problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
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Find the number of ordered pairs of integers $(a, b) \in\{1,2, \ldots, 35\}^{2}$ (not necessarily distinct) such that $a x+b$ is a "quadratic residue modulo $x^{2}+1$ and 35 ", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds: - there exist po... | 225 |
What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers? | -5 |
Wendy is a fruit vendor, and she sells an apple at $1.50 each and one orange at $1. In the morning, she was able to sell an average of 40 apples and 30 oranges. In the afternoon, she was able to sell 50 apples and 40 oranges. How much are her total sales for the day? | There were an average of 40 + 50 = <<40+50=90>>90 apples.
And, there were 30 + 40 = <<30+40=70>>70 oranges.
Thus, Wendy gained 90 x $1.50 = $<<90*1.5=135>>135 for selling apples.
And, she gained an additional 70 x $1 = $<<70*1=70>>70 for selling oranges.
Therefore, her total sales for the day is $135 + 70 = $<<135+70=2... |
Given that the median of the numbers $3, 5, 7, 23,$ and $x$ is equal to the mean of those five numbers, calculate the sum of all real numbers $x$. | -13 |
Given the function f(x) = 2/(x+1) for a positive number x, calculate the sum of f(100) + f(99) + f(98) + ... + f(2) + f(1) + f(1/2) + ... + f(1/98) + f(1/99) + f(1/100). | 199 |
The measure of each exterior angle of a regular polygon is $30$ degrees. What is the sum of the measures of the interior angles, in degrees? | 1800 |
One interior angle of a convex polygon is 160 degrees. The rest of the interior angles of the polygon are each 112 degrees. How many sides does the polygon have? | 6 |
There exists a constant $k$ so that the minimum value of
\[4x^2 - 6kxy + (3k^2 + 2) y^2 - 4x - 4y + 6\]over all real numbers $x$ and $y$ is 0. Find $k.$ | 2 |
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 |
Rebecca makes her own earrings out of buttons, magnets, and gemstones. For every earring, she uses two magnets, half as many buttons as magnets, and three times as many gemstones as buttons. If Rebecca wants to make 4 sets of earrings, how many gemstones will she need? | Four sets of earrings is 2*4=<<2*4=8>>8 earrings.
Since for one earring, she uses two magnets, for eight earrings she uses 2*8=<<2*8=16>>16 magnets.
Since she uses half as many buttons as magnets, then for 8 earrings she would use 16/2=8 buttons.
Since she uses three times as many gemstones as buttons, then in 8 earrin... |
Find the number of all natural numbers in which each subsequent digit is less than the previous one. | 1013 |
$\triangle ABC\sim\triangle DBE$, $BC=20\text{ cm}.$ How many centimeters long is $DE$? Express your answer as a decimal to the nearest tenth. [asy]
draw((0,0)--(20,0)--(20,12)--cycle);
draw((13,0)--(13,7.8));
label("$B$",(0,0),SW);
label("$E$",(13,0),S);
label("$D$",(13,7.8),NW);
label("$A$",(20,12),NE);
label("$C$",(... | 7.8 |
Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins t... | \frac{5}{36} |
Given algebraic expressions $A=2m^{2}+3my+2y-1$ and $B=m^{2}-my$. Find:<br/>
$(1)$ Simplify $3A-2\left(A+B\right)$.<br/>
$(2)$ If $\left(m-1\right)^{2}+|y+2|=0$, find the value of $3A-2\left(A+B\right)$.<br/>
$(3)$ If the value of $3A-2\left(A+B\right)$ is independent of $y$, find the value of $m$. | -0.4 |
The price of buying a wooden toy at the new Craftee And Best store is $20, and the cost of buying a hat is $10. If Kendra went to the shop with a $100 bill and bought two wooden toys and three hats, calculate the change she received. | When Kendra bought 2 toys, she paid 2*$20 = $<<2*20=40>>40
Since the price of a hat is $10, when Kendra bought 3 hats, she paid 3*$10 = $<<3*10=30>>30
The total costs for the hats and wooden toys Kendra bought is $40+$30 = $<<40+30=70>>70
From the $100 bill, Kendra received change worth $100-$70 =$<<100-70=30>>30
#### ... |
What is the 43rd digit after the decimal point in the decimal representation of $\frac{1}{13}$? | 0 |
Jerry has ten distinguishable coins, each of which currently has heads facing up. He chooses one coin and flips it over, so it now has tails facing up. Then he picks another coin (possibly the same one as before) and flips it over. How many configurations of heads and tails are possible after these two flips? | 46 |
Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors.
(M Levin) | 11250, 4050, 7500, 1620, 1200, 720 |
Given the scores of the other five students are $83$, $86$, $88$, $91$, $93$, and Xiaoming's score is both the mode and the median among these six scores, find Xiaoming's score. | 88 |
How many whole numbers between $100$ and $400$ contain the digit $2$? | 138 |
Prejean's speed in a race was three-quarters that of Rickey. If they both took a total of 70 minutes to run the race, calculate the total number of minutes that Rickey took to finish the race. | Let's say Rickey took t minutes to finish the race.
If Prejean's speed in the race was three-quarters that of Rickey, and Rickey took t minutes to finish the race, the time that Prejean took to finish the race is 3/4*t=3/4t
Together, Rickey and Prejean took t+3/4t = 70
They took 1 3/4 t hours=70 to finish the race
This... |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\le x\le 4$ and $1\le y\le 4$?
$\text{(A) } 496\quad \text{(B) } 500\quad \text{(C) } 512\quad \text{(D) } 516\quad \text{(E) } 560$
| 516 |
In a blackboard, it's written the following expression $ 1-2-2^2-2^3-2^4-2^5-2^6-2^7-2^8-2^9-2^{10}$ We put parenthesis by different ways and then we calculate the result. For example: $ 1-2-\left(2^2-2^3\right)-2^4-\left(2^5-2^6-2^7\right)-2^8-\left( 2^9-2^{10}\right)= 403$ and $ 1-\left(2-2^2 \left(-2^3-2^4 \right)-... | 1024 |
When $\sqrt[3]{7200}$ is simplified, the result is $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ is as small as possible. What is $c+d$? | 452 |
The sum of Alice's weight and Clara's weight is 220 pounds. If you subtract Alice's weight from Clara's weight, you get one-third of Clara's weight. How many pounds does Clara weigh? | 88 |
Markus is twice the age of his son, and Markus's son is twice the age of Markus's grandson. If the sum of the ages of Markus, his son, and his grandson is 140 years, then how many years old is Markus's grandson? | Let "x" be the age of Markus's grandson.
If Markus's son is twice the age of Markus's grandson, then Markus's son is 2*x.
If Markus is twice the age of his son, then Markus is 2*2*x.
Therefore, if the sum of the ages of Markus, his son, and his grandson is 140 years, then x+(2*x)+(2*2*x)=140 years.
Simplifying the equa... |
Mia has $20 more than twice as much money Darwin has. If Darwin has $45, how much money does Mia have? | Twice the money of Darwin is 2 * $45 = $<<2*45=90>>90.
The money that Mia have is $20 + $90 = $<<20+90=110>>110.
#### 110 |
What is the positive difference between the median and the mode of the data given in the stem and leaf plot below? In this plot $5|8$ represents $58.$
\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
1 & $2 \hspace{2mm} 3 \hspace{2mm} 4 \hspace{2mm} 5 \hspace{2mm} 5$ \\ \hline
2 & $2 \hspace{2mm} ... | 9 |
In a right triangle \( A B C \) (with right angle at \( C \)), the medians \( A M \) and \( B N \) are drawn with lengths 19 and 22, respectively. Find the length of the hypotenuse of this triangle. | 29 |
What is the area, in square units, of the square with the four vertices at $A\ (0, 0)$, $B\ (-5, -1)$, $C\ (-4, -6)$ and $D\ (1, -5)$? | 26 |
In the sequence ${a_{n}}$, $a_{n+1}=\begin{cases} 2a_{n}\left(a_{n} < \frac{1}{2}\right) \\ 2a_{n}-1\left(a_{n}\geqslant \frac{1}{2}\right) \end{cases}$, if $a_{1}=\frac{4}{5}$, then the value of $a_{20}$ is $\_\_\_\_\_\_$. | \frac{2}{5} |
The country of HMMTLand has 8 cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly 3 other cities via a single road, and from any pair of distinct cities, either exactly 0 or 2 other cities can b... | 875 |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and for $x \geqslant 0$, $f(x) = \begin{cases} \log_{\frac{1}{2}}(x+1), & 0 \leqslant x < 1 \\ 1-|x-3|, & x \geqslant 1 \end{cases}$, determine the sum of all zeros of the function $y = f(x) + \frac{1}{2}$. | \sqrt{2} - 1 |
There is one odd integer \(N\) between 400 and 600 that is divisible by both 5 and 11. What is the sum of the digits of \(N\)? | 18 |
Kimberly borrows $1000$ dollars from Lucy, who charged interest of $5\%$ per month (which compounds monthly). What is the least integer number of months after which Kimberly will owe more than twice as much as she borrowed? | 15 |
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{1}{4}} \frac{\sqrt[3]{\frac{x}{16}}-\frac{1}{4}}{\sqrt{\frac{1}{4}+x}-\sqrt{2x}}$ | -\frac{2\sqrt{2}}{6} |
Consider the 100th, 101st, and 102nd rows of Pascal's triangle, denoted as sequences $(p_i)$, $(q_i)$, and $(r_i)$ respectively. Calculate:
\[
\sum_{i = 0}^{100} \frac{q_i}{r_i} - \sum_{i = 0}^{99} \frac{p_i}{q_i}.
\] | \frac{1}{2} |
In the Cartesian coordinate plane, a polar coordinate system is established with the origin as the pole and the non-negative half of the x-axis as the polar axis. It is known that point A has polar coordinates $$( \sqrt{2}, \frac{\pi}{4})$$, and the parametric equation of line $l$ is:
$$\begin{cases} x= \frac{3}{2} - \... | \frac{4\sqrt{2}}{5} |
When a car's brakes are applied, it travels 7 feet less in each second than the previous second until it comes to a complete stop. A car goes 28 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time the car stops? | 70 |
The diagram shows a square \(PQRS\) with sides of length 2. The point \(T\) is the midpoint of \(RS\), and \(U\) lies on \(QR\) so that \(\angle SPT = \angle TPU\). What is the length of \(UR\)? | 1/2 |
A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | 441 |
An exchange point conducts two types of operations:
1) Give 2 euros - receive 3 dollars and a candy as a gift;
2) Give 5 dollars - receive 3 euros and a candy as a gift.
When the wealthy Buratino came to the exchange point, he had only dollars. When he left, he had fewer dollars, he did not acquire any euros, but he ... | 10 |
On the first day, Barry Sotter used his magic to make an object's length increase by $\frac{1}{3}$, so if the original length was $x$, it became $x + \frac{1}{3} x$. On the second day, he increased the new length by $\frac{1}{4}$; on the third day by $\frac{1}{5}$; and so on. On the $n^{\text{th}}$ day, Barry made the ... | 147 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and point $A(1, \frac{3}{2})$ on ellipse $C$ has a sum of distances to these two points equal to $4$.
(I) Find the equation of the ellipse $C$ and the coordinates of its foci.
(II) Let point... | \sqrt{5} |
On March 12, 2016, the fourth Beijing Agriculture Carnival opened in Changping. The event was divided into seven sections: "Three Pavilions, Two Gardens, One Belt, and One Valley." The "Three Pavilions" refer to the Boutique Agriculture Pavilion, the Creative Agriculture Pavilion, and the Smart Agriculture Pavilion; th... | 144 |
Alan counted how many chairs the office canteen has. It has 2 round tables and 2 rectangular tables. Each round table has 6 chairs, and each rectangular table has 7 chairs. How many chairs are there in all? | The number of chairs at the round tables is 2 tables × 6 chairs/table = <<2*6=12>>12.
The number of chairs at rectangular tables is 2 tables × 7 chairs/table = <<2*7=14>>14.
There are 12 chairs + 14 chairs = <<12+14=26>>26 chairs in the canteen.
#### 26 |
Two students, A and B, are preparing to have a table tennis match during their physical education class. Assuming that the probability of A winning against B in each game is $\frac{1}{3}$, the match follows a best-of-three format (the first player to win two games wins the match). What is the probability of A winning t... | \frac{7}{27} |
Tim plans a weeklong prank to repeatedly steal Nathan's fork during lunch. He involves different people each day:
- On Monday, he convinces Joe to do it.
- On Tuesday, either Betty or John could undertake the prank.
- On Wednesday, there are only three friends from whom he can seek help, as Joe, Betty, and John are not... | 48 |
Natalie bought some food for a party she is organizing. She bought two cheesecakes, an apple pie, and a six-pack of muffins. The six-pack of muffins was two times more expensive than the cheesecake, and one cheesecake was only 25% cheaper than the apple pie. If the apple pie cost $12, how much did Natalie pay for all h... | One cheesecake was 30% cheaper than the apple pie, which means, it was 25/100 * 12 = $3 cheaper.
So for one cheesecake, Natalie needed to pay 12 - 3 = $<<12-3=9>>9.
The six-pack of muffins was two times more expensive than the cheesecake, which means its price was 9 * 2 = $<<9*2=18>>18.
So for two cheesecakes, Natalie ... |
The number obtained from the last two nonzero digits of $70!$ is equal to $n$. Calculate the value of $n$. | 12 |
Consider a revised dataset given in the following stem-and-leaf plot, where $7|1$ represents 71:
\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
2 & $0 \hspace{2mm} 0 \hspace{2mm} 1 \hspace{2mm} 1 \hspace{2mm} 2$ \\ \hline
3 & $3 \hspace{2mm} 6 \hspace{2mm} 6 \hspace{2mm} 7$ \\ \hline
4 & $3 \hsp... | 23 |
A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$? | $3\sqrt{2}+\sqrt{6}$ |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 |
Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{24}/R_4$. | 15 |
Find the product of all $x$ such that the expression $\frac{x^2+2x+1}{x^2+2x-3}$ is undefined. | -3 |
The perimeter of a square with side length $x$ units is equal to the circumference of a circle with radius 2 units. What is the value of $x$? Express your answer as a decimal to the nearest hundredth. | 3.14 |
When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is calculated. | 18185 |
In $\triangle{ABC}, AB=10, \angle{A}=30^\circ$ , and $\angle{C=45^\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Then $AP^2=\dfrac{m}{n}$, where ... | 77 |
Adam and Bettie are playing a game. They take turns generating a random number between $0$ and $127$ inclusive. The numbers they generate are scored as follows: $\bullet$ If the number is zero, it receives no points. $\bullet$ If the number is odd, it receives one more point than the number one less than it. $\bu... | 429 |
Given that a certain product requires $6$ processing steps, where $2$ of these steps must be consecutive and another $2$ steps cannot be consecutive, calculate the number of possible processing sequences. | 144 |
Given that in a class test, $15\%$ of the students scored $60$ points, $50\%$ scored $75$ points, $20\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test. | 2.75 |
The formula for the total surface area of a cylinder is $SA = 2\pi r^2 + 2\pi rh,$ where $r$ is the radius and $h$ is the height. A particular solid right cylinder of radius 2 feet has a total surface area of $12\pi$ square feet. What is the height of this cylinder? | 1 |
Bob knows that Alice has 2021 secret positive integers $x_{1}, \ldots, x_{2021}$ that are pairwise relatively prime. Bob would like to figure out Alice's integers. He is allowed to choose a set $S \subseteq\{1,2, \ldots, 2021\}$ and ask her for the product of $x_{i}$ over $i \in S$. Alice must answer each of Bob's quer... | 11 |
A positive integer $n$ not exceeding $120$ is chosen such that if $n\le 60$, then the probability of choosing $n$ is $p$, and if $n > 60$, then the probability of choosing $n$ is $2p$. The probability that a perfect square is chosen is?
A) $\frac{1}{180}$
B) $\frac{7}{180}$
C) $\frac{13}{180}$
D) $\frac{1}{120}$
E) $\f... | \frac{13}{180} |
What is the value of $f(-1)$ if $f(x)=x^{2}-2x$? | 3 |
The value of \( 2 \frac{1}{10} + 3 \frac{11}{100} \) can be calculated. | 5.21 |
Determine the residue of $-998\pmod{28}$. Your answer should be an integer in the range $0,1,2,\ldots,25,26,27$. | 10 |
Find the area of the shaded region. | 6\dfrac{1}{2} |
Simplify $$(x^3+4x^2-7x+11)+(-4x^4-x^3+x^2+7x+3).$$ Express your answer as a polynomial with the terms in order by decreasing degree. | -4x^4+5x^2+14 |
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\text{o}$. The block is then sliced in half al... | 53 |
A man loves to go hiking. He knows that he has to pack about .5 pounds of supplies for every mile he hikes. He plans to go for a very long hike. He knows that he can get 1 resupply during the hike that will be 25% as large as his first pack. He can hike at a rate of 2.5 mph. He is going to hike 8 hours a day for 5... | He is going to hike 8*=40 hours
That means he will be hiking for 40*2.5=<<40*2.5=100>>100 miles
So he needs 100*.5=<<100*.5=50>>50 pounds of supplies
With the resupply he can get up to 1+.25=<<1+.25=1.25>>1.25 times his initial pack
That means his initial pack has to weigh 50/1.25=<<50/1.25=40>>40 pounds
#### 40 |
Define a function $A(m, n)$ by \[ A(m,n) = \left\{ \begin{aligned} &n+1& \text{ if } m = 0 \\ &A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\ &A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0. \end{aligned} \right.\]Compute $A(2, 1).$ | 5 |
Find the smallest positive multiple of 9 that can be written using only the digits: (a) 0 and 1; (b) 1 and 2. | 12222 |
Screws are sold in packs of $10$ and $12$ . Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of $k$ ? | 60 |
The line joining the midpoints of the diagonals of a trapezoid has length $3$. If the longer base is $97,$ then the shorter base is: | 91 |
Let \( a \) be a positive integer that is a multiple of 5 such that \( a+1 \) is a multiple of 7, \( a+2 \) is a multiple of 9, and \( a+3 \) is a multiple of 11. Determine the smallest possible value of \( a \). | 1735 |
Consider a "Modulo $m$ graph paper," with $m = 17$, forming a grid of $17^2$ points. Each point $(x, y)$ represents integer pairs where $0 \leq x, y < 17$. We graph the congruence $$5x \equiv 3y + 2 \pmod{17}.$$ The graph has single $x$-intercept $(x_0, 0)$ and single $y$-intercept $(0, y_0)$, where $0 \leq x_0, y_0 < ... | 19 |
Find all $a,$ $0^\circ < a < 360^\circ,$ such that $\cos a,$ $\cos 2a,$ and $\cos 3a$ form an arithmetic sequence, in that order. Enter the solutions, separated by commas, in degrees. | 45^\circ, 135^\circ, 225^\circ, 315^\circ |
Set \( S \) satisfies the following conditions:
1. The elements of \( S \) are positive integers not exceeding 100.
2. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) = 1\).
3. For any \( a, b \in S \) where \( a \neq b \), there ex... | 50 |
Given real numbers $x$, $y$ satisfying $x > y > 0$, and $x + y \leqslant 2$, the minimum value of $\dfrac{2}{x+3y}+\dfrac{1}{x-y}$ is | \dfrac {3+2 \sqrt {2}}{4} |
If $a=\log_8 225$ and $b=\log_2 15$, then | $a=2b/3$ |
Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$. | (3, 2, 6) |
In $\triangle ABC$, $\angle C= \frac{\pi}{2}$, $\angle B= \frac{\pi}{6}$, and $AC=2$. $M$ is the midpoint of $AB$. $\triangle ACM$ is folded along $CM$ such that the distance between $A$ and $B$ is $2\sqrt{2}$. The surface area of the circumscribed sphere of the tetrahedron $M-ABC$ is \_\_\_\_\_\_. | 16\pi |
A production team in a factory is manufacturing a batch of parts. Initially, when each worker is on their own original position, the task can be completed in 9 hours. If the positions of workers $A$ and $B$ are swapped, and other workers' efficiency remains the same, the task can be completed one hour earlier. Similarl... | 108 |
Compute
\[\sum_{k = 1}^\infty \frac{6^k}{(3^k - 2^k)(3^{k + 1} - 2^{k + 1})}.\] | 2 |
Gabriel is 3 years younger than Frank. The current total of their ages right now is 17. What is Frank's age? | Let X be Frank's age.
So Gabriel is X-3 years old.
The total of their ages is X + (X-3) = 17 years old.
2X-3 = 17
2X = 20
X = <<10=10>>10
So Frank is X = <<10=10>>10 years old.
#### 10 |
Ed booked a hotel while he was on vacation. Staying at the hotel cost was $1.50 per hour every night and $2 per hour every morning. If Ed had $80 and he stayed in the hotel for 6 hours last night and 4 hours this morning, how much money was he left with after paying for his stay at the hotel? | Ed spent $1.50 x 6 = $<<1.5*6=9>>9 for staying in the hotel last night.
He spent $2 x 4 = $<<2*4=8>>8 for staying in the hotel this morning
So, the total money he spent in the hotel is $9 + $8 = $<<9+8=17>>17.
Therefore, he still has $80 - $17 = $<<80-17=63>>63 left after staying in the hotel.
#### 63 |
Given any two positive integers, a certain operation (denoted by the operator $\oplus$) is defined as follows: when $m$ and $n$ are both positive even numbers or both positive odd numbers, $m \oplus n = m + n$; when one of $m$ and $n$ is a positive even number and the other is a positive odd number, $m \oplus n = m \cd... | 15 |
The 15th number in a regularly arranged sequence of numbers 2, 1, 4, 3, 6, 5, 8, 7, … is 16. What is the sum of the first 15 numbers? | 121 |
Let $P$ be a point not on line $XY$, and $Q$ a point on line $XY$ such that $PQ \perp XY$. Meanwhile, $R$ is a point on line $PY$ such that $XR \perp PY$. Given $XR = 6$, $PQ = 12$, and $XY = 7$, find the length of $PY$. | 14 |
Lines $L_1, L_2, \dots, L_{100}$ are distinct. All lines $L_{4n}$, where $n$ is a positive integer, are parallel to each other. All lines $L_{4n-3}$, where $n$ is a positive integer, pass through a given point $A$. The maximum number of points of intersection of pairs of lines from the complete set $\{L_1, L_2, \dots, ... | 4351 |
Define $||x||$ $(x\in R)$ as the integer closest to $x$ (when $x$ is the arithmetic mean of two adjacent integers, $||x||$ takes the larger integer). Let $G(x)=||x||$. If $G(\frac{4}{3})=1$, $G(\frac{5}{3})=2$, $G(2)=2$, and $G(2.5)=3$, then $\frac{1}{G(1)}+\frac{1}{G(2)}+\frac{1}{G(3)}+\frac{1}{G(4)}=$______; $\frac{1... | \frac{1334}{15} |
The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?
$\begin{array}{c@{}c@{}c@{}c@{}c}
& & 6 & 4 & 1 \\
& & 8 & 5 & 2 \\
& + & 9 & 7 & 3 \\
\hline
& 2 & 4 & 5 & 6
\end{array}$ | 7 |
My mother celebrated her birthday with a total of 60 guests. Half of the guests are women, 15 are men, and the rest are children. In the middle of the celebration, 1/3 of the men and 5 children left. How many people stayed and enjoyed the birthday celebration? | There were 60 guests / 2 = <<60/2=30>>30 women at my mother's birthday celebration.
There were 30 women + 15 men = <<30+15=45>>45 men and women.
So, 60 people - 45 men and women = <<60-45=15>>15 were children.
Then, 15 men / 3 = <<15/3=5>>5 men left.
Thus, 5 men + 5 children = 10 people left.
Therefore, 60 people - 10 ... |
In a Cartesian coordinate system, the parametric equation for curve $C_1$ is
$$
\left\{ \begin{aligned}
x &= 2\cos\alpha, \\
y &= \sqrt{2}\sin\alpha
\end{aligned} \right.
$$
with $\alpha$ as the parameter. Using the origin as the pole, the positive half of the $x$-axis as the polar axis, and the same unit length as ... | \frac{\sqrt{7} - 1}{2} |
Tony made a sandwich with two slices of bread for lunch every day this week. On Saturday, he was extra hungry from doing yard work and made two sandwiches. How many slices of bread are left from the 22-slice loaf Tony started with? | There are 7 days in a week, so Tony made 1 * 7 = <<1*7=7>>7 sandwiches.
On Saturday, he made 1 extra sandwich, so he made 7 + 1 = <<7+1=8>>8 sandwiches this week.
A sandwich has 2 slices of bread, so he used 8 * 2 = <<8*2=16>>16 slices.
Thus, Tony has 22 - 16 = <<22-16=6>>6 slices of bread left.
#### 6 |
Yvon has 4 different notebooks and 5 different pens. Determine the number of different possible combinations of notebooks and pens he could bring. | 20 |
Bob rolls a fair six-sided die each morning. If Bob rolls a composite number, he eats sweetened cereal. If he rolls a prime number, he eats unsweetened cereal. If he rolls a 1, then he rolls again. In a non-leap year, what is the expected number of times Bob will roll his die? | 438 |
All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$. What is the area of trapezoid $DBCE$? | 20 |
A boy is riding a scooter from one bus stop to another and looking in the mirror to see if a bus appears behind him. As soon as the boy notices the bus, he can change the direction of his movement. What is the maximum distance between the bus stops so that the boy is guaranteed not to miss the bus, given that he rides ... | 1.5 |
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