All Questions ⁺⁰234574 Questions

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Geometry

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 ..

Rangcr897 Jul 19, 2024

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5

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+601

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 $S$?

AnswerscorrectIy Jul 19, 2024

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+601

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 ..

AnswerscorrectIy Jul 19, 2024

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+601

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}

AnswerscorrectIy Jul 19, 2024

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+601

Number Theory

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

AnswerscorrectIy Jul 19, 2024

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+165

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 ..

BRAlNBOLT Jul 19, 2024

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+165

Number Theory

A positive integer is called terrific if it has exactly $10$ positive divisors. What is the smallest terrific positive integer?

BRAlNBOLT Jul 19, 2024

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I might know the answer...

A positive integer is called terrific if it has exactly $10$ positive divisors. What is the smallest number of primes that could divide a terrific positive integer?

BRAINBOLT Jul 19, 2024

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Counting

At a meeting, $5$ scientists, $2$ mathematicians, and a journalist are to be seated around a circular table. How many different arrangements are possible if each scientist must sit next to a mathematician? (Two seatings are considered equivalent read more ..

AnswerscorrectIy Jul 19, 2024

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+601

Counting

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 ..

AnswerscorrectIy Jul 19, 2024

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+601

Counting

I have $4$ different mathematics textbooks and $3$ different psychology textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row, if there must be a mathematics textbook exactly in the middle, and all the psychology textbooks are read more ..

AnswerscorrectIy Jul 19, 2024

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Help!

Solve for the variable in terms of and :

MellowJello Jul 19, 2024

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+8

Trigonometric Identities

I need help solving this problem!

Ishiaki Jul 19, 2024

Jul 18, 2024

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+467

Algebra

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 2 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator read more ..

ChiIIBill Jul 18, 2024

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Algebra

Sophie's favorite (positive) number is a two-digit number. If she reverses the digits, the result is 72 less than her favorite number. Also, one digit is 1 less than double the other digit. What is Sophie's favorite number?

NotThatSmart ●

ChiIIBill Jul 18, 2024

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Precalc

abc12345678 Jul 18, 2024

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+601

Algebra

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

AnswerscorrectIy Jul 18, 2024

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+601

Algebra

Find the largest value of $x$ such that $3x^2 + 17x + 15 = 2x^2 + 21x + 12 - 5x^2 + 17x + 34.$

AnswerscorrectIy Jul 18, 2024

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+601

Algebra

Fill in the blanks to make a quadratic whose roots are $1$ and $-1.$
x^2 + ___ x + ___

AnswerscorrectIy Jul 18, 2024

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+601

Algebra

Write the equation of the line below in the form Ax+By=C, where A, B, and C are integers with greatest common divisor 1 and A is positive.
The line passes through (-2,5) and (9,1).

AnswerscorrectIy Jul 18, 2024

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+601

Algebra

The system of equations
\frac{xy}{x + y} = 1, \quad \frac{xz}{x + z} = 1, \quad \frac{yz}{y + z} = 2
has exactly one solution. What is $z$ in this solution?

AnswerscorrectIy Jul 18, 2024

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