problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Every time she goes to the store, Felicity gets a lollipop. After she finishes them, she uses the sticks to build a fort. The fort needs 400 sticks to finish it. Her family goes to the store three times a week and she always goes. If the fort is 60% complete, how many weeks has Felicity been collecting lollipops for? | She has 240 sticks because 400 x .6 = <<400*.6=240>>240
She has been going to the store for 80 weeks because 240 / 3 = <<240/3=80>>80
#### 80 |
Vann is a veterinarian. Today he is going to be doing dental cleanings only. Dogs have 42 teeth, cats have 30 teeth and pigs have 28 teeth. If he is to do 5 dogs, 10 cats and 7 pigs, how many total teeth will Vann clean today? | First let's find the number for each animal separately, 42 teeth per dog * 5 dogs = <<42*5=210>>210 teeth for dogs.
Then cats total will be 30 teeth per cat * 10 cats = <<30*10=300>>300 teeth in total for cats.
Finally, the number for pigs is 28 teeth per pig * 7 pigs = <<28*7=196>>196 teeth in total for pigs.
So the t... |
The scent of blooming lily of the valley bushes spreads within a radius of 20 meters around them. How many blooming lily of the valley bushes need to be planted along a straight 400-meter-long alley so that every point along the alley can smell the lily of the valley? | 10 |
Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$. | f(x) = \begin{cases}
\frac{2}{2 - x}, & 0 \leq x < 2, \\
0, & x \geq 2.
\end{cases} |
Given positive numbers $x$ and $y$ satisfying $2x+y=2$, the minimum value of $\frac{1}{x}-y$ is achieved when $x=$ ______, and the minimum value is ______. | 2\sqrt{2}-2 |
Jose had 400 tabs opened up in his windows browser. After about one hour of browsing, he closed 1/4 of the tabs to allows easy browsing. He read some news from different pages, then closed 2/5 of the remaining tabs. If he closed half of the remaining tabs after further analyzing some pages, how many windows tabs did he... | When he closed 1/4 of the tabs, he closed 1/4*400 = <<400*1/4=100>>100 tabs.
The number of windows tabs that remained open is 400-100 = <<400-100=300>>300
After reading some news from different pages, he closed 2/5 of the remaining tabs, a total of 2/5*300 = <<2/5*300=120>>120 pages.
The number of tabs that remained op... |
Evaluate $i^6+i^{16}+i^{-26}$. | -1 |
Let $a,$ $b,$ and $c$ be the roots of $x^3 - 7x^2 + 5x + 2 = 0.$ Find
\[\frac{a}{bc + 1} + \frac{b}{ac + 1} + \frac{c}{ab + 1}.\] | \frac{15}{2} |
Adam and Simon start on bicycle trips from the same point at the same time. Adam travels east at 8mph and Simon travels south at 6mph. After how many hours are they 60 miles apart? | 6 |
How many ordered triples \((x, y, z)\) satisfy the following conditions:
\[ x^2 + y^2 + z^2 = 9, \]
\[ x^4 + y^4 + z^4 = 33, \]
\[ xyz = -4? \] | 12 |
Arrange the sequence $\{2n+1\}$ ($n\in\mathbb{N}^*$), sequentially in brackets such that the first bracket contains one number, the second bracket two numbers, the third bracket three numbers, the fourth bracket four numbers, the fifth bracket one number, and so on in a cycle: $(3)$, $(5, 7)$, $(9, 11, 13)$, $(15, 17, ... | 403 |
In how many ways can one arrange the natural numbers from 1 to 9 in a $3 \times 3$ square table so that the sum of the numbers in each row and each column is odd? (Numbers can repeat) | 6 * 4^6 * 5^3 + 9 * 4^4 * 5^5 + 5^9 |
(1) Given the complex number $z=3+bi$ ($i$ is the imaginary unit, $b$ is a positive real number), and $(z-2)^{2}$ is a pure imaginary number, find the complex number $z$;
(2) Given that the sum of all binomial coefficients in the expansion of $(3x+ \frac{1}{ \sqrt{x}})^{n}$ is $16$, find the coefficient of the $x$ term... | 54 |
Given that five boys, A, B, C, D, and E, are randomly assigned to stay in 3 standard rooms (with at most two people per room), calculate the probability that A and B stay in the same standard room. | \frac{1}{5} |
Every year, Tabitha adds a new color to her hair. She started this tradition when she was 15 years old, which was the year she added her second hair color. In three years, Tabitha will have 8 different colors in the hair. Currently, how old is Tabitha? | Since, three years from now, Tabitha will have 8 different colors in her hair, this year she has 8 - 3 = <<8-3=5>>5 colors in her hair.
Since she had 2 colors in her hair at the age of 15, she has added 5-2=3 colors since she was 15 years old.
Since she ads one color per year, 3 added colors = <<3=3>>3 added years
Thus... |
In a regular hexagon \(ABCDEF\), points \(M\) and \(K\) are taken on the diagonals \(AC\) and \(CE\) respectively, such that \(AM : AC = CK : CE = n\). Points \(B, M,\) and \(K\) are collinear. Find \(n\). | \frac{\sqrt{3}}{3} |
Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\overline{AB}$, and the second is tangent to $\overline{DE}$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle? | 81 |
What is the least positive multiple of 45 for which the product of its digits is also a positive multiple of 45? | 945 |
When flipping a fair coin, what is the probability that the first two flips are both heads? Express your answer as a common fraction. | \frac{1}{4} |
The first row of a triangle is given as:
$$
1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{1993}
$$
Each element of the following rows is calculated as the difference between two elements that are above it. The 1993rd row contains only one element. Find this element. | \frac{1}{1993} |
Joe and Adam built a garden wall with three courses of bricks. They realized the wall was too low and added 2 more courses. If each course of the wall had 400 bricks, and they took out half of the bricks in the last course to allow easy checkup of the garden, calculate the total number of bricks the wall has. | When they added two more courses for the wall, the number of courses became 3+2 = <<3+2=5>>5
Since each course has 400 bricks, the total number of bricks for 5 courses is 5*400 = <<5*400=2000>>2000
To allow for an easy checkup of the wall, they removed 1/2*400 = <<1/2*400=200>>200 bricks from the last course of the wal... |
Given the sequence $\{a_n\}$ with the sum of its first $n$ terms $S_n = 6n - n^2$, find the sum of the first $20$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}}\right\}$. | -\frac{4}{35} |
Heath spent his weekend helping at his uncle’s farm planting carrots. He planted 400 rows of carrots with 300 plants in each row. He used a machine to get the planting done and it took him 20 hours. How many carrots did he plant each hour? | Heath planted 400 rows x 300 plants = <<400*300=120000>>120,000 carrot plants.
He planted 120,000 plants in 20 hours, or 120,000 / 20 = <<120000/20=6000>>6,000 plants per hour.
#### 6,000 |
Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$ .
*Proposed by Andy Xu* | 42 |
If $h(x)$ is a function whose domain is $[-8,8]$, and $g(x)=h\left(\frac x2\right)$, then the domain of $g(x)$ is an interval of what width? | 32 |
How many three-digit whole numbers have at least one 7 or at least one 9 as digits? | 452 |
Given $\triangle ABC$ with the sides opposite to angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, and it satisfies $\frac {\sin (2A+B)}{\sin A}=2+2\cos (A+B)$.
(I) Find the value of $\frac {b}{a}$;
(II) If $a=1$ and $c= \sqrt {7}$, find the area of $\triangle ABC$. | \frac { \sqrt {3}}{2} |
How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? | 42 |
There is a card game called "Twelve Months" that is played only during the Chinese New Year. The rules are as follows:
Step 1: Take a brand new deck of playing cards, remove the two jokers and the four Kings, leaving 48 cards. Shuffle the remaining cards.
Step 2: Lay out the shuffled cards face down into 12 columns, ... | 1/12 |
In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\angle C K L = \angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$. | 12 |
Distribute 4 college students to three factories A, B, and C for internship activities. Factory A can only arrange for 1 college student, the other factories must arrange for at least 1 student each, and student A cannot be assigned to factory C. The number of different distribution schemes is ______. | 12 |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 |
Twenty percent less than 60 is one-third more than what number? | 36 |
Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$ . | 13 |
Segment $AB$ has midpoint $C$, and segment $BC$ has midpoint $D$. Semi-circles are constructed with diameters $\overline{AB}$ and $\overline{BC}$ to form the entire region shown. Segment $CP$ splits the region into two sections of equal area. What is the degree measure of angle $ACP$? Express your answer as a decimal t... | 112.5 |
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\] | \[
\boxed{\left(\frac{c^2+c-1}{c^2}, \frac{c}{c-1}, c\right)}
\] |
Consider a right rectangular prism \(B\) with edge lengths \(2,\ 5,\) and \(6\), including its interior. For any real \(r \geq 0\), let \(T(r)\) be the set of points in 3D space within a distance \(r\) from some point in \(B\). The volume of \(T(r)\) is expressed as \(ar^{3} + br^{2} + cr + d\), where \(a,\) \(b,\) \(c... | \frac{8112}{240} |
Calculate
$$
\operatorname{tg} \frac{\pi}{43} \cdot \operatorname{tg} \frac{2 \pi}{43}+\operatorname{tg} \frac{2 \pi}{43} \cdot \operatorname{tg} \frac{3 \pi}{43}+\ldots+\operatorname{tg} \frac{k \pi}{43} \cdot \operatorname{tg} \frac{(k+1) \pi}{43}+\ldots+\operatorname{tg} \frac{2019 \pi}{43} \cdot \operatorname{tg} ... | -2021 |
In the expansion of \((x + y + z)^8\), determine the sum of the coefficients of all terms of the form \(x^2 y^a z^b\) (\(a, b \in \mathbf{N}\)). | 1792 |
At the feline sanctuary, there were 12 lions, 14 tigers, and several cougars. If there were half as many cougars as lions and tigers combined, then what was the total number of big cats at the feline sanctuary? | Half as many cougars as lions and tigers combined is (12+14)/2=13.
Then the total number of big cats at the feline sanctuary is 12+14+13=<<12+14+13=39>>39.
#### 39 |
The graph of the function y=sin(2x+φ) is shifted to the left by π/6 units along the x-axis, resulting in an even function graph. Determine the value of φ such that the equation 2(x + π/6) + φ = -x + 2πk is satisfied for some integer k. | \frac{\pi}{6} |
Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same
list... | 10 |
Trapezoid $A B C D$ is inscribed in the parabola $y=x^{2}$ such that $A=\left(a, a^{2}\right), B=\left(b, b^{2}\right)$, $C=\left(-b, b^{2}\right)$, and $D=\left(-a, a^{2}\right)$ for some positive reals $a, b$ with $a>b$. If $A D+B C=A B+C D$, and $A B=\frac{3}{4}$, what is $a$? | \frac{27}{40} |
A retail store wants to hire 50 new phone reps to assist with the increased call volume that they will experience over the holiday. Each phone rep will work 8 hours a day and will be paid $14.00 an hour. After 5 days, how much will the company pay all 50 new employees? | There are 50 reps and they will work 8 hour days so that's 50*8 = <<50*8=400>>400 hours
They will work 5 days a week so that's 5*400 = <<5*400=2000>>2,000 hours
Each worker will be paid $14.00 an hour and in 5 days they will have worked 2,000 hours so that's 14*2000 = $<<14*2000=28000>>28,000
#### 28000 |
A plane takes off at 6:00 a.m. and flies for 4 hours from New York City to Chicago. The plane stays at the port in Chicago for 1 hour and then departs for Miami. If the aircraft took three times as many hours to fly to Miami than it took to fly from New York to Chicago, calculate the total time to travel from New York ... | Before departing for Miami from Chicago, the total time the plane had taken for the journey is 4+1 = <<4+1=5>>5 hours.
It took 3*4 = <<3*4=12>>12 hours to fly to Miami
The total flying time from New York to Miami is 12+5 = <<12+5=17>>17 hours
#### 17 |
It is known that \( b^{16} - 1 \) has four distinct prime factors. Determine the largest one, denoted by \( c \). | 257 |
Let $n$ be a positive integer. Initially, a $2n \times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed :
(1) If a $2\times 2$ square has exactly three black cells, the fourth is changed to a black cell;
(2) If there are exactly two black cells in a $2 \times 2$ square, t... | n^2 + n + 1 |
For any real number $x$ , we let $\lfloor x \rfloor$ be the unique integer $n$ such that $n \leq x < n+1$ . For example. $\lfloor 31.415 \rfloor = 31$ . Compute \[2020^{2021} - \left\lfloor\frac{2020^{2021}}{2021} \right \rfloor (2021).\]
*2021 CCA Math Bonanza Team Round #3* | 2020 |
A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? | 9 |
The points $(2, 9), (12, 14)$, and $(4, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? | 20 |
You have a rectangular prism box with length $x+5$ units, width $x-5$ units, and height $x^{2}+25$ units. For how many positive integer values of $x$ is the volume of the box less than 700 units? | 1 |
Given $f(\alpha) = \frac{\sin(\frac{\pi}{2} + \alpha) + 3\sin(-\pi - \alpha)}{2\cos(\frac{11\pi}{2} - \alpha) - \cos(5\pi - \alpha)}$.
(I) Simplify $f(\alpha)$;
(II) If $\tan \alpha = 3$, find the value of $f(\alpha)$. | -2 |
Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. | (0,9) |
A boss plans a business meeting at Starbucks with the two engineers below him. However, he fails to set a time, and all three arrive at Starbucks at a random time between 2:00 and 4:00 p.m. When the boss shows up, if both engineers are not already there, he storms out and cancels the meeting. Each engineer is willing t... | \frac{7}{24} |
Jerry has an interesting novel he borrowed from a friend to read over the weekend. The book has 93 pages. On Saturday, he reads 30 pages. On Sunday, he goes to church and comes back, sits down, and reads 20 pages of the book. How many pages are remaining before Jerry can finish the book? | When Jerry reads 30 pages on Saturday, he would be remaining with 93-30 = <<93-30=63>>63 pages.
After reading an additional 20 pages on Sunday, he would be remaining with 63 – 20 = <<63-20=43>>43 pages.
#### 43 |
The rational numbers $x$ and $y$, when written in lowest terms, have denominators 60 and 70 , respectively. What is the smallest possible denominator of $x+y$ ? | 84 |
Derek has $40. He spends $14 on lunch for himself, $11 for lunch for his dad, and $5 on more lunch for himself. His brother Dave has $50 and only spends $7 on lunch for his mom. How much more money does Dave have left than Derek? | Derek has 40-14-11-5 = <<40-14-11-5=10>>10 dollars left
Dave has 50-7 = <<50-7=43>>43 dollars left.
Dave has 43-10 = <<43-10=33>>33 more dollars left than Derek.
#### 33 |
There are five positive integers that are common divisors of each number in the list $$36, 72, -24, 120, 96.$$ Find the sum of these five positive integers. | 16 |
Archie needs to lay sod in his backyard that measures 20 yards by 13 yards. He has a shed on it that measures 3 yards by 5 yards. How many square yards of sod will Archie need for his backyard? | 20 * 13 = <<20*13=260>>260 square yards
3 * 5 = <<3*5=15>>15 square yards
260 - 15 = <<260-15=245>>245 square yards
Archie needs 245 square yards of sod.
#### 245 |
Janet uses her horses' manure as fertilizer. One horse produces 5 gallons of fertilizer per day. Once Janet has collected enough fertilizer, she'll spread it over 20 acres of farmland. Each acre needs 400 gallons of fertilizer and Janet can spread fertilizer over 4 acres per day. If Janet has 80 horses, how long will i... | First find the total amount of fertilizer the horses produce per day: 5 gallons/horse * 80 horses = <<5*80=400>>400 gallons
Then multiply the number of acres of farmland by the number of gallons per acre to find the total amount of fertilizer needed: 400 gallons/acre * 20 acres = <<400*20=8000>>8000 gallons
Then divide... |
Todd has $20. He buys 4 candy bars that cost $2 each. How much money in dollars does Todd have left? | The candy bars cost 4*2= $<<4*2=8>>8.
Todd has 20-8= $<<20-8=12>>12 left.
#### 12 |
The side length of the regular hexagon is 10 cm. What is the number of square centimeters in the area of the shaded region? Express your answer in simplest radical form.
[asy]
size(100);
pair A,B,C,D,E,F;
A = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240); F = dir(300);
fill(B--C--E--F--cycle,heavycya... | 100\sqrt{3} |
Let $Q(x) = 0$ be the polynomial equation of the least possible degree, with rational coefficients, having $\sqrt[4]{13} + \sqrt[4]{169}$ as a root. Compute the product of all of the roots of $Q(x) = 0.$ | -13 |
On each horizontal line in the figure below, the five large dots indicate the populations of cities $A, B, C, D$ and $E$ in the year indicated.
Which city had the greatest percentage increase in population from $1970$ to $1980$? | C |
Given a set of seven positive integers with the unique mode being 6 and the median being 4, find the minimum possible sum of these seven integers. | 26 |
Three different numbers are chosen at random from the list \(1, 3, 5, 7, 9, 11, 13, 15, 17, 19\). The probability that one of them is the mean of the other two is \(p\). What is the value of \(\frac{120}{p}\) ? | 720 |
Compute
\[\frac{\lfloor \sqrt[4]{1} \rfloor \cdot \lfloor \sqrt[4]{3} \rfloor \cdot \lfloor \sqrt[4]{5} \rfloor \dotsm \lfloor \sqrt[4]{2015} \rfloor}{\lfloor \sqrt[4]{2} \rfloor \cdot \lfloor \sqrt[4]{4} \rfloor \cdot \lfloor \sqrt[4]{6} \rfloor \dotsm \lfloor \sqrt[4]{2016} \rfloor}.\] | \frac{5}{16} |
Frank is making hamburgers and he wants to sell them to make $50. Frank is selling each hamburger for $5 and 2 people purchased 4 and another 2 customers purchased 2 hamburgers. How many more hamburgers does Frank need to sell to make $50? | Frank sold 4 hamburgers and then sold 2 more, so all together Frank already sold 4+2= <<4+2=6>>6 hamburgers
If each hamburger is $5 and he sold 6 hamburgers already, Frank has made 5*6= <<5*6=30>>30 dollars.
Frank wants to make $50 and has already made $30, so Frank still needs 50-30= <<50-30=20>>20 dollars.
Since he n... |
In a regular pentagon $PQRST$, what is the measure of $\angle PRS$? | 72^{\circ} |
The company's data entry team had 5 employees working on a large project. Rudy types 64 words per minute, Joyce types 76 words per minute, Gladys types 91 words per minute, Lisa types 80 words per minute and Mike types 89 words per minute. What is the team's average typed words per minute? | Rudy types 64, Joyce types 76, Gladys types 91, Lisa types 80 and Mike types 89 so 64+76+91+80+89 = <<400=400>>400
There are 5 team members, so they type at an average speed of 400/5 = <<400/5=80>>80 words per minute
#### 80 |
Val has three times as many dimes as nickels. If she accidentally finds twice as many nickels as she has in her older brother's treasure box, and takes them for herself, what would be the value of money she has, in dollars, if she had 20 nickels before finding the new ones from her brother's treasure box? | If Val currently has 20 nickels and finds twice as many nickels as she has in her older brother's treasure box, she will have 2*20=40 more nickels.
The total number of nickels she'll have is 40+20=<<40+20=60>>60
Since nickel is worth $0.05, the value of the nickels Val is having is 60*0.05=$<<60*0.05=3>>3
Val had three... |
Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$. | 20 |
3 people run for president. John manages to capture 150 votes. James captures 70% of the remaining vote. If there were 1150 people voting, how many more votes did the third guy get than John? | There were 1150-150=<<1150-150=1000>>1000 people who didn't vote for John
That means James got 1000*.7=<<1000*.7=700>>700 votes
So the other candidate got 1000-700=<<1000-700=300>>300 votes
This means he got 300-150=<<300-150=150>>150 more votes than John.
#### 150 |
Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=5$ while $HQ=2$, then calculate $(BP)(PC)-(AQ)(QC)$. [asy]
size(150); defaultpen(linewidth(0.8));
pair B = (0,0), C = (3,0), A = (2,2), P = foot(A,B,C), Q = foot(B,A,C),H = intersectionpoint(B--Q,A--P);
dr... | 21 |
The cities of Coco da Selva and Quixajuba are connected by a bus line. From Coco da Selva, buses leave for Quixajuba every hour starting at midnight. From Quixajuba, buses leave for Coco da Selva every hour starting at half past midnight. The bus journey takes exactly 5 hours.
If a bus leaves Coco da Selva at noon, ho... | 10 |
What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps? | 6 |
A local music festival is held every year for three days. The three day attendance this year was 2700 people. The second day was rainy so only half the number of people that showed up the first day showed up the second day. The third day was the finale, so attendance was triple the original day. How many people att... | Let x represent the attendance the first day
Second day: x/2
Third day: 3x
Total:x+(x/2)+3x=<<2700=2700>>2700
(9/2)x=2700
9x=5400
x=<<600=600>>600
Second day:600/2=<<600/2=300>>300 people
#### 300 |
A certain intelligence station has four different kinds of passwords $A$, $B$, $C$, and $D$. Each week, one of these passwords is used, and each week a password is chosen uniformly at random from the three passwords that were not used the previous week. Given that password $A$ is used in the first week, what is the pro... | 61/243 |
What is the largest 2-digit prime factor of the integer $n = {180 \choose 90}$? | 59 |
How many distinct $x$-intercepts does the graph of $y = (x-5)(x^2+5x+6)$ have? | 3 |
Compute: $55\times1212-15\times1212$ . | 48480 |
What is the coefficient of $x^8$ in the expansion of $(x-1)^9$? | -9 |
In an isosceles right triangle $ABC$ with $\angle A = 90^{\circ}$ and $AB = AC = 2$, calculate the projection of the vector $\vec{AB}$ in the direction of $\vec{BC}$. | -\sqrt{2} |
During a long voyage of a passenger ship, it was observed that at each dock, a quarter of the passenger composition is renewed, that among the passengers leaving the ship, only one out of ten boarded at the previous dock, and finally, that the ship is always fully loaded.
Determine the proportion of passengers at any ... | 21/40 |
In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$. | 147 |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 5,$ and $\|\mathbf{a} \times \mathbf{b}\| = 8.$ Find $|\mathbf{a} \cdot \mathbf{b}|.$ | 6 |
Borya and Vova play the following game on an initially white $8 \times 8$ board. Borya goes first and, on each of his turns, colors any four white cells black. After each of his turns, Vova colors an entire row or column white. Borya aims to color as many cells black as possible, while Vova tries to hinder him. What is... | 25 |
Annie plants 3 pots of basil, 9 pots of rosemary, and 6 pots of thyme. Each basil plant has 4 leaves, each rosemary plant has 18 leaves, and each thyme plant has 30 leaves. How many leaves are there total? | First find the total number of basil leaves: 3 pots * 4 leaves/pot = <<3*4=12>>12 leaves
Then find the total number of rosemary leaves: 9 pots * 18 leaves/pot = <<9*18=162>>162 leaves
Then find the total number of thyme leaves: 6 pots * 30 leaves/pot = <<6*30=180>>180 leaves
Then add the number of each type of leaf to ... |
In triangle $ABC$, $\angle C=90^\circ$, $AC=6$ and $BC=8$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED=90^\circ$. If $DE=4$, then what is the length of $BD$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
draw(origin--(6,0)--(6,8)--cycle);... | \frac{20}{3} |
On the ellipse $\frac {x^{2}}{3}+ \frac {y^{2}}{2}=1$, the distance from a point P to the left focus is $\frac { \sqrt {3}}{2}$. Find the distance from P to the right directrix. | \frac{9}{2} |
Given that $\cos ( \frac {π}{6}+α) \cdot \cos ( \frac {π}{3}-α)=- \frac {1}{4}$, where $α \in ( \frac {π}{3}, \frac {π}{2})$, find the value of $\sin 2α$ and the value of $\tan α - \frac {1}{\tan α}$. | \frac{2\sqrt{3}}{3} |
A gymnastics team consists of 48 members. To form a square formation, they need to add at least ____ people or remove at least ____ people. | 12 |
In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Y = \frac34$. If $YZ = 30$, then what is $XY$? | 24 |
Nadia walked 18 kilometers, which was twice as far as Hannah walked. How many kilometers did the two girls walk in total? | Hannah = (1/2) 18 = 9
9 + 18 = <<9+18=27>>27 km
Together they walked 27 kilometers.
#### 27 |
Let $a$ and $b$ be the roots of $k(x^2 - x) + x + 5 = 0.$ Let $k_1$ and $k_2$ be the values of $k$ for which $a$ and $b$ satisfy
\[\frac{a}{b} + \frac{b}{a} = \frac{4}{5}.\]Find
\[\frac{k_1}{k_2} + \frac{k_2}{k_1}.\] | 254 |
Jerry can run from his house to his school and back in the time it takes his brother Carson to run to the school. If it takes Jerry 15 minutes to make a one-way trip from his house to his school and the school is 4 miles away, how fast does Carson run in miles per hour? | We know that Carson takes twice as long as Jerry to get to the school, so we can find the time it takes him by multiplying Jerry's time by 2: 15 minutes * 2 = <<15*2=30>>30 minutes
Then convert that time to house by dividing by 60 minutes/hour = 30 minutes / 60 minutes/hour = <<30/60=.5>>.5 hour
Then divide the distanc... |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$ Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\] | 96 |
Given a sequence $\{a_n\}$ satisfying $a_1=81$ and $a_n= \begin{cases} -1+\log_{3}a_{n-1}, & n=2k \\ 3^{a_{n-1}}, & n=2k+1 \end{cases}$ (where $k\in\mathbb{N}^*$), find the maximum value of the sum of the first $n$ terms of the sequence, $S_n$. | 127 |
A sequence \(a_1\), \(a_2\), \(\ldots\) of non-negative integers is defined by the rule \(a_{n+2}=|a_{n+1}-a_n|\) for \(n\geq1\). If \(a_1=1010\), \(a_2<1010\), and \(a_{2023}=0\), how many different values of \(a_2\) are possible? | 399 |
In a bag containing 12 green marbles and 8 purple marbles, Phil draws a marble at random, records its color, replaces it, and repeats this process until he has drawn 10 marbles. What is the probability that exactly five of the marbles he draws are green? Express your answer as a decimal rounded to the nearest thousandt... | 0.201 |
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