All Questions +0 237417 Questions

Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $11,$ and the slope of line segment $\overline{OQ}$ is $13.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: read more ..

Ibrahin097YT 7 hours ago

0

1

0

+56

Geometry

Ibrahin097YT 7 hours ago

0

1

+56

Geometry

A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.
The circle is x^2 + y^2 = 1, and the line is y = x.

Chords UV, WX, and YZ of a circle are parallel. The distance between chords UV and WX is 1, and the distance between chords WX and YZ is also 1. If UV =6 and YZ = 4, then find WX.

Ibrahin097YT May 13, 2025

0

1

0

+56

Geometry

Ibrahin097YT May 13, 2025

0

1

0

+56

Geometry

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

Ibrahin097YT May 13, 2025

0

1

0

+56

Geometry

In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $CE =3$ and $DF = 3$, then what is $BD$?

Ibrahin097YT May 13, 2025

0

1

2

+456

Counting

How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

The following cards are split into four piles at random, so that every pile contains the same number of cards. What is the probability that every pile contains an Ace?
The cards are:
Ace of clubs
Ace of diamonds read more ..

Jasmo May 10, 2025

0

1

0

+458

Counting

bIueb3rry May 10, 2025

0

1

0

+458

Counting

A bag contains red and blue tiles. Each tile has a number from the set \{-1, 0, 1\} written on it. I want to arrange 7 of these tiles in a row, so that the numbers on any three consecutive tiles sum to 3. In how many ways can this be done, assuming that read more ..

bIueb3rry May 10, 2025

0

1

0

+56

i need answer now

Find the number of ways of filling in the squares of a 3 \times 3 grid so that:
* Each square contains a 0 or a 1.
* The sum of the numbers in each row and each column is at most 1.
An example is shown below.

Ibrahin097YT May 10, 2025

0

1

0

+56

plz help now

Find the number of paths from A to B in the grid below, so that
* Each step is down or to the right.
* The path cannot pass through any point more than once.

An example path is shown.
The grid is 3 by 3, with read more ..

Ibrahin097YT May 10, 2025

0

1

+56

help me help help

If an ant crawls from one corner to the other corner of a 3 \times 5 rectangle, then it will cross through seven squares.
If the ant crawls from one corner to the other corner of a 15 \times 16 rectangle, then how many squares will read more ..

A survey conducted among 150 high school students revealed that

* 68 students like Math
* 85 students like English
* 55 students like History
* 20 students like both Math and English
* 15 students like both Math and read more ..

Ibrahin097YT May 10, 2025

0

1

2

+56

help really need help

What is the value of the expression. Express your asnwer as a common fraction.

Ibrahin097YT May 10, 2025

0

1

+9

HELP PLEASE ASAP!!!!

Let ABC be a triangle with side lengths AB = 5, BC = 6, and AC = 9. What is the area of the triangle with side lengths \tan A, \tan B, and \tan C?

Ibrahim097YT May 10, 2025

-1

1

2

+1800

Geometry

Acute triangle ABC has \sin A = \dfrac{2}{5} and \sin B = \dfrac{2}{\sqrt{7}}. What is \sin C?

blackpanther May 9, 2025

-1

1

+1800

Geometry

Find the area of the triangle with side lengths 4, 5, and 8.

blackpanther May 9, 2025

0

1

+1800

Geometry

Triangle ABC has side lengths AB = 8, BC = 10, and AC = 12. Find \dfrac{\cos A}{\cos B}.

blackpanther May 9, 2025

0

1

0

+1800

Geometry

blackpanther May 9, 2025

0

1

0

+608

Counting

A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (spades, hearts, diamonds, and clubs), such that there is exactly one card for any given rank and suit.
You are dealt a hand of read more ..

onyuIee May 9, 2025

0

1

0

+608

Counting

Find the number of sequences (a_1, a_2, a_3, \dots, a_8) such that:
* a_i \in \{1, 2, 3, 4, 5, 6, 7, 8\} for all 1 \le i \le 8.
* Every number1, 2, 3, 4, 5, 6, 7, 8 appears at least once in the sequence.

onyuIee May 9, 2025

0

1

0

+608

Counting

A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3) is a permutation of (1,2,3,4,5).
Find the number of permutations (x_1, x_2, \dots, x_8) read more ..

onyuIee May 9, 2025

0

1

0

+56

How many subsets of \{0, 1, \dots, 9\} have the property

How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4?

Ibrahin097YT May 9, 2025

0

1

0

+56

We choose a positive divisor of 20^{20} at random (with all divisors equally likely to be chosen).

We choose a positive divisor of 20^{20} at random (with all divisors equally likely to be chosen). What is the probability that we chose a multiple of 5?

Ibrahin097YT May 9, 2025

0

1

0

+56

Find the number of ways of arranging one A, two Bs, three Cs, and four Ds

Find the number of ways of arranging one A, two Bs, three Cs, and four Ds, so that no two Bs are next to each other, no two Cs are next to each other, and no two Ds are next to each other.

Ibrahin097YT May 9, 2025

0

1

0

+4

Trigonometric Laws and Identities

Two students are calculating the exact value of tan(7pi/12)? Choose one of the following to methods to solve.
-Student A says 7pi/12 is half of 7pi/6, and they will use the half-angle identity and the unit of circle to solve.
-Student B says read more ..

blueberiii May 9, 2025

0

1

0

+56

In the diagram below, \overline{AD} and \overline{BE} are angle bisectors

In the diagram below, \overline{AD} and \overline{BE} are angle bisectors of \angle BAC and \angle ABC, respectively, and they intersect at T. We know that BC = 12, AC = 18 and BF = 3 \cdot AF. Find AB.

Ibrahin097YT May 9, 2025

+1

1

+9

What is the value of the expression shown? Express your answer as a common fraction.

How many 7-digit sequences have a digit that appears at least 6 times?
(For example, 3339333 and 0200000 are two such sequences. A sequence is allowed to begin with 0.)

Ibrahim097YT May 9, 2025

+1

1

0

+1271

Counting

Akhain1 May 1, 2025

0

1

+829

Counting

How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?

How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

cooIcooIcooI17 May 1, 2025

0

1

0

+829

Counting

cooIcooIcooI17 May 1, 2025

0

1

0

+56

Geometry

In the diagram, \overline{AD} is an altitude, \overline{BE} is a median, and \overline{AD}, \overline{BE}, and \overline{CF} are concurrent. If AE = 11, BE = 9, and AD = 6\sqrt{2}, what is AF?

PikaPika989 Apr 26, 2025

0

1

0

+56

Geometry

In the diagram below, \overline{AD} and \overline{BE} are angle bisectors of \angle BAC and \angle ABC, respectively, and they intersect at T. We know that BD=12, AE=8, and BF=3+AE. Find AB.

PikaPika989 Apr 26, 2025

0

1

+4

Calculate the best ISO for drone night scenes

If a drone has Max ISO 44000, 12-Bit DNG, how will its night 6K photos be?

How many squares in the plane have at least two points in the lattice below as vertices?

rainyxyb Apr 24, 2025

0

1

+1216

Counting

At a meeting, two scientists, two mathematicians, two historians, and two artists are to be seated around a circular table. In how many ways can they be seated so that all four pairs of people from the same discipline are seated together?

AnswerscorrectIy Apr 23, 2025

0

1

2

+1216

Counting

What is the equation of the reflection of the line y = 2x + 1 across the line y = 5x - 3? Give your answer in slope-intercept form.

AnswerscorrectIy Apr 23, 2025

0

1

+56

Help me plz

In \triangle ABC, we have AB = AC = 4 and \angle BAC = 45^\circ. If M is the midpoint of BC, then find AM^2.

PikaPika989 Apr 23, 2025

0

1

0

+56

Help need help

PikaPika989 Apr 23, 2025

0

2

1

+56

Help geometry

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of triangle ABC?
A is at (0,0), B is at (1,0), and C is at (1,1).

In the complex plane, z, z2 , z3 form, in some order, three of the vertices of a non-degenerate square. Enter all possible areas of the square, separated by commas.

PikaPika989 Apr 23, 2025

+1

1

+16

Super Hard Algebra 2/Precalculus Question Help Please

Without knowing the relationship between h and x, how is this possible?

PikaPika999 Apr 23, 2025

0

1

+4

Any help appreciated - preferably in simple language for a non mathematician

Let S be a set of distinct integers. What is the smallest number of elements that S must contain, to ensure that S has a nonempty subset, where the sum of the elements in the subset is divisible by 2?

dadtryingtoHelp Apr 22, 2025

0

1

+56

Counting

Vera has 20 white socks, 21 black socks, 22 brown socks, 23 blue socks, 24 red socks, and 25 green socks. How many socks (at a minimum) must she pull out of her sock drawer to ensure at least six matching pairs of different colors?

PikaPika989 Apr 22, 2025

0

1

+56

Counting

Recall that a partition of a positive integer n means a way of writing n as the sum of some positive integers, where the order of the parts does not matter. For example, there are five partitions of 4:
4
3 + 1
2 + 2 read more ..

PikaPika989 Apr 21, 2025

0

1

0

+977

Counting

magenta Apr 20, 2025

0

1

0

+977

Counting

Find the number of ways filling in a 4 \times 4 grid, such that
* Each cell contains a 0 or a 1.
* The sum of the numbers in each row and each column is at least 2.

An example is shown below.
0110 read more ..

magenta Apr 20, 2025

0

1

0

+977

Counting

Consider the set
S = {1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 18, 23, 24, ..., 12345678},
which consists of all positive integers whose digits strictly increase from left to right, and the digits are from 1 to 8. This set is read more ..

magenta Apr 20, 2025

0

1

0

+458

Counting

Find the number of subsets of
S = \{1, 3, 8, 17, 30, 36, 47, 58\},
so that the sum of the elements in the subset is less than 20. (Note that for the empty subset, we take the sum of the elements as 0.)

bIueb3rry Apr 19, 2025

0

3

0

+458

Counting

Find the number of subsets of
S = \{1, 3, 8, 17, 30, 36, 47, 58\},
so that the sum of the elements in the subset is a multiple of 5. (Note that for the empty subset, we take the sum of the elements as 0.)

bIueb3rry Apr 19, 2025

0

3

0

+608

Counting

Adam the ant starts at (0,0). Each minute, he flips a fair coin. If he flips heads, he moves 1 unit up; if he flips tails, he moves 1 unit right.
Betty the beetle starts at (1,1). Each minute, she flips a fair coin. If she flips heads, read more ..

onyuIee Apr 19, 2025

0

3

0

+608

Counting

Consider the sequence
1, 5, 6, 25, 26, 30, 31, ...
which consists of every positive integer that can be expressed as a sum of distinct powers of 5.
What is the first term that is greater than 50?

onyuIee Apr 19, 2025

0

3

0

+608

Counting

Consider the sequence
1, 3, 4, 9, 10, 12, 13, ...
which consists of every positive integer that can be expressed as a sum of distinct powers of 3.
What is the first term that is greater than 20?

onyuIee Apr 19, 2025

0

2

0

+1800

Counting

Find the number of sequences containing three terms, such that
* The second term is equal to the sum of the first term plus one.
* The third term is equal to twice the second term.
* Each term is an integer in \{0, 1, 2, \dots, 100\} read more ..

blackpanther Apr 19, 2025

0

1

+56

Counting

Find the number of arithmetic sequences such that:
* The arithmetic sequence contains three terms
* All the terms are integers in \{0, 1, 2, \dots, 10\}
* The sum of the terms is 9.

Find the number of paths from A to B in the grid below, where each step is down or to the right.
The grid is 3 by 3, with A in the upper-left and B in the lower-right.

PikaPika989 Apr 18, 2025

0

2

0

+56

Counting

PikaPika989 Apr 18, 2025

0

1

+56

Counting

A basket contains 7 green marbles and 6 red marbles. Two marbles are taken from the basket at random without replacement (i.e., the first marble is not put back before the second marble is drawn). What is the probability that the first marble is green, read more ..

In the United States, there are 50 states. Each state is represented by 2 senators. In how many ways can we form a committee with 5 senators, in which no two of the senators are from the same state?

PikaPika989 Apr 18, 2025

0

2

0

+357

Counting

cwimie Apr 17, 2025

0

2

0

+357

Counting

Suppose for this problem (though it may not be accurate in real life) that the Senate has 48 Republicans and 52 Democrats. In how many ways can we form a committee with 5 senators, in which neither party holds all 5 seats?

cwimie Apr 17, 2025

0

2

1

+56

help with counting

Now suppose that not only must Sir Lancelot and Sir Gawain be next to each other, but Sir Galahad and Sir Percival also demand to be next to each other. How many seatings of the 8 knights are possible?

King Arthur's round table has 8 evenly spaced chairs. In how many ways can 8 knights be seated in the chairs, if Sir Lancelot and Sir Gawain insist on being seated next to each other?

PikaPika989 Apr 17, 2025

0

2

0

+56

Counting

PikaPika989 Apr 17, 2025

0

1

0

+56

Counting

In how many ways can 5 balls be placed in 4 boxes, if 2 balls are white and 3 balls are black? (Balls of the same color are indistinguishable. The boxes are indistinguishable.)

PikaPika989 Apr 17, 2025

0

2

1

+56

help with counting

How many zeroes do we write when we write all the integers from 1 to 625 in base 4?

The range of the function is . Let . Find the range of the function .
Thanks!

PikaPika989 Apr 17, 2025

+1

2

3

+16

Help Please Intermediate Algebra

Given a regular octagon, in how many ways can we color one diagonal red and another diagonal blue so that the two colored diagonals intersect at an endpoint? Consider rotations and reflections distinct.

PikaPika999 Apr 17, 2025

0

2

1

+806

Counting

A survey conducted among 150 high school students revealed that

* 68 students like Math
* 85 students like English
* 55 students like History
* 20 students like both Math and English
* 15 students like both Math and read more ..

crimefightingvigiI Apr 17, 2025

0

1

0

+958

Counting

gnistory Apr 17, 2025

0

2

0

+958

Counting

If an ant crawls from one corner to the other corner of a 3 \times 5 rectangle, then it will cross through seven squares.
If the ant crawls from one corner to the other corner of a 15 \times 16 rectangle, then how many squares will read more ..

gnistory Apr 17, 2025

0

3

1

+22

Using Geo/Trig on an Algebra Problem

What is $xy+yz+xz$ if $x^2+xy+y^2=1$, $y^2+yz+z^2=2$, and $x^2+xz+z^2=3$?

Find the number of paths from A to B in the grid below, so that
* Each step is down or to the right.
* The path cannot pass through any point more than once.

An example path is shown.
The grid is 3 by 3, with read more ..

Xenosmilus Apr 17, 2025

0

2

0

+458

Counting

bIueb3rry Apr 17, 2025

0

2

0

+458

Counting

Find the number of ways of filling in the squares of a 3 \times 3 grid so that:
* Each square contains a 0 or a 1.
* The sum of the numbers in each row and each column is at most 1.
An example is shown below.

bIueb3rry Apr 17, 2025

0

2

0

+458

Counting

How many ordered pairs of positive integers (m,n) satisfy \text{lcm}[m,n] = 360 and \gcd(m,n) = 360?

bIueb3rry Apr 17, 2025

0

2

0

+458

Counting

A bag contains red and blue tiles. Each tile has a number from the set \{-1, 0, 1\} written on it. I want to arrange 7 of these tiles in a row, so that the numbers on any three consecutive tiles sum to 3. In how many ways can this be done, assuming that read more ..

bIueb3rry Apr 17, 2025

0

2

0

+400

Counting

The following cards are split into four piles at random, so that every pile contains the same number of cards. What is the probability that every pile contains an Ace?
The cards are:
Ace of clubs
Ace of diamonds read more ..

Stanry Apr 17, 2025

0

1

0

+400

Counting

How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

Stanry Apr 16, 2025

0

1

+400

Counting

How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?

How many 7-digit sequences have a digit that appears at least 6 times?
(For example, 3339333 and 0200000 are two such sequences. A sequence is allowed to begin with 0.)

Stanry Apr 16, 2025

0

1

+400

Counting

Find the number of ways of arranging one A, two Bs, three Cs, and four Ds, so that no two Bs are next to each other, no two Cs are next to each other, and no two Ds are next to each other.

Stanry Apr 15, 2025

0

1

0

+400

Counting

Stanry Apr 15, 2025

0

4

0

+400

Counting

We choose a positive divisor of 20^{20} at random (with all divisors equally likely to be chosen). What is the probability that we chose a multiple of 5?

Stanry Apr 14, 2025

0

2

0

+400

Counting

How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4?

Stanry Apr 14, 2025

0

1

0

+400

Counting

A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3) is a permutation of (1,2,3,4,5).
Find the number of permutations (x_1, x_2, \dots, x_8) read more ..

Stanry Apr 13, 2025

0

1

0

+400

Counting

Find the number of sequences (a_1, a_2, a_3, \dots, a_8) such that:
* a_i \in \{1, 2, 3, 4, 5, 6, 7, 8\} for all 1 \le i \le 8.
* Every number1, 2, 3, 4, 5, 6, 7, 8 appears at least once in the sequence.

Stanry Apr 13, 2025

0

2

0

+400

Counting

A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (spades, hearts, diamonds, and clubs), such that there is exactly one card for any given rank and suit.
You are dealt a hand of read more ..

Stanry Apr 13, 2025

0

3

1

+400

Counting

In the array below, in how many different ways can we start with the letter and move from letter to letter (horizontally, vertically, or diagonally), to spell the word "ARCS"?
A read more ..

How many squares in the plane have at least two points in the lattice below as vertices?

Stanry Apr 13, 2025

0

3

0

+400

Counting

Stanry Apr 13, 2025

0

1

2

+400

Counting

At a meeting, two scientists, two mathematicians, two historians, and two artists are to be seated around a circular table. In how many ways can they be seated so that all four pairs of people from the same discipline are seated together?

Triangle ABC has AB = 6. Let D lie on BC such that \overline{AD} bisects \angle BAC. If BD = 3 and CD = 5, what is AC?

Stanry Apr 12, 2025

+1

1

3

+1518

Geometry

CPhill ●●●

learnmgcat Apr 12, 2025

0

1

0

+1518

Geometry

Find an acute angle A such that \sin 4A = \sin A + \sin 2A. Express your answer in degrees.

learnmgcat Apr 12, 2025

0

1

0

+1518

Geometry

What is the distance between the points (\cos 37^{\circ}, \sin 37^{\circ}) and (\cos 127^{\circ}, \sin 127^{\circ})?

learnmgcat Apr 12, 2025

0

1

0

+1518

Geometry

Chris places an orange cone at his current location. Then, he faces west, walks 40 meters, turns 30^{\circ} to his right, and walks 20 meters. How far is Chris from the cone, in meters? Round your answer to the nearest whole number.
(You will need read more ..

learnmgcat Apr 12, 2025

0

1

0

+806

Counting

In a basketball tournament, there are four teams, and each team plays against every other team exactly twice. (So each team plays six games. Also, each team is equally likely to win a game, and there are no ties.) Find the probability read more ..

crimefightingvigiI Apr 12, 2025

0

1

0

+806

Counting

Five standard 6-sided dice are rolled, and the resulting numbers are multiplied together. What is the probability that the product is divisible by 12?

crimefightingvigiI Apr 12, 2025

0

2

0

+806

Counting

Five people check identical suitcases before boarding an airplane. At the baggage claim, each person takes one of the five suitcases at random. What is the probability that exactly one person ends up with the wrong suitcase?

crimefightingvigiI Apr 12, 2025

0

3

0

+806

Counting

Find the number of positive integers that are divisors of at least one of 6^{6}, 10^{10}, 15^{15}, and 30^{30}.

crimefightingvigiI Apr 12, 2025

0

2

0

+806

Counting

I can only remember a seven-digit telephone number if the first three digits (the "prefix") are equal to the next three digits or the last three digits. For example, I can remember 389-3892 and 274-9274.
How many seven-digit telephone numbers can read more ..

crimefightingvigiI Apr 12, 2025

0

2

0

+1800

Counting

In how many ways can we seat 3 pairs of siblings in a row of 10 chairs, so that nobody sits next to their sibling? (Two chairs will be left empty, of course.)

blackpanther Apr 12, 2025

0

1

0

+1800

Counting

A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3)$ is a permutation of (1,2,3,4,5).
Let \pi = (x_1,x_2,x_3,\dots,x_n) be a permutation read more ..

blackpanther Apr 12, 2025

0

3

0

+1800

Counting

If an ant crawls from one corner to the other corner of a 3 \times 5 rectangle, then it will cross through seven squares.
If the ant crawls from one corner to the other corner of a 15 \times 16 rectangle, then how many squares will read more ..

blackpanther Apr 12, 2025

0

1

0

+1800

Counting

A survey conducted among 150 high school students revealed that

* 68 students like Math
* 85 students like English
* 55 students like History
* 20 students like both Math and English
* 15 students like both Math and read more ..

blackpanther Apr 11, 2025

+1

3

1

+608

Counting

Let P be a point chosen uniformly at random inside ABC. Extend ray BP to hit side AC at D. What is the probability that BD < 4?
The sides of triangle ABC are 3, 5, and 7.

Given a regular octagon, in how many ways can we color one diagonal red and another diagonal blue so that the two colored diagonals intersect at an endpoint? Consider rotations and reflections distinct.

onyuIee Apr 10, 2025

0

2

0

+608

Counting

onyuIee Apr 10, 2025

0

4

0

+608

Counting

Find the number of ways of placing three As, three Bs, and three Cs in a 3 \times 3 grid, so that every square contains one letter, and each diagonals contains one A, one B, and one C.

onyuIee Apr 10, 2025

0

2

1

+164

Counting

How many zeroes do we write when we write all the integers from 1 to 625 in base 4?

In how many ways can 5 balls be placed in 4 boxes, if 2$balls are white and 3 balls are black? (Balls of the same color are indistinguishable. The boxes are indistinguishable.)

HumanBemg Apr 9, 2025

0

2

1

+164

Counting

Now suppose that not only must Sir Lancelot and Sir Gawain be next to each other, but Sir Galahad and Sir Percival also demand to be next to each other. How many seatings of the 8 knights are possible?

HumanBemg Apr 9, 2025

0

1

0

+164

Counting

HumanBemg Apr 8, 2025

0

1

0

+164

Counting

King Arthur's round table has 8 evenly spaced chairs. In how many ways can 8 knights be seated in the chairs, if Sir Lancelot and Sir Gawain insist on being seated next to each other?

HumanBemg Apr 8, 2025

0

1

0

+921

Counting

Suppose for this problem (though it may not be accurate in real life) that the Senate has 48 Republicans and 52 Democrats. In how many ways can we form a committee with 5 senators, in which neither party holds all 5 seats?

booboo44 Apr 8, 2025

0

1

0

+921

Counting

In the United States, there are 50 states. Each state is represented by 2 senators. In how many ways can we form a committee with 5 senators, in which no two of the senators are from the same state?

booboo44 Apr 8, 2025

0

2

1

+921

Counting

A basket contains 7 green marbles and 6 red marbles. Two marbles are taken from the basket at random without replacement (i.e., the first marble is not put back before the second marble is drawn). What is the probability that the first marble is green, read more ..

A basket contains 7 green marbles and 6 red marbles. A marble is taken from the basket at random; its color is recorded, then the marble is returned to the basket. A second marble is then taken from the basket at random, and its color is recorded. read more ..

booboo44 Apr 8, 2025

0

2

1

+921

Counting

Let a_1, a_2, a_3, \dots be an infinite geometric series with positive terms. If a_2 = 10, then find the smallest possible value of
a_1 + a_2 + a_3.

booboo44 Apr 8, 2025

0

2

1

+958

Algebra

Compute
1.11111 + 0.11111 + 0.01111 + 0.00111 + 0.00011 + 0.00001

gnistory Apr 7, 2025

+1

1

2

+958

Algebra

Tackling Friction in Free body diagrams, would appreciate your generous help.
Problem: A block with a mass of 10 kg is placed on a rough inclined plane that makes a 30° angle with the horizontal. The coefficient of static read more ..

gnistory Apr 7, 2025

0

2

1

+448

Free Body Diagram

A and B are two points on a sphere of radius 2. We know the space distance between A and B is 2, What is distance from A to B along the (minor) arc of a great circle?

HiylinLink Apr 7, 2025

+1

1

2

+400

Geometry

Bosco ●●

Stanry Apr 7, 2025

-1

2

+164

Geometry

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of triangle ABC?
A is at (0,0), B is at (1,0), and C is at (1,1).

In \triangle ABC, we have AB = AC = 4 and \angle BAC = 45^\circ. If M is the midpoint of BC, then find AM^2.

HumanBemg Apr 5, 2025

+1

2

1

+164

Geometry

What is the equation of the reflection of the line y = 2x + 1 across the line y = 5x - 3? Give your answer in slope-intercept form.

HumanBemg Apr 5, 2025

+1

2

1

+164

Geometry

In the diagram below, ABCD is a rectangle with AD=6 and B=10. Point M is the intersection of its diagonals and point E lies on \overline{AD} such that BE=2. Find CE^2. (Diagram not to scale)

HumanBemg Apr 5, 2025

-1

4

1

+958

Geometry

Let R=(a,b) be the image of rotating point P=(7,2) clockwise by 150^\circ degrees around Q=(12,-5). What is b?

gnistory Apr 5, 2025

+1

2

1

+958

Geometry

Let R be the image of rotating point P=(4,0) counterclockwise by 60^\circ$ degrees around Q=(12,-7). What is PR?

gnistory Apr 5, 2025

0

5

0

+958

Geometry

gnistory Apr 5, 2025

0

3

0

+958

Geometry

Point B is the reflection of point A=(2,-1) over the line y=1. Point C is the reflection of point B over the line x + 2y = a. If C=(-5,8), what is a?

gnistory Apr 4, 2025

0

1

+958

Geometry

The area of a cross section of a sphere is 64\% of the largest possible cross sectional area of the sphere. If the sphere has radius 1/2, what is the area of the cross section?

The base of right pyramid ABCDE is a rhombus with side 5. We also know that \triangle ABD \cong \triangle CBD and EA=BA=2. Find the volume of the pyramid.

gnistory Apr 4, 2025

0

2

1

+921

Geometry

In the diagram, \overline{AD} is an altitude, \overline{BE} is a median, and \overline{AD}, \overline{BE}, and \overline{CF} are concurrent. If AB = 11, AC = 12, and AD = 6, what is AF?

booboo44 Apr 4, 2025

0

2

1

+290

Geometry

In the diagram below, \overline{AD} and \overline{BE} are angle bisectors of \angle BAC and \angle ABC, respectively, and they intersect at T. We know that BC = 12, AC = 18 and BF = 3 \cdot AF. Find AB.

AUnVerifedTaxPayer Apr 4, 2025

0

2

1

+400

i need help

The left column contains pairs of triangles with different side-length and angle-measure congruences marked. Match each diagram in the left column with a congruence/similarity criterion in the right column that justifies why the two triangles are congruent/similar. read more ..

Stanry Apr 4, 2025

0

1

+400

help me plz

The left column contains pairs of triangles with different side-length and angle-measure congruences marked. Match each diagram in the left column with a congruence/similarity criterion in the right column that justifies why the two triangles are read more ..

Stanry Apr 4, 2025

0

1

2

+400

need help

A stick is broken at two points, chosen at random. If the length of the stick is 6 units, then what is the probability that all three resulting pieces are shorter than 5 units?
What I've tried so far:
I've treated the stick as read more ..

Stanry Apr 4, 2025

0

2

+71

Counting and Probability, Melody or CPhill help ;-;

In the diagram below, we know \tan \theta = \frac{3}{4}. Find the area of the triangle.
The triangle is right, and the hypotenuse is 80.

Starry Apr 4, 2025

0

3

0

+1518

Geometry

learnmgcat Apr 4, 2025

0

2

1

+1518

Geometry

Find the value of \sin 40^\circ \cdot \cos 50^\circ \cdot \cos 40^\circ \cdot \sin 50^\circ.

Find sin B.
BD = 15, CD = 10, AC = 6

learnmgcat Apr 4, 2025

0

2

0

+458

Geometry

bIueb3rry Apr 4, 2025

0

2

0

+458

Geometry

Find AC in the diagram to the nearest integer.
(You will need to use a calculator. Make sure to use "Degree" instead of "Radian" as the angle unit when computing trigonometric functions.)
AB = 31, angle A = 90 degrees, read more ..

bIueb3rry Apr 4, 2025

0

1

+458

Geometry

In \triangle PQR, we have \angle P = 90^\circ, PQ=3, and QR = 5. Find \tan R.

For a positive integer n, let f(n) denote the integer that is closest to \sqrt[4]{n}. Find the integer m so that
\sum_{n = 1}^m f(n) = 100.

bIueb3rry Apr 4, 2025

0

4

0

+806

Algebra

crimefightingvigiI Apr 3, 2025

0

2

0

+806

Algebra

For how many integer values of a does the equation
x^2 + ax + 12a = 5x + 8
have integer solutions for x?

crimefightingvigiI Apr 2, 2025

0

1

0

+1729

Algebra

Let a and b be positive real numbers. Let
m = \min \left\{ a, \frac{1}{b}, b^2 + \frac{1}{a + 1} \right\}.
Find the largest possible value of m.

parmen Apr 1, 2025

0

3

0

+164

Algebra

The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies
xf(x) + f(1 - x)/x = x^3 + 3x^2 + 14x - 13
for all real x. Find f(x).

HumanBemg Apr 1, 2025

+2

4

1

+6

Hello, guys and gals.

안녕하세요, 여러분. 저는 여러분이 파이에 대해 배우도록 이 글을 찾아오겠습니다. 파이는 정당한 스러울 수 있는 동그라미입니다. 그럼요. 이 문자 그대로 무한한 숫자의 여행을 시작하면 됩니다. 그리고, 위의 경고를 꼭 읽어주세요. 피자는 날씨가 좋으니까요.
1. PI란 무엇인가?
파이는 원의 직경과 원둘레이의 경우입니다. 흥미로운 점은 우주만큼 크든 원자핵만큼 작든 모든 원에 대해 동일하다는 것입니다.
원의 read more ..

Find all complex solutions to the equation z^3 = 8i - (15 + 2i)z^2 + (7 - 4i)z.

xhxmsja Mar 31, 2025

0

5

1

+44

Algebra

Find all complex numbers z such that |z - 1| = |z + 3| + |z - 2i|.

XenosmiIus Mar 30, 2025

0

2

0

+44

Algebra

XenosmiIus Mar 30, 2025

0

1

+44

Algebra

For parts (a) through (d), let f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7.
What is the vertex of the parabola y = f(x)?

What is the equation of the line of symmetry of the parabola $y = f(x)$.
f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7.

XenosmiIus Mar 30, 2025

0

1

2

+44

Algebra

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

XenosmiIus Mar 30, 2025

-1

2

3

+478

Geometry

In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $CE =3$ and $DF = 3$, then what is $BD$?

MeIdHunter Mar 30, 2025

-1

1

+478

Geometry

In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

MeIdHunter Mar 30, 2025

0

1

+478

Geometry

Find the area of triangle ABC
[asy]
unitsize (2 cm);
pair A, B, C, D, E, F, H;
A = (0,2);
B = (-1,0);<#> C = (1,0);<#> D = (0,0);<#> E = (B + reflect(A,C)*(B))/2;<#> F = (C + reflect(A,B)*(C))/2;<#> read more ..

MeIdHunter Mar 30, 2025

0

2

1

+4

help please

Let x_1, x_2, \dots, x_{100} be real numbers. If
x_1 + 2x_2 + \dots + 100x_{100} = 1,
then find the minimum value of x_1/1 + x_2/2 + \dots + x_{100}/100.

whatthedonut Mar 30, 2025

0

2

0

+44

Algebra

XenosmiIus Mar 30, 2025

0

2

0

+44

Algebra

Let x, y, and z be real numbers. If x^2 + y^2 + z^2 = 1, then find the maximum value of
3x + 4y + 5z + x^3 + \frac{4x^2 y}{z} + \frac{z^5}{xy^2}

XenosmiIus Mar 30, 2025

0

3

0

+44

Algebra

Let x_1, x_2, \dots, x_n be real numbers. If
x_1^2 + 2x_2^2 + \dots + nx_n^2 = 1,
then find the maximum value of (x_1 + x_2/2 + \dots + x_n/n)^2, in terms of n.

XenosmiIus Mar 30, 2025

0

6

0

+44

Help algebra help

Let x_1, x_2, \dots, x_n be real numbers. If
x_1 + 2x_2 + \dots + nx_n = 1,
then find the minimum value of x_1^2/1 + x_2^2/2 + \dots + x_n^2/n, in terms of n.

XenosmiIus Mar 30, 2025

0

2

0

+44

Help me algebra

Let a, b, and c be positive real numbers. Find the minimum value of
(a + 1)^2 + \left( \frac{b}{a} + a + 1 \right)^2 + \left( \frac{c}{b} + abc \right)^2 + \left( \frac{4}{c} + \frac{c}{a} \right)^2.

XenosmiIus Mar 30, 2025

0

4

0

+44

Algebra question

Let x and y be positive real numbers. If
\frac{x^2}{9} + \frac{y^2}{25} = 1,
then find the maximum value of \frac{x^2}{3} \cdot \frac{y^2}{5}.

XenosmiIus Mar 30, 2025

0

3

0

+44

Help with Algebra

Let a, b, and c be positive real numbers. If \frac{a}{2} + b + 3c = 12, then find the maximum value of
\min \left\{ \frac{ab}{2}, ac, 12bc \right\}.

XenosmiIus Mar 30, 2025

+1

4

3

+22

Interesting Range Question; Can use polynomial long division to solve

Find the range of the function

Enter your answer in interval notation.

Find all x such that
(2x)^{\log_{10} 2} = (9x)*{\log_{10} 9} + (5x)^{\log_{10} 9} + \log_x 243

Xenosmilus Mar 29, 2025

0

3

0

+164

Algebra

HumanBemg Mar 29, 2025

0

3

0

+164

Algebra

Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)

HumanBemg Mar 29, 2025

0

2

0

+1800

Algebra

Suppose that f(x) and g(x) are functions which satisfy
f(g(x)) = x^2 \quad \text{and} \quad g(f(x)) = x^4
for all x \ge 1. If g(16) = 1, then compute \log_2 g(2).

blackpanther Mar 27, 2025

0

4

0

+164

Algebra

Suppose f(x) is a rational function such that
3 f \left( \frac{1}{x} \right) - \frac{f(x)}{x} = x
for all $x \neq 0$. Find f(-2).

HumanBemg Mar 27, 2025

0

7

0

+164

Algebra

Let F(x) be the real-valued function defined for all real x except for x = 1 and x = 2 and satisfying the functional equation
F(x) + F \left( \frac{2x - 3}{x - 1} \right) + F \left( \frac{1}{x} \right) = x.
Find the function F(x) satisfying read more ..

HumanBemg Mar 27, 2025

0

4

0

+164

Algebra

The function f(n) is defined for all integers n, such that
f(x) + f(y) = f(x + y) - 4xy - 1 + f(x^2) + f(y^2)
for all integers x and y, and f(1) = 1. Find f(n).

HumanBemg Mar 27, 2025

0

4

0

+164

Algebra

The function f(n) takes the integers to the real numbers such that
f(m + n) + f(m - n) = 2f(m) + 2f(n) + mn
for all integers m and n, and f(1) = 2. Find f(n).

HumanBemg Mar 27, 2025

0

3

1

+1800

Algebra

The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies
f(x) f(y) - f(xy) = -2x - 6y + 10
for all x, y \in \mathbb{R}. Find f(x).

Let f be a function defined on the positive integers, such that
f(xy) = f(x) + f(y)
for all positive integers x and y. Given that f(5) = 6, f(65) = 7, f(86) = 9, f(93) = 10, find (120).

blackpanther Mar 26, 2025

0

4

2

+1800

Algebra

The function f has the following properties:
* f(a,b) is defined for all positive integers a and b
* f(a,1) = a
* f(a,b) = 1 if b > a
* f(a + 1,b) = b[f(a,b) - f(a,b - 1)] read more ..

blackpanther Mar 26, 2025

0

3

1

+1800

Algebra

Fill in the blanks, to make a true equation:
(8x^3 + 24x^2 + 15x + 1)/((x^2 - 1)(x^2 + 3x)) = ___/(x - 1) + ___/(x + 3) + ___/x + ___/(x + 1)

blackpanther Mar 25, 2025

0

2

0

+1800

Algebra

blackpanther Mar 25, 2025

0

1

0

+1800

Algebra

Find the all real numbers that are not in the domain of f(g(x)), where
f(x) = \frac{3x^2 - 10x - 25}{x + 1} and g(x) = \frac{14x - 6}{3x^2 + 5x + 15}

blackpanther Mar 25, 2025

0

2

0

+1800

Algebra

Give a polynomial g(x) so that f(x) + g(x) has a horizontal asymptote of y = 0 as x approaches positive infinity, where
f(x) = \frac{2x^4 - 3x^3 - 8x^2 + 4x - 4}{x^2 + x}.

blackpanther Mar 25, 2025

0

1

0

+1800

Algebra

Fill in the blanks, to make a true equation:
\frac{2x^4 - 3x^3 - x^2 + 4x - 4}{x^2 + x} = ___x^2 + ___x + ___ + ___/x + ___/(x + 1).

blackpanther Mar 25, 2025

0

4

0

+164

Algebra

Find the largest value of x where the plots of
f(x) = - \frac{2x + 5}{x + 3} and g(x) = \frac{12}{x - 1}
intersect.

HumanBemg Mar 25, 2025

0

2

0

+164

Algebra

Fill in the blanks with constants, to make a true equation:
\frac{x^2 - 6x - 3}{x^3 - 4x} = ___/x + ___/(x - 2) + ___/(x + 2)

HumanBemg Mar 25, 2025

0

7

0

+164

Algebra

Let (x,y,z) be the real solution to the system of equations
x + y = \sqrt{4z + 3}
y + z = \sqrt{4x - 1}
z + x = \sqrt{4y + 5}
Find x + y + z.

HumanBemg Mar 25, 2025

0

7

0

+164

Algebra

Find all x such that
sqrt{3x^2 + 2x + 1} + sqrt{3x^2 + 2x - 3} = 20.

HumanBemg Mar 25, 2025

0

6

1

+8

For part b, we solve the answer as 11. The answer given by the book is 44. May i know why is it 44?

Eddie has some white, blue and yellow marbles. He has 4/5 as many white marbles as blue marbles and 120% as many blue marbles as yellow marbles.
(a) Find the ratio of the number of white marbles to the number of blue marbles to the number read more ..

Simplify \frac{6 \sqrt{2]}{\sqrt{2} - \sqrt{3} - \sqrt{5}}

sPpBCdIT Mar 25, 2025

0

6

0

+339

Algebra

jaekg Mar 24, 2025

0

3

1

+339

Algebra

Fill in the blanks.
If f(x) is an even function and g(x) is an even function and h(x) is an even function, then f(x) + g(x) + h(x) is an ___ function.
If f(x) is an odd function and g(x) is an odd function and h(x) read more ..

Find the value of x that maximizes
f(x) = \log (-20x + 16 \sqrt{x} - x).

jaekg Mar 24, 2025

0

3

1

+339

Algebra

If 2^{16^x} = 128*4^x, then find 2^x.

jaekg Mar 24, 2025

0

3

0

+339

Algebra

jaekg Mar 24, 2025

0

3

2

+194

someone helpp

Find an expression for the number of nonnegative integer triples (x, y, z) which are solutions of the equation x + y + z = n, where n is a positive integer.
I believe it has something to do with the hockey stick identity, but read more ..

The function f has the following properties:
* f(x) is defined for x > 0
* f(x) > 0 for all x > 0
* f(x - y) = \sqrt{f(xy) + 4x + 4y + 8} for all x > y > 0

Determine f(1).

HumenBeing Mar 23, 2025

0

5

0

+806

Algebra

crimefightingvigiI Mar 22, 2025

0

3

0

+806

Algebra

Let f be a function such that
f(x) + f(2x + y) + 5xy = f(4x - y) - x^2 + 5xy - 8x + 17y + 1
for all real numbers x and y. Find f(10).

crimefightingvigiI Mar 22, 2025

0

6

0

+164

Algebra

Let f be a function such that
f(xy) + x = xf(y) + f(x) + xy^2
for all real numbers x and y. If f(-1) = 3, then compute f(100).

HumanBemg Mar 21, 2025

0

4

0

+164

Algebra

Let a, b, and c be positive real numbers. If a + b + c = 1, then find the minimum value of
\frac{1}{a} + \frac{1}{b} + \frac{4}{c*a^2} + \frac{16}{b^4*a} + \frac{32}{a*b^3}

HumanBemg Mar 21, 2025

0

5

0

+164

Algebra

Let a, b, and c be positive real numbers. If a + b + c = 1, then find the minimum value of
\frac{1}{a} + \frac{1}{b} + \frac{1}{c*a^2} + \frac{2}{ab^2} + \frac{8}{c^3}.

HumanBemg Mar 21, 2025

+1

5

3

+116

Algebra

The function f satisfies
f(m + n) = f(m) + f(n) - 2f(mn + m + n + 1) + m^2 + n^2
for all nonnegative integers m and n, and f(1) = 0. Compute f(123).