All Questions ⁺⁰237717 Questions

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+52

Math

Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if all the children are next to each other, and all the adults are next to each other? (Two seatings are considered the same if read more ..

XenosmiIus Aug 28, 2025

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+52

Math

At a meeting, four scientists, two mathematicians, and a journalist are to be seated around a circular table. How many different arrangements are possible if the mathematicians must sit next to each other? (Two seatings are considered equivalent if read more ..

XenosmiIus Aug 28, 2025

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+805

Math

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

bingboy Aug 28, 2025

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+805

Math

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 $BD$?

bingboy Aug 28, 2025

Aug 27, 2025

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+460

Math

Let f be a cubic polynomial such that f(0) = 5, f(2) = 8, f(3)=13, and f(7) = -5. What is the sum of the coefficients of f?

MEMEG0D Aug 27, 2025

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+460

Math

Let b be a constant. What is the smallest possible degree of the polynomial f(x) + b \cdot g(x), where f(x) = 2x^5 - 6x^4 - 4x^3 + 12x^2 + 7x - 5 and g(x) = x^6 - 17x^5 + 2x^4 + 6x^3 + 11x^2 - 8x + 1.

MEMEG0D Aug 27, 2025

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+460

Math

Suppose $a(x) = 2x + 5 - 3x^2 + x^3$ and $b(x) = 4 - x^2 + 4x^4$. Find $(a\circ b)(3) - (b\circ a)(3).$

MEMEG0D Aug 27, 2025

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+498

Math

Find the ordered quintuplet (a,b,c,d,e) that satisfies the system of equations
a+2b+3c+4d+5e=47
2a+3b+4c+5d=15
3a+4b+5c=34 read more ..

MeIdHunter Aug 27, 2025

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+498

Math

For a certain value of $k$, the system
x + y + 3z &= 10, \\
6x + 4y + z &= 18 - 5y - x, \\
kx + 7z &= 8 - 14y
has no solutions. What is this value of $k$?

MeIdHunter Aug 27, 2025

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+202

Math

Compute the sum
\frac{1}{\sqrt{36} + \sqrt{39}} + \frac{1}{\sqrt{36} + \sqrt{45}} + \frac{1}{\sqrt{39} +\sqrt{96}}

HumanBemg Aug 27, 2025

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+202

Math

Compute
\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 12}

HumanBemg Aug 27, 2025

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+202

Math

Compute
1 + \frac{3}{6} + \frac{5}{6^2}

HumanBemg Aug 27, 2025

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+1820

Math

Let $ABC$ be a triangle. Let $D$ be a point on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 30^\circ$, $\angle C = 90^\circ$, and $AC = 12$, then find the area of triangle $ABD$. Give your answer in exact read more ..

blackpanther Aug 27, 2025

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+1820

Math

Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$. Find the area of triangle $BET$ if $AB = 20$, $AC = 20$,
<read more ..

blackpanther Aug 27, 2025

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+1820

Math

Let $M$, $N$, and $P$ be the midpoints of sides $\overline{TU}$, $\overline{US}$, and $\overline{ST}$ of triangle $STU$, respectively. Let $\overline{UZ}$ be an altitude of the triangle. If $\angle TSU = 60^\circ$ and $\angle STU = 45^\circ$, read more ..

blackpanther Aug 27, 2025

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+971

Math

When the same constant is added to the numbers $a,$ $b,$ and $c,$ a three-term geometric sequence arises. If $a = 60,$ $b = 100,$ and $c = 140,$ what is the common ratio of the resulting sequence?

booboo44 Aug 27, 2025

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+971

Math

The graph of the equation
4x^2 - 12x + 4y^2 + 16y + 17 = -4x + 2y + 5
is a circle. Find the radius of the circle.

booboo44 Aug 27, 2025

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+971

Math

Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If $A = (7,-11),$ $B = (10,13),$ and $C = (18,-22)$, then find the constant read more ..

booboo44 Aug 27, 2025

Aug 26, 2025

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+312

I need help

What are all solutions to the equation x^2 + 3x - 10 = 18x - 15?

CInsbstnkng Aug 26, 2025

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+312

Help help

The solutions to the equation
6x^2 + 10x = -2x^2 - 12x - 17
can be written in the form $x=\frac{P \pm \sqrt Q}{R}$, where $P$ and $R$ are relatively prime integers and $R > 0$.

CInsbstnkng Aug 26, 2025

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+1564

Math

Given that
5a - 7b + 3c = 19
4a + 2b - 8c = 4
7a + 5b + 3c = -11 read more ..

kittykat Aug 26, 2025

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+1564

Math

Find all values of z such that z^4 - 4z^2 + 3 = 5z^2 + 11z^3 - 40.

kittykat Aug 26, 2025

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+306

Math

All members of our painting team paint at the same rate. If $10$ members can paint a $1000$ square foot wall in $10$ minutes, then how long would it take for $5$ members to paint a $1000$ square foot wall, in minutes?

AUnVerifedTaxPayer Aug 26, 2025

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