All Questions ⁺⁰236320 Questions

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+1234

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence. If
\[a_n = a_{n - 1} + a_{n - 2}\]
for all $n \ge 3,$ and $a_{11} = 1$ and $a_{10} = 4,$ then find $a_6.$

Akhain1 Jun 22, 2024

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20

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+1234

Algebra

Evaluate $a^3 - \dfrac{1}{a^3}$ if $a - \dfrac{1}{a} = 0$.

Akhain1 Jun 22, 2024

0

10

1

+52

Help with counting

Franklin the fly starts at the point (0, 0) in the coordinate plane. At each point, Franklin takes a step to the right, left, up, or down. After 12 steps, how many different points could Franklin end up at?

MEMEGOD Jun 22, 2024

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12

2

+1234

Algebra

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

Akhain1 Jun 22, 2024

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11

2

+1379

Geometry

In triangle $ABC$, point $D$ is on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 45^\circ$, $\angle C = 45^\circ$, and $AC = 12$, then find the area of triangle $ABD$.

learnmgcat Jun 22, 2024

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12

2

+1379

Geometry

In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $BC = 15,$ and $CX = 5$. What is $BX$?

learnmgcat Jun 22, 2024

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16

2

+52

Counting

Find the number of positive integers that satisfy both the following conditions:
Each digit is a 1 or a 3.
The sum of the digits is 11.

MEMEGOD Jun 22, 2024

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32

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+1520

Counting

Each square below is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the squares? read more ..

kittykat Jun 22, 2024

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35

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+1520

Counting

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?

kittykat Jun 22, 2024

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16

1

+1839

Terminal zeros

A terminal zero is a $0$ that appears at the end of a number. For example, the number $3,800$ has two terminal zeros.
How many terminal zeroes does $40 \cdot 6 \cdot 75 \cdot 12$ have?

ABJeIIy Jun 22, 2024

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15

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+1520

Counting

I have $3$ different mathematics textbooks, $2$ different psychology textbooks, and $2$ different chemistry textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row?

kittykat Jun 22, 2024

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17

1

+1839

Square Divisors

The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$
What is the smallest positive integer that has exactly $2$ perfect square divisors?

ABJeIIy Jun 22, 2024

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12

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+1839

Number Theory

For a positive integer $n$, $\phi(n)$ denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$.
What is $\phi(191)$?

ABJeIIy Jun 22, 2024

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15

1

+1839

Number Theory

For a positive integer $n$, $\phi(n)$ denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$.
What is $\phi(1200)$?

ABJeIIy Jun 22, 2024

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16

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+845

Number Theory

Find the $4000$th digit following the decimal point in the expansion of $\frac{1}{17}$.
Be sure to include complete explanations with your answer, using complete sentences. Imagine you were going to show your solution to a classmate, read more ..

gnistory Jun 22, 2024

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18

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+1520

Probability

Raina places the following balls into a bag. She then draws three balls out of the bag, one at a time, without replacement. What is the probability that the colors of the balls alternate?
There are 4 orange balls and 4 purple read more ..

kittykat Jun 22, 2024

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21

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+845

Number Theory

A positive integer is called nice if it is a multiple of $8.$
A certain nice positive integer $n$ has exactly $9$ positive divisors. What is the smallest possible value of $n?$

gnistory Jun 22, 2024

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15

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+845

Number Theory

A positive integer is called nice if it is a multiple of $8.$
A certain nice positive integer $n$ has exactly $9$ positive divisors. How many prime numbers are divisors of $n?$

gnistory Jun 22, 2024

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24

1

+219

Algebra

Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 4/3 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?

EnormousBighead ●

BRAlNBOLT Jun 22, 2024

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17

0

+1379

Number Theory

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

learnmgcat Jun 22, 2024

0

18

0

+1379

Number Theory

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

learnmgcat Jun 22, 2024

-2

13

1

+219

Algebra

Seven years ago, Grogg's dad was $9$ times as old as Grogg. Four years ago, Grogg's dad was $6$ times as old as Grogg. How old is Grogg's dad currently?

BRAlNBOLT Jun 22, 2024

0

21

1

+1379

Number Theory

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?

learnmgcat Jun 22, 2024

0

15

1

+845

Euclidean Algorithm

You have a total supply of $1000$ pieces of candy, and an empty vat. You also have a machine that can add exactly $5$ pieces of candy per scoop to the vat, and another machine that can remove exactly $3$ pieces of candy with a different scoop from read more ..

gnistory Jun 22, 2024

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