All Questions ⁺⁰236282 Questions

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Geometry

In triangle $ABC,$ $AB = 3,$ $AC = 5,$ $BC = 7,$ and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Find $\cos \angle BAD.$

LiIIiam0216 Jun 19, 2024

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19

1

+976

Algebra

Simplify \frac{4}{1 - \sqrt[3]{3}} + \sqrt[3]{9}.

Hi6942O Jun 19, 2024

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22

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+976

Algebra

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2ab - 12a + 3a + 3b - 8b - ab + ___

Tottenham10 ●

Hi6942O Jun 19, 2024

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11

1

+2653

Algebra

Find all values of c such that 3(2c + 1) = 28*(3c) - 9 + 45(3c) - 18. If you find more than one value of c, then list your values in increasing order, separated by commas.

NotThatSmart ●

LiIIiam0216 Jun 19, 2024

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14

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+2653

Algebra

Compute
1 + \frac{3}{6} + \frac{5}{6^2} + \frac{7}{6^3} + 2 + \frac{2}{5} + \frac{2}{5^3} + \frac{2}{5^4}

LiIIiam0216 Jun 19, 2024

0

12

2

+2653

Algebra

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

LiIIiam0216 Jun 19, 2024

0

19

1

+2653

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 ..

LiIIiam0216 Jun 19, 2024

0

15

1

+355

Shapes

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 ..

Lilliam0216 Jun 19, 2024

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10

1

+355

Numbers

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?

Lilliam0216 Jun 19, 2024

0

5

4

+355

Socks

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 ..

Lilliam0216 Jun 19, 2024

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8

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+355

Triangles color

Sam wants to color the three sides of an equilateral triangle. He has five 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 read more ..

Lilliam0216 Jun 19, 2024

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25

1

+2653

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?

LiIIiam0216 Jun 19, 2024

0

19

1

+2653

Algebra

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

NotThatSmart ●

LiIIiam0216 Jun 19, 2024

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19

1

+2653

Integers

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

LiIIiam0216 Jun 19, 2024

0

16

1

+2653

Maximum

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

LiIIiam0216 Jun 19, 2024

+1

14

3

+85

Quadratics Problem

Square A and Square B are both 2009 by 2009 squares. Square A has both its length and width increased by an amount x, while Square B has its length and width decreased by the same amount x. What is the minimum value of x such that the difference read more ..

PurpleWasp ●●●

PurpleWasp Jun 19, 2024

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19

2

+52

counting and probability help please

A piece of cord is laid out along the path shown on the left. At each of the six points where the path intersects itself, by random choice the cord can either pass over or under where it has already been laid. One possible combination of choices is shown read more ..

newsss Jun 19, 2024

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12

1

+2653

Minimum

For a real number x, find the minimum value of x^4 - 2x^2 + 3x^4 - 8x^2.

LiIIiam0216 Jun 19, 2024

-1

18

1

+1839

Euclidean Algorithm

Use the Euclidean Algorithm to find gcd(972, 1118).

ABJeIIy Jun 19, 2024

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16

1

+2653

Algebraic Inequality

Find the number of positive integers n that satisfy 2^{200} < n^{1000} < (130n)^{5000}.

LiIIiam0216 Jun 19, 2024

0

11

1

+355

...

Does itself have any square roots among the complex numbers?

Lilliam0216 Jun 19, 2024

0

14

1

+2653

Minimum

For all real numbers x, find the minimum value of (x + 12)^2 - (x + 7)^2 + (x + 13)^2 - (x - 14)^2 + (x - 18)^2

LiIIiam0216 Jun 19, 2024

0

25

1

+2653

Maximum

Let -4 <= x <= -2 and 2 <= y <= 4. Find the largest possible value of x/(x - 2y).

LiIIiam0216 Jun 19, 2024

0

18

1

+2653

Hard Algebra

For 0 <= x <= 1, let f(x) = max{ x^2 - x + 1, 1/2*x^2 + 1/2*x}.Find the minimum value of f(x) for 0 <= x <= 1. Note: For real numbers a, b, c,...,
max {a, b, c, . . . } denotes the maximum read more ..

LiIIiam0216 Jun 19, 2024

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