problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
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Let \( a \), \( b \), and \( c \) be the roots of the polynomial \( x^3 - 15x^2 + 25x - 10 = 0 \). Compute
\[
(a-b)^2 + (b-c)^2 + (c-a)^2.
\] | 125 |
Laura conducted a survey in her neighborhood about pest awareness. She found that $75.4\%$ of the people surveyed believed that mice caused electrical fires. Of these, $52.3\%$ incorrectly thought that mice commonly carried the Hantavirus. Given that these 31 people were misinformed, how many total people did Laura sur... | 78 |
Ilya has a one-liter bottle filled with freshly squeezed orange juice and a 19-liter empty jug. Ilya pours half of the bottle's contents into the jug, then refills the bottle with half a liter of water and mixes everything thoroughly. He repeats this operation a total of 10 times. Afterward, he pours all that is left i... | 0.05 |
Trace has five shopping bags that weigh the same amount as Gordon’s two shopping bags. One of Gordon’s shopping bags weighs three pounds and the other weighs seven. Trace’s shopping bags all weigh the same amount. How many pounds does one of Trace’s bags weigh? | Gordon’s bags weigh 3 + 7 = <<3+7=10>>10 pounds.
Trace’s five bags all weigh the same amount, so each bag weighs 10 / 5 = <<10/5=2>>2 pounds.
#### 2 |
In soccer, players receive yellow cards when they are cautioned and red cards when they are sent off. Coach Tim has a team of 11 players, 5 of them didn't receive cautions, the rest received one yellow card each. How many red cards would the whole team collect, knowing that each red card corresponds to 2 yellow cards? | In coach Tim's team, 5 out of 11 players did not receive any caution, so 11 - 5 = <<11-5=6>>6 players were cautioned.
Each cautioned player received a yellow card, so all the cautioned players received 6 * 1 =<<6*1=6>>6 yellow cards.
Knowing that each red card corresponds to 2 yellow cards, the team would have 6 / 2 = ... |
Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1\le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is: | n_1n_2 |
A telephone number has the form \text{ABC-DEF-GHIJ}, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive o... | 8 |
Liezl prepared four sets of 2-dozen paper cups for her daughter's birthday party. If 5 cups were damaged and 30 were not used, how many paper cups were used? | Since 1 dozen is equal to 12 then 2 dozens of cups is equal to 2 x 12 = <<2*12=24>>24 cups.
So, Liezl prepared a total of 24 cups x 4 sets = <<24*4=96>>96 paper cups.
A total of 5 + 30 = <<5+30=35>>35 paper cups were not used.
Hence, 96 - 35 = <<96-35=61>>61 paper cups were used.
#### 61 |
25% of oysters have pearls in them. Jamie can collect 16 oysters during each dive. How many dives does he have to make to collect 56 pearls? | First find how many pearls Jamie collects per dive by multiplying the number of oysters by the chance each oyster has a pearl: .25 pearls/oyster * 16 oysters/dive = 4 pearls/dive
Then divide the number of pearls he wants by the number of pearls per dive to find the number of dives: 56 pearls / 4 pearls/dive = <<56/4=14... |
A box has 2 dozen water bottles and half a dozen more apple bottles than water bottles. How many bottles are in the box? | Since a dozen has 12 items, the box has 2 dozen water bottles, a total of 2*12=<<2*12=24>>24 water bottles.
The box also has half a dozen more apple bottles than water bottles, meaning there are 1/2*12= 6 more apple bottles than water bottles.
The total number of apple bottles in the box is 24+6 = <<24+6=30>>30
In the ... |
The temperature in New York in June 2020 was 80 degrees. If the temperature in Miami on this day was 10 degrees hotter than the temperature in New York, and 25 degrees cooler than the temperature in San Diego, what was the average temperature for the three cities? | If it was 10 degrees hotter on this day in Miami than in New York, then the temperature in Miami was 80+10 = 90 degrees.
If it was 25 degrees cooler in Miami than in San Diego, then the temperature in San Diego was 90+25 = <<90+25=115>>115 degrees.
The total temperature for all the cities is 115+90+80 = <<115+90+80=285... |
The graph of \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]has its foci at $(0,\pm 4),$ while the graph of \[\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1\]has its foci at $(\pm 6,0).$ Compute the value of $|ab|.$ | 2 \sqrt{65} |
Given $sin2θ=-\frac{1}{3}$, if $\frac{π}{4}＜θ＜\frac{3π}{4}$, then $\tan \theta =$____. | -3-2\sqrt{2} |
For an integer $n>2$, the tuple $(1, 2, \ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\{1, 2,\ldots, n\}$ can appear on the blackboard i... | n \geq 14 |
Let $w, x, y$, and $z$ be positive real numbers such that $0 \neq \cos w \cos x \cos y \cos z$, $2 \pi =w+x+y+z$, $3 \tan w =k(1+\sec w)$, $4 \tan x =k(1+\sec x)$, $5 \tan y =k(1+\sec y)$, $6 \tan z =k(1+\sec z)$. Find $k$. | \sqrt{19} |
Suppose a number $m$ is randomly selected from the set $\{7, 9, 12, 18, 21\}$, and a number $n$ is randomly selected from $\{2005, 2006, 2007, \ldots, 2025\}$. Calculate the probability that the last digit of $m^n$ is $6$. | \frac{8}{105} |
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 5, 6, 6\}$ if no digit may be used more times than it appears in the given set of available digits? | 43 |
Let $T$ be a triangle with side lengths $1, 1, \sqrt{2}$. Two points are chosen independently at random on the sides of $T$. The probability that the straight-line distance between the points is at least $\dfrac{\sqrt{2}}{2}$ is $\dfrac{d-e\pi}{f}$, where $d$, $e$, and $f$ are positive integers with $\gcd(d,e,f)=1$. Wh... | 17 |
Given a sequence $\{a_{n}\}$ such that $a_{1}+2a_{2}+\cdots +na_{n}=n$, and a sequence $\{b_{n}\}$ such that ${b_{m-1}}+{b_m}=\frac{1}{{{a_m}}}({m∈N,m≥2})$. Find:<br/>
$(1)$ The general formula for $\{a_{n}\}$;<br/>
$(2)$ The sum of the first $20$ terms of $\{b_{n}\}$. | 110 |
The average of five distinct natural numbers is 15, and the median is 18. What is the maximum possible value of the largest number among these five numbers? | 37 |
Find the distance between the vertices of the hyperbola
\[\frac{y^2}{27} - \frac{x^2}{11} = 1.\] | 6 \sqrt{3} |
Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$. Enter all the solutions, separated by commas. | 3 |
A point is randomly dropped onto the interval $[8, 13]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2k-35\right)x^{2}+(3k-9)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. | 0.6 |
At Beaumont High School, there are 12 players on the baseball team. All 12 players are taking at least one of biology or chemistry. If 7 players are taking biology and 2 players are taking both sciences, how many players are taking chemistry? | 7 |
When $1 - i \sqrt{3}$ is converted to the exponential form $re^{i \theta}$, what is $\theta$? | \frac{5\pi}{3} |
Solve in positive integers the following equation:
\[{1\over n^2}-{3\over 2n^3}={1\over m^2}\] | (m, n) = (4, 2) |
Kelvin the Frog is trying to hop across a river. The river has 10 lilypads on it, and he must hop on them in a specific order (the order is unknown to Kelvin). If Kelvin hops to the wrong lilypad at any point, he will be thrown back to the wrong side of the river and will have to start over. Assuming Kelvin is infinite... | 176 |
Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$ | 16 |
68% of all pies are eaten with forks. If there are 2000 pies of all kinds, how many of the pies are not eaten with forks? | If there are 2000 pies of all kinds, and 68% of all pies are eaten with forks, there are 68/100*2000= <<68/100*2000=1360>>1360 pies eaten with forks.
The number of pies that are not eaten with pie is 2000-1360=<<2000-1360=640>>640
#### 640 |
Given that $a, b, c$ are integers with $a b c=60$, and that complex number $\omega \neq 1$ satisfies $\omega^{3}=1$, find the minimum possible value of $\left|a+b \omega+c \omega^{2}\right|$. | \sqrt{3} |
Francie saves up her allowance for several weeks. She receives an allowance of $5 a week for 8 weeks. Then her dad raises her allowance, and she receives $6 a week for 6 weeks. Francie uses half of the money to buy new clothes. With the remaining money, she buys a video game that costs $35. How much money does Francie ... | When her allowance is $5 a week, Francie gets a total of $5 * 8 = $<<5*8=40>>40
When her allowance is $6 a week, Francie gets a total of $6 * 6 = $<<6*6=36>>36
The total amount of money Francie gets is $40 + $36 = $<<40+36=76>>76
After purchasing new clothes, she has $76 / 2 = $<<76/2=38>>38 remaining
After buying the ... |
How many positive two-digit integers are there in which each of the two digits is prime? | 16 |
Given the complex numbers $z\_1=a^2-2-3ai$ and $z\_2=a+(a^2+2)i$, if $z\_1+z\_2$ is a purely imaginary number, determine the value of the real number $a$. | -2 |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? | \dfrac{9}{16} |
Among the following four propositions:
(1) If line $a$ is parallel to line $b$, and line $a$ is parallel to plane $\alpha$, then line $b$ is parallel to plane $\alpha$.
(2) If line $a$ is parallel to plane $\alpha$, and line $b$ is contained in plane $\alpha$, then plane $\alpha$ is parallel to line $b$.
(3) If l... | (4) |
Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter of 6 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles? | 10\text{ inches} |
On a spherical surface with an area of $60\pi$, there are four points $S$, $A$, $B$, and $C$, and $\triangle ABC$ is an equilateral triangle. The distance from the center $O$ of the sphere to the plane $ABC$ is $\sqrt{3}$. If the plane $SAB$ is perpendicular to the plane $ABC$, then the maximum volume of the pyramid $S... | 27 |
In the second year of junior high school this semester, a basketball game was held. In order to better train the participating athletes, the sports department plans to purchase two brands of basketballs, brand A and brand B. It is known that the unit price of brand A basketball is $40$ yuan lower than the unit price of... | m = 20 |
In a game of Fish, R2 and R3 are each holding a positive number of cards so that they are collectively holding a total of 24 cards. Each player gives an integer estimate for the number of cards he is holding, such that each estimate is an integer between $80 \%$ of his actual number of cards and $120 \%$ of his actual ... | 20 |
A company has 200 employees. 60% of the employees drive to work. Of the employees who don't drive to work, half take public transportation. How many more employees drive to work than take public transportation? | Drive to work:200(.60)=<<200*.60=120>>120
Don't Drive to work:200-120=<<200-120=80>>80
Public Transportation:80(.50)=40 employees
80-40=<<80-40=40>>40 employees
#### 40 |
Evaluate $(2-w)(2-w^2)\cdots(2-w^{10})$ where $w=e^{2\pi i/11}.$ | 2047 |
John reads his bible every day. He reads for 2 hours a day and reads at a rate of 50 pages an hour. If the bible is 2800 pages long how many weeks will it take him to read it all? | He reads 2*50=<<2*50=100>>100 pages a day
So he reads 7*100=<<7*100=700>>700 pages a week
So it takes 2800/700=<<2800/700=4>>4 weeks
#### 4 |
Suppose a point P(m, n) is formed by using the numbers m and n obtained from rolling a dice twice as its horizontal and vertical coordinates, respectively. The probability that point P(m, n) falls below the line x+y=4 is \_\_\_\_\_\_. | \frac{1}{12} |
Place the arithmetic operation signs and parentheses between the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ so that the resulting expression equals 100. | 100 |
At peak season, 6 packs of tuna fish are sold per hour, while in a low season 4 tuna packs are sold per hour. If each tuna pack is sold at $60, How much more money is made in a day during a high season than a low season if the fish are sold for 15 hours? | 6 packs of tuna fish are sold per hour. If the sales are made for 15 hours, the total number of packs sold is 4=6*15 = <<6*15=90>>90 packs.
At $60 per pack, during a high season, the total sales in 15 hours are 90*$60 = $<<90*60=5400>>5400
4 packs of tuna fish are sold per hour. If the sales are made for 15 hours, the ... |
Compute the number of positive real numbers $x$ that satisfy $\left(3 \cdot 2^{\left\lfloor\log _{2} x\right\rfloor}-x\right)^{16}=2022 x^{13}$. | 9 |
What is the average of all the integer values of $M$ such that $\frac{M}{56}$ is strictly between $\frac{3}{7}$ and $\frac{1}{4}$? | 19 |
Compute $\tan\left(\frac{\pi}{7}\right)\tan\left(\frac{2\pi}{7}\right)\tan\left(\frac{3\pi}{7}\right)$. | \sqrt{7} |
Given a rectangular grid of dots with 5 rows and 6 columns, determine how many different squares can be formed using these dots as vertices. | 40 |
Given the complex number $z$ satisfies $|z+3-\sqrt{3} i|=\sqrt{3}$, what is the minimum value of $\arg z$? | $\frac{5}{6} \pi$ |
In a certain business district parking lot, the temporary parking fee is charged by time period. The fee is 6 yuan for each car if the parking duration does not exceed 1 hour, and each additional hour (or a fraction thereof is rounded up to a full hour) costs 8 yuan. It is known that two individuals, A and B, parked in... | \frac{1}{4} |
Find the smallest positive real number $d,$ such that for all nonnegative real numbers $x, y,$ and $z,$
\[
\sqrt{xyz} + d |x^2 - y^2 + z^2| \ge \frac{x + y + z}{3}.
\] | \frac{1}{3} |
What is the maximum number of parts into which the coordinate plane \(xOy\) can be divided by the graphs of 100 quadratic polynomials of the form
\[ y = a_{n} x^{2} + b_{n} x + c_{n} \quad (n=1, 2, \ldots, 100) ? \] | 10001 |
The value of $999 + 999$ is | 1998 |
Tree Elementary School is raising money for a new playground. Mrs. Johnson’s class raised $2300, which is twice the amount that Mrs. Sutton’s class raised. Mrs. Sutton’s class raised 8 times less than Miss Rollin’s class. Miss Rollin’s class raised a third of the total amount raised by the school. How much money did th... | Find the amount Mrs. Sutton’s class raised by dividing $2300 by 2. $2300/2 = $<<2300/2=1150>>1150
Find the amount Miss Rollin’s class raised by multiplying $1150 by 8. $1150 x 8 = $<<1150*8=9200>>9200
Multiply $9200 by 3 to find the total amount raised. $9200 x 3 = $<<9200*3=27600>>27600
Convert 2% to decimal. 2/100 = ... |
Here are two functions: $$\begin{array}{ccc}
f(x) & = & 3x^2-2x+ 4\\
g(x) & = & x^2-kx-6
\end{array}$$ If $f(10) - g(10) = 10,$ what is the value of $k?$ | -18 |
Given $$\overrightarrow {m} = (\sin \omega x + \cos \omega x, \sqrt {3} \cos \omega x)$$, $$\overrightarrow {n} = (\cos \omega x - \sin \omega x, 2\sin \omega x)$$ ($\omega > 0$), and the function $f(x) = \overrightarrow {m} \cdot \overrightarrow {n}$, if the distance between two adjacent axes of symmetry of $f(x)$ is ... | \sqrt {3} |
Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than $15$ such that $BC \cdot CD = AB \cdot DA$. What is the largest possible value of $BD$? | \sqrt{\dfrac{425}{2}} |
A phone number \( d_{1} d_{2} d_{3}-d_{4} d_{5} d_{6} d_{7} \) is called "legal" if the number \( d_{1} d_{2} d_{3} \) is equal to \( d_{4} d_{5} d_{6} \) or to \( d_{5} d_{6} d_{7} \). For example, \( 234-2347 \) is a legal phone number. Assume each \( d_{i} \) can be any digit from 0 to 9. How many legal phone numbe... | 19990 |
Arrange the $n$ consecutive positive integers from 1 to $n$ (where $n > 1$) in a sequence such that the sum of each pair of adjacent terms is a perfect square. Find the minimum value of $n$. | 15 |
There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds. | 9 |
Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\{0,1,2\}$. How many equilateral triangles have all their vertices in $S$? | 80 |
A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$.
| 73 |
Suppose $a,$ $b,$ and $c$ are real numbers such that
\[\frac{ac}{a + b} + \frac{ba}{b + c} + \frac{cb}{c + a} = -9\]and
\[\frac{bc}{a + b} + \frac{ca}{b + c} + \frac{ab}{c + a} = 10.\]Compute the value of
\[\frac{b}{a + b} + \frac{c}{b + c} + \frac{a}{c + a}.\] | 11 |
Given $\overrightarrow{a}=(\tan (\theta+ \frac {\pi}{12}),1)$ and $\overrightarrow{b}=(1,-2)$, where $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $\tan (2\theta+ \frac {5\pi}{12})$. | - \frac{1}{7} |
Given positive numbers $a$, $b$, $c$ satisfying: $a^2+ab+ac+bc=6+2\sqrt{5}$, find the minimum value of $3a+b+2c$. | 2\sqrt{10}+2\sqrt{2} |
At a painting club meeting, 7 friends are present. They need to create two separate teams: one team of 4 members, and another team of 2 members for different competitions. How many distinct ways can they form these teams? | 105 |
If $x^{2y}= 4$ and $x = 4$, what is the value of $y$? Express your answer as a common fraction. | \frac{1}{2} |
Using the same Rotokas alphabet, how many license plates of five letters are possible that begin with G, K, or P, end with T, cannot contain R, and have no letters that repeat? | 630 |
The expression $\cos x + \cos 3x + \cos 7x + \cos 9x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\]for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$ | 13 |
What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 9? | 4096 |
Calculate the probability that the line $y=kx+k$ intersects with the circle ${{\left( x-1 \right)}^{2}}+{{y}^{2}}=1$. | \dfrac{1}{3} |
Scenario: In a math activity class, the teacher presented a set of questions and asked the students to explore the pattern by reading the following solution process:
$\sqrt{1+\frac{5}{4}}=\sqrt{\frac{9}{4}}=\sqrt{{(\frac{3}{2})}^{2}}=\frac{3}{2}$;
$\sqrt{1+\frac{7}{9}}=\sqrt{\frac{16}{9}}=\sqrt{{(\frac{4}{3})}^{2}}=\f... | \frac{1012}{1011} |
Heather made four times as many pizzas as Craig made on their first day at work at Mr. Pizza Inns. On their second day, Heather made 20 fewer pizzas than Craig's number. If Craig made 40 pizzas on their first day and 60 more pizzas on their second day than their first day, calculate the total number of pizzas the two m... | Craig made 40 pizzas on their first day at work, while heather made four times that number, a total of 4*40= 160 pizzas.
Together, they made 160+40 = <<160+40=200>>200 pizzas on their first day.
On the second day, Craig made 60 more pizzas than his first day at work, which totals 60+40 = <<60+40=100>>100 pizzas.
On the... |
What is the probability that Fatima gets fewer heads than tails if she flips 10 coins? | \dfrac{193}{512} |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product $abc = 72$. | \frac{1}{24} |
Express the quotient $1021_3 \div 11_3$ in base $3$. | 22_3 |
The entire exterior of a solid $6 \times 6 \times 3$ rectangular prism is painted. Then, the prism is cut into $1 \times 1 \times 1$ cubes. How many of these cubes have no painted faces? | 16 |
Sandy bought 1 million Safe Moon tokens. She has 4 siblings. She wants to keep half of them to herself and divide the remaining tokens among her siblings. After splitting it up, how many more tokens will she have than any of her siblings? | She will keep 1000000 / 2 = <<1000000/2=500000>>500000 Safe Moon tokens for herself.
For the remaining siblings, they will each receive 500000 / 4 = <<500000/4=125000>>125000 tokens.
This means that Sandy will have 500000 - 125000 = <<500000-125000=375000>>375000 tokens more than any of her siblings.
#### 375000 |
Three times the sum of marbles that Atticus, Jensen, and Cruz have is equal to 60. If Atticus has half as many marbles as Jensen, and Atticus has 4 marbles, how many marbles does Cruz have? | Three times the sum of marbles that Atticus, Jensen, and Cruz have is equal to 60, meaning together they have 60/3=<<60/3=20>>20 marbles.
If Atticus has half as many marbles as Jensen, and Atticus has four marbles, Jensen has 2*4=<<4*2=8>>8 marbles.
Together, Atticus and Jensen have 8+4=<<8+4=12>>12 marbles.
Since the ... |
Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{... | 17 |
How many four-digit numbers are there in which at least one digit occurs more than once? | 4464 |
The number of points common to the graphs of
$(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$ is: | 4 |
Given the function $f(x)=2\sin(2x-\frac{\pi}{3})-1$, find the probability that a real number $a$ randomly selected from the interval $\left[0,\frac{\pi}{2}\right]$ satisfies $f(a) > 0$. | \frac{1}{2} |
Define $x\otimes y=x^3-y$. What is $h\otimes (h\otimes h)$? | h |
The tripodasaurus has three legs. In a flock of tripodasauruses, there is a total of 20 heads and legs. How many tripodasauruses are in the flock? | Each animal has 1+3=<<1+3=4>>4 total heads and legs.
Then there are 20/4=<<20/4=5>>5 in the flock.
#### 5 |
Suppose $f$ and $g$ are polynomials, and that $h(x)=f(g(x))+g(x)$. Find the degree of $g(x)$ given that the degree of $h(x)$ is $6$ and the degree of $f(x)$ is $2$. | 3 |
Given the curve
\[
(x - \arcsin \alpha)(x - \arccos \alpha) + (y - \arcsin \alpha)(y + \arccos \alpha) = 0
\]
is intersected by the line \( x = \frac{\pi}{4} \), determine the minimum value of the length of the chord intercepted as \( \alpha \) varies. | \frac{\pi}{2} |
A delicious circular pie with diameter $12\text{ cm}$ is cut into three equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $l^2$? | 108 |
Let $\zeta=e^{2 \pi i / 99}$ and $\omega=e^{2 \pi i / 101}$. The polynomial $$x^{9999}+a_{9998} x^{9998}+\cdots+a_{1} x+a_{0}$$ has roots $\zeta^{m}+\omega^{n}$ for all pairs of integers $(m, n)$ with $0 \leq m<99$ and $0 \leq n<101$. Compute $a_{9799}+a_{9800}+\cdots+a_{9998}$. | 14849-\frac{9999}{200}\binom{200}{99} |
Remi prepared a tomato nursery and planted tomato seedlings. After 20 days, the seedlings were ready to be transferred. On the first day, he planted 200 seedlings on the farm. On the second day, while working alongside his father, he planted twice the number of seedlings he planted on the first day. If the total number... | On the second day, he planted 2 * 200 seedlings = <<2*200=400>>400 seedlings.
The total number of seedlings Remi planted on the two days is 400 seedlings + 200 seedlings = <<400+200=600>>600 seedlings.
If the total number of seedlings transferred from the nursery was 1200 after the second day, Remi's father planted 120... |
Tony lifts weights as a form of exercise. He can lift 90 pounds with one arm in the exercise known as "the curl." In an exercise known as "the military press," he can lift over his head twice the weight that he can curl. His favorite exercise is known as "the squat" and he can squat 5 times the weight that he can li... | If Tony can curl 90 pounds, he can military press 2*90=<<90*2=180>>180 pounds.
Therefore, Tony can squat 5*180=<<5*180=900>>900 pounds.
#### 900 |
A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$ . Heracle defeats a hydra by cutting it into two part... | 10 |
Let $a$ and $b$ be nonzero real numbers. Find the minimum value of
\[a^2 + b^2 + \frac{1}{a^2} + \frac{b}{a}.\] | \sqrt{3} |
Ewan writes out a sequence where he counts by 11s starting at 3. The resulting sequence is $3, 14, 25, 36, \ldots$. What is a number that will appear in Ewan's sequence? | 113 |
Consider a $4 \times 4$ grid of squares. Aziraphale and Crowley play a game on this grid, alternating turns, with Aziraphale going first. On Aziraphales turn, he may color any uncolored square red, and on Crowleys turn, he may color any uncolored square blue. The game ends when all the squares are colored, and Azirapha... | \[ 6 \] |
Find the roots of
\[6x^4 - 35x^3 + 62x^2 - 35x + 6 = 0.\]Enter the roots, separated by commas. | 2, 3, \frac{1}{2}, \frac{1}{3} |
If a rectangle has a width of 42 inches and an area of 1638, how many rectangles of the same size would reach a length of 390 inches? | The rectangle has a length of 1638 in/ 42in = <<1638/42=39>>39 in.
The number of rectangles required to reach 390 inches is 390/39 = <<390/39=10>>10 rectangles.
#### 10 |
Peter Ivanovich, along with 49 other men and 50 women, are seated in a random order around a round table. We call a man satisfied if a woman is sitting next to him. Find:
a) The probability that Peter Ivanovich is satisfied.
b) The expected number of satisfied men. | \frac{1250}{33} |
Consider all ordered pairs of integers $(a, b)$ such that $1 \leq a \leq b \leq 100$ and $$\frac{(a+b)(a+b+1)}{a b}$$ is an integer. Among these pairs, find the one with largest value of $b$. If multiple pairs have this maximal value of $b$, choose the one with largest $a$. For example choose $(3,85)$ over $(2,85)$ ove... | (35,90) |
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