All Questions ⁺⁰235702 Questions

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How does one solve this problem?

Points A, B, and C are given in the coordinate plane. There exists a point Q and a constant k such that for any point P,
PA² + PB² + PC² = 3PQ² + k.
If A = (4,-4), B = (3,8), and C = (-1, 2), then find the constant k.

NinvisibleYT Aug 9, 2024

0

6

1

+306

Probability

The numbers $x_1,$ $x_2,$ $x_3,$ $x_4$ are chosen at random in the interval $[0,1].$ Let $I$ be the interval between $x_1$ and $x_2,$ and let $J$ be the interval between $x_3$ and $x_4.$ Find the probability that intervals $I$ and $J$ both contain read more ..

nathanl6656 Aug 9, 2024

0

12

1

+306

Counting

The entries in a certain row of Pascal's triangle are
1, n, \dots, n, 1.
The average of the entries in this row is 2. Find $n$.

nathanl6656 Aug 9, 2024

0

8

1

+250

Counting

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?

MEMEG0D Aug 9, 2024

0

12

0

+2653

Algebra

The real numbers $a$ and $b$ satisfy $a - b = 1$ and $a^3 - b^3 = 1.$
(a) Find all possible values of $ab.$
(b) Find all possible values of $a + b.$
(c) Find all possible values of $a$ and $b.$

LiIIiam0216 Aug 9, 2024

0

3

1

+2653

Algebra

Find the largest integer $k$ such that the equation
5x^2 - kx + 8 - 13x^2 + 39 = 0
has no real solutions.

LiIIiam0216 Aug 9, 2024

0

6

2

+544

Algebra

Will and Grace are canoeing on a lake. Will rows at $50$ meters per minute and Grace rows at $30$ meters per minute. Will starts rowing at $2$ p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at $2{:}45$ p.m. read more ..

cooIcooIcooI17 Aug 9, 2024

0

10

2

+544

Algebra

I have to paint one side of a wall. The wall is $15$ meters tall and $60$ meters long. Each gallon of paint covers $400$ square feet. If a foot is approximately $0.3048$ meters, then what is the smallest whole number of gallons I can buy and read more ..

cooIcooIcooI17 Aug 9, 2024

0

9

2

+544

Algebra

I drove to the beach at a rate of $40$ miles per hour. If I had driven at a rate of $55$ miles per hour instead, then I would have arrived $25$ minutes earlier. How many miles did I drive?

cooIcooIcooI17 Aug 9, 2024

0

4

1

+85

Imaginary Numbers

For all complex numbers z, let Find f(f(f(f(1+i)))).

PurpleWasp Aug 9, 2024

0

6

1

+448

Algebra

When the same constant is added to the numbers $60,$ $120,$ and $140,$ a three-term geometric sequence arises. What is the common ratio of the resulting sequence?

onyuIee Aug 9, 2024

0

12

1

+448

Algebra

A geometric sequence has $400$ terms. The first term is $1000$ and the common ratio is $-\frac{8}{9}.$ How many terms of this sequence are greater than $1?$

onyuIee Aug 9, 2024

0

16

1

+448

Algebra

Find the sum of all positive integers less than 1000 ending in 3 or 4 or 6 or 9.

onyuIee Aug 9, 2024

0

11

1

+637

Counting

Find the number of ways of choosing three circles below, so that no two circles are next to each other.

siIviajendeukie Aug 9, 2024

0

2

0

+637

Counting

Annville Junior High School has a math club and a science club. There are $3$ students in the math club and $3$ students who are members of both clubs. There are twice as many students in the science club as in the math club. Find read more ..

siIviajendeukie Aug 9, 2024

0

4

0

+250

Counting

I have $3$ different mathematics textbooks, $2$ different psychology textbooks, and $2$ different chemistry textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row?

MEMEG0D Aug 9, 2024

0

7

0

+250

Probability

Raina places the following balls into a bag. She then draws three balls out of the bag, one at a time, without replacement. What is the probability that the colors of the balls alternate?
There are 3 orange balls and 2 purple read more ..

MEMEG0D Aug 9, 2024

0

10

0

+834

Counting

Miyu is giving out $8$ identical chocolates to her $5$ friends, including Dhruv. All possible distributions are equally likely. What is the probability that Dhruv gets at least $6$ chocolates?

RedDragonl Aug 9, 2024

0

5

0

+834

Counting

Find the number of solutions to
x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 \le 10
in nonnegative integers, where x_2 is even and x_5 is odd.

RedDragonl Aug 9, 2024

0

6

0

+279

Algebra

Find all ordered pairs (x,y) of real numbers such that x + y = 10 and x^2 + y^2 = 62 + 2xy.
For example, to enter the solutions $(2,4)$ and $(-3,9)$, you would enter "(2,4),(-3,9)" (without the quotation marks).

fjeihqoO Aug 9, 2024

0

11

1

+279

Algebra

Our badminton team has finished 60% of its season. So far, we have won 30% of the games we played. What percent of the remainder of our games must we win in order to finish the season with the same number of wins as losses?

fjeihqoO Aug 9, 2024

-2

7

0

+331

Help algebra

For what real values of c is x^2 - 8x - 16x + c the square of a binomial?

Stanry Aug 9, 2024

-2

19

0

+331

Help me plz

Find all real values of s such that x^2 + sx + 144 - 140 is the square of a binomial.

Stanry Aug 9, 2024

-2

9

0

+331

Need help

Find one ordered pair (x,y) of real numbers such that x + y = 8 and x^3 + y^3 = 200 + x^2 + y^2.

Stanry Aug 9, 2024

0

4

2

+58

I’m not sure what to do and my diagram keeps looking wonky

Points T and U lie on a circle centered at O, and point P is outside the circle such that PT and PU are tangent to the circle. If angle TPO = 33 degrees, then what is the measure of minor arc TU, in degrees?

Starry Aug 9, 2024

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