All Questions ⁺⁰232716 Questions

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Number Theory

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

ABJeIIy Jun 16, 2024

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+1213

Number Theory

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

ABJeIIy Jun 16, 2024

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+1213

Number Theory

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

ABJeIIy Jun 16, 2024

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+437

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 $DE$?

Rangcr897 Jun 16, 2024

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+437

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 $BD$?

Rangcr897 Jun 16, 2024

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+437

Geometry

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

Rangcr897 Jun 16, 2024

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+437

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?

Rangcr897 Jun 16, 2024

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+283

We introduced ${}i$ as a square root of $-1.$ Does ${}i$ itself have any square roots among the complex numbers? If so, name one. If not, an

We introduced i as a square root of -1. Does i itself have any square roots among the complex numbers? If so, name one. If not, answer with "None."

Lilliam0216 Jun 16, 2024

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+624

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}.

learnmgcat Jun 16, 2024

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+624

Algebra

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

learnmgcat Jun 16, 2024

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+624

Algebra

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

learnmgcat Jun 16, 2024

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+641

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?

eramsby1O1O Jun 16, 2024

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+641

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?

eramsby1O1O Jun 16, 2024

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+641

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots,$ $a_{10},$ $a_{11},$ $a_{12}$ be an arithmetic sequence. If $a_1 + a_3 + a_5 = 22$ and $a_2 + a_4 = 17$, then find $a_1$.

eramsby1O1O Jun 16, 2024

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+641

Algebra

If x and y are positive integers for which 3x + 2y + xy = 115 + 11x + 2y, then what is x + y?

eramsby1O1O Jun 16, 2024

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+641

Algebra

A geometric sequence has 400 terms. The first term is 1000 and the common ratio is -\frac{8}{9}$ How many terms of this sequence are greater than 1?

eramsby1O1O Jun 16, 2024

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+641

Algebra

In an arithmetic sequence, the 23rd term is $\frac{1}{4},$ and the $33$rd term is $-\frac{1}{5}$. What is the $43$rd term?

eramsby1O1O Jun 16, 2024

Jun 15, 2024

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+641

Algebra

Find the ordered pair (s, t) that satisfies the system
(s/2) + 5t =3 - 7t + 8s
3t - 6s = 9 - 2t

NotThatSmart ●

eramsby1O1O Jun 15, 2024

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+641

Algebra

For certain values of $k$ and $m,$ the system
a + 2b = -3 - 7a + b,
4a + 2b = k - 5a - mb
has infinitely many solutions $(a,b).$ What are $k$ and $m?$

eramsby1O1O Jun 15, 2024

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+140

Number Theory

Find the sum of the distinct prime factors of 123 - 47.

NotThatSmart ●

BRAlNBOLT Jun 15, 2024

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+140

Number Theory

Some of the following integers are prime. Select all the prime integers in the list.
1 2 3 4 5 6 7 8 9 10

NotThatSmart ●

BRAlNBOLT Jun 15, 2024

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+140

Number Theory

What is the fifth smallest prime number greater than 2000?

NotThatSmart ●

BRAlNBOLT Jun 15, 2024

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+140

Number Theory

Jon teaches a fourth grade class at an elementary school where class sizes are always at least 100 students and at most 200. One day Jon decides that he wants to arrange the students in their desks in a rectangular grid with no gaps. Unfortunately read more ..

BRAlNBOLT Jun 15, 2024

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+437

Number Theory

What are the first 5 digits after the decimal point (technically the hexadecimal point...) when the fraction \frac{2}{23} is written in base 16?

Rangcr897 Jun 15, 2024

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+437

Number Theory

Express 2.\overline{57} as a base 10 fraction in reduced form.

Rangcr897 Jun 15, 2024

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