problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. Calculate the probability that the sum of the die rolls is odd. | \frac{7}{16} |
If $\log_6 x=2.5$, the value of $x$ is: | 36\sqrt{6} |
There are 120 crayons in a box. One third of the crayons are new, 20% are broken, and the rest are slightly used. How many are slightly used? | New:120/3=<<120/3=40>>40 crayons
Broken:120(.20)=24 crayons
Slightly Used:120-40-24=<<120-40-24=56>>56 crayons
#### 56 |
Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. What is $\tfrac{AB}{BC}$?
$\textbf{(A) } 1+\sqrt... | 2+\sqrt{5} |
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? | 5\sqrt{2} - 7 |
For breakfast, Daisy bought a muffin for $2 and a cup of coffee for $4. For lunch, Daisy had soup, a salad, and lemonade. The soup cost $3, the salad cost $5.25, and the lemonade cost $0.75. How much more money did Daisy spend on lunch than on breakfast? | For breakfast Daisy spent $2 + $4 = $<<2+4=6>>6.
For lunch Daisy spent $3 + $5.25 + $0.75 = $<<3+5.25+0.75=9>>9.
Daisy spent $9 - $6 = $<<9-6=3>>3 more on lunch than on breakfast.
#### 3 |
Cloud 9 Diving Company has taken individual bookings worth $12,000 and group bookings worth $16,000. Some people have cancelled at the last minute. $1600 has had to be returned to them. How much money has the sky diving company taken altogether? | Cloud 9 earned $12,000 + $16,000 = $<<12000+16000=28000>>28,000 from bookings.
After returning money due to cancellations, they had a final total of $28,000 - $1600 = $<<28000-1600=26400>>26,400.
#### 26400 |
Let a binary operation $\star$ on ordered pairs of integers be defined by $(a,b)\star (c,d)=(a-c,b+d)$. Then, if $(3,3)\star (0,0)$ and $(x,y)\star (3,2)$ represent identical pairs, $x$ equals: | $6$ |
The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars? | 10.00 |
Olaf collects colorful toy cars. At first, his collection consisted of 150 cars. His family, knowing his hobby, decided to give him some toy cars. Grandpa gave Olaf twice as many toy cars as the uncle. Dad gave Olaf 10 toy cars, 5 less than Mum. Auntie gave Olaf 6 toy cars, 1 more than the uncle. How many toy cars does... | Dad gave Olaf 10 toy cars,
Mom has given Olaf 5 more toy cars than Dad, so 10 + 5 = <<10+5=15>>15 toy cars
Auntie gave Olaf 6 toy cars,
Uncle has given 1 less toy than Auntie, so 6 - 1 = <<6-1=5>>5 toy cars
Grandpa gave Olaf 2 * 5 = <<2*5=10>>10 toy cars.
All the family together gave Olaf 10 +15 + 6 + 5 + 10 = <<10+15+... |
An equilateral triangle is placed on top of a square with each side of the square equal to one side of the triangle, forming a pentagon. What percent of the area of the pentagon is the area of the equilateral triangle? | \frac{4\sqrt{3} - 3}{13} \times 100\% |
Janet buys a multi-flavor pack of cheese sticks. 15 of the sticks are cheddar, 30 are mozzarella, and 45 are pepperjack. If Janet picks a cheese stick at random, what is the percentage chance it will be pepperjack? | First find the total number of cheesesticks: 15 cheddar + 30 mozzarella + 45 pepperjack = <<15+30+45=90>>90 cheese sticks
Then divide the number of pepperjack sticks by the total number of cheese sticks and multiply by 100% to express the answer as a percentage: 45 pepperjack sticks / 90 cheese sticks * 100% = 50%
####... |
While shopping, Greg spent 300$ on a shirt and shoes. If Greg spent 9 more than twice as much on shoes as he did a shirt, how much did Greg spend on a shirt? | Let x be the amount spent on a shirt.
Greg spent 2x+9 dollars on shoes.
300=x+(2x+9)
300=3x+9
291=3x
x=<<97=97>>97$
Greg spent 97$ on a shirt.
#### 97 |
A bag of pistachios has 80 pistachios in it. 95 percent have shells, and 75 percent of those have shells that are opened. How many pistachios in the bag have shells and have an opened shell? | Shells:80(.95)=76
Opened Shells:76(.75)=57
#### 57 |
A pet shop has 2 puppies and some kittens. A puppy costs $20, and a kitten costs $15. If the stock is worth $100, how many kittens does the pet shop have? | The 2 puppies cost 2 * 20 = <<2*20=40>>40 dollars
The stock is worth 100 dollars, meaning that the kittens cost 100 - 40 = <<100-40=60>>60 dollars
Since the cost of a kitten is 15 dollars, the pet shop has 60/15 = <<60/15=4>>4 kittens
#### 4 |
Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$. | 231 |
Find the greatest value of $b$ such that $-b^2+7b-10 \ge 0$. | 5 |
Let $x,$ $y,$ and $z$ be angles such that
\begin{align*}
\cos x &= \tan y, \\
\cos y &= \tan z, \\
\cos z &= \tan x.
\end{align*}Find the largest possible value of $\sin x.$ | \frac{\sqrt{5} - 1}{2} |
Given points $a$ and $b$ in the plane, let $a \oplus b$ be the unique point $c$ such that $a b c$ is an equilateral triangle with $a, b, c$ in the clockwise orientation. Solve $(x \oplus(0,0)) \oplus(1,1)=(1,-1)$ for $x$. | \left(\frac{1-\sqrt{3}}{2}, \frac{3-\sqrt{3}}{2}\right) |
Compute the sum of all positive integers $n$ such that $n^{2}-3000$ is a perfect square. | 1872 |
The expression $\cos 2x + \cos 6x + \cos 10x + \cos 14x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\] for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$ | 18 |
Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 1),$ find $n.$ | 12 |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 24 of the sides are congruent. The perimeter of the polygon is 48. Find the area of the polygon. | 48 |
Consider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\frac{z^{}_{}}{40}$ and $\frac{40^{}_{}}{\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$, inclusive. Find the area of $A.$ | 1200 - 200 \pi |
James took a job delivering groceries in his neighborhood. He can carry 10 bags on each trip. If he takes 20 trips a day, how many bags does he deliver in 5 days? | James delivers 10 x 20 = <<10*20=200>>200 bags a day.
So, he can deliver 200 x 5 = <<200*5=1000>>1000 bags in 5 days.
#### 1000 |
If the roots of the quadratic equation $\frac32x^2+11x+c=0$ are $x=\frac{-11\pm\sqrt{7}}{3}$, then what is the value of $c$? | 19 |
From the point $(x, y)$, a legal move is a move to $\left(\frac{x}{3}+u, \frac{y}{3}+v\right)$, where $u$ and $v$ are real numbers such that $u^{2}+v^{2} \leq 1$. What is the area of the set of points that can be reached from $(0,0)$ in a finite number of legal moves? | \frac{9 \pi}{4} |
Two pedestrians departed simultaneously from point A in the same direction. The first pedestrian met a tourist heading towards point A 20 minutes after leaving point A, and the second pedestrian met the tourist 5 minutes after the first pedestrian. The tourist arrived at point A 10 minutes after the second meeting. Fin... | 15/8 |
Simplify $(3-2i)-(5-2i)$. | -2 |
There are 400 students. 120 students take dance as their elective. 200 students take art as their elective. The rest take music. What percentage of students take music? | There are 400-120-200=<<400-120-200=80>>80 students in music.
Thus, students in music make up (80/400)*100=<<80/400*100=20>>20% of the students.
#### 20 |
Alice wants to compare the percentage increase in area when her pizza size increases first from an 8-inch pizza to a 10-inch pizza, and then from the 10-inch pizza to a 14-inch pizza. Calculate the percent increase in area for both size changes. | 96\% |
Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $ | 165 |
Benjie is five years older than Margo. If Benjie is 6 years old, how old will Margo be in three years? | Margo is 6 - 5 = <<6-5=1>>1 year old now.
So, Margo will be 1 + 3 = 4 years old in three years.
#### 4 |
Four siblings inherited a plot of land shaped like a convex quadrilateral. By connecting the midpoints of the opposite sides of the plot, they divided the inheritance into four quadrilaterals. The first three siblings received plots of $360 \, \mathrm{m}^{2}$, $720 \, \mathrm{m}^{2}$, and $900 \, \mathrm{m}^{2}$ respec... | 540 |
Given the sequence $\left\{a_{n}\right\}$ defined by $a_{0} = \frac{1}{2}$ and $a_{n+1} = a_{n} + \frac{a_{n}^{2}}{2023}$ for $n=0,1,2,\ldots$, find the integer $k$ such that $a_{k} < 1 < a_{k+1}$. | 2023 |
Amerigo Vespucci has a map of America drawn on the complex plane. The map does not distort distances. Los Angeles corresponds to $0$ on this complex plane, and Boston corresponds to $2600i$. Meanwhile, Knoxville corresponds to the point $780+1040i$. With these city-point correspondences, how far is it from Knoxville to... | 1300 |
What is the constant term of the expansion of $\left(6x+\dfrac{1}{3x}\right)^6$? | 160 |
Kevin has an elm tree in his yard that is $11\frac{2}{3}$ feet tall and an oak tree that is $17\frac{5}{6}$ feet tall. How much taller is the oak tree than the elm tree? Express your answer as a simplified mixed number. | 6\frac{1}{6}\text{ feet} |
Given two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of 120° between them, find the projection of $\overrightarrow{a} + \overrightarrow{b}$ onto the direction of $\overrightarrow{b}$. | \frac{1}{2} |
Given that $a > 0$, $b > 0$, and $4a - b \geq 2$, find the maximum value of $\frac{1}{a} - \frac{1}{b}$. | \frac{1}{2} |
Let $\mathbf{a} = \begin{pmatrix} -3 \\ 10 \\ 1 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} 5 \\ \pi \\ 0 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} -2 \\ -2 \\ 7 \end{pmatrix}.$ Compute
\[(\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})].\] | 0 |
If one side of a triangle is $12$ inches and the opposite angle is $30^{\circ}$, then the diameter of the circumscribed circle is: | 24 |
Circle $C$ has radius 6 cm. How many square centimeters are in the area of the largest possible inscribed triangle having one side as a diameter of circle $C$? | 36 |
Two 8-sided dice are tossed (each die has faces numbered 1 to 8). What is the probability that the sum of the numbers shown on the dice is either a prime or a multiple of 4? | \frac{39}{64} |
Hendricks buys a guitar for $200, which is 20% less than what Gerald bought the same guitar for. How much did Gerald pay for his guitar? | Let G be the price Gerald paid for his guitar.
Then 0.8 * G = $200
So G = $200 / 0.8 = $<<200/0.8=250>>250
#### 250 |
A certain kind of wild mushroom has either spots or gills, but never both. Gilled mushrooms are rare, with only one growing for every nine spotted mushrooms that grow. A fallen log had 30 mushrooms growing on its side. How many gilled mushrooms were on the fallen log’s side? | There is 1 gilled mushroom for every 1 + 9 = <<1+9=10>>10 mushrooms.
Thus, there were 30 / 10 = <<30/10=3>>3 gilled mushrooms on the fallen log’s side.
#### 3 |
The blue parabola shown is the graph of the equation \( x = ay^2 + by + c \). The vertex of the parabola is at \( (5, 3) \), and it passes through the point \( (3, 5) \). Find \( c \). | \frac{1}{2} |
Let \[f(x) = \left\{
\begin{array}{cl}
-x + 3 & \text{if } x \le 0, \\
2x - 5 & \text{if } x > 0.
\end{array}
\right.\]How many solutions does the equation $f(f(x)) = 4$ have? | 3 |
In the expression \(5 * 4 * 3 * 2 * 1 = 0\), replace the asterisks with arithmetic operators \(+, -, \times, \div\), using each operator exactly once, so that the equality holds true (note: \(2 + 2 \times 2 = 6\)). | 5 - 4 \times 3 : 2 + 1 |
There were 80 cars in a parking lot. At lunchtime, 13 cars left the parking lot but 5 more cars went in than left. How many cars are in the parking lot now? | There are 80 - 13 = <<80-13=67>>67 cars in the parking lot after 13 went out.
13 + 5 = <<13+5=18>>18 cars went in the parking lot at lunchtime.
So, there are 67 + 18 = <<67+18=85>>85 cars in the parking lot now.
#### 85 |
A wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into sixteen 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number?
[asy]
size(4cm,4cm);
pair A,B,C,D,E,F,G,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r;
A=(0.5,0... | 8 |
Dana goes first and Carl's coin lands heads with probability $\frac{2}{7}$, and Dana's coin lands heads with probability $\frac{3}{8}$. Find the probability that Carl wins the game. | \frac{10}{31} |
Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$. | 6 |
Evaluate $2000^3-1999\cdot 2000^2-1999^2\cdot 2000+1999^3$ | 3999 |
Two integers are relatively prime if they have no common factors other than 1 or -1. What is the probability that a positive integer less than or equal to 30 is relatively prime to 30? Express your answer as a common fraction. | \frac{4}{15} |
Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king? | \frac{17}{884} |
Five friends eat at Wendy's and ordered the following: a platter of Taco Salad that cost $10, 5 orders of Dave's Single hamburger that cost $5 each, 4 sets of french fries that cost $2.50, and 5 cups of peach lemonade that cost $2 each. How much will each of them pay if they will split the bill equally? | The cost of 5 pieces of Dave's Single hamburger is $5 x 5 = $<<5*5=25>>25.
The cost of 4 sets of french fries is $2.50 x 4 = $<<2.5*4=10>>10.
The cost of 5 cups of peach lemonade is $2 x 5 = $<<2*5=10>>10.
So, their total bill is $10 + $25 + $10 +$10 = $<<10+25+10+10=55>>55.
Therefore, each of them will contribute $55/... |
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | -16 |
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian must occupy chair $1$ and an Earthl... | 346 |
Calculate $\sum_{n=1}^{2001} n^{3}$. | 4012013006001 |
Let $P(x) = 3\sqrt[3]{x}$, and $Q(x) = x^3$. Determine $P(Q(P(Q(P(Q(4))))))$. | 108 |
How many digits are located to the right of the decimal point when $\frac{3^6}{6^4\cdot625}$ is expressed as a decimal? | 4 |
Given that in $\triangle ABC$, $\sin A + 2 \sin B \cos C = 0$, find the maximum value of $\tan A$. | \frac{\sqrt{3}}{3} |
Last week, the price of a movie ticket was $100. This year the price is down by 20%. What's the new price of the movie ticket? | The price of the book is down by: 100*0.2 = $<<100*0.2=20>>20.
So the new price of the movie ticket is: 100 - 20 = $<<100-20=80>>80.
#### 80 |
In triangle $\triangle ABC$, point $G$ satisfies $\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\overrightarrow{0}$. The line passing through $G$ intersects $AB$ and $AC$ at points $M$ and $N$ respectively. If $\overrightarrow{AM}=m\overrightarrow{AB}$ $(m>0)$ and $\overrightarrow{AN}=n\overrightarrow{AC}... | \frac{4}{3}+\frac{2\sqrt{3}}{3} |
Becky has 50 necklaces in her jewelry collection. 3 of the necklaces have broken beads so she collects the unbroken beads for crafting and throws the other parts of the 3 the necklaces out. Becky buys 5 new necklaces that week. She decides to give 15 of her old necklaces to her friends as gifts. How many necklaces does... | Becky started with 50 necklaces and then got rid of 3, 50 - 3 = <<50-3=47>>47 necklaces.
Becky buys 5 new necklaces that week, 47 + 5 = <<47+5=52>>52 necklaces.
She gives 17 of her necklaces as gifts to friends, 52 - 15 = <<37=37>>37 necklaces Becky owns now.
#### 37 |
Mary has 400 sheep on her farm. She gave a quarter of her sheep to her sister, and half of the remaining sheep to her brother. How many sheep remain with Mary? | Mary gives 1/4 * 400 = <<400*1/4=100>>100 sheep to her sister.
She now has 400 - 100 = <<400-100=300>>300 sheep remaining.
She then gives 1/2 * 300 = <<300/2=150>>150 sheep to her brother.
So 300 - 150 = <<300-150=150>>150 sheep remain.
#### 150 |
David, Ellie, Natasha, and Lucy are tutors in their school science lab. Their working schedule is as follows: David works every fourth school day, Ellie works every fifth school day, Natasha works every sixth school day, and Lucy works every eighth school day. Today, they all happened to be working together. In how man... | 120 |
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of th... | \frac{37}{35} |
A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?
$\textbf{(A)}\ \sqrt{13}\qquad \textbf{(B)}\ \sqrt{14}\qquad \textbf{(C)}\ \sqrt{15}\qquad \textbf{(D)}\ \sqrt{16}... | \sqrt{13} |
Let $A B C$ be an acute triangle with $A$-excircle $\Gamma$. Let the line through $A$ perpendicular to $B C$ intersect $B C$ at $D$ and intersect $\Gamma$ at $E$ and $F$. Suppose that $A D=D E=E F$. If the maximum value of $\sin B$ can be expressed as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a, b$, and $c$,... | 705 |
A trapezoid has one base equal to twice its height, $x$, and the other base is three times as long as the height. Write the expression for the area of the trapezoid as a common fraction in terms of the height $x$. | \dfrac{5x^2}{2} |
In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\angle A = 37^\circ$, $\angle D = 53^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$. | 504 |
For a complex number $z \neq 3$ , $4$ , let $F(z)$ denote the real part of $\tfrac{1}{(3-z)(4-z)}$ . If \[
\int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn
\] for relatively prime positive integers $m$ and $n$ , find $100m+n$ .
*Proposed by Evan Chen* | 100 |
If a class of 30 students is seated in a movie theater, then in any case at least two classmates will be in the same row. If the same is done with a class of 26 students, then at least three rows will be empty. How many rows are in the theater?
| 29 |
The union of sets \( A \) and \( B \) is \( A \cup B = \left\{a_{1}, a_{2}, a_{3}\right\} \). When \( A \neq B \), the pairs \((A, B)\) and \((B, A)\) are considered different. How many such pairs \((A, B)\) are there? | 27 |
Xiao Zhang departs from point A to point B at 8:00 AM, traveling at a speed of 60 km/h. At 9:00 AM, Xiao Wang departs from point B to point A. After arriving at point B, Xiao Zhang immediately returns along the same route and arrives at point A at 12:00 PM, at the same time as Xiao Wang. How many kilometers from point ... | 96 |
In the diagram below, points $A$, $B$, $C$, and $P$ are situated so that $PA=2$, $PB=3$, $PC=4$, and $BC=5$. What is the maximum possible area of $\triangle ABC$? [asy]
defaultpen(linewidth(0.8)); size(150);
pair B = (0,0), C = (5,0), A = (2,3), P = (2.2,2);
draw(A--B--C--cycle^^B--P^^C--P^^A--P);
label("$A$",A,N); lab... | 11 |
Given that Liliane has $30\%$ more cookies than Jasmine and Oliver has $10\%$ less cookies than Jasmine, and the total number of cookies in the group is $120$, calculate the percentage by which Liliane has more cookies than Oliver. | 44.44\% |
In $\triangle ABC$, it is given that $\cos A= \frac{5}{13}$, $\tan \frac{B}{2}+\cot \frac{B}{2}= \frac{10}{3}$, and $c=21$.
1. Find the value of $\cos (A-B)$;
2. Find the area of $\triangle ABC$. | 126 |
Given that the positive numbers $x$ and $y$ satisfy the equation $$3x+y+ \frac {1}{x}+ \frac {2}{y}= \frac {13}{2}$$, find the minimum value of $$x- \frac {1}{y}$$. | - \frac {1}{2} |
Consider a square where each side measures 1 unit. At each vertex of the square, a quarter circle is drawn outward such that each side of the square serves as the radius for two adjoining quarter circles. Calculate the total perimeter formed by these quarter circles. | 2\pi |
Find the smallest positive real number $c,$ such that for all nonnegative real numbers $x$ and $y,$
\[\sqrt{xy} + c |x - y| \ge \frac{x + y}{2}.\] | \frac{1}{2} |
Jerry is making cherry syrup. He needs 500 cherries per quart of syrup. It takes him 2 hours to pick 300 cherries and 3 hours to make the syrup. How long will it take him to make 9 quarts of syrup? | First find how many cherries Jerry can pick in one hour: 300 cherries / 2 hours = <<300/2=150>>150 cherries/hour
Then multiply the number of quarts of syrup by the number of cherries per quart to find the total number of quarts Jerry needs: 500 cherries/quart * 9 quarts = 4500 cherries
Then divide the total number of c... |
A necklace has a total of 99 beads. Among them, the first bead is white, the 2nd and 3rd beads are red, the 4th bead is white, the 5th, 6th, 7th, and 8th beads are red, the 9th bead is white, and so on. Determine the total number of red beads on this necklace. | 90 |
Inside the cube \( ABCD A_1 B_1 C_1 D_1 \) is located the center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( A A_1 D_1 D \) in a circle with a radius of 1, the face \( A_1 B_1 C_1 D_1 \) in a circle with a radius of 1, and the face \( C D D_1 C_1 \) in a circle with a radius of 3. Find th... | 17 |
How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? | 60 |
The cost of five pencils and one pen is $\$2.50$, and the cost of one pencil and two pens is $\$1.85$. What is the cost of two pencils and one pen? | 1.45 |
Given the product sequence $\frac{5}{3} \cdot \frac{6}{5} \cdot \frac{7}{6} \cdot \ldots \cdot \frac{a}{b} = 12$, determine the sum of $a$ and $b$. | 71 |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 |
Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point ... | 73 |
Let $\mathbf{a} = \begin{pmatrix} 7 \\ -4 \\ -4 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} -2 \\ -1 \\ 2 \end{pmatrix}.$ Find the vector $\mathbf{b}$ such that $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are collinear, and $\mathbf{b}$ bisects the angle between $\mathbf{a}$ and $\mathbf{c}.$
[asy]
unitsize(0.5... | \begin{pmatrix} 1/4 \\ -7/4 \\ 1/2 \end{pmatrix} |
If six people decide to come to a basketball game, but three of them are only 2/5 sure that they will stay for the entire time (the other three are sure they'll stay the whole time), what is the probability that at the end, at least 5 people stayed the entire time? | \frac{44}{125} |
A company offers its employees a salary increase, provided they increase their work productivity by 2% per week. If the company operates 5 days a week, by what percentage per day must employees increase their productivity to achieve the desired goal? | 0.4 |
The sum of sides \( AB \) and \( BC \) of triangle \( ABC \) is 11, angle \( B \) is \( 60^\circ \), and the radius of the inscribed circle is \(\frac{2}{\sqrt{3}}\). It is also known that side \( AB \) is longer than side \( BC \). Find the height of the triangle dropped from vertex \( A \). | 4\sqrt{3} |
The first tank is 300 liters filled while the second tank is 450 liters filled. The second tank is only 45% filled. If the two tanks have the same capacity, how many more liters of water are needed to fill the two tanks? | Since 450 liters represents 45%, then each 1% is equal to 450 liters/45 = <<450/45=10>>10 liters.
Hence, the capacity of each tank is 10 liters x 100 = <<10*100=1000>>1000 liters.
So, the first tank needs 1000 - 300 = <<1000-300=700>>700 liters of water more.
While the second tank needs 1000 - 450 = <<1000-450=550>>550... |
From the following infinite list of numbers, how many are integers? $$\sqrt{4096},\sqrt[3]{4096},\sqrt[4]{4096},\sqrt[5]{4096},\sqrt[6]{4096},\ldots$$ | 5 |
A sphere is inscribed in a cone whose axial cross-section is an equilateral triangle. Find the volume of the cone if the volume of the sphere is \( \frac{32\pi}{3} \ \text{cm}^3 \). | 24 \pi |
Jimmy decides to make sandwiches for a picnic. He makes 8 sandwiches in total, using two slices of bread each. How many packs of bread does he need to buy to make these sandwiches, assuming he starts with no bread and each pack has 4 slices of bread in it? | First, we need to determine how many slices of bread Jimmy will use to make the sandwiches. We determine this by performing 8*2=<<8*2=16>>16 slices of bread needed.
We then divide the number of slices needed by the number of slices per bread pack, performing 16/4=<<16/4=4>>4 packs of bread needed.
#### 4 |
A circle is circumscribed around a unit square \(ABCD\), and a point \(M\) is selected on the circle.
What is the maximum value that the product \(MA \cdot MB \cdot MC \cdot MD\) can take? | 0.5 |
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