nathanl6656

47 Questions
14 Answers

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Algebra

Let $A$, $B$, and $C$ be constants so that
\frac{1}{x^3-2x^2-5x+9}=\frac{A}{x+5}+\frac{B}{x-3} + \frac{C}{(x - 3)^2}
holds for all real numbers $x$ other than $-5$ and $3.$ What is $A$?

nathanl6656 9 hours ago

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1

2

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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

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1

1

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Algebra

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

nathanl6656 Aug 27, 2024

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1

1

+200

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

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2

1

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Probability

The numbers $x_1,$ $x_2,$ $x_3,$ $x_4$ are chosen at random in the interval $[0,1].$ Let $I$ be the interval between $x_1$ and $x_2,$ and let $J$ be the interval between $x_3$ and $x_4.$ Find the probability that intervals $I$ and $J$ both contain read more ..

nathanl6656 Aug 9, 2024

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3

1

+200

Counting

The entries in a certain row of Pascal's triangle are
1, n, \dots, n, 1.
The average of the entries in this row is 2. Find $n$.

nathanl6656 Aug 9, 2024

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4

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Number Theory

Find the largest positive integer n such that
\frac{(n + 1)^2}{n + 2}
is an integer.

nathanl6656 Jul 5, 2024

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7

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+200

Number Theory

Find the largest positive integer n such that
\frac{(n + 1)^2}{n + 2}
is an integer.

nathanl6656 Jul 5, 2024

0

8

0

+200

Number Theory

Let $a$, $b$, $c$, and $n$ be positive integers. If $a + b + c = 19 \cdot 97$ and
a + 3n = b - 2n = \frac{c}{5n},
compute the value of $a$.

nathanl6656 Jul 5, 2024

0

7

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+200

Number Theory

How many positive four-digit integers $N$ have the property that the three-digit number obtained by removing the leftmost digit is equal to $\frac{N}{10}?$

nathanl6656 Jul 5, 2024

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4

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Counting

Starting with the A moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?
A
RRR
CCCCC read more ..

nathanl6656 Jul 5, 2024

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7

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Number Theory

Let $m$ and $n$ be positive integers. If $m$ has exactly $10$ positive divisors, $n$ has exactly $16$ positive divisors, and $mn$ has exactly $21$ positive divisors. How many divisors does $m^2 n^2$ have?

nathanl6656 Jul 5, 2024

0

7

0

+200

Number Theory

Let $m$ and $n$ be positive integers. If $m$ has exactly $10$ positive divisors, $n$ has exactly $16$ positive divisors, and $mn$ has exactly $21$ positive divisors. How many divisors does $m^2 n^2$ have?

nathanl6656 Jul 5, 2024

0

8

0

+200

Number Theory

Let $M$ be the least common multiple of $1,$ $2,$ $\dots,$ $12$, $13$, $14$, $15$, $16$. How many positive divisors does $M$ have?

nathanl6656 Jul 5, 2024

0

4

0

+200

Number Theory

Let $m$ and $n$ be positive integers. If $m$ has exactly $10$ positive divisors, $n$ has exactly $16$ positive divisors, and $mn$ has exactly $21$ positive divisors. How many divisors does $m^2 n^2$ have?

nathanl6656 Jul 5, 2024

0

3

1

+200

Graphing

Assume that f(3) = 4. Name a point that must be on the graph of y= (5 + f(2x/3))/11$.

nathanl6656 Jun 13, 2024

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5

1

+200

Geometry

Find all points (x,y) that are 5 units away from the point (2,7) and that lie on the line x + y = 13.

nathanl6656 Jun 13, 2024