All Questions ⁺⁰232960 Questions

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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 18, it has the same three digits, in reverse order. What is N? (Express your answer in decimal read more ..

eramsby1O1O Jul 1, 2024

Jun 30, 2024

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+21

Combinatorics

Erica and Alan each flip a fair coin 5 times. Suppose that Erica has flipped more heads than Alan after each has flipped the coin 2 times. The probability that Erica has flipped more heads than Alan after each has flipped the coin 5 times is m n , where read more ..

Sxchen1110 Jun 30, 2024

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+526

Number Theory

For a certain positive integer $n$, the number $n^{6873}$ leaves a remainder of $3$ when divided by $23.$ What remainder does $n$ leave when divided by $23$?

Rangcr897 Jun 30, 2024

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+526

Number Theory

For an integer $n,$ let $f(n)$ be the remainder when $n^8 + n^{16}$ is divided by $5.$ Compute $f(0) + f(1) + f(2) + f(3) + f(4).$

Rangcr897 Jun 30, 2024

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+526

Algebra

Fill in the coefficients:
n(n - 1)(n + 17)(n - 21)(n - 33) = n^5 + ___ n^4 + ___ n^3 + ___ n^2 + ___n + ___

Rangcr897 Jun 30, 2024

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+526

Algebra

The Fibonacci sequence is defined by F_1 = F_2 = 1 and F_{n + 2} = F_{n + 1} + F_n. Find the remainder when F_{1000} is divided by 17.

Rangcr897 Jun 30, 2024

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+246

Geometry

In triangle $PQR$, let $M$ be the midpoint of $\overline{PQ}$, let $N$ be the midpoint of $\overline{PR}$, and let $O$ be the intersection of $\overline{QN}$ and $\overline{RM}$, as shown. If $\overline{QN}\perp\overline{PR}$, $QN = 15$, and $PR = 20$, read more ..

ABJelly Jun 30, 2024

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+1206

Geometry

In triangle ABC, angle ACB = 90 degrees. Let H be the foot of the altitude from C to side AB.
If BC = 1 and AH = 2, find CH.

blackpanther Jun 30, 2024

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+526

Algebra

The sequence $a_1$, $a_2$, $a_3,$ $\dots$ is defined by $a_1 = 1,$ $a_2 = 3,$ and
a_n = 2a_{n - 1} + a_{n - 2}.

Thus, the next few terms are $a_3 = 2a_2 + a_2 = 6 + 1 = 7$ and $a_4 = 2a_3 + a_2 = 14 + 3 = 17$.
Find read more ..

Rangcr897 Jun 30, 2024

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+1206

Geometry

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

blackpanther Jun 30, 2024

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Algebra

How many three-digit palindromes (numbers that read the same forward and backward) satisfy the following property: the sum of three things -- the hundreds digit, the units digit, and the product of the units and tens digits -- is eight more than the tens read more ..

DeluluSunshine Jun 30, 2024

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+526

Number Theory

The number $N$ is a multiple of $7$. The base $2$ representation of $N$ is
10011010011ABC110_2.
Compute the ordered triple of digits $(A,B,C)$.

Rangcr897 Jun 30, 2024

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+526

Number Theory

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.

Rangcr897 Jun 30, 2024

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+526

Counting

Find the number of $7$-digit numbers, where the sum of the digits is divisible by $11.$

Rangcr897 Jun 30, 2024

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+526

Counting

In how many ways can the numbers 1, 2, 3, 4, 5, 6 be arranged in a row, so that the product of any two adjacent numbers is at least 5?

Rangcr897 Jun 30, 2024

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+354

Algebra

Solve the system of equations
y = \log_2 (2x)
y = log_4 (16 + x)

gnistory Jun 30, 2024

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+354

Algebra

Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

gnistory Jun 30, 2024

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+354

Algebra

Let $f(x) = 3x - 8 + 4x^2$ and $g(x) = 15x + c - 3x^2$. Find $c$ if $(f \circ g)(x) = (g \circ f)(x)$ for all $x$.

gnistory Jun 30, 2024

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+354

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be an arithmetic sequence. Let $S_n$ denote the sum of the first $n$ terms. If $S_{20} = \frac{1}{5}$ and $S_{10} = 0,$ then find $S_{70}.$

gnistory Jun 30, 2024

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+354

Algebra

Laverne starts counting out loud by 5's. She starts with 2. As Laverne counts, Shirley sums the numbers Laverne says. When the sum finally exceeds 20, Shirley runs screaming from the room. What number does Laverne say that sends read more ..

gnistory Jun 30, 2024

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solve this pls its kinda hard

Let be the roots of . Let . Then the product can be expressed as a polynomial in . Find this polynomial.

asdasdasddwasdwasd Jun 30, 2024

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+354

Algebra

Let a_1, a_2, a_3, \dots, a_{10}, a_{11}, a_{12} be an arithmetic sequence. If $a_1 + a_3 + a_5 + a_7 + a_9 + a_{11} = 0$ and $a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12} = 0$, then find $a_1$.

gnistory Jun 30, 2024

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+354

Algebra

When the same constant is added to the numbers 60, 120, and 160, a three-term geometric sequence arises. What is the common ratio of the resulting sequence?

gnistory Jun 30, 2024

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help hard algebra

Let $x$ and $y$ be real numbers such that $x^3 + 3xy^2 = 679$ and $x^3 - 3xy^2 = 615.$ Find $x - y.$

gnistory Jun 30, 2024

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+354

Algebra

Fill in the blanks to get an identity. (Each blank should contain an integer.)
(n - k + 2)^2 = __ n^2 + ___ k^2 + ___ n + ___k + ___ nk + ___

gnistory Jun 30, 2024

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