All Questions ⁺⁰235558 Questions

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+297

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 $5$ 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 ..

rtsdylifdts Oct 31, 2024

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+1525

Algebra

Let $x$ and $y$ be nonnegative real numbers. If $xy = \frac{2}{3}$, then find the minimum value of $4x \cdot \frac{2}{3} y^2.$

parmen Oct 31, 2024

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+518

Algebra

The roots of Ax^2+Bx+1 are the same as the roots of x^2- 3x + 5 + x^2 - 2x + 1. What is A+B?

cooIcooIcooI17 Oct 31, 2024

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+518

Algebra

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

cooIcooIcooI17 Oct 31, 2024

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+1427

Algebra

Find the number of ordered pairs $(a,b)$ of integers such that
\frac{a + 2}{a + 1} = \frac{b}{12}.

blackpanther Oct 31, 2024

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+1427

Algebra

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

blackpanther Oct 31, 2024

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+1427

Algebra

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 $n$.

blackpanther Oct 31, 2024

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3

+12

HELP!!

Idek what to do with the problem: Let P(x) be a nonconstant polynomial, where all the coefficients are nonnegative integers. Prove that there exist infinitely many positive integers n such that P(n) is composite.

crimefightingvigil Oct 31, 2024

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+227

Algebra

Simplify (1 - 3i)(2 - i)(2 + i + 1 + 3i).

jaekg Oct 31, 2024

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+227

Algebra

Find the number of ordered pairs (m,n) of integers that satisfy
mn = 3m + 6n + 2m - 5n + 1.

jaekg Oct 31, 2024

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+227

Algebra

Find the largest real number $c$ such that $1$ is in the range of $f(x)=x^2-5x+c-3x+8$.

jaekg Oct 31, 2024

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+227

Algebra

Find the sum of all values of $k$ for which $x^2 + kx - 9x + 25 - 9$ is the square of a binomial.

jaekg Oct 31, 2024

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+227

Algebra

Find the domain of the function $f(x) = \sqrt{6-x-x^2-2x^2}$.

jaekg Oct 31, 2024

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+6

Solve the following system of equations

Solve the following system of equations:
{2x+3y=124x−y=5\begin{cases} 2x + 3y = 12 \\ 4x - y = 5 \end{cases}{2x+3y=124x−y=5
Find the values of x and y.

PaulineMahone Oct 31, 2024

Oct 30, 2024

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+285

Algebra

For a real number t, let M(t) denote the maximum value of f(x) = 4x^2 - x + t for 11 \le x \le 1. What is the smallest possible value of M(t)?

MeIdHunter Oct 30, 2024

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+306

Polygons

A regular a-gon, a regular b-gon, a regular c-gon, and a regular d-gon fit perfectly around a point. What is the largest possible value of c?

nathanl6656 Oct 30, 2024

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+306

Number Theory

In the triangular array below, n different prime numbers p_1, p_2, \dots, p_n are written in the bottom row. Then in each box, we write the product of the two numbers below it. The number $K = p_1^{\alpha_1} p_2^{\alpha_2} \dotsm p_n^{\alpha_n}$ read more ..

nathanl6656 Oct 30, 2024

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+238

Number Theory

Carl writes eight consecutive three-digit numbers on a blackboard. Each number that Carl writes is divisible by 2, 3, 4, or 5. What is the sum of the digits of the smallest number that Carl wrote?

MEMEG0D Oct 30, 2024

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+238

Number Theory

When a prime is divided by 60, the remainder is a composite number. When a second prime is divided by 60, the remainder is a prime. Find the smallest possible value of the second prime.

MEMEG0D Oct 30, 2024

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+238

Number Theory

How many three-digit numbers are equal to five times the sum of their digits?

MEMEG0D Oct 30, 2024

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+140

Counting

Given a sequence which contains the numbers 1 through 8, a move consists of moving one of the numbers to the front of the sequence. For example, for the sequence $(1, 5, 8, 3, 6, 2, 7, 4),$ we can move the number $6$ to the front, which results in read more ..

AUnVerifedTaxPayer Oct 30, 2024

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+741

Counting

A sequence of ten letters has the following properties:
* Each letter is A, B, C, D, or E.
* The only letters that can come directly after A are B and C.
* The only letters that can come directly after B are C and D and E.
* The read more ..

gnistory Oct 30, 2024

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+741

Counting

The map below displays the train network of Beastville, including the stations X and $Y,$ and the lines $a,$ $b,$ $c,$ $d,$ $e,$ and $f.$ There is a station where any two lines intersect, and on every line, trains stop at each station.
Grogg read more ..

gnistory Oct 30, 2024

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+741

Counting

Each vertex of cube ABCDEFGH is colored either black or white. Find the number of colorings in which all four vertices of at least two faces have the same color.

gnistory Oct 30, 2024

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+741

Counting

Ava, Lucero, and Vinh are playing a game involving a single $6$-sided die. The rules are as follows:
* Ava rolls first. If she rolls a $1,$ or $2,$ then she wins.
* Otherwise, Lucero rolls next. If Lucero rolls a $4$ or a $5,$ read more ..

gnistory Oct 30, 2024

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