All Questions ⁺⁰236075 Questions

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Algebra

The equation of the line below can be expressed in the form
\frac{x}{a} + \frac{y}{b} = 1.
Find a.

gnistory Dec 1, 2024

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+809

Algebra

The parabola $y = ax^2 + bx + c$ is graphed below. Find $a \cdot b \cdot c.$ (The grid lines are one unit apart.)

gnistory Dec 1, 2024

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+809

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 $A$?

gnistory Dec 1, 2024

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+809

Algebra

Suppose $f(x)$ is a function defined for all real $x$, and suppose $f$ is invertible (that is, $f^{-1}(x)$ exists for all $x$ in the range of $f$). If the graphs of $y=f(x)$ and $y=f(x^3)$ are drawn, at how many points do they intersect?

gnistory Dec 1, 2024

Nov 30, 2024

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+441

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 $\frac{N}{24}?$

crimefightingvigiI Nov 30, 2024

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+441

Number Theory

The product of a set of distinct positive integers is $630$. If one of the numbers is 42, what are the other two numbers?

crimefightingvigiI Nov 30, 2024

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+441

Number Theory

On planet Enigma, the residents use a currency called the confusion. There are only 2 confusion bills on Enigma, one worth 2 confusions and the other worth 3 confusions. There are also some coins of smaller value, but each weighs over 10 read more ..

crimefightingvigiI Nov 30, 2024

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+441

Number Theory

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

crimefightingvigiI Nov 30, 2024

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+441

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 $a$.

crimefightingvigiI Nov 30, 2024

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+441

Number Theory

Let $m$ and $n$ be positive integers. If $m$ has exactly $7$ positive divisors, $n$ has exactly $11$ positive divisors, and $mn$ has exactly $28$ positive divisors, then how many divisors does $m^2 n^2$ have?

crimefightingvigiI Nov 30, 2024

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+441

Number Theory

When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is
0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.
When read more ..

crimefightingvigiI Nov 30, 2024

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+441

Number Theory

You have a total supply of $1000$ pieces of candy, and an empty vat. You also have a machine that can add exactly $52$ pieces of candy per scoop to the vat, and another machine that can remove exactly $39$ pieces of candy with a different scoop from read more ..

crimefightingvigiI Nov 30, 2024

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+441

Algebra

Let a, b, c, be the numbers 1, 2, 3, in some order. Find the largest possible value of ab + 2ac + 3bc.

crimefightingvigiI Nov 23, 2024

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+441

Algebra

Find the number of integers n that satisfy 64 <= n^2 <= 100 + 20n.

crimefightingvigiI Nov 23, 2024

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+441

Algebra

Find the number of ordered pairs (x,y) of real numbers such that (2x^2 + 4x + 2)(3y^2 - 6y + 3) = 0.

crimefightingvigiI Nov 23, 2024

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+441

Algebra

If y = x/(x^2 + 2x + 1) and x is any real number, then what is the sum of the maximum and minimum possible values of y?

crimefightingvigiI Nov 23, 2024

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+868

Counting

Sam wants to color the three sides of an equilateral triangle. He has two different colors to choose from. In how many different ways can Sam color the sides of the triangle? (Two colorings are considered the same if one coloring can be rotated and/or reflected read more ..

eramsby1O1O Nov 23, 2024

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+868

Counting

My sock drawer contains two red socks, two white socks, and two blue socks. For three days in a row, I pull two socks out of my sock drawer at random. (I do not replace the socks in the drawer before drawing on the next day.) What is the probability that read more ..

eramsby1O1O Nov 23, 2024

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+868

Counting

Cai writes down the list of positive integers, excluding squares and cubes. His sequence starts
2, 3, 4, 6, 7, 10, 11, ...
What is the 1000th term in Cai's list?

eramsby1O1O Nov 23, 2024

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+868

Counting

We define a bow-tie quadrilateral as a quadrilateral where two sides cross each other.
Seven distinct points are chosen on a circle. We draw all 7C2=21 chords that connect two of these points. Four of these 21 chords are selected read more ..

eramsby1O1O Nov 23, 2024

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+441

Algebra

Determine all of the following for $f(x) \cdot g(x)$, where $f(x) = -x^2+ 8x - 5$ and $g(x) = x^3 - 11x^2 + 2x$.
Leading term
Leading coefficient
Degree read more ..

crimefightingvigiI Nov 23, 2024

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Algebra

Let $x$ and $y$ be complex numbers. If $x + y = 2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?

crimefightingvigiI Nov 23, 2024

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+310

Algebra

The distance between the points A and B is \sqrt{34}. If A = (a,4) and B = (-4,5), then find the sum of all possible values of a.

MEMEG0D Nov 23, 2024

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Algebra

A line passes through the points $A,$ $B,$ and $C.$ Find $y.$
A = (7,2), B = (1,3), and C = (6,y)

MEMEG0D Nov 23, 2024

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Algebra

Find all values of a such that \frac{a - 5}{\sqrt{a}} = -\sqrt{a} + a^2.

MEMEG0D Nov 23, 2024

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