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How many positive integers $n$ less than 150 have a corresponding integer $m$ not divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers?
|
50
|
The positions of cyclists in the race are determined by the total time across all stages: the first place goes to the cyclist with the shortest total time, and the last place goes to the cyclist with the longest total time. There were 500 cyclists, the race consisted of 15 stages, and no cyclists had the same times either on individual stages or in total across all stages. Vasya finished in seventh place every time. What is the lowest position (i.e., position with the highest number) he could have taken?
|
91
|
Given the function $f(x)= \sqrt {3}\sin 2x+2\cos ^{2}x-1$.
(I) Find the smallest positive period of $f(x)$:
(II) Find the maximum and minimum values of $f(x)$ in the interval $\[- \dfrac {π}{6}, \dfrac {π}{4}\]$.
|
-1
|
The roots of $(x^{2}-3x+2)(x)(x-4)=0$ are:
|
0, 1, 2 and 4
|
Let $x,$ $y,$ and $z$ be real numbers such that
\[x^3 + y^3 + z^3 - 3xyz = 8.\]
Find the minimum value of $x^2 + y^2 + z^2.$
|
\frac{40}{7}
|
Six 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal?
|
\frac{5}{16}
|
In trapezoid EFGH, sides EF and GH are equal. It is known that EF = 12 units and GH = 10 units. Additionally, each of the non-parallel sides forms a right-angled triangle with half of the difference in lengths of EF and GH and a given leg of 6 units. Determine the perimeter of trapezoid EFGH.
|
22 + 2\sqrt{37}
|
In daily life, specific times are usually expressed using the 24-hour clock system. There are a total of 24 time zones globally, with adjacent time zones differing by 1 hour. With the Prime Meridian located in Greenwich, England as the reference point, in areas east of Greenwich, the time difference is marked with a "+", while in areas west of Greenwich, the time difference is marked with a "-". The table below shows the time differences of various cities with respect to Greenwich:
| City | Beijing | New York | Sydney | Moscow |
|--------|---------|----------|--------|--------|
| Time Difference with Greenwich (hours) | +8 | -4 | +11 | +3 |
For example, when it is 12:00 in Greenwich, it is 20:00 in Beijing and 15:00 in Moscow.
$(1)$ What is the time difference between Beijing and New York?
$(2)$ If Xiao Ming in Sydney calls Xiao Liang in New York at 21:00, what time is it in New York?
$(3)$ Xiao Ming takes a direct flight from Beijing to Sydney at 23:00 on October 27th. After 12 hours, he arrives. What is the local time in Sydney when he arrives?
$(4)$ Xiao Hong went on a study tour to Moscow. After arriving in Moscow, he calls his father in Beijing at an exact hour. At that moment, his father's time in Beijing is exactly twice his time in Moscow. What is the specific time in Beijing when the call is connected?
|
10:00
|
A mole has chewed a hole in a carpet in the shape of a rectangle with sides of 10 cm and 4 cm. Find the smallest size of a square patch that can cover this hole (a patch covers the hole if all points of the rectangle lie inside the square or on its boundary).
|
\sqrt{58}
|
What is the number of radians in the smaller angle formed by the hour and minute hands of a clock at 3:40? Express your answer as a decimal rounded to three decimal places.
|
2.278
|
Nathan bought one large box of bananas. He saw that there are six bunches with eight bananas in a bunch and five bunches with seven bananas in a bunch. How many bananas did Nathan have?
|
There were 6 x 8 = <<6*8=48>>48 bananas from the bunches with 8 bananas in a bunch.
There were 5 x 7 = <<5*7=35>>35 bananas from the bunches with 7 bananas in a bunch.
Therefore, Nathan had 48 + 35 = <<48+35=83>>83 bananas.
#### 83
|
Two players, A and B, take turns shooting baskets. The probability of A making a basket on each shot is $\frac{1}{2}$, while the probability of B making a basket is $\frac{1}{3}$. The rules are as follows: A goes first, and if A makes a basket, A continues to shoot; otherwise, B shoots. If B makes a basket, B continues to shoot; otherwise, A shoots. They continue to shoot according to these rules. What is the probability that the fifth shot is taken by player A?
|
\frac{247}{432}
|
Brenda is going from $(-4,5)$ to $(5,-4)$, but she needs to stop by the origin on the way. How far does she have to travel?
|
2\sqrt{41}
|
Let \( x, y, \) and \( z \) be real numbers such that \(\frac{4}{x} + \frac{2}{y} + \frac{1}{z} = 1\). Find the minimum of \( x + 8y + 4z \).
|
64
|
When a polynomial is divided by $2x^2 - 7x + 18,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas.
|
0,1
|
Define the function $g$ on the set of integers such that \[g(n)= \begin{cases} n-4 & \mbox{if } n \geq 2000 \\ g(g(n+6)) & \mbox{if } n < 2000. \end{cases}\] Determine $g(172)$.
|
2000
|
What is the greatest common factor of 120, 180, and 300?
|
60
|
A set of $25$ square blocks is arranged into a $5 \times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column?
|
600
|
How many real numbers $x^{}_{}$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$?
|
159
|
Given a circle $O: x^{2}+y^{2}=4$.<br/>$(1)$ A tangent line is drawn from point $P(2,1)$ to circle $O$, find the equation of the tangent line $l$;<br/>$(2)$ Let $A$ and $B$ be the points where circle $O$ intersects the positive $x$-axis and positive $y$-axis, respectively. A moving point $Q$ satisfies $QA=\sqrt{2}QB$. Is the locus of the moving point $Q$ intersecting circle $O$ at two points? If yes, find the length of the common chord; if not, explain the reason.
|
\frac{8\sqrt{5}}{5}
|
On a quiz, Nicole answered 3 fewer questions correctly than Kim, and Kim answered 8 more questions correctly than Cherry. If Nicole answered 22 correctly, how many did Cherry answer correctly?
|
Kim answered 22+3=<<22+3=25>>25 questions correctly.
Cherry answered 25-8=<<25-8=17>>17 questions correctly.
#### 17
|
Point $A$ has coordinates $(x,6)$. When Point $A$ is reflected over the $y$-axis it lands on Point $B$. What is the sum of the four coordinate values of points $A$ and $B$?
|
12
|
Jason borrowed money from his parents to buy a new surfboard. His parents have agreed to let him work off his debt by babysitting under the following conditions: his first hour of babysitting is worth $\$1$, the second hour worth $\$2$, the third hour $\$3$, the fourth hour $\$4$, the fifth hour $\$5$, the sixth hour $\$6$, the seventh hour $\$1$, the eighth hour $\$2$, etc. If he repays his debt by babysitting for 39 hours, how many dollars did he borrow?
|
\$132
|
The difference between the larger root and the smaller root of $x^2 - px + \frac{p^2 - 1}{4} = 0$ is:
|
1
|
In regular hexagon $ABCDEF$, diagonal $AD$ is drawn. Given that each interior angle of a regular hexagon measures 120 degrees, calculate the measure of angle $DAB$.
|
30
|
Given that positive real numbers \( a \) and \( b \) satisfy \( ab(a+b)=4 \), find the minimum value of \( 2a + b \).
|
2\sqrt{3}
|
A natural number plus 13 is a multiple of 5, and its difference with 13 is a multiple of 6. What is the smallest natural number that satisfies these conditions?
|
37
|
If person A has either a height or weight greater than person B, then person A is considered not inferior to person B. Among 100 young boys, if a person is not inferior to the other 99, he is called an outstanding boy. What is the maximum number of outstanding boys among the 100 boys?
|
100
|
An iterative process is used to find an average of the numbers -1, 0, 5, 10, and 15. Arrange the five numbers in a certain sequence. Find the average of the first two numbers, then the average of the result with the third number, and so on until the fifth number is included. What is the difference between the largest and smallest possible final results of this iterative average process?
|
8.875
|
In the polar coordinate system, curve $C$: $\rho =2a\cos \theta (a > 0)$, line $l$: $\rho \cos \left( \theta -\frac{\pi }{3} \right)=\frac{3}{2}$, $C$ and $l$ have exactly one common point. $O$ is the pole, $A$ and $B$ are two points on $C$, and $\angle AOB=\frac{\pi }{3}$, then the maximum value of $|OA|+|OB|$ is __________.
|
2 \sqrt{3}
|
To stabilize housing prices, a local government decided to build a batch of affordable housing for the community. The plan is to purchase a piece of land for 16 million yuan and build a residential community with 10 buildings on it. Each building has the same number of floors, and each floor has a construction area of 1000 square meters. The construction cost per square meter is related to the floor level, with the cost for the x-th floor being (kx+800) yuan (where k is a constant). After calculation, if each building has 5 floors, the average comprehensive cost per square meter of the community is 1270 yuan.
(The average comprehensive cost per square meter = $$\frac {\text{land purchase cost} + \text{total construction cost}}{\text{total construction area}}$$).
(1) Find the value of k;
(2) To minimize the average comprehensive cost per square meter of the community, how many floors should each of the 10 buildings have? What is the average comprehensive cost per square meter at this time?
|
1225
|
Can you multiply 993 and 879 in your head? Interestingly, if we have two two-digit numbers containing the same number of tens, and the sum of the digits of their units place equals 10, then such numbers can always be multiplied mentally as follows:
Suppose we need to multiply 97 by 93. Multiply 7 by 3 and write down the result, then add 1 to 9 and multiply it by another 9, $9 \times 10=90$. Thus, $97 \times 93=9021$.
This rule turns out to be very useful when squaring numbers ending in 5, for example, $85^{2}=7225$. There is also a simple rule for multiplying two fractions whose whole parts are the same and whose fractional parts sum to one. For example, $7 \frac{1}{4} \times 7 \frac{3}{4} = 56 \frac{3}{16}$. Multiplying the fractional parts, we get $\frac{3}{16}$; adding 1 to 7 and multiplying the result by another 7, we get $7 \times 8 = 56$.
|
872847
|
If 5 times a number is 2, then 100 times the reciprocal of the number is
|
50
|
In a set of 15 different-colored markers, how many ways can Jane select five markers if the order of selection does not matter?
|
3003
|
Suppose that we have a right triangle $ABC$ with the right angle at $B$ such that $AC = \sqrt{61}$ and $AB = 5.$ A circle is drawn with its center on $AB$ such that the circle is tangent to $AC$ and $BC.$ If $P$ is the point where the circle and side $AC$ meet, then what is $CP$?
|
6
|
Given the function \( f(x) = \frac{2+x}{1+x} \), let \( f(1) + f(2) + \cdots + f(1000) = m \) and \( f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + \cdots + f\left(\frac{1}{1000}\right) = n \). What is the value of \( m + n \)?
|
2998.5
|
How can you cut 50 cm from a string that is $2 / 3$ meters long without any measuring tools?
|
50
|
Triangle $DEF$ has vertices $D(0,10)$, $E(4,0)$, $F(10,0)$. A vertical line intersects $DF$ at $P$ and $\overline{EF}$ at $Q$, forming triangle $PQF$. If the area of $\triangle PQF$ is 16, determine the positive difference of the $x$ and $y$ coordinates of point $P$.
|
8\sqrt{2}-10
|
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. For how many values of $n$ is $n+S(n)+S(S(n))=2007$?
|
4
|
Acute-angled $\triangle ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 120^\circ$ and $\stackrel \frown {BC} = 72^\circ$.
A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Then the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$ is:
$\textbf{(A)}\ \frac{5}{18}\qquad \textbf{(B)}\ \frac{2}{9}\qquad \textbf{(C)}\ \frac{1}{4}\qquad \textbf{(D)}\ \frac{1}{3}\qquad \textbf{(E)}\ \frac{4}{9}$
|
\frac{1}{3}
|
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors such that $\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = 0,$ and the angle between $\mathbf{b}$ and $\mathbf{c}$ is $\frac{\pi}{4}.$ Then
\[\mathbf{a} = k (\mathbf{b} \times \mathbf{c})\]for some constant $k.$ Enter all the possible values of $k,$ separated by commas.
|
\sqrt{2}, -\sqrt{2}
|
Cody goes to the store and buys $40 worth of stuff. The taxes were 5%. After taxes, he got an $8 discount. Cody and his friend split the final price equally. How much did Cody pay?
|
The taxes were 40*.05=$<<40*.05=2>>2.
So the price was 40+2=$<<40+2=42>>42.
He got a discount so the price he paid was 42-8=$<<42-8=34>>34.
Since he paid half his price was 34/2=$<<34/2=17>>17.
#### 17
|
Let $T$ be the set of all complex numbers $z$ where $z = w - \frac{1}{w}$ for some complex number $w$ of absolute value $2$. Determine the area inside the curve formed by $T$ in the complex plane.
|
\frac{9}{4} \pi
|
Jillian, Savannah, and Clayton were collecting shells on the beach. Jillian collected 29, Savannah collected 17, and Clayton collected 8. They decided that they wanted to give the shells to two of their friends who had just arrived. They put their shells together and distributed them evenly to each friend. How many shells did each friend get?
|
The kids collected 29 + 17 + 8 = <<29+17+8=54>>54 shells.
If they distribute the shells evenly between 2 friends, 54 / 2 = <<54/2=27>>27 shells for each friend.
#### 27
|
Given an ellipse M: $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ (where $a>b>0$) whose eccentricity is the reciprocal of the eccentricity of the hyperbola $x^{2}-y^{2}=1$, and the major axis of the ellipse is 4.
(1) Find the equation of ellipse M;
(2) If the line $y= \sqrt {2}x+m$ intersects ellipse M at points A and B, and P$(1, \sqrt {2})$ is a point on ellipse M, find the maximum area of $\triangle PAB$.
|
\sqrt {2}
|
Solve the equation \(\sqrt{8x+5} + 2 \{x\} = 2x + 2\). Here, \(\{x\}\) denotes the fractional part of \(x\), i.e., \(\{x\} = x - \lfloor x \rfloor\). Write down the sum of all solutions.
|
0.75
|
If $m, n$ and $p$ are positive integers with $m+\frac{1}{n+\frac{1}{p}}=\frac{17}{3}$, what is the value of $n$?
|
1
|
Jack is a soccer player. He needs to buy two pairs of socks and a pair of soccer shoes. Each pair of socks cost $9.50, and the shoes cost $92. Jack has $40. How much more money does Jack need?
|
The total cost of two pairs of socks is $9.50 x 2 = $<<9.5*2=19>>19.
The total cost of the socks and the shoes is $19 + $92 = $<<19+92=111>>111.
Jack need $111 - $40 = $<<111-40=71>>71 more.
#### 71
|
A shipbuilding company has an annual shipbuilding capacity of 20 ships. The output function of building $x$ ships is $R(x) = 3700x + 45x^2 - 10x^3$ (unit: ten thousand yuan), and the cost function is $C(x) = 460x + 5000$ (unit: ten thousand yuan). In economics, the marginal function $Mf(x)$ of a function $f(x)$ is defined as $Mf(x) = f(x+1) - f(x)$.
(1) Find the profit function $P(x)$ and the marginal profit function $MP(x)$; (Hint: Profit = Output - Cost)
(2) How many ships should be built annually to maximize the company's annual profit?
|
12
|
The price of a bottle of "Komfort" fabric softener used to be 13.70 Ft, and half a capful was needed for 15 liters of water. The new composition of "Komfort" now costs 49 Ft, and 1 capful is needed for 8 liters of water. By what percentage has the price of the fabric softener increased?
|
1240
|
A nine-digit number is formed by repeating a three-digit number three times; for example, $256256256$. Determine the common factor that divides any number of this form exactly.
|
1001001
|
The sum of three numbers $x$ ,$y$, $z$ is 165. When the smallest number $x$ is multiplied by 7, the result is $n$. The value $n$ is obtained by subtracting 9 from the largest number $y$. This number $n$ also results by adding 9 to the third number $z$. What is the product of the three numbers?
|
64,\!328
|
Twelve coworkers go out for lunch together and order three pizzas. Each pizza is cut into eight slices. If each person gets the same number of slices, how many slices will each person get?
|
The total number of slices is 3 × 8 = <<3*8=24>>24.
Every one of the colleagues gets 24 / 12 = <<24/12=2>>2 slices per person.
#### 2
|
In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.
\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200\]What is the median of the numbers in this list?
|
142
|
a) Vanya flips a coin 3 times, and Tanya flips a coin 2 times. What is the probability that Vanya gets more heads than Tanya?
b) Vanya flips a coin $n+1$ times, and Tanya flips a coin $n$ times. What is the probability that Vanya gets more heads than Tanya?
|
\frac{1}{2}
|
My three friends and I have dinner together every weekend. Each weekend, two of us cook and the other two clean up afterwards. How many different ways are there for us to choose who cooks and who cleans?
|
6
|
Sasha has $\$3.20$ in U.S. coins. She has the same number of quarters and nickels. What is the greatest number of quarters she could have?
|
10
|
Find the polynomial $p(x),$ with real coefficients, such that
\[p(x^3) - p(x^3 - 2) = [p(x)]^2 + 12\]for all real numbers $x.$
|
6x^3 - 6
|
Express the following as a common fraction: $\sqrt[3]{4\div 13.5}$.
|
\frac23
|
Rectangle $ABCD$ has $AB=8$ and $BC=6$. Point $M$ is the midpoint of diagonal $\overline{AC}$, and $E$ is on $AB$ with $\overline{ME} \perp \overline{AC}$. What is the area of $\triangle AME$?
|
\frac{75}{8}
|
Two lines touch a circle with center \(O\) at points \(A\) and \(B\) and intersect at point \(C\). Find the angle between these lines if \(\angle ABO = 40^{\circ}\).
|
80
|
Evaluate the polynomial \[ x^3 - 3x^2 - 12x + 9, \] where \(x\) is the positive number such that \(x^2 - 3x - 12 = 0\) and \(x \neq -2\).
|
-23
|
In a $10 \times 5$ grid, an ant starts from point $A$ and can only move right or up along the grid lines but is not allowed to pass through point $C$. How many different paths are there from point $A$ to point $B$?
|
1827
|
Tony's dad is very strict about the washing machine and family members are only allowed to wash 50 total ounces of clothing at a time. Tony doesn't want to break the rules, so he weighs his clothes and finds that a pair of socks weighs 2 ounces, underwear weighs 4 ounces, a shirt weighs 5 ounces, shorts weigh 8 ounces, and pants weigh 10 ounces. Tony is washing a pair of pants, 2 shirts, a pair of shorts, and 3 pairs of socks. How many more pairs of underwear can he add to the wash and not break the rule?
|
He is washing 10 ounces of shirts because 2 x 5 = <<2*5=10>>10
He is washing 6 ounces of socks because 3 x 2 = <<3*2=6>>6
He is already washing 34 ounces of clothes because 10 + 10 + 8 + 6 = <<10+10+8+6=34>>34
He can wash 16 more ounces because 50 - 34 = <<50-34=16>>16
He can wash 4 pairs of underwear because 16 / 4 = <<16/4=4>>4
#### 4
|
Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per blob. The cost of the peanut butter and jam to make all the sandwiches is $\$2.53$. Assume that $B$, $J$, and $N$ are positive integers with $N>1$. What is the cost, in dollars, of the jam Elmo uses to make the sandwiches?
|
\$1.65
|
Meghan had money in the following denomination: 2 $100 bills, 5 $50 bills, and 10 $10 bills. How much money did he have altogether?
|
Total value of $100 bills is 2 × $100 = $<<2*100=200>>200
Total value of $50 bills is 5 × $50 = $<<5*50=250>>250
Total value of $10 bills is 10 × $10 = $<<10*10=100>>100
Final total value is $200 + $250 + $100 = $<<200+250+100=550>>550
#### 550
|
A cube is inscribed in a regular octahedron in such a way that its vertices lie on the edges of the octahedron. By what factor is the surface area of the octahedron greater than the surface area of the inscribed cube?
|
\frac{2\sqrt{3}}{3}
|
Jose invested $\$50,\!000$ for $2$ years at an annual interest rate of $4$ percent compounded yearly. Patricia invested $\$50,\!000$ for the same period of time, at the same interest rate, but the interest was compounded quarterly. To the nearest dollar, how much more money did Patricia's investment earn than that of Jose?
|
63
|
Jordan wants to divide his $\frac{48}{5}$ pounds of chocolate into $4$ piles of equal weight. If he gives one of these piles to his friend Shaina, how many pounds of chocolate will Shaina get?
|
\frac{12}{5}
|
From the set \( \{1, 2, 3, \ldots, 999, 1000\} \), select \( k \) numbers. If among the selected numbers, there are always three numbers that can form the side lengths of a triangle, what is the smallest value of \( k \)? Explain why.
|
16
|
The segments of two lines, enclosed between two parallel planes, are in the ratio of \( 5:9 \), and the acute angles between these lines and one of the planes are in the ratio of \( 2:1 \), respectively. Find the cosine of the smaller angle.
|
0.9
|
Points \( A, B, C, \) and \( D \) lie on a straight line in that order. For a point \( E \) outside the line,
\[ \angle AEB = \angle BEC = \angle CED = 45^\circ. \]
Let \( F \) be the midpoint of segment \( AC \), and \( G \) be the midpoint of segment \( BD \). What is the measure of angle \( FEG \)?
|
90
|
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a 50-cent piece. What is the probability that at least 40 cents worth of coins come up heads?
|
\frac{9}{16}
|
Sara and Joe have a combined height of 120 inches. Joe is 6 inches more than double Sara's height. How tall is Joe?
|
Let x be Sara's height in inches
Joe is 6+2x inches tall
120=x+(6+2x)
120=3x+6
114=3x
x=<<38=38>>38 inches
Joe is 6+2(38)=82 inches
#### 82
|
Let p, q, r, s, and t be distinct integers such that (9-p)(9-q)(9-r)(9-s)(9-t) = -120. Calculate the value of p+q+r+s+t.
|
32
|
On the complex plane, the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ has an area of $\frac{12}{13}.$ If the real part of $z$ is positive, compute the smallest possible value of $\left| z + \frac{1}{z} \right|^2.$
|
\frac{36}{13}
|
While Greg was camping with his family for a week, it rained for 3 days. When he looked at the weather records, he saw that the amount of rain was 3 mm, 6 mm, and 5 mm on the three days. During the same week, it rained 26 mm at his house. How much less rain did Greg experience while camping?
|
While camping there was 3 + 6 + 5 = <<3+6+5=14>>14 mm of rain in 3 days.
Greg experienced 26 – 14 = <<26-14=12>>12 mm less rain at the campsite.
#### 12
|
In the rectangular coordinate system \( xOy \), the equation of ellipse \( C \) is \( \frac{x^2}{9} + \frac{y^2}{10} = 1 \). Let \( F \) and \( A \) be the upper focus and the right vertex of ellipse \( C \), respectively. If \( P \) is a point on ellipse \( C \) located in the first quadrant, find the maximum value of the area of quadrilateral \( OAPF \).
|
\frac{3\sqrt{11}}{2}
|
Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \( (i+1) \) does not immediately follow \( i \) for all \( i \) that runs from 1 to 7.
|
16687
|
Jason is counting the number of cars that drive by his window. He counted four times as many green cars as red cars, and 6 more red cars than purple cars. If he counted 312 cars total, how many of them were purple?
|
Let g be the number of green cars, r be the number of red cars, and p be the number of purple cars. We know that g + r + p = 315, r = p + 6, and g = 4r
Substitute the second equation into the third equation to get: g = 4r = 4(p + 6) = 4p + 24
Now substitute the equations that express g and r in terms of p in the equation for the total number of cars: 4p + 24 + p + 6 + p = 315
Now combine like terms to get 6p + 30 = 312
Now subtract 30 from both sides to get 6p = 282
Now divide both sides by 6 to get p = 47
#### 47
|
We have a cube with 4 blue faces and 2 red faces. What's the probability that when it is rolled, a blue face will be facing up?
|
\frac{2}{3}
|
Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 7$ and $x, y, z \geq 2.$ Find the maximum value of
\[\sqrt{2x + 3} + \sqrt{2y + 3} + \sqrt{2z + 3}.\]
|
\sqrt{69}
|
If $5(\cos a + \cos b) + 4(\cos a \cos b + 1) = 0,$ then find all possible values of
\[\tan \frac{a}{2} \tan \frac{b}{2}.\]Enter all the possible values, separated by commas.
|
3,-3
|
Mark buys a loaf of bread for $4.20 and some cheese for $2.05. He gives the cashier $7.00. If the cashier only has 1 quarter and 1 dime in his till, plus a bunch of nickels, how many nickels does Mark get in his change?
|
First subtract the cost of Mark's groceries from the amount he gives the cashier to find how much he gets in change: $7.00 - $4.20 - $2.05 = $<<7-4.2-2.05=0.75>>0.75
Then subtract the value of a quarter in cents (25) and the value of a dime in cents (10) from the change amount to find how much Mark gets paid in nickels: $0.75 - $0.25 - $0.10 = $<<0.75-0.25-0.10=0.40>>0.40
Now divide the amount Mark gets in nickels by the value per nickel in cents (5) to find how many nickels Mark gets: $0.40 / $0.05/nickel = <<0.40/0.05=8>>8 nickels
#### 8
|
Given the hyperbola $C$: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a > 0, b > 0$) with its left and right foci being $F_1$ and $F_2$ respectively, and $P$ is a point on hyperbola $C$ in the second quadrant. If the line $y=\dfrac{b}{a}x$ is exactly the perpendicular bisector of the segment $PF_2$, then find the eccentricity of the hyperbola $C$.
|
\sqrt{5}
|
Calculate $7 \cdot 9\frac{2}{5}$.
|
65\frac{4}{5}
|
Find the positive solution to
$\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0$
|
13
|
Nathan plays amateur baseball. He played for 3 hours for two weeks, every day. His friend Tobias played for 5 hours every day, but only for one week. How many hours did Nathan and Tobias play in total?
|
Two weeks are 14 days, so Nathan played for 3 * 14 = <<14*3=42>>42 hours.
Tobias played for 7 days, so he played a total of 5 * 7 = <<5*7=35>>35 hours.
Nathan and Tobias played together for 42 + 35 = <<42+35=77>>77 hours.
#### 77
|
Three cards are dealt successively without replacement from a standard deck of 52 cards. What is the probability that the first card is a $\heartsuit$, the second card is a King, and the third card is a $\spadesuit$?
|
\frac{13}{2550}
|
Quadrilateral $ABCD$ has right angles at $A$ and $C$, with diagonal $AC = 5$. If $AB = BC$ and sides $AD$ and $DC$ are of distinct integer lengths, what is the area of quadrilateral $ABCD$? Express your answer in simplest radical form.
|
12.25
|
In the diagram, two pairs of identical isosceles triangles are cut off of square $ABCD$, leaving rectangle $PQRS$. The total area cut off is $200 \text{ m}^2$. What is the length of $PR$, in meters? [asy]
size(5cm);
pair a = (0, 1); pair b = (1, 1); pair c = (1, 0); pair d = (0, 0);
pair s = (0, 0.333); pair p = (0.667, 1); pair q = (1, 0.667); pair r = (0.333, 0);
// Thicken pen
defaultpen(linewidth(1));
// Fill triangles
path tri1 = a--p--s--cycle;
path tri2 = p--q--b--cycle;
path tri3 = q--c--r--cycle;
path tri4 = s--r--d--cycle;
fill(tri1, gray(0.75));fill(tri2, gray(0.75));
fill(tri3, gray(0.75));fill(tri4, gray(0.75));
// Draw rectangles
draw(a--b--c--d--cycle); draw(p--q--r--s--cycle);
// Labels
label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, SE); label("$D$", d, SW);
label("$P$", p, N); label("$Q$", q, E); label("$R$", r, S); label("$S$", s, W);
[/asy]
|
20
|
Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$.)
|
550
|
The rules for a race require that all runners start at $A$, touch any part of the 1200-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. [asy]
import olympiad; import geometry; size(250);
defaultpen(linewidth(0.8));
draw((0,3)--origin--(12,0)--(12,5));
label("300 m",(0,3)--origin,W); label("1200 m",(0,0)--(12,0),S); label("500 m",(12,0)--(12,5),E);
draw((0,3)--(6,0)--(12,5),linetype("3 3")+linewidth(0.7));
label("$A$",(0,3),N); label("$B$",(12,5),N);
[/asy]
|
1442
|
Compute the exact value of the expression $\left|\pi - | \pi - 7 | \right|$. Write your answer using only integers and $\pi$, without any absolute value signs.
|
7-2\pi
|
Given the sequence $\{a\_n\}$, where $a\_1=a\_2=1$ and $a_{n+2}-a_{n}=1$, find the sum of the first 100 terms of the sequence.
|
2550
|
Which of the following words has the largest value, given that the first five letters of the alphabet are assigned the values $A=1, B=2, C=3, D=4, E=5$?
|
BEE
|
Consider the system of equations:
\[
8x - 6y = a,
\]
\[
12y - 18x = b.
\]
If there's a solution $(x, y)$ where both $x$ and $y$ are nonzero, determine $\frac{a}{b}$, assuming $b$ is nonzero.
|
-\frac{4}{9}
|
When Jayson is 10 his dad is four times his age and his mom is 2 years younger than his dad. How old was Jayson's mom when he was born?
|
When Jayson is 10 his dad is 10 x 4 = <<10*4=40>>40 years old.
His mom is 40 - 2 = <<40-2=38>>38 years old.
When Jayson is born, his mom is 38 - 10 = <<38-10=28>>28 years old.
#### 28
|
Given that the terminal side of angle $\alpha$ passes through the point $(3a, 4a)$ ($a < 0$), then $\sin\alpha=$ ______, $\tan(\pi-2\alpha)=$ ______.
|
\frac{24}{7}
|
$2(81+83+85+87+89+91+93+95+97+99)= $
|
1800
|
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