All Questions ⁺⁰234657 Questions

0

2

1

+673

Number Theory

Find the smallest positive integer $N$ such that
N &\equiv 2 \pmod{5}, \\
N &\equiv 2 \pmod{7}.

magenta Aug 14, 2024

+1

3

1

+673

Number Theory

When $N$ is divided by $10$, the remainder is $a$. When $N$ is divided by $2$, the remainder is $b$. What is $N$ modulo $20$, in terms of $a$ and $b$?

magenta Aug 14, 2024

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3

1

+16

Question:

What is the product of the smallest two prime factors of ?

CelsiusSolving Aug 14, 2024

Aug 13, 2024

+2

4

2

+16

Difficult Counting Problem

Gracie starts at her house and walkes one block east. After each block she walks, she flips a coin. She walks one block east if the coin lands heads; otherwise one block west. She stops her walk the first time she lands back home. What is the probability read more ..

CelsiusSolving Aug 13, 2024

0

2

0

+625

Number Theory

Brandon spends his afternoon picking apples from an orchard. He notices that if he groups the apples he picked by fives, six, or sevens, then he ends up with $1$ left over. If he groups by eights, he has $1$ left over. If Brandon read more ..

AnswerscorrectIy Aug 13, 2024

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2

0

+625

Number Theory

Find the number of solutions to
N &\equiv 2 \pmod{5}, \\
N &\equiv 2 \pmod{6}, \\
N &\equiv 2 \pmod{7}
in the interval 0 \le N < 1000.

AnswerscorrectIy Aug 13, 2024

0

4

0

+772

Coefficient

There exists a polynomial $f(x)$ and a constant $k$ such that
(x^2 - 2x - 5) f(x) = 2x^4 + 19x^3 + kx^2 - 15x - 1.
What is $k?$

RedDragonl Aug 13, 2024

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3

1

+772

Coefficient

What is the coefficient of $x$ in $(x^3 + x^2 + x + 1)(x^4 - 8x^3 + 17x^2 - 23x + 14)$?

RedDragonl Aug 13, 2024

0

3

7

+9

Help please! Simon's Favorite Factoring Trick

How many integers with can be written as for at least one ordered pair of positive integers ?
This question really confused me because I don't understand what it's asking.
♥️ read more ..

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anonymousSingularity Aug 13, 2024

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1

1

+328

Algebra

Find the sum of all positive integers less than 1000 ending in 3 or 4 or 5.

onyuIee Aug 13, 2024

0

2

1

+328

Algebra

Will and Grace are canoeing on a lake. Will rows at $50$ meters per minute and Grace rows at $30$ meters per minute. Will starts rowing at $2$ p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at $2{:}45$ p.m. read more ..

onyuIee Aug 13, 2024

0

3

1

+2651

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 $BX$?

LiIIiam0216 Aug 13, 2024

0

4

1

+2651

Algebra

Let $x$ and $y$ be nonnegative real numbers. If $x + y = 25$, then find the minimum value of $6x + 3y.$

LiIIiam0216 Aug 13, 2024

0

2

1

+1706

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

ABJeIIy Aug 13, 2024

0

2

0

+1706

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 ..

ABJeIIy Aug 13, 2024

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2

1

+1475

Algebra

Find one ordered pair $(x,y)$ of real numbers such that $x + y = 10$ and $x^3 + y^3 = 162 + x^2 + y^2.$

parmen Aug 13, 2024

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2

1

+1475

Algebra

Find the minimum value of 9^x - 2 \cdot 3^x + 1 over all real numbers x.

parmen Aug 13, 2024

0

3

0

+1706

Algebra

I own a large truck, and my neighbor owns four small trucks that are all identical. My truck can carry a load of at least 600 pounds more than each of her trucks, but no more than 1/2 of the total load her four trucks combined can carry. Based on these read more ..

ABJeIIy Aug 13, 2024

0

5

0

+1706

Algebra

The product $(3-\sqrt{5})(4+\sqrt{5})(1+6\sqrt{5})$ can be expressed in the form $a+b\sqrt{5}$, where $a$ and $b$ are integers. Find $a+b$.

ABJeIIy Aug 13, 2024

0

2

2

+38

PLEASE HELP PLEASE

Find the exponential form of the complex number
e^17πi/60+e^27πi/60+e^37πi/60+e^47πi/60+e^57πi/60
with proof.
Could someone help me on this and type out the solving process and the proof please?
Thanks! read more ..

JRFPLSHelp Aug 13, 2024

0

1

2

+32

Triangle

Triangle ABC has altitudes AD, BE, and CF. and If AD= 12, BE=16, and CF is a positive integer, then find the largest possible value of CF.

surfz Aug 13, 2024

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2

1

+32

Question about triangles

All the sides of a triangle are integers. If the perimeter of the triangle is 16, then how many different possible triangles are there? (Assume that the triangle is non-degenerate. Two triangles are considered the same if they are congruent.) read more ..

surfz Aug 13, 2024

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2

0

+1341

Algebra

My street hockey team plays three games each week. My team lost all $9$ games in the first three weeks. Then, my team won one game and lost two games in the fourth week, bringing our record to $1$ win and $11$ losses. Each week after that, my team read more ..

kittykat Aug 13, 2024

0

1

1

+1341

Algebra

When the same constant is added to the numbers $a,$ $b,$ and $c,$ a three-term geometric sequence arises. If $a = 60,$ $b = 100,$ and $c = 140,$ what is the common ratio of the resulting sequence?

kittykat Aug 13, 2024

0

2

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+1341

Algebra

Let a_1, a_2, a_3, \dots, a_8, a_9, a_{10} be an arithmetic sequence. If $a_1 + a_3 = 5$ and $a_2 + a_4 = 6$, then find $a_1$.

kittykat Aug 13, 2024

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