All Questions ⁺⁰237119 Questions

0

23

3

+1493

Geometry

Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If $A = (7,-11),$ $B = (10,13),$ and $C = (18,-22)$, then find the constant read more ..

learnmgcat Jul 11, 2024

0

28

1

+1493

Geometry

Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $4,$ and the slope of line segment $\overline{OQ}$ is $5.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: read more ..

learnmgcat Jul 11, 2024

0

14

1

+1493

Geometry

Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$. The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has slope $2$. Find $a + b$.

learnmgcat Jul 11, 2024

-3

22

3

+216

Algebra

Find the sum of the squares of the roots of $2x^2+4x-1=x^2-8x+3$.

BRAlNBOLT Jul 11, 2024

-3

33

0

+216

Algebra

Let $m$ be a real number. If the quadratic equation $x^2+mx+4 = 2x^2 + 17x + 8$ has two distinct real roots, then what are the possible values of $m$? Express your answer in interval notation.

BRAlNBOLT Jul 11, 2024

-3

9

1

+216

Algebra

If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 131$, what is the minimal possible value of $|a - b|$?

BRAlNBOLT Jul 11, 2024

-3

31

0

+216

Algebra

Let $a$ and $b$ be complex numbers. If $a + b = 4$ and $a^2 + b^2 = 6,$ then what is $a^3 + b^3?$

BRAlNBOLT Jul 11, 2024

+1

27

1

+280

Algebra

Evaluate $a^3 - \dfrac{1}{a^3}$ if $a - \dfrac{1}{a} = 0$.

BRAINBOLT Jul 11, 2024

+1

20

1

+280

Number Theory

How many of the $1000$ smallest positive integers are congruent to $5$ modulo $171?$

BRAINBOLT Jul 11, 2024

0

21

0

+1000

Geometry

In triangle $ABC,$ the angle bisector of $\angle BAC$ meets $\overline{BC}$ at $D.$ If $\angle BAC = 60^\circ,$ $\angle CAD = 45^\circ,$ and $AD = 24,$ then find the area of triangle $ABC.$

Hi6942O Jul 11, 2024

0

32

0

+1000

Geometry

In triangle $ABC,$ $AB = 15,$ $BC = 9,$ and $AC = 10.$ Find the length of the shortest altitude in this triangle.

Hi6942O Jul 11, 2024

0

16

1

+1000

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 $BX$?

Hi6942O Jul 11, 2024

0

15

1

+1000

Geometry

In triangle $ABC$, point $D$ is on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 45^\circ$, $\angle C = 45^\circ$, and $AC = 12$, then find the area of triangle $ABD$.

Hi6942O Jul 11, 2024

Jul 10, 2024

0

31

1

+1493

Number Theory

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?

learnmgcat Jul 10, 2024

0

20

0

+1493

Number Theory

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

learnmgcat Jul 10, 2024

0

31

2

+1493

Number Theory

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

learnmgcat Jul 10, 2024

0

18

0

+1493

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 $(A,B,C)$.

learnmgcat Jul 10, 2024

0

26

2

+4

plz help

Find the number of digit numbers, where the sum of the digits is divisible by

JUICYTON Jul 10, 2024

0

25

1

+1000

Algebra

Let
f(x) = \sqrt{x - \sqrt{x}}.
Find the largest three-digit value of $x$ such that $f(x)$ is an integer.

Hi6942O Jul 10, 2024

0

29

1

+1000

Algebra

Evaluate $a^3 - \dfrac{1}{a^3}$ if $a - \dfrac{1}{a} = 0$.

Hi6942O Jul 10, 2024

0

23

0

+1721

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence. If
\[a_n = a_{n - 1} + a_{n - 2}\]
for all $n \ge 3,$ and $a_{11} = 1$ and $a_{10} = 4,$ then find $a_6.$

parmen Jul 10, 2024

0

14

1

+1721

Algebra

Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$

parmen Jul 10, 2024

0

30

1

+1721

Algebra

Find the constant k such that the quadratic 2x^2 + 3x + 8x - x^2 + 4x + k has a double root.

parmen Jul 10, 2024

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