magenta

178 Questions
23 Answers

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Counting

I run a book club with $n$ people, not including myself. Every day, for $400$ days, I invite $2$ members in the club to review a book. What is the smallest positive integer $n$ so that I can avoid ever having the exact same group of $2$ read more ..

magenta Oct 17, 2024

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1

+748

Counting

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

magenta Oct 17, 2024

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+748

Counting

Joanna has seven beads that she wants to assemble into a bracelet. Three of the beads have the same color, and the other four all have different colors. In how many different ways can Joanna assemble her bracelet? (Two bracelets are considered identical read more ..

magenta Oct 16, 2024

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+748

Counting

Starting with the A moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?
A
RRR
CCCCC read more ..

magenta Oct 15, 2024

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+748

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, so that at least one square is yellow, and at least one square read more ..

magenta Oct 10, 2024

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1

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+748

Counting

In the Olympic women's skating competition, the gold medal goes to first place, silver to second, and bronze to third. If there are 19 skaters, including 2 Americans and 5 Canadians, in how many ways can the medals be awarded to three of the 19 skaters read more ..

magenta Oct 10, 2024

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+748

Algebra

What is the coefficient of $x$ in $(x^3 + x^2 + x + 1)(x^4 - 8x^3 + 17x^2 - 23x + 14)$?

magenta Oct 8, 2024

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+748

Algebra

There exists a polynomial $f(x)$ and a constant $k$ such that
(x^2 - 2x - 5) f(x) = 2x^4 + 19x^3 + kx^2 - 15x - 1.
What is $k?$

magenta Oct 8, 2024

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+748

Algebra

Find $t$ if the expansion of the product of $x^3$ and $x^2 + tx$ has no $x^2$ term.

magenta Oct 8, 2024

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1

1

+748

Geometry

The interior angles of a polygon form an arithmetic sequence. The difference between the largest angle and smallest angle is $56^\circ$. If the polygon has $3$ sides, then find the smallest angle, in degrees.

magenta Oct 5, 2024

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1

+748

Geometry

Let $IJKLMN$ be a hexagon with side lengths $IJ = LM = 3,$ $JK = MN = 3,$ and $KL = NI = 3$. Also, all the interior angles of the hexagon are equal. Find the area of hexagon $IJKLMN$.

magenta Oct 5, 2024

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2

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+748

Geometry

The tangent to the circumcircle of triangle $WXY$ at $X$ is drawn, and the line through $W$ that is parallel to this tangent intersects $\overline{XY}$ at $Z.$ If $XY = 15$ and $WY = 10,$ find $WX.$

magenta Oct 5, 2024

0

2

2

+748

Geometry

A square is inscribed in a right triangle, as shown below. Find the area of the square.

I don't know how to add an image but the right triangle as a base of 1 and a height of 3.

magenta Sep 30, 2024

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2

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+748

Geometry

Point $B$ is the image of point $A$ under a dilation with center $O$. If $OB = 30$, and the scale factor of the dilation is $5$, then what is $OA$?

magenta Sep 30, 2024

+1

2

1

+748

Geometry

The perimeter of a rectangle is $40,$ and the length of one of its diagonals is $10 \sqrt{2}.$ Find the area of the rectangle.

magenta Sep 29, 2024

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2

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+748

Algebra

Find the smallest positive real number such that
\lfloor x^2 \rfloor - x \lfloor x \rfloor = 1.

magenta Sep 28, 2024

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1

1

+748

Algebra

At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15. Gary orders a bagel and a muffin, which comes out to $3.50. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15. How many cents does read more ..

magenta Sep 25, 2024

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4

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+748

Algebra

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

magenta Sep 14, 2024

+748

0

The answer is 2000 * 1.0015^60.

magentaSep 28, 2024

+748

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The answer is (B) 27/16.

magentaSep 28, 2024