All Questions ⁺⁰236027 Questions

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Tan 2θ (Theta) = 3/4 Without A Calculator

Hello! Me and my teacher both discussed the problem and we don't know how to solve it so I came to this forum.
The question is
The only catch is that it has to be solved without a calculator using trig rules discussed in read more ..

ryme12 May 22, 2024

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22

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+2653

Algebra

Let $f(x) = x^2 + bx + 12 - 3x + 10$ for all real numbers $x$. Find the greatest integer value of $b$ such that $-4$ is not in the range of $f(x)$.

LiIIiam0216 May 22, 2024

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30

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+2653

Algebra

Let $f(x)$ be a polynomial with integer coefficients. There exist distinct integers $p,$ $q,$ $r,$ $s,$ $t$ such that
\[f(p) = f(q) = f(r) = f(s) = 1\]
and $f(t) > 1.$ What is the smallest possible value of $f(t)?$

LiIIiam0216 May 22, 2024

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23

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+2653

Geometry

The circles $x^2 + y^2 = 4$ and $(x - 5)^2 + (y - 8)^2 = 60$ intersect in two points $A$ and $B.$ Find the distance $AB$.

LiIIiam0216 May 22, 2024

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20

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+1310

Algebra

Solve the inequality
\frac{3-z}{z+1} \ge 2(z + 4).
Write your answer in interval notation.

learnmgcat May 22, 2024

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22

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+1310

Algebra

Solve the inequality 4t^2 \le -9t + 12 - 4t + 23 + (2t - 1)(2t + 1). Write your answer in interval notation.

learnmgcat May 22, 2024

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20

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+1310

Algebra

Solve the inequality x(x + 6) > 16 - x + 14 + x^2. Write your answer in interval notation.

learnmgcat May 22, 2024

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22

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+1310

Algebra

Solve the inequality (x - 2)(x + 6) \le (x - 2)(x + 5). Write your answer in interval notation.

learnmgcat May 22, 2024

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6

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+27

Expected Value

A standard deck of cards contains 52 cards. These 52 cards are arranged in a circle, at random. Find the expected number of pairs of adjacent cards that are both hearts.
Please help quickly!
Any help is appreciated!

colinqiu May 22, 2024

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20

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+2653

Geometry

Triangle $ABC$ has altitudes $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$ If $AD = 12,$ $BE = 20,$ and $CF$ is a positive integer, then find the largest possible value of $CF.$

LiIIiam0216 May 22, 2024

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27

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+2653

Geometry

There are four cities: Capital City, Little Village, Mytown, and Yourtown. The distance from Capital City to Little Village is $5$ miles. The distance from Capital City to Mytown is $2$ miles. The distance from Mytown to Yourtown is $3$ read more ..

LiIIiam0216 May 22, 2024

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28

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+2653

Geometry

All the sides of a triangle are integers. If the perimeter of the triangle is $3,$ then how many different possible triangles are there? (Assume that the triangle is non-degenerate. Two triangles are considered the same if they are co read more ..

LiIIiam0216 May 22, 2024

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11

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Casework

A standard deck of playing cards has 52 cards divided into four suits (clubs, diamonds, hearts, spades). Each suit consists of nine "number cards," each containing a different number from 2 to 10 , and four "face cards" that include a jack, a queen, read more ..

eee333 May 22, 2024

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18

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+2653

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 $ABD$.

LiIIiam0216 May 22, 2024

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21

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+2653

Geometry

A quadrilateral has angles of $q$, $4q + 17^\circ $, $8q + 21^\circ$, and $15q + 148^\circ $. Find the largest angle of the quadrilateral, in degrees.

LiIIiam0216 May 22, 2024

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31

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+2653

Geometry

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.

LiIIiam0216 May 22, 2024

May 21, 2024

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34

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+2653

Geometry

In right triangle $ABC,$ $\angle C = 90^\circ$. Median $\overline{AM}$ has a length of 1, and median $\overline{BN}$ has a length of 1. What is the length of the hypotenuse of the triangle?

LiIIiam0216 May 21, 2024

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20

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+2653

Geometry

In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

LiIIiam0216 May 21, 2024

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36

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+2653

Geometry

In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $CE =3$ and $DF = 3$, then what is $BD$?

LiIIiam0216 May 21, 2024

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26

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+2653

Geometry

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

LiIIiam0216 May 21, 2024

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29

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+2653

Geometry

Triangle ABC is rotated completely around side BC, sweeping out a solid in space. Find the volume of the solid.
AB = 5, AC = 5, CB = 5

LiIIiam0216 May 21, 2024

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15

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+27

CPhill, Please!

What is the sum of all positive integers "q" such that (n^7 - n^3)/q is an integer for every positive integer n < 2021?
Thanks in advance!

colinqiu May 21, 2024

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22

1

+355

Binomial's

For nonnnegative integers a and b, where a >= 2 and b >= 2, simplify
binom{a + b}{2} + binom{2a + b}{2} + binom{a + 2b}{2} - binom{a - b}{2}

Lilliam0216 May 21, 2024

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23

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+868

Counting

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?

eramsby1O1O May 21, 2024

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