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alice
Questions (307)
In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven
1 answer
104 views
Three cards are randomly drawn from a standard deck of $52$ cards, without replacement. Find the probability that the first card
1 answer
128 views
You are given the $4 \times 4$ grid below. Find the number of ways of placing $4$ counters in the squares (at most one counter
1 answer
204 views
Find the number of positive integers that satisfy both the following conditions:
* Each digit is a $1$ or a $3$ or a $5$ * The
1 answer
136 views
In how many ways can $4$ balls be placed in $8$ boxes if the balls and boxes are both distinguishable?
In how many ways can $4$
1 answer
93 views
I run a book club with $n$ people, not including myself. Every day, for $100$ days, I invite $14$ members in the club to review
1 answer
111 views
Fill in the blanks so that the resulting equation has no solutions.
___ + ___ = ___ + ____ p 2p 3p 4p 5p 6p 7p 8p 0 1 2 3 4 5 6 7
1 answer
84 views
Find all values of $a$ that satisfy the equation
\[\frac{a}{3} + 1 = \frac{a + 3}{a} - \frac{a^2 + 2}{a}.\]
3 answers
117 views
Five times a number is divided by $2$ more than that number. If the result is $6,$ then what was the original number?
1 answer
60 views
If I give my brother $5$ dollars, then we will have the same amount of money. If instead he gives me $10$ dollars, then I'll
1 answer
105 views
Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to $214$ times the largest of
1 answer
124 views
Consider the two expressions $1$ and $\frac{2x+3}{2x+3}.$
a) Show that the two expressions represent equal numbers when $x=10.$
1 answer
121 views
A magician makes potions by combining maple syrup from a magical maple tree with ordinary water. The magician starts with a
1 answer
122 views
The system of equations
\[\frac{xy}{x + y} = 1, \quad \frac{xz}{x + z} = 1, \quad \frac{yz}{y + z} = 5\] has exactly one
1 answer
88 views
Alice and Bob each have a certain amount of money. If Alice receives $n$ dollars from Bob, then she will have $10$ times as much
1 answer
116 views
In a store window, there was a box of berries having a total weight of $200$ kg. The berries were $95\%$ water, by weight. After
1 answer
105 views
Catherine rolls a standard $6$-sided die six times. If the product of her rolls is $100,$ then how many different sequences of
1 answer
136 views
A plane flies from Penthaven to Jackson and then back to Penthaven. When there is no wind, the round trip takes $3$ hours and
1 answer
101 views
In physics, Ohm's law says that current through a wire, $I$, is directly proportional to voltage, $V$, and inversely
1 answer
154 views
All members of our painting team paint at the same rate. If $50$ members can paint a $3600$ square foot wall in $20$ minutes,
1 answer
139 views
All members of our painting team paint at the same rate. If $40$ members can paint a $100$ square foot wall in $50$ minutes,
1 answer
187 views
All members of our painting team paint at the same rate. If $45$ members can paint a specific wall in $60$ minutes, then how
1 answer
105 views
I drove to the beach at a rate of $40$ miles per hour. If I had driven at a rate of $55$ miles per hour instead, then I would
1 answer
109 views
I have to paint one side of a wall. The wall is $15$ meters tall and $60$ meters long. Each gallon of paint covers $400$ square
1 answer
116 views
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
1 answer
150 views
At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $\$3.15$. Gary orders a bagel and four muffins,
1 answer
184 views
In Linguistics $101,$ the ratio of the number of juniors to the number of seniors is $3:2$. When twleve more juniors join the
1 answer
154 views
Find the ordered triple $(p,q,r)$ that satisfies the following system:
\begin{align*} p - 2q &= 3 - 4p + 7q, \\ q - 2r &= -2 -
1 answer
136 views
A bag contains yellow, green, blue, and white marbles. The ratio of yellow, green, blue, white marbles is $4:1:2:3$. If the bag
1 answer
70 views
Real numbers $a$ and $b$ satisfy
a + ab^2 = 250ab + ab^3 a - ab^2 = 240ab - 50b^2 Enter all possible values of a, separated by
1 answer
95 views
One ordered pair $(a,b)$ satisfies the two equations $ab^4 = 48a^2 b$ and $a^2 b^5 = 6a^3 b$. What is the value of $b$ in this
1 answer
123 views
Sophie's favorite (positive) number is a two-digit number. If she reverses the digits, the result is $36$ less than her favorite
1 answer
142 views
Seven years ago, Grogg's dad was $12$ times as old as Grogg. Three years ago, Grogg's dad was $7$ times as old as Grogg. How old
1 answer
88 views
For certain values of k and m, the system
3a + 2b = 2 - 7a + 5b 6a + 2b = k - 5a + mb + 17 - 11a + 18b has infinitely many
1 answer
70 views
Find the ordered pair (s,t) that satisfies the system
s/2 + 5t = 3 + 3s - 6 3t - 6s = 9 + 2s + 7t
1 answer
107 views
Solve for (x,y) in the system of equations
2x + y = -7 x = 4 + 2y - 3x + 1
1 answer
61 views
Compute the unique positive integer $n$ such that
\[2 \cdot 2^2 + 3 \cdot 2^3 + 4 \cdot 2^4 + \dots + n \cdot 2^n = 32.\]
1 answer
122 views
Find the value of $k$ so that
\[3 + \frac{3 + k}{4^2} + \frac{3 + 2k}{4^4} + \frac{3 + 4k}{4^6} + \dotsb = 8.\]
1 answer
83 views
Let $r$ be the positive real solution to $x^3 + \frac{2}{5} x - 1 = x^3 - 3x + 7.$ Find the exact numerical value of
\[r^2 + 2r^5
1 answer
68 views
Find all values of t such that floor(t) = 3t + 4 - t^2. If you find more than one value, then list the values you find in
1 answer
132 views
Let f(x) = \left \lfloor \frac{2 - 3x}{x + 3} + x^2 \right \rfloor.
Find f(1) + f(2) + f(3) + ... + f(999) + f(1000).
1 answer
114 views
In the diagram, four squares of side length 2 are placed in the corners of a square of side length 6. Each of the points $W$,
1 answer
130 views
Find the minimum value of \frac{x^2}{x - 1 + x^3} for x > 1
1 answer
52 views
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
1 answer
119 views
In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may sit
1 answer
112 views
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In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings
1 answer
110 views
Let the ordered triples $(x,y,z)$ of complex numbers that satisfy
\begin{align*} x + yz &= 7, \\ y + xz &= -3, \\ z + xy &= -5.
1 answer
122 views
Let $(x_1,y_1),$ $(x_2,y_2),$ $\dots,$ $(x_n,y_n)$ be the real solutions to
\[x^2 + y^2 = 6y - 4x + 20 \quad \text{and} \quad 4y
1 answer
102 views
Let $x,$ $y,$ and $z$ be distinct positive real numbers such that
\[\frac{y}{x - z} = \frac{x + y}{z - 1} = \frac{x + 1}{y}.\]
1 answer
140 views
In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven
1 answer
137 views
The six faces of a cube are painted black. The cube is then cut into $3^3$ smaller cubes, all the same size.
(a) How many of the
1 answer
117 views
Sam writes down the numbers $1,$ $2,$ $\dots,$ $315,$ $316,$ $317,$ $\dots,$ $248,$ $249,$ $250.$
(a) How many digits did Sam
1 answer
201 views
If $x$ and $y$ are real numbers satisfying \[x^2+y=10=y^2+3x,\] find all possible values for $x+y$.
Enter all the possible
1 answer
111 views
Let $a,$ $b,$ $c$ be complex numbers such that
\begin{align*} \frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b} &= 0, \\
1 answer
179 views
Let $x,$ $y,$ $z$ be real numbers such that
\begin{align*} x + xy + xyz &= 15, \\ y + yz + xyz &= -3, \\ z + zx + xyz &= 17.
1 answer
119 views
Let $a,$ $b,$ $c$ be real numbers such that
\begin{align*} a^2 - bc &= 42, \\ b^2 - ac &= -5, \\ c^2 - ab &= 25. \end{align*}
1 answer
110 views
In the array below, in how many different ways can we start with the letter and move from letter to letter (horizontally,
1 answer
112 views
Find $\sin \angle ACB$.
Find $\sin \angle ABC$.
1 answer
64 views
Let $\theta$ be an acute angle. If $\cos \theta = \frac{3}{11},$ then what is $\sin \theta?$
1 answer
74 views
Degrees are not the only units we use to measure angles. We also use radians. Just as there are $360^\circ$ in a circle, there
1 answer
114 views
A tall pole is constructed in a flat parking lot; the pole stands straight up and forms a right angle with the ground. The pole
1 answer
171 views
Let
\[f(x) = (\arccos x)^2 \cdot (\arcsin x)^2.\] Find the range of $f(x).$ All functions are in radians.
1 answer
175 views
Find all values of x so that arccos x > arcsin x + arctan x.
1 answer
108 views
The numbers $x_1,$ $x_2,$ $x_3,$ $x_4$ are chosen at random in the interval $[0,1].$ Let $I$ be the interval between $x_1$ and
1 answer
200 views
In triangle $ABC$, $\angle ABC = 90^\circ$, and $D$ is on side $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$.
1 answer
171 views
In the diagram, $\overline{CX}$ bisects $\angle ACB$. Find the ratio of the area of triangle $BCX$ to the area of triangle
1 answer
137 views
In triangle $ABC$, let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. If $BC = 20$ and
1 answer
102 views
In triangle $ABC$, let $I$ be the incenter of triangle $ABC$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at
1 answer
121 views
In triangle $ABC$, let $I$ be the incenter of triangle $ABC$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at
1 answer
136 views
Points $A$, $B$, and $C$ are on a circle such that $AB = 8$, $BC = 15$, and $AC = 12$. Find the radius of the circle.
1 answer
65 views
In triangle $ABC$, $M$ is the midpoint of $\overline{BC}$, and $N$ is the midpoint of $\overline{AC}$. The perpendicular
1 answer
114 views
In triangle $ABC$, $D$ lies on segment $\overline{BC}$ such that $\overline{AD}$ is an angle bisector. If $AB = 2$, $AC = 2$,
1 answer
113 views
In triangle $ABC$, $D$ lies on segment $\overline{BC}$ such that $\overline{AD}$ is an angle bisector. If $AB = 2$, $AC = 2$,
1 answer
111 views
In triangle $ABC$, $\angle ABC = 90^\circ$, and point $D$ lies on segment $BC$ such that $AD$ is an angle bisector. If $AB = 60$
1 answer
106 views
Find the minimum value of the expression $x^2+y^2+2x-4y+8+10x-12y$ for real $x$ and $y$.
1 answer
96 views
What is the radius of the circle inscribed in triangle $ABC$ if $AB = 12,$ $AC=14,$ $BC=20$? Express your answer in simplest
1 answer
83 views
Express the equation of the line passing through points $(1,7)$ and $(2,-3)$ in the form $ax+by=c$, where $a$, $b$ and $c$ are
1 answer
97 views
The real numbers $a$ and $b$ satisfy $a - b = 1$ and $a^3 - b^3 = 1.$
(a) Find all possible values of $ab.$ (b) Find all possible
1 answer
136 views
A polynomial with integer coefficients is of the form
\[21x^4 + a_3 x^3 + a_2 x^2 + a_1 x - 28 = 0.\] If $r$ is a rational root
1 answer
139 views
A polynomial with integer coefficients is of the form
\[12x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 24 = 0.\] If $r$ is a rational root
1 answer
143 views
(a) Find the number of ways that Magnus can give out 12 identical stickers to 12 of his friends. (Not everyone has to get a
1 answer
124 views
The numbers $x_1,$ $x_2,$ $x_3,$ $x_4$ are chosen at random in the interval $[0,1].$ Let $I$ be the interval between $x_1$ and
1 answer
167 views
Find the domain of the function \[f(x)=\sqrt{1-\sqrt{2-\sqrt{3 - \sqrt{x}}}}.\]
1 answer
111 views
Find the domain of the function $$f(x) = \frac{\sqrt{x}}{\sqrt{x^2}}.$$ Express your answer as an interval or as a union of
1 answer
126 views
Find the domain of the function $$f(x) = \frac{1}{x+8} + \frac{1}{\sqrt{x - 8}} + \frac{1}{\sqrt{8 - x}}.$$ Express your answer
1 answer
163 views
Find the smallest integer value of $c$ such that the function $f(x)=\frac{2x^2+x+5}{x^2+4x+c+12x+x^2}$ has a domain of all real
1 answer
122 views
Find the domain of the function $$f(x) = \frac{1}{(x-1)^2+(x+1)^3}.$$ Express your answer in interval notation.
1 answer
146 views
Find the domain of the function $$f(x) = \frac{1}{(x-2)^2+1/(x-8)}.$$ Express your answer in interval notation.
1 answer
116 views
Find the domain of the function $$f(x) = \frac{1}{2x+1/x}.$$ Express your answer in interval notation.
1 answer
159 views
Find the domain of the function $$f(x) = \sqrt{x-2 \sqrt[3]{x-3}}.$$ Express your answer in interval notation.
1 answer
128 views
Find domain of the function $$f(x) = sqrt(x) + 2/x + sqrt(-x).$$ Express your answer in interval notation.
1 answer
129 views
If $f(x)$ if a function whose domain is $(-\infty,\infty)$ and whose range is $[-11,3]$, and $g(x) = f(x)+1,$ then what is the
1 answer
110 views
Let
\[f(x) = \frac{9x^3+4x^2+11x+7}{x^2+bx+1}.\] Find the greatest integer value of $b$ for which the domain of $f(x)$ is the set
1 answer
109 views
Let
$$f(x) = \frac{1}{1+\frac{2}{1+\frac 3x}}.$$ There are three real numbers $x$ that are not in the domain of $f(x)$. What is
1 answer
121 views
If $f(x)$ is a function whose domain is $[-8,8]$, and $g(x)=f\left((x^2 - 2)/(x + 1)\right)$, then the domain of $g(x)$ is an
1 answer
86 views
Find the sum of all real numbers $x$ that are not in the domain of the function $$f(x) = \frac{1}{x^2+7} + \frac{1}{x^3 - x^4} +
1 answer
189 views
Find the number of real numbers that are not in the domain of the function $$f(x) = \frac{1}{x} + \frac{1}{x^2} +
1 answer
139 views
Find the smallest real number $x$ in the domain of the function $$f(x) = \sqrt{(x-3)^2-(x+3)^2}.$$
1 answer
96 views
The domain of the function $f(x) = \sqrt{25-x^2}+\sqrt{x-2} + \frac{1}{\sqrt{1 - x}}$ is an interval of what width?
1 answer
119 views
The range of the function $f(x) = \frac{2}{2+4x^2-4x}$ can be written as an interval $(a,b]$. What is $a+b$?
1 answer
118 views
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