All Questions ⁺⁰237313 Questions

0

15

1

+1000

Algebra

Let x and y be complex numbers. If x + y = 2 and xy = 5 - x^2 - y^2, then what is x^2 + y^2?

Hi6942O Aug 28, 2024

0

24

1

+1000

Algebra

Let a and b be complex numbers. If a + b = 4 and a^2 + b^2 = 6 + ab, then what is a^3 + b^3?

Hi6942O Aug 28, 2024

Aug 27, 2024

0

29

1

+582

Number Theory

Compute
4 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4}}}}

onyuIee Aug 27, 2024

0

25

1

+582

Number Theory

In base $b,$ the square of the number $17_b$ is $211_b$. Find the base $b$.

onyuIee Aug 27, 2024

0

29

2

+413

Algebra

Let f(x) = \sqrt{x}
Find the largest three-digit value of x such that f(x) is an integer.

nathanl6656 Aug 27, 2024

0

22

1

+413

Algebra

Compute
\frac{1}{1 \times 4} + \frac{1}{7 \times 10}

nathanl6656 Aug 27, 2024

0

18

1

+413

Algebra

Let a be a real number such that a - \frac{1}{a} = 4 + a^2 + \frac{1}{a^2}. Compute a^3 - \frac{1}{a^3}.

nathanl6656 Aug 27, 2024

0

20

2

+4

HELP!!

For a certain value of , the system:
x+y+3z=10
-4x+2y+5z=7
kx+z=3 read more ..

twinkletoe Aug 27, 2024

0

33

0

+1729

Geometry

In right triangle $ABC,$ $\angle C = 90^\circ$. Median $\overline{AM}$ has a length of 1, and median $\overline{BN}$ has a length of 1. What is the length of the hypotenuse of the triangle?

parmen Aug 27, 2024

0

31

0

+1729

Geometry

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

parmen Aug 27, 2024

0

21

0

+750

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 $BD$?

siIviajendeukie Aug 27, 2024

0

29

0

+750

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 $DE$?

siIviajendeukie Aug 27, 2024

0

33

1

+582

Number Theory

Let $S$ be the set of numbers of the form
n(n + 1)(n + 3)(n + 4)(n + 8)(n + 10)(n + 12)
where $n$ is any positive integer.
What is the GCD of the elements of $S$?

onyuIee Aug 27, 2024

0

30

1

+582

Number Theory

There is a single sequence of integers $a_2,$ $a_3,$ $a_4,$ $a_5,$ $a_6,$ $a_7$ such that
\frac{a_2}{2!} + \frac{a_3}{3!} + \frac{a_4}{4!} + \frac{a_5}{5!} + \frac{a_6}{6!} + \frac{a_7}{7!} = \frac{3}{128}
and $0 \le a_i < i$ for $i = 2,$ read more ..

onyuIee Aug 27, 2024

0

23

1

+582

Number Theory

Find the unique triple $(x,y,z)$ of positive integers such that $x < y < z$ and
\frac{1}{x} - \frac{1}{xy} - \frac{1}{xyz} = \frac{1}{3}

onyuIee Aug 27, 2024

Aug 26, 2024

0

34

0

+582

Number Theory

Find the largest positive integer $n$ such that $n^3$ divides 15.

onyuIee Aug 26, 2024

0

23

1

+582

Number Theory

When expanded as a decimal, the fraction \frac{1}{977} has a repetend (the repeating part of the decimal) that begins right after the decimal point, and is $976$ digits long. Find the first three digits after the decimal point.

onyuIee Aug 26, 2024

0

26

2

+70

help

The letters A, B, C, and D represent four distinct integers from 0 through 9. When A is added to B the result is C. When B is subtracted from A the result is D. How many possible ways are there to assign values to A, B, C, and D?

Hamburger Aug 26, 2024

0

21

1

+1782

Number Theory

How many bases $b \ge 2$ are there such that $100_b + 1_b$ is prime?

blackpanther Aug 26, 2024

0

26

1

+1782

Number Theory

Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in base $12$, it has the same three digits, in reverse order. What is $N$? (Express your answer in decim read more ..

blackpanther Aug 26, 2024

0

15

1

+750

Number Theory

What is the smallest prime divisor of 5^{19} * 7^{13} * 3^{31}?

siIviajendeukie Aug 26, 2024

0

23

1

+750

Number Theory

Find the 4000th digit following the decimal point in the expansion of \frac{1}{17}.

siIviajendeukie Aug 26, 2024

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