problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Let $x$ and $y$ be two positive real numbers such that $x + y = 35.$ Enter the ordered pair $(x,y)$ for which $x^5 y^2$ is maximized. | (25,10) |
Chris is trying to sell his car for $5200 and has gotten two price offers. One buyer offered to pay the full price if Chris would pay for the car maintenance inspection, which cost a tenth of Chris’s asking price. The other buyer agreed to pay the price if Chris replaced the headlights for $80 and the tires for three t... | Chris would earn 5200 - 5200 / 10 = 5200 - 520 = $<<5200-5200/10=4680>>4680 from the first buyer.
He would earn 5200 - 80 - 80 * 3 = 5200 - 80 * 4 = 5200 - 320 = $<<5200-80-80*3=4880>>4880 from the second buyer.
The difference between Chris’s earnings would be 4880 - 4680 = $<<4880-4680=200>>200.
#### 200 |
The digits $1,2,3,4,5,6$ are randomly chosen (without replacement) to form the three-digit numbers $M=\overline{A B C}$ and $N=\overline{D E F}$. For example, we could have $M=413$ and $N=256$. Find the expected value of $M \cdot N$. | 143745 |
If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 30 |
Tommy is making 12 loaves of bread. He needs 4 pounds of flour per loaf. A 10-pound bag of flour costs $10 and a 12-pound bag costs $13. When he is done making his bread, he has no use for flour and so he will throw away whatever is left. How much does he spend on flour if he buys the cheapest flour to get enough? | He needs 48 pounds of flour because 12 x 4 = <<12*4=48>>48
He needs 4.8 bags of 10 pound bags because 48 / 10 = <<48/10=4.8>>4.8
He therefore needs to buy 5 bags of flour because 4 < 4.8 < 5
He spends $50 if he buys this flour because 5 x 10 = <<5*10=50>>50
He needs 4 bags of the 12-pound flour because 48 / 12 = <<48/1... |
1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996= | 0 |
We call a positive integer $t$ good if there is a sequence $a_{0}, a_{1}, \ldots$ of positive integers satisfying $a_{0}=15, a_{1}=t$, and $a_{n-1} a_{n+1}=\left(a_{n}-1\right)\left(a_{n}+1\right)$ for all positive integers $n$. Find the sum of all good numbers. | 296 |
Find the point where the line passing through $(3,4,1)$ and $(5,1,6)$ intersects the $xy$-plane. | \left( \frac{13}{5}, \frac{23}{5}, 0 \right) |
In a square, 20 points were marked and connected by non-intersecting segments with each other and with the vertices of the square, dividing the square into triangles. How many triangles were formed? | 42 |
In an isosceles triangle \(ABC\), the base \(AC\) is equal to \(x\), and the lateral side is equal to 12. On the ray \(AC\), point \(D\) is marked such that \(AD = 24\). From point \(D\), a perpendicular \(DE\) is dropped to the line \(AB\). Find \(x\) given that \(BE = 6\). | 18 |
In a certain city, vehicle license plates are numbered consecutively from "10000" to "99999". How many license plates out of these 90,000 have the digit 9 appearing at least once and where the sum of the digits is a multiple of 9? | 4168 |
Yvonne and Janna were writing their 1000-word pair research paper. Yvonne was able to write 400 words while Janna wrote 150 more words than Yvonne. When they edited their paper, they removed 20 words and added twice as many words as they removed. How many more words should they add to reach the research paper requireme... | Janna wrote 400 + 150 = <<400+150=550>>550 words.
Together, they wrote 400 + 550 = <<400+550=950>>950 words.
They have 950 - 20 = <<950-20=930>>930 words left after omitting 20 words.
Then 20 x 2 = <<20*2=40>>40 words were added during the editing.
Overall, they have 930 + 40 = <<930+40=970>>970 words.
Thus, they shoul... |
Eastbound traffic flows at 80 miles per hour and westbound traffic flows at 60 miles per hour. An eastbound driver observes 30 westbound vehicles in a 10-minute period. Calculate the number of westbound vehicles in a 150-mile section of the highway. | 193 |
The ratio of the sums of the first \( n \) terms of two arithmetic sequences is \(\frac{9n+2}{n+7}\). Find the ratio of their 5th terms. | \frac{83}{16} |
For each ordered pair of real numbers $(x,y)$ satisfying \[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that \[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$. | 189 |
Nick has 35 quarters. 2/5 of the quarters are state quarters, and 50 percent of the state quarters are Pennsylvania. How many Pennsylvania state quarters does Nick have? | State quarters:35(2/5)=14 quarters
Pennsylvania:14(.50)=7 quarters
#### 7 |
When the height of a cylinder is doubled and its radius is increased by $200\%$, the cylinder's volume is multiplied by a factor of $X$. What is the value of $X$? | 18 |
The graph of $r = \cos \theta$ is a circle. Find the smallest value of $t$ so that when $r = \cos \theta$ is plotted for $0 \le \theta \le t,$ the resulting graph is the entire circle. | \pi |
What is the greatest common factor of 40 and 48? | 8 |
Given the sequence \(\left\{a_{n}\right\}\) with the general term
\[ a_{n} = n^{4} + 6n^{3} + 11n^{2} + 6n, \]
find the sum of the first 12 terms \( S_{12} \). | 104832 |
Rosie runs 6 miles per hour. She runs for 1 hour on Monday, 30 minutes on Tuesday, 1 hour on Wednesday, and 20 minutes on Thursday. If she wants to run 20 miles for the week, how many minutes should she run on Friday? | On Monday and Wednesday, Rosie runs 6*1= <<6*1=6>>6 miles.
On Tuesday, Rosie runs 6*(30 minutes/60 minutes) = <<6*(30/60)=3>>3 miles.
On Thursday, Rosie runs 6*(20 minutes/60 minutes)=<<6*(20/60)=2>>2 miles.
So far this week Rosie has run 6+3+6+2=<<6+3+6+2=17>>17 miles.
To reach her goal, she must run 20-17=<<20-17=3>>... |
An urn contains $k$ balls labeled with $k$, for all $k = 1, 2, \ldots, 2016$. What is the minimum number of balls we must draw, without replacement and without looking at the balls, to ensure that we have 12 balls with the same number? | 22122 |
Jan's three-eyed lizard has 3 times more wrinkles than eyes, and seven times more spots than wrinkles. How many fewer eyes does the lizard have than the combined number of spots and wrinkles? | Three times more wrinkles than three eyes is 3*3=<<3*3=9>>9 wrinkles.
Seven times more spots than wrinkles is 7*9=<<7*9=63>>63 spots.
Therefore, the combined number of spots and wrinkles on the lizard is 63+9=<<63+9=72>>72 spots and wrinkles.
Thus, the lizard has 72-3=69 fewer eyes than spots and wrinkles
#### 69 |
Given that $\operatorname{tg} \theta$ and $\operatorname{ctg} \theta$ are the real roots of the equation $2x^{2} - 2kx = 3 - k^{2}$, and $\alpha < \theta < \frac{5 \pi}{4}$, find the value of $\cos \theta - \sin \theta$. | -\sqrt{\frac{5 - 2\sqrt{5}}{5}} |
A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thickness.... | 593 |
What is the integer formed by the rightmost two digits of the integer equal to \(4^{127} + 5^{129} + 7^{131}\)? | 52 |
If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 26$ and\[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1,\] compute $pqr$.
| 576 |
A square with a side length of 100 cm is drawn on a board. Alexei intersected it with two lines parallel to one pair of its sides. After that, Danil intersected the square with two lines parallel to the other pair of sides. As a result, the square was divided into 9 rectangles, with the dimensions of the central rectan... | 2400 |
Anna has a certain number of phone chargers and five times more laptop chargers than phone chargers. If she has 24 chargers total, how many phone chargers does she have? | Let p be the number of phone chargers Anna has and l be the number of laptop chargers. We know that l = 5p and l + p = 24.
Substituting the first equation into the second equation, we get 5p + p = 24
Combining like terms, we get 6p = 24
Dividing both sides of the equation by 6, we get p = 4
#### 4 |
Determine the product of the solutions of the equation $-21 = -x^2 + 4x$. | -21 |
How many of the numbers in Grace's sequence, starting from 43 and each number being 4 less than the previous one, are positive? | 11 |
In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals: | 15 |
For how many integers $n$ with $1 \le n \le 2023$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]equal to zero, where $n$ needs to be an even multiple of $5$? | 202 |
Find the area of rhombus $EFGH$ given that the radii of the circles circumscribed around triangles $EFG$ and $EGH$ are $15$ and $30$, respectively. | 60 |
There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth... | 1009 |
Alice places a coin, heads up, on a table then turns off the light and leaves the room. Bill enters the room with 2 coins and flips them onto the table and leaves. Carl enters the room, in the dark, and removes a coin at random. Alice reenters the room, turns on the light and notices that both coins are heads. What is ... | 3/5 |
$908 \times 501 - [731 \times 1389 - (547 \times 236 + 842 \times 731 - 495 \times 361)] =$ | 5448 |
How many positive integer multiples of $1001$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0\leq i < j \leq 99$?
| 784 |
Yanni has 5 paintings that combined, take up 200 square feet. 3 of the paintings are 5 feet by 5 feet. 1 painting is 10 feet by 8 feet. If the final painting is 5 feet tall, how wide is it? | Each 5 by 5 painting is 25 square feet because 5 x 5 = <<5*5=25>>25
All three of them take up 75 square feet because 3 x 25 = <<3*25=75>>75
The 10 by 8 painting takes up 80 square feet because 10 x 8 = <<10*8=80>>80
These four paintings take up 155 square feet because 80 + 75 = <<80+75=155>>155
The final painting is 45... |
Linda makes $10.00 an hour babysitting. There is a $25.00 application fee for each college application she submits. If she is applying to 6 colleges, how many hours will she need to babysit to cover the application fees? | The application fee is $25.00 per college and she is applying to 6 colleges so that's 25*6 = $<<25*6=150.00>>150.00
She makes $10.00 an hour babysitting. Her application fees total $150.00 so she needs to work 150/10 = <<150/10=15>>15 hours to cover the cost
#### 15 |
What is the median number of moons per planet? (Include Pluto, although arguments rage on about Pluto's status...) \begin{tabular}{c|c}
Planet & $\#$ of Moons\\
\hline
Mercury&0\\
Venus &0\\
Earth &1\\
Mars &2\\
Jupiter&16\\
Saturn&23\\
Uranus&15\\
Neptune&2\\
Pluto&5\\
\end{tabular} | 2 |
Let $m$ and $n$ satisfy $mn = 6$ and $m+n = 7$. Additionally, suppose $m^2 - n^2 = 13$. Find the value of $|m-n|$. | \frac{13}{7} |
Define a sequence of convex polygons \( P_n \) as follows. \( P_0 \) is an equilateral triangle with side length 1. \( P_{n+1} \) is obtained from \( P_n \) by cutting off the corners one-third of the way along each side (for example, \( P_1 \) is a regular hexagon with side length \(\frac{1}{3}\)). Find \( \lim_{n \to... | \frac{\sqrt{3}}{7} |
A regular decagon is formed by connecting three sequentially adjacent vertices of the decagon. Find the probability that all three sides of the triangle are also sides of the decagon. | \frac{1}{12} |
Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\sqrt{113}$ units and diagonals that differ by 10 units? | 72 |
Choose the largest of the following sums, and express it as a fraction in simplest form:
$$\frac{1}{4} + \frac{1}{5}, \ \ \frac{1}{4} + \frac{1}{6}, \ \ \frac{1}{4} + \frac{1}{3}, \ \ \frac{1}{4} + \frac{1}{8}, \ \ \frac{1}{4} + \frac{1}{7}$$ | \frac{7}{12} |
The function $y=f(x)$ is an even function with the smallest positive period of 4, and when $x \in [-2, 0]$, $f(x) = 2x + 1$. If there exist $x_1, x_2, \ldots, x_n$ satisfying $0 \leq x_1 < x_2 < \ldots < x_n$, and $|f(x_1) - f(x_2)| + |f(x_2) - f(x_3)| + \ldots + |f(x_{n-1}) - f(x_n)| = 2016$, then the minimum value of... | 1513 |
In a right prism with triangular bases, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base) is 24, find the maximum volume of the prism.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F;
A = (0,0);
B = (3,-1);
C = (-1,-2);
D = A + (0,-4);
E = B + (0,-4);
F = C + (0,-4)... | 16 |
Let \(x\) and \(y\) be positive real numbers such that
\[
\frac{1}{x + 1} + \frac{1}{y + 1} = \frac{1}{2}.
\]
Find the minimum value of \(x + 3y.\) | 4 + 4 \sqrt{3} |
A ''super ball'' is dropped from a window 16 meters above the ground. On each bounce it rises $\frac34$ the distance of the preceding high point. The ball is caught when it reached the high point after hitting the ground for the third time. To the nearest meter, how far has it travelled? | 65 |
Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.
Hint
\[\color{red}\boxed{\boxed{\color{blue}\textbf{Use Vieta's Formulae!}}}\] | 420 |
(1) If the terminal side of angle $\theta$ passes through $P(-4t, 3t)$ ($t>0$), find the value of $2\sin\theta + \cos\theta$.
(2) Given that a point $P$ on the terminal side of angle $\alpha$ has coordinates $(x, -\sqrt{3})$ ($x\neq 0$), and $\cos\alpha = \frac{\sqrt{2}}{4}x$, find $\sin\alpha$ and $\tan\alpha$. | \frac{2}{5} |
How many five-digit numbers are there that are divisible by 5 and do not contain repeating digits? | 5712 |
Two distinct natural numbers end with 8 zeros and have exactly 90 divisors. Find their sum. | 700000000 |
When each edge of a cube is increased by $50\%$, by what percent is the surface area of the cube increased? | 125\% |
The Grey's bought several chickens at a sale. John took 5 more of the chickens than Mary took. Ray took 6 chickens less than Mary. If Ray took 10 chickens, how many more chickens did John take than Ray? | Mary took 10+6 = <<10+6=16>>16 chickens.
John took 16+5 = <<16+5=21>>21 chickens
John took 21-10 = <<21-10=11>>11 more chickens than Ray.
#### 11 |
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer stan... | \frac{3}{16} |
Aston has accidentally torn apart his comics and needs to put all the pages back together. Each comic has 25 pages and Aston has found 150 pages on the floor. He puts his comics back together and adds them back into his box of comics. If there were already 5 untorn comics in the box, how many comics are now in the box ... | Aston had torn and has now fixed a total of 150 pages on the floor / 25 pages per comic = <<150/25=6>>6 comics.
Adding this to the comics that were never torn means there are 6 fixed comics + 5 untorn comics = <<6+5=11>>11 comics in the box of comics.
#### 11 |
What is the smallest possible sum of two consecutive integers whose product is greater than 420? | 43 |
How many different two-person sub-committees can be selected from a committee of six people (the order of choosing the people does not matter)? | 15 |
Sophie does 4 loads of laundry a week and uses 1 dryer sheet per load. A box of dryer sheets costs $5.50 and has 104 dryer sheets in a box. On her birthday, she was given wool dryer balls to use instead of dryer sheets. How much money does she save in a year not buying dryer sheets? | She does 4 loads of laundry a week and used 1 dryer sheet per load for a total of 4*1 = <<4*1=4>>4 dryer sheets in a week
She used 4 dryer sheets per week for 52 weeks for a total of 4*52 = <<4*52=208>>208 dryer sheets
Her box of dryer sheets has 104 sheets so she needs 208/104 = 2 boxes of dryer sheets
Each box costs ... |
Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to defi... | \frac{29}{120} |
If the internal angles of $\triangle ABC$ satisfy $\sin A + 2\sin B = 3\sin C$, then the minimum value of $\cos C$ is | \frac{2 \sqrt{10} - 2}{9} |
Triangle $ABC$ with vertices $A(-2, 0)$, $B(1, 4)$ and $C(-3, 2)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $C$ to $C'$? | 6 |
Dad is $a$ years old this year, which is 4 times plus 3 years more than Xiao Hong's age this year. Xiao Hong's age expressed in an algebraic expression is ____. If Xiao Hong is 7 years old this year, then Dad's age is ____ years old. | 31 |
Peggy is moving and is looking to get rid of her record collection. Sammy says that he will buy all of them for 4 dollars each. Bryan is only interested in half of the records but will offer 6 dollars each for the half that he is interested in and 1 dollar each for the remaining half that he is not interested in with t... | Sammy is offering to take the whole collection of 200 records and pay Peggy 4 dollars each for them which would net Peggy 200 * 4=<<200*4=800>>800 dollars for her entire record collection.
Bryan is willing to buy Peggy's entire record collection but at two different price points, half at one point and half at another. ... |
Jane is sewing sequins onto her trapeze artist costume. She sews 6 rows of 8 blue sequins each, 5 rows of 12 purple sequins each, and 9 rows of 6 green sequins each. How many sequins does she add total? | First find the total number of blue sequins: 6 rows * 8 sequins/row = <<6*8=48>>48 sequins
Then find the total number of purple sequins: 5 rows * 12 sequins/row = <<5*12=60>>60 sequins
Then find the total number of green sequins: 9 rows * 6 sequins/row = <<9*6=54>>54 sequins
Then add the number of each color of sequin ... |
John worked 8 hours a day every day from the 3rd to the 8th, including the 3rd and not including the 8th. How many hours did he work? | He worked for 8-3=<<8-3=5>>5 days
So he worked for 5*8=<<5*8=40>>40 hours
#### 40 |
What is the product of all real numbers that are tripled when added to their reciprocals? | -\frac{1}{2} |
For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$. | 1 |
Given a periodic sequence $\left\{x_{n}\right\}$ that satisfies $x_{n}=\left|x_{n-1}-x_{n-2}\right|(n \geqslant 3)$, if $x_{1}=1$ and $x_{2}=a \geqslant 0$, calculate the sum of the first 2002 terms when the period of the sequence is minimized. | 1335 |
If $x$ is a real number and $x^2 = 16$, what is the sum of all possible values of $x$? | 0 |
Find the minimum value of
\[
\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}
\]
over all real numbers $x$. | 2\sqrt{5} |
\[
\frac{\left(\left(4.625 - \frac{13}{18} \cdot \frac{9}{26}\right) : \frac{9}{4} + 2.5 : 1.25 : 6.75\right) : 1 \frac{53}{68}}{\left(\frac{1}{2} - 0.375\right) : 0.125 + \left(\frac{5}{6} - \frac{7}{12}\right) : (0.358 - 1.4796 : 13.7)}
\]
| \frac{17}{27} |
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads? | \frac{1}{2} |
Roll two dice consecutively. Let the number on the first die be $m$, and the number on the second die be $n$. Calculate the probability that:
(1) $m+n=7$;
(2) $m=n$;
(3) The point $P(m,n)$ is inside the circle $x^2+y^2=16$. | \frac{2}{9} |
The average age of the three Wilson children is 7 years. If the two younger children are 4 years old and 7 years old, how many years old is the oldest child? | 10 |
There are 2020 quadratic equations written on the board:
$$
\begin{gathered}
2020 x^{2}+b x+2021=0 \\
2019 x^{2}+b x+2020=0 \\
2018 x^{2}+b x+2019=0 \\
\ldots \\
x^{2}+b x+2=0
\end{gathered}
$$
(each subsequent equation is obtained from the previous one by decreasing the leading coefficient and the constant term by o... | 2021 |
A certain school is actively preparing for the "Sunshine Sports" activity and has decided to purchase a batch of basketballs and soccer balls totaling $30$ items. At a sports equipment store, each basketball costs $80$ yuan, and each soccer ball costs $60$ yuan. During the purchase period at the school, there is a prom... | 2080 |
Given a point P(3, 2) outside the circle $x^2+y^2-2x-2y+1=0$, find the cosine of the angle between the two tangents drawn from this point to the circle. | \frac{3}{5} |
There are five students, A, B, C, D, and E, arranged to participate in the volunteer services for the Shanghai World Expo. Each student is assigned one of four jobs: translator, guide, etiquette, or driver. Each job must be filled by at least one person. Students A and B cannot drive but can do the other three jobs, wh... | 108 |
Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number. | 343 |
Given non-negative numbers $x$, $y$, and $z$ such that $x+y+z=2$, determine the minimum value of $\frac{1}{3}x^3+y^2+z$. | \frac{13}{12} |
The first term of a geometric sequence is 729, and the 7th term is 64. What is the positive, real value for the 5th term? | 144 |
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe that there are two HH, three HT, four TH, a... | 560 |
Simplify the expression: $\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)^{6} + \left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)^{6}$ using DeMoivre's Theorem. | 2 |
Solve the system of equations:
\begin{cases}
\frac{m}{3} + \frac{n}{2} = 1 \\
m - 2n = 2
\end{cases} | \frac{2}{7} |
Let $p,$ $q,$ $r$ be positive real numbers. Find the smallest possible value of
\[4p^3 + 6q^3 + 24r^3 + \frac{8}{3pqr}.\] | 16 |
Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red?
[i]Proposed by M... | 2/5 |
Mr. and Mrs. Boyden take their 3 children to a leisure park. They buy tickets for the whole family. The cost of an adult ticket is $6 more than the cost of a child ticket. The total cost of the 5 tickets is $77. What is the cost of an adult ticket? | Let X be the cost of an adult ticket.
So the cost of a child ticket is X-6.
The total cost of the 5 tickets is X*2 + 3*(X-6) = 77.
X*2 + 3*X - 3*6 = 77.
5*X - 18 = 77.
5*X = 77 + 18 = 95
X = <<19=19>>19
#### 19 |
Square $ABCD$ has center $O,\ AB=900,\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\angle EOF =45^\circ,$ and $EF=400.$ Given that $BF=p+q\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$
| 307 |
The line $y = b-x$ with $0 < b < 4$ intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 9:25, what is the value of $b$? Express the answer as a decimal to the nearest tenth.
[asy]
draw((0,-3)--(0,5.5),Arrows);
draw((4,-3.5)--(4,5),Arrows... | 2.5 |
Find the greatest constant $\lambda$ such that for any doubly stochastic matrix of order 100, we can pick $150$ entries such that if the other $9850$ entries were replaced by $0$, the sum of entries in each row and each column is at least $\lambda$.
Note: A doubly stochastic matrix of order $n$ is a $n\times n$ matrix... | \frac{17}{1900} |
In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure below shows four of the entries of a magic square. Find $x$.
[asy]
size(2cm);
for (int i=0; i<=3; ++i) draw((i,0)--(i,3)^^(0,i)--(3,i));
label("$x$",(0.5,2.5));label("$19$",(1.5,2.5));
label("$96$",(2.5,2.5));l... | 200 |
Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34$. What is $x+y$? | 2 |
Given a connected simple graph \( G \) with a known number of edges \( e \), where each vertex has some number of pieces placed on it (each piece can only be placed on one vertex of \( G \)). The only operation allowed is when a vertex \( v \) has a number of pieces not less than the number of its adjacent vertices \( ... | e |
An ice cream shop offers 6 kinds of ice cream. What is the greatest number of two scoop sundaes that can be made such that each sundae contains two types of ice cream and no two sundaes are the same combination? | 15 |
Consider a dark rectangle created by merging two adjacent unit squares in an array of unit squares; part as shown below. If the first ring of squares around this center rectangle contains 10 unit squares, how many unit squares would be in the $100^{th}$ ring? | 802 |
The set of points $(x,y,z)$ that are equidistant to $(1,2,-5)$ and point $P$ satisfy an equation of the form
\[10x - 4y + 24z = 55.\]Find the point $P.$ | (6,0,7) |
How many interior diagonals does an icosahedron have? (An $\emph{icosahedron}$ is a 3-dimensional figure with 20 triangular faces and 12 vertices, with 5 faces meeting at each vertex. An $\emph{interior}$ diagonal is a segment connecting two vertices which do not lie on a common face.) | 36 |
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