All Questions ⁺⁰234662 Questions

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Tangents

Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{PA}$ is $2,$ and the circle has a radius of $2.$ Find the length $AB.$ read more ..

tomtom Dec 24, 2023

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35

0

+1887

Semicircle

In triangle $ABC,$ $\angle C = 90^\circ.$ A semicircle is constructed along side $\overline{AC}$ that is tangent to $\overline{BC}$ and $\overline{AB}.$ If the radius of the semicircle is equal to $3,$ then find $\frac{AC}{BC}.$

tomtom Dec 24, 2023

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20

1

+1887

Tangent circles

The two circles below are externally tangent. A common external tangent intersects line PQ at R. Find QR.

tomtom Dec 24, 2023

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35

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+1348

Geometry

A rectangle contains a strip of width $1,$ as shown below. Find the area of the strip.

sandwich Dec 24, 2023

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44

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+1887

Circles

Points $T$ and $U$ lie on a circle centered at $O$, and point $P$ is outside the circle such that $\overline{PT}$ and $\overline{PU}$ are tangent to the circle. If $\angle TOP = 45^{\circ}$, then what is the measure of minor arc $TU$, in deg read more ..

tomtom Dec 24, 2023

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27

1

+180

Find Cos C

itsash Dec 24, 2023

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41

1

+180

Find sin B

itsash Dec 24, 2023

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19

1

+1348

help counting

In how many ways can you distribute $8$ indistinguishable balls among $2$ distinguishable boxes, if at least one of the boxes must be empty?

sandwich Dec 24, 2023

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25

1

+1348

Counting

There is a group of five children, where two of the children are twins. In how many ways can I distribute $3$ identical pieces of candy to the children, if the twins must get an equal amount of candy?

sandwich Dec 24, 2023

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23

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+1348

Counting

How many solutions are there to the equation
\[u + v + w + x + y + z = 2,\]
where $u,$ $v,$ $w,$ $x,$ $y,$ and $z$ are nonnegative integers, and $x$ is at most $1?$

sandwich Dec 24, 2023

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31

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+1348

probability

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 exactly $7$ chocolates?

sandwich Dec 24, 2023

0

41

0

+1348

help counting

Find the number of ways that Magnus can give out $12$ identical stickers to $2$ of his friends, if every friend gets at least one sticker.

sandwich Dec 24, 2023

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40

0

+1348

help with counting

Find the number of ways that Magnus can give out $12$ identical stickers to $2$ of his friends. (Not everyone has to get a sticker.)

sandwich Dec 24, 2023

Dec 23, 2023

-1

25

1

+1323

Geometry

In square ABCD, P is on BC such that BP = 5 and PC = 2, and Q is on CD such that DQ = 4 and QC = 3. Find sin angle PAQ.

blackpanther Dec 23, 2023

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