the larger one? Express your answer as a common fraction.

Guest Jul 3, 2022

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123

0

Parabola

Find the vertex of the graph of the equation x - y^2 + 8y = 13 + 4y - 8.

Guest Jul 3, 2022

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206

1

circles

In the diagram, each circle is divided into two equal areas and $O$ is the center of the larger circle. The area of the larger circle is $64\pi.$ What is the total perimeter of the shaded regions?

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Guest Jul 3, 2022

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128

0

geometry

To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. Her small kite is drawn on a one-inch grid, as shown below.

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Guest Jul 3, 2022

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191

1

help digits

My co-worker Larry only likes numbers that are divisible by 4, such as 20, or 4,004. How many different tens digits are possible in numbers that Larry likes?

Guest Jul 3, 2022

+2

212

2

+248

What is the product of the real parts of the roots of z^2 - z = 5 - 5i?

What is the product of the real parts of the roots of z^2 - z = 5 - 5i?

Blizzardshine Jul 3, 2022

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203

2

help probability

A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits such that there is exactly one card for any given rank and suit. Two of the suits are black and the other two suits are read more ..

Guest Jul 3, 2022

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132

0

counting question

There are 200 students enrolled at Memorial Middle School. Seventy of the students are in band and 85 are in chorus. If only 150 students are in band and/or chorus, how many students are in both band and chorus?

Guest Jul 3, 2022

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126

0

geometry

A square 10cm on each side has four quarter circles drawn with centers at the four corners. How many square centimeters are in the area of the shaded region? Express your answer in terms of pi.

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Guest Jul 3, 2022

+1

215

1

+248

Find all solutions, including the ones that are not real, to the equation z^3 + 8 = 0.

Find all solutions, including the ones that are not real, to the equation z^3 + 8 = 0.

Blizzardshine Jul 3, 2022

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137

0

help counting

Everyone in a class of 35 students takes math and history. 8 students received an A in history, 15 received an A in math and 8 received an A in both courses. How many students did not receive an A in either course?

Guest Jul 3, 2022

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137

0

Coordinates

Find the equation whose graph is the red line shown below. Write your answer in standard form. (Standard form is Ax+By = C, where A is positive, and A, B, and C are integers with greatest common divisor 1.)

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Guest Jul 3, 2022

+1

199

2

+248

Find the sums of the squares of the solutions of the equation 3z^2 - 5z + 8 = 0.

Find the sums of the squares of the solutions of the equation 3z^2 - 5z + 8 = 0.

Blizzardshine Jul 3, 2022

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132

0

rectangles

Rectangle PQRS is divided into eight squares, as shown.
The side length of each shaded square is 10. What is the length of the side of the largest square?

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Guest Jul 3, 2022

0

192

1

Proportion

Suppose that $\alpha$ is inversely proportional to $\beta$. If $\alpha = -3$ when $\beta = 8$, find $\alpha$ when $\beta = 16$. Express your answer as a fraction.

Guest Jul 3, 2022

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129

0

Coordinates

Write the equation y = 4/5*x - 7/10 + 2/3*y in standard form. (Standard form is Ax + By = C where A is positive, and A, B, and C are integers with greatest common divisor 1).

Guest Jul 3, 2022

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