All Questions ⁺⁰233017 Questions

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Algebra

Five workers have been hired to complete a job. If one additional worker is hired, they could complete the job $15$ days earlier. If the job needs to be completed $40$ days earlier, how many additional workers should be hired?

LiIIiam0216 May 17, 2024

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5

1

+1858

Algebra

Three runners, Dirk, Edith, and Foley all start at the same time for a $24$ km race, and each of them runs at a constant speed. When Dirk finishes the race, Edith is $10$ km behind, and Foley is $15$ km behind. When Edith finishes the race, how far read more ..

LiIIiam0216 May 17, 2024

0

6

1

+1858

Algebra

For a certain value of $k$, the system
x + y + 3z = 10,
x - 2y + 8z = 7,
kx + 5z = 3
has no solutions. What is this value of k?

LiIIiam0216 May 17, 2024

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7

2

+1858

Algebra

Ruth has a beaker containing a solution of $800$ mL of acid and $200$ mL of water. She thinks the solution is a little strong, so she drains $150$ mL from the beaker, adds $150$ mL of water, and stirs the solution. Ruth thinks the solution is read more ..

LiIIiam0216 May 17, 2024

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5

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+1858

Algebra

In a store window, there was a box of berries having a total weight of $200$ kg. The berries were $95\%$ water, by weight. After two days in the sun, the water content of the berries was only $90\%$, by weight. What was the total weight read more ..

LiIIiam0216 May 17, 2024

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4

1

+1858

Algebra

The function f(x) is defined for 1 \le x \le 5 as follows:
f(x) = 2x + 8 if 1 \le x \le 2
f(x) = 13 - 5x if 2 < x \le 3
f(x) = 20 - 14x if 3 < x \le 4 read more ..

LiIIiam0216 May 17, 2024

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2

1

+1858

Algebra

Find all values of t such that floor(t) = 3t + 4 - t^2. If you find more than one value, then list the values you find in increasing order, separated by commas.

LiIIiam0216 May 17, 2024

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4

1

+1858

Algebra

Let f(x) = \left \lfloor \frac{2 - 3x}{x + 3} + x^2 \right \rfloor.
Find f(1) + f(2) + f(3) + ... + f(999) + f(1000).

LiIIiam0216 May 17, 2024

0

6

1

+1858

Algebra

Find the minimum value of \frac{x^2}{x - 1 + x^3} for x > 1

LiIIiam0216 May 17, 2024

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4

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+1858

Algebra

Find $x$ if $\log_2 (\log_3 x) = \log_3 (\log_2 x).$

LiIIiam0216 May 17, 2024

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4

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+1858

Algebra

Find all $c$ such that $|c + 5| - 3|c + 7| = 10 + 2|c - 8|.$ Enter all the solutions, separated by commas.

LiIIiam0216 May 17, 2024

May 16, 2024

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5

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+1858

Algebra

Let $x$, $y$, and $z$ be nonzero real numbers. Find all possible values of
\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{x + y + z}{|x + y + z|}

LiIIiam0216 May 16, 2024

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5

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+1858

Counting

Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if each child must be next to two adults? (Two seatings are considered the same if one can be rotated to form the other.)

LiIIiam0216 May 16, 2024

0

3

0

+1858

Counting

In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven numbers (1, 2, 3, 4, 5, 6, and 7). Among all the cards of each color, there is exactly one card labeled with each number. read more ..

LiIIiam0216 May 16, 2024

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5

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+1858

Counting

At a meeting, 4 mathematicians, 3 scientists, and 2 journalists are to be seated around a circular table. How many different arrangements are possible if the mathematicians must all sit together (in 4 consecutive seats) and the 3 scientists must read more ..

LiIIiam0216 May 16, 2024

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6

1

+1858

Counting

My square patio is tiled with square tiles, all the same size. All the tiles are gray, except the tiles along the two diagonals, which are all yellow. (The corners are yellow, the center is yellow, and all the tiles along the diagonal in between read more ..

LiIIiam0216 May 16, 2024

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5

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+1858

Counting

Catherine rolls a standard $6$-sided die six times. If the product of her rolls is $300,$ then how many different sequences of rolls could there have been? (The order of the rolls matters.)

LiIIiam0216 May 16, 2024

0

4

1

+287

Area and probability

Right triangle XYZ has legs of length XY=12 and YZ=6. Point D is chosen at random within the triangle XYZ. What is the probability that the area of triangle XYD is at most 24?

Lilliam0216 May 16, 2024

0

7

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+697

Counting

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

eramsby1O1O May 16, 2024

0

5

2

+697

Counting

Franklin the fly starts at the point $(0,0)$ in the coordinate plane. At each point, Franklin takes a step to the right, left, up, or down. After $2$ steps, how many different points could Franklin end up at?

eramsby1O1O May 16, 2024

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9

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+730

help with trig

Let a be a real number such that 0 < a < \frac{\pi}{2}. Show that \left(\sin(a)\right)^7 + \left(\cos ( a)\right)^7< 2 sin(a)^4 cos(a)^4.

learnmgcat May 16, 2024

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3

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+730

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?

learnmgcat May 16, 2024

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7

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+730

Counting

In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may sit next to each other in the same row, but no child may sit directly in front of their sibling?

learnmgcat May 16, 2024

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6

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+697

Probability

In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven numbers ($1,$ $2,$ $3,$ $4,$ $5,$ $6,$ and $7$). Among all the cards of each color, there is exactly one card labeled with each number. read more ..

eramsby1O1O May 16, 2024

0

7

0

+730

Probability

The six faces of a cube are painted black. The cube is then cut into $3^3$ smaller cubes, all the same size.
(a) How many of the smaller cubes have exactly one black face?
(b) How many of the smaller cubes do not have any read more ..

learnmgcat May 16, 2024

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