All Questions ⁺⁰236027 Questions

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no graph, but i labeled steps to draw the graph

As shown in the graph, the area of regular hexagon ABCDEF is 72, P is the midpoint of EF. What is the area of the shadow?
Graph: Step 1: Draw regular hexagon ABCDEF. Step 2: Connect AD. Step 3: Mark the midpoint of EF as P, and connect BP. Name the read more ..

celestial Oct 20, 2024

0

3

2

+267

Number Theory

Let $m$ be a positive integer such that $m$ has exactly $8$ positive divisors. How many distinct prime factors does $m$ have?

bIueb3rry Oct 20, 2024

0

3

1

+267

Number Theory

Let $m$ and $n$ be positive integers such that $m$ has exactly $8$ positive divisors, $n$ has exactly $6$ positive divisors, and $mn$ has exactly $12$ positive divisors. What is the smallest possible value of $mn?$

bIueb3rry Oct 20, 2024

+1

5

2

+13

HELPPPP

In the diagram, $AOB$ is a sector of a circle with $\angle AOB=60^\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ What is the length of $XY ?$

tomorspain Oct 20, 2024

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5

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+330

Counting

In how many ways can you distribute $8$ indistinguishable balls among $5$ distinguishable boxes, if at least three of the boxes must be empty?

Stanry Oct 20, 2024

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8

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+330

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?

Stanry Oct 20, 2024

Oct 19, 2024

0

6

1

+330

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 ..

Stanry Oct 19, 2024

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7

1

+330

Counting

A standard six-sided die is rolled $8$ times. You are told that among the rolls, there was two $1$'s, two $2$'s, two $3$'s, and two $4$s. How many possible sequences of rolls could there have been? (For example, $2,$ $1,$ $3,$ $4,$ $2,$ read more ..

Stanry Oct 19, 2024

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6

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+330

Counting

We call a number cozy if every digit in the number is either a $3$ or next to a $3.$ For example, the numbers $333,$ $83,$ $303,$ and $3773$ are all cozy, but the numbers $32423,$ $786,$ $340,$ and $3999$ are not cozy.
How many read more ..

Stanry Oct 19, 2024

0

5

0

+1310

Number Theory

Let $m$ be a positive integer. If $m$ has exactly $18$ positive divisors, then how many positive divisors does $m^2$ have?

learnmgcat Oct 19, 2024

0

5

0

+1310

Number Theory

Let $m$ and $n$ be positive integers. If $m$ has exactly $10$ positive divisors, $n$ has exactly $16$ positive divisors, and $mn$ has exactly $21$ positive divisors. How many divisors does $m^2 n^2$ have?

learnmgcat Oct 19, 2024

0

7

0

+1310

Number Theory

Let $M$ be the least common multiple of $1,$ $2,$ $\dots,$ $12$, $13$, $14$, $15$, $16$. How many positive divisors does $M$ have?

learnmgcat Oct 19, 2024

0

9

0

+1310

Number Theory

Let $m$ and $n$ be positive integers. If $m$ has exactly $10$ positive divisors, $n$ has exactly $16$ positive divisors, and $mn$ has exactly $21$ positive divisors. How many divisors does $m^2 n^2$ have?

learnmgcat Oct 19, 2024

0

8

0

+363

Number Theory

As shown in class, the Euclidean algorithm can be used to find solutions to equations of the form
ax + by = c
Use the Euclidean algorithm to find integers $x$ and $y$ such that $5x + 2y = 1,$ with the smallest possible positive read more ..

MeIdHunter Oct 19, 2024

0

3

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+363

Number Theory

You have a total supply of $1000$ pieces of candy, and an empty vat. You also have a machine that can add exactly $7$ pieces of candy per scoop to the vat, and another machine that can remove exactly $6$ pieces of candy with a different scoop from read more ..

MeIdHunter Oct 19, 2024

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8

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+363

Number Theory

Determine all positive integers that have exactly 8 positive divisors.

MeIdHunter Oct 19, 2024

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9

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+785

Counting

Joanna has several beads that she wants to assemble into a bracelet. There are seven beads: five of the beads have the same color, and the other two all have different colors. Using all seven beads, in how many different ways can Joanna assemble read more ..

gnistory Oct 19, 2024

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+785

Counting

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

gnistory Oct 19, 2024

0

6

0

+785

Counting

Find the number of positive integers that satisfy both the following conditions:
Each digit is a 1, 2, 3, or 4.
The sum of the digits is 11.

gnistory Oct 19, 2024

0

6

0

+785

Counting

In a series of coin flips, a run is a series of one or more consecutive coin flips that all have the same result. For example, in the sequence
\[TT \textcolor{red}{HHH} TTHHHTH,\]

the red letters form a run of length $3$. (A run read more ..

gnistory Oct 19, 2024

0

6

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+1521

Algebra

Let a and b be real numbers such that a^3 + 3ab^2 = 679 and 3a^3 - ab^2 = 615. Find a - b.

blackpanther Oct 19, 2024

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6

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+1521

Algebra

Find the coefficient of u^2*v^9 in (2u - 3*v^3)*5*(3u^3 - 2v^5)^3.

blackpanther Oct 19, 2024

0

3

1

+4

i need some help

Let IJKLMN be a hexagon with side lengths IJ = LM = 3, JK = LM = 4, and KL = IN = 6. Also, all the interior angles of the hexagon are equal. Find the area of hexagon IJKLMN.

themia Oct 19, 2024

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