All Questions ⁺⁰235714 Questions

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Geometry

Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$. Find the area of triangle $BET$ if $AB = 20$, $AC = 20$, read more ..

blackpanther Dec 23, 2023

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28

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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 $\overline{QN}\perp\overline{PR}$, $QN = 1$, and $PR =1$, read more ..

blackpanther Dec 23, 2023

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31

1

+1911

Inequality

For an integer n, the inequality
x^2 + nx + 15 < -21 - 7x + x^2
has no real solutions in x. Find the number of different possible values of n.

tomtom Dec 23, 2023

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22

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+1911

Inequality

Solve the inequality x^3 + 4x > 5x^2 + 28x + x^3. Write your answer in interval notation.

tomtom Dec 23, 2023

0

32

1

+1911

quadratic

Let $a$ and $b$ be the roots of $x^2 + 7x - 4 = -x^2 + 9x + 6$. Find $(a + 3) + (b + 3).$

tomtom Dec 23, 2023

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21

1

+1911

quadratic

One of the five quadratics below has a repeated root. (The other four have distinct roots.) What is the repeated root?
x^2 - 2x + 1
x^2 - 4x - 4
x^2 + 6x - 9 read more ..

tomtom Dec 23, 2023

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18

1

+4

base number arithmetic

Suppose that $A$, $B$, and $C$ are non-zero distinct digits less than $6$, and suppose we have ${AB_6}+{C_6}={C}0_6$ and ${AB_6}+{BA_6}={CC_6}$. Find the three-digit number ${ABC}$. (Interpret $AB_6$ as a base-6 number with digits $A$ and $B$, not as $A$ read more ..

nouser Dec 23, 2023

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21

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+1678

Average

The entries in a certain row of Pascal's triangle are
\[1, n, \dots, n, 1.\]
The average of the entries in this row is $4/3$. Find $n$.

wiseowl Dec 23, 2023

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36

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+1678

Counting paths

Find the number of paths from $A$ to $B,$ if each step must go towards the right. One example path is shown below.

wiseowl Dec 23, 2023

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31

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+1678

Expected value

The grid below is made up of line segments, like the line segment in red. There are a number of paths that go from A to B in the grid, where every step goes to the right or up. If we choose a line segment at random, then what is the expected number of squares read more ..

wiseowl Dec 23, 2023

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38

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+1678

Math problem

Let PQR be an equilateral triangle, centered at O. A point X is chosen at random inside the triangle. Find the probability that X is closer to O than to any of the sides. (In other words, find the probability that XO is read more ..

wiseowl Dec 23, 2023

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20

1

+1768

Probability

Two points on a circle of radius $1$ are chosen at random. Find the probability that the distance between the two points is at most $1.5.$

bader Dec 23, 2023

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29

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+1768

Probability

The following cards are split into three piles, so that every pile contains the same number of cards. What is the probability that every pile has an Ace?

bader Dec 23, 2023

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40

1

+1768

help help

Joanna has seven beads that she wants to assemble into a bracelet. Five of the beads have the same color, and the other two all have different colors. In how many different ways can Joanna assemble her bracelet? (Two bracelets are considered identical read more ..

bader Dec 23, 2023

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27

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help me plz

Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if all the children are next to each other, and all the adults are next to each other? (Two seatings are considered the same if read more ..

bader Dec 23, 2023

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41

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+1768

help with probability

Raina places the following balls into a bag. She draws three balls out of the bag, one at a time, without replacement. What is the probability that the colors of the balls alternate?
There are 6 orange balls and 6 purple read more ..

bader Dec 23, 2023

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37

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plz help

A standard six-sided die is rolled $7$ times. You are told that among the rolls, there was one $1,$ one $2$, one $3$, one $4$, one $5$, and two $6$s. How many possible sequences of rolls could there have been? (For example, 2, 3, read more ..

bader Dec 23, 2023

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21

1

+1678

Counting

I have $4$ different mathematics textbooks and $3$ different psychology textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row, if there must be a psychology textbook exactly in the middle, and there must be a mathematics textbook read more ..

wiseowl Dec 23, 2023

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29

1

+1678

Counting

A speech class has $4$ freshmen and $3$ sophomores. (Everyone is distinguishable.) In how many ways can they stand in line, so that all the freshmen are standing together?

wiseowl Dec 23, 2023

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31

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+1678

Counting

Each square below 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 squares? read more ..

wiseowl Dec 23, 2023

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43

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+1678

Counting

North High School has a math club and a science club. There are a students in the math club and b students who are members of both clubs. The total number of students is c. Find an expression for the total number of students read more ..

wiseowl Dec 23, 2023

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44

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+1678

Counting

North High School has a math club and a science club. There are a students in the math club and b students who are members of both clubs. The total number of students is c. Find an expression for the total number of students read more ..

wiseowl Dec 23, 2023

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49

0

+1678

Counting

We call a number cozy if every digit in the number is either even or next to an even digit. For example, a number is cozy if every digit in the number is either a 2 or next to a 2. For example, the number 222, 72, 45, and 810 and read more ..

wiseowl Dec 23, 2023

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38

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+1678

Counting

We call a number cozy if every digit in the number is either even or next to an even digit. For example, a number is cozy if every digit in the number is either a 2 or next to a 2. For example, the number 222, 72, 45, and 810 and read more ..

wiseowl Dec 23, 2023

0

33

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+1678

Counting

We call a number cozy if every digit in the number is either even or next to an even digit. For example, a number is cozy if every digit in the number is either a 2 or next to a 2. For example, the number 222, 72, 45, and 810 and read more ..

wiseowl Dec 23, 2023

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