problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
The integer 119 is a multiple of which number? | 7 | The ones digit of 119 is not even, so 119 is not a multiple of 2. The ones digit of 119 is not 0 or 5, so 119 is not a multiple of 5. Since $120=3 \times 40$, then 119 is 1 less than a multiple of 3 so is not itself a multiple of 3. Since $110=11 \times 10$ and $121=11 \times 11$, then 119 is between two consecutive mu... | 0.75 | 607.375 | 613.833333 | 588 |
Given that $a$ and $b$ are both positive real numbers, and $\frac{1}{a} + \frac{1}{b} = 2$, find the maximum value of $\frac{1}{b}(\frac{2}{a} + 1)$. | \frac{25}{8} | 0.875 | 5,053.0625 | 4,604.642857 | 8,192 | |
Let \( M_n \) be the set of \( n \)-digit pure decimals in decimal notation \( 0.\overline{a_1a_2\cdots a_n} \), where \( a_i \) can only take the values 0 or 1 for \( i=1,2,\cdots,n-1 \), and \( a_n = 1 \). Let \( T_n \) be the number of elements in \( M_n \), and \( S_n \) be the sum of all elements in \( M_n \). Fin... | \frac{1}{18} | 0.25 | 7,425.6875 | 6,030.75 | 7,890.666667 | |
An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Ea... | 180 | The total number of green cubes is given by $12a=20b\Longrightarrow a=\frac{5}{3}b$.
Let $r$ be the number of red cubes on each one of the $b$ layers then the total number of red cubes is $9a=br$. Substitute $a=\frac{5}{3}b$ gives $r=15$.
Repeating the procedure on the number of yellow cubes $y$ on each of the $a$ la... | 0.1875 | 7,700.4375 | 5,570.333333 | 8,192 |
Two couriers start from two locations $A$ and $B$; the first heading towards $B$, and the second towards $A$. When they meet, the first courier has traveled 12 miles more than the second. If they continue their journeys at their original speeds, the first courier will reach their destination 9 days after the meeting, a... | 84 | 0.625 | 5,550.5625 | 4,116.4 | 7,940.833333 | |
Given: $\cos\left(\alpha+ \frac{\pi}{4}\right) = \frac{3}{5}$, $\frac{\pi}{2} < \alpha < \frac{3\pi}{2}$, find $\cos\left(2\alpha+ \frac{\pi}{4}\right)$. | -\frac{31\sqrt{2}}{50} | 0 | 7,541.875 | -1 | 7,541.875 | |
What is the smallest positive integer $n$ such that $17n \equiv 1234 \pmod{7}?$ | 3 | 1 | 2,342.0625 | 2,342.0625 | -1 | |
In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right? | 372 | For each dot in the diagram, we can count the number of paths from January 1 to it by adding the number of ways to get to the dots to the left of it, above it, and above and to the left of it, starting from the topmost leftmost dot. This yields the following numbers of paths: 372. | 0 | 7,852.8125 | -1 | 7,852.8125 |
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD=\angle CMD$. What is the degree measure of $\angle AMD$? | 75 | 1. **Identify the Given Information**:
- Rectangle $ABCD$ has sides $AB = 6$ and $BC = 3$.
- Point $M$ is on side $AB$ such that $\angle AMD = \angle CMD$.
2. **Analyze the Angles**:
- Since $AB \parallel CD$ (as $ABCD$ is a rectangle), by the Alternate Interior Angles Theorem, $\angle AMD = \angle CMD$.
... | 0.5 | 6,381.0625 | 4,570.125 | 8,192 |
Sarah multiplied an integer by itself. Which of the following could be the result? | 36 | 0 | 6,224.375 | -1 | 6,224.375 | |
A community organization begins with twenty members, among which five are leaders. The leaders are replaced annually. Each remaining member persuades three new members to join the organization every year. Additionally, five new leaders are elected from outside the community each year. Determine the total number of memb... | 15365 | 0.0625 | 5,992.3125 | 5,023 | 6,056.933333 | |
Given a tetrahedron \( P-ABCD \) where the edges \( AB \) and \( BC \) each have a length of \(\sqrt{2}\), and all other edges have a length of 1, find the volume of the tetrahedron. | \frac{\sqrt{2}}{6} | 0 | 8,192 | -1 | 8,192 | |
Express \( 0.3\overline{45} \) as a common fraction. | \frac{83}{110} | 0 | 3,904.375 | -1 | 3,904.375 | |
If 25,197,624 hot dogs are packaged in sets of 4, how many will be left over? | 0 | 0.875 | 2,496.625 | 2,243.785714 | 4,266.5 | |
Provide a negative integer solution that satisfies the inequality $3x + 13 \geq 0$. | -1 | 0.3125 | 496.1875 | 514.4 | 487.909091 | |
Twenty tiles are numbered 1 through 20 and are placed into box $A$. Twenty other tiles numbered 11 through 30 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than 15 and the tile from box $B$ is either even or greater than 25? Express your an... | \frac{21}{50} | 0.9375 | 2,388.375 | 2,438.066667 | 1,643 | |
Given an arithmetic sequence ${a_{n}}$ with the sum of its first $n$ terms denoted as $S_{n}$, if $S_{5}$, $S_{4}$, and $S_{6}$ form an arithmetic sequence, then determine the common ratio of the sequence ${a_{n}}$, denoted as $q$. | -2 | 0.25 | 5,964.625 | 4,993 | 6,288.5 | |
A string of 33 pearls has its middle pearl as the largest and most valuable. The values of the remaining pearls decrease by $3000 \mathrm{Ft}$ per pearl towards one end and by $4500 \mathrm{Ft}$ per pearl towards the other end. How much is the middle pearl worth if the total value of the string is 25 times the value of... | 90000 | 0.125 | 6,108.75 | 4,644 | 6,318 | |
How many whole numbers between 1 and 1000 do not contain the digit 1? | 728 | 1. **Identify the Range and Condition**: We need to find how many whole numbers between 1 and 1000 do not contain the digit 1.
2. **Consider the Number of Digits**: Numbers between 1 and 1000 can have 1, 2, or 3 digits. We will consider each case separately.
3. **Counting 1-Digit Numbers**:
- Possible digits: 0, 2... | 0.125 | 7,298.3125 | 4,921.5 | 7,637.857143 |
Car $A$ departs from Station $J$ towards Station $Y$, while cars $B$ and $C$ depart from Station $Y$ towards Station $J$ simultaneously, and move in opposite directions towards car $A$. Car $A$ meets car $B$ first, then 20 minutes later it meets car $C$. Given the speeds of cars $A$, $B$, and $C$ are $90 \text{ km/h}$,... | 425 | 0.5 | 6,078.1875 | 4,254 | 7,902.375 | |
A bridge needs to be constructed over a river with a required elevation of 800 feet from one side to the other. Determine the additional bridge length required if the gradient is reduced from 2% to 1.5%. | 13333.33 | 0.375 | 5,968.5 | 4,177.666667 | 7,043 | |
In a town of $351$ adults, every adult owns a car, motorcycle, or both. If $331$ adults own cars and $45$ adults own motorcycles, how many of the car owners do not own a motorcycle? | 306 | 1. **Identify the total number of adults and their vehicle ownership**:
- Total number of adults: $351$
- Adults owning cars: $331$
- Adults owning motorcycles: $45$
2. **Apply the Principle of Inclusion-Exclusion (PIE)**:
- The formula for PIE in this context is:
\[
|A \cup B| = |A| + |B| - |A \... | 1 | 1,408.4375 | 1,408.4375 | -1 |
In the diagram, all rows, columns, and diagonals have the sum 12. What is the sum of the four corner numbers? | 16 | 0.75 | 5,845.375 | 5,063.166667 | 8,192 | |
For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$? | 16 | 0.0625 | 7,669.3125 | 5,394 | 7,821 | |
How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? | 240 | 0 | 8,192 | -1 | 8,192 | |
Jia and his four friends each have a private car. The last digit of Jia's license plate is 0, and the last digits of his four friends' license plates are 0, 2, 1, 5, respectively. To comply with the local traffic restrictions from April 1st to 5th (cars with odd-numbered last digits are allowed on odd days, and cars wi... | 64 | 0 | 8,092.8125 | -1 | 8,092.8125 | |
Find $\frac{9}{10}+\frac{5}{6}$. Express your answer as a fraction in simplest form. | \frac{26}{15} | 1 | 1,934.625 | 1,934.625 | -1 | |
A teacher received letters on Monday through Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the variance (${s^{2}} =$) of this data set. | 3.2 | 0.125 | 2,104.125 | 1,891.5 | 2,134.5 | |
The line $l: x - 2y + 2 = 0$ passes through the left focus F<sub>1</sub> and a vertex B of an ellipse. Find the eccentricity of the ellipse. | \frac{2\sqrt{5}}{5} | 0 | 6,664.75 | -1 | 6,664.75 | |
Let $\frac{1}{1-x-x^{2}-x^{3}}=\sum_{i=0}^{\infty} a_{n} x^{n}$, for what positive integers $n$ does $a_{n-1}=n^{2}$ ? | 1, 9 | Multiplying both sides by $1-x-x^{2}-x^{3}$ the right hand side becomes $a_{0}+\left(a_{1}-a_{0}\right) x+\left(a_{2}-a_{1}-a_{0}\right) x^{2}+\ldots$, and setting coefficients of $x^{n}$ equal to each other we find that $a_{0}=1, a_{1}=1, a_{2}=2$, and $a_{n}=a_{n-1}+a_{n-2}+a_{n-3}$ for $n \geq 3$. Thus the sequence ... | 0 | 7,300.6875 | -1 | 7,300.6875 |
Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 98.$ Find $\frac n{100}.$
| 196 | 0.9375 | 3,585.1875 | 3,278.066667 | 8,192 | |
From the consecutive natural numbers 1, 2, 3, …, 2014, select $n$ numbers such that for any two numbers chosen, one is not five times the other. Find the maximum value of $n$ and explain the reason. | 1665 | 0 | 8,102.5 | -1 | 8,102.5 | |
What is the area of the polygon with vertices at $(2, 1)$, $(4, 3)$, $(6, 1)$, $(5, -2)$, and $(3, -2)$? | 13 | 0.75 | 6,838.0625 | 6,386.75 | 8,192 | |
If $x+y=\frac{7}{13}$ and $x-y=\frac{1}{91}$, what is the value of $x^2-y^2$? Express your answer as a common fraction. | \frac{1}{169} | 0.9375 | 2,691.5625 | 2,775.333333 | 1,435 | |
Given that $y < 1$ and \[(\log_{10} y)^2 - \log_{10}(y^3) = 75,\] compute the value of \[(\log_{10}y)^3 - \log_{10}(y^4).\] | \frac{2808 - 336\sqrt{309}}{8} - 6 + 2\sqrt{309} | 0 | 7,090.375 | -1 | 7,090.375 | |
How many positive two-digit integers are there in which each of the two digits is prime? | 16 | 1 | 1,613.5625 | 1,613.5625 | -1 | |
To obtain the graph of the function $y=\sin\left(2x+\frac{\pi}{3}\right)$, find the transformation required to obtain the graph of the function $y=\cos\left(2x-\frac{\pi}{3}\right)$. | \left(\frac{\pi}{12}\right) | 0 | 6,904.5 | -1 | 6,904.5 | |
Three marbles are randomly selected, without replacement, from a bag containing two red, two blue and two green marbles. What is the probability that one marble of each color is selected? Express your answer as a common fraction. | \frac{2}{5} | 0.875 | 4,401.1875 | 3,859.642857 | 8,192 | |
The price of Type A remote control car is 46.5 yuan, and the price of Type B remote control car is 54.5 yuan. Lele has 120 yuan. If he buys both types of remote control cars, will he have enough money? If so, how much money will he have left after the purchase? | 19 | 0.625 | 447.125 | 446.2 | 448.666667 | |
The set $H$ is defined by the points $(x, y)$ with integer coordinates, $-8 \leq x \leq 8$, $-8 \leq y \leq 8$. Calculate the number of squares of side length at least $9$ that have their four vertices in $H$. | 81 | 0 | 8,192 | -1 | 8,192 | |
Given $(x+1)^4(x+4)^8 = a + a_1(x+3) + a_2(x+3)^2 + \ldots + a_{12}(x+3)^{12}$, find the value of $a_2 + a_4 + \ldots + a_{12}$. | 112 | 0.3125 | 5,341.4375 | 4,488.6 | 5,729.090909 | |
There is a five-digit number that, when divided by each of the 12 natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13, gives different remainders. What is this five-digit number? | 83159 | 0 | 7,376.0625 | -1 | 7,376.0625 | |
The expression $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to: | \frac{\sqrt{3}}{6} | 1. **Simplify $\sqrt{\frac{4}{3}}$:**
\[
\sqrt{\frac{4}{3}} = \frac{\sqrt{4}}{\sqrt{3}} = \frac{2}{\sqrt{3}}
\]
Rationalizing the denominator:
\[
\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\]
2. **Simplify $\sqrt{\frac{3}{4}}$:**
\[
\sqrt{\frac{3}{4}} = \frac{\s... | 0 | 2,263.3125 | -1 | 2,263.3125 |
In triangle $ABC$, $AB=125$, $AC=117$ and $BC=120$. The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$, respectively. Find ... | 56 | 0 | 8,192 | -1 | 8,192 | |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the... | \frac{2\sqrt{3}+1}{3} | 0 | 6,777.875 | -1 | 6,777.875 | |
Find the largest real number \(x\) such that
\[
\frac{x^{2} + x - 1 + \left|x^{2} - (x - 1)\right|}{2} = 35x - 250.
\] | 25 | 0.9375 | 3,817.4375 | 3,525.8 | 8,192 | |
Let $f(x) = 3x + 3$ and $g(x) = 4x + 3.$ What is $f(g(f(2)))$? | 120 | 1 | 2,236.3125 | 2,236.3125 | -1 | |
Camille the snail lives on the surface of a regular dodecahedron. Right now he is on vertex $P_{1}$ of the face with vertices $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$. This face has a perimeter of 5. Camille wants to get to the point on the dodecahedron farthest away from $P_{1}$. To do so, he must travel along the surface ... | \frac{17+7 \sqrt{5}}{2} | Consider the net of the dodecahedron. It suffices to look at three pentagons $A B C D E, E D F G H$, and $G F I J K$, where $A J=L$. This can be found by the law of cosines on triangle $A E J$. We have $A E=1$, $E J=\tan 72^{\circ}$, and $\angle A E J=162^{\circ}$. Thus $L^{2}=1+\tan ^{2} 72^{\circ}+2 \cdot \tan 72^{\c... | 0 | 4,905.375 | -1 | 4,905.375 |
Given an ellipse $C$ with its center at the origin and its foci on the $x$-axis, and its eccentricity equal to $\frac{1}{2}$. One of its vertices is exactly the focus of the parabola $x^{2}=8\sqrt{3}y$.
(Ⅰ) Find the standard equation of the ellipse $C$;
(Ⅱ) If the line $x=-2$ intersects the ellipse at points $P$ and ... | \frac{1}{2} | 0.0625 | 8,164.125 | 7,746 | 8,192 | |
What is the largest $2$-digit prime factor of the integer $n = {300\choose 150}$? | 97 | 0.75 | 5,268.625 | 5,228 | 5,390.5 | |
Let $f_{1}(x)=\sqrt{1-x}$, and for integers $n \geq 2$, let \[f_{n}(x)=f_{n-1}\left(\sqrt{n^2 - x}\right).\]Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N,$ the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$ | -231 | 0.75 | 5,274.6875 | 5,175.916667 | 5,571 | |
Three positive integers have a sum of 72 and are in the ratio 1:3:4. What is the least of these three integers? | 9 | 1 | 1,375.25 | 1,375.25 | -1 | |
The line passing through the points (3,9) and (-1,1) has an x-intercept of ( ). | -\frac{3}{2} | 0.5625 | 2,755.1875 | 3,133.888889 | 2,268.285714 | |
The ratio $\frac{2^{2001} \cdot 3^{2003}}{6^{2002}}$ is: | \frac{3}{2} | 1. **Rewrite the given expression using the properties of exponents:**
\[
\frac{2^{2001} \cdot 3^{2003}}{6^{2002}}
\]
2. **Express $6^{2002}$ in terms of $2$ and $3$:**
\[
6^{2002} = (2 \cdot 3)^{2002} = 2^{2002} \cdot 3^{2002}
\]
3. **Substitute this expression back into the original ratio:**
\[... | 1 | 3,161.75 | 3,161.75 | -1 |
In a kindergarten's junior group, there are two (small) Christmas trees and five children. The caregivers want to divide the children into two round dances around each of the trees, with each round dance having at least one child. The caregivers distinguish between children but do not distinguish between the trees: two... | 50 | 0.1875 | 7,682.0625 | 6,862.666667 | 7,871.153846 | |
Margaret started a stamp collection. She collected 8 stamps the first day. Each subsequent day she collected 8 more stamps than she had collected the previous day. If she collected stamps for 5 consecutive days, what was the average number of stamps collected per day? | 24 | 1 | 1,507.3125 | 1,507.3125 | -1 | |
A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps. After a total of $n$ jumps, the position of the grasshopper is 162 cm to the west and 158 cm to the south of its original position... | 22 | Each group of four jumps takes the grasshopper 1 cm to the east and 3 cm to the west, which is a net movement of 2 cm to the west, and 2 cm to the north and 4 cm to the south, which is a net movement of 2 cm to the south. We note that $158=2 \times 79$. Thus, after 79 groups of four jumps, the grasshopper is $79 \times... | 0.125 | 7,830.5625 | 5,300.5 | 8,192 |
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$... | \mathbb{Z} |
To find all rootiful sets of integers \( S \) that contain all numbers of the form \( 2^a - 2^b \) for positive integers \( a \) and \( b \), we need to analyze the properties of such sets.
### Step 1: Understand the Definition
A set \( S \) is rootiful if, for any positive integer \( n \) and any integers \( a_0, a... | 0.25 | 7,617.9375 | 6,423.75 | 8,016 |
In triangle $ABC$, $\angle A = 60^\circ$ and $\angle B = 45^\circ$. A line $DE$, with $D$ on $AB$ and $\angle ADE = 45^\circ$, divides $\triangle ABC$ into two pieces of equal area. Determine the ratio $\frac{AD}{AB}$.
A) $\frac{\sqrt{6}}{4}$
B) $\frac{\sqrt{7}}{4}$
C) $\frac{\sqrt{8}}{4}$
D) $\frac{\sqrt{6} + \s... | \frac{\sqrt{6} + \sqrt{2}}{4\sqrt{2}} | 0 | 8,192 | -1 | 8,192 | |
Determine the ratio of the shorter side to the longer side of the rectangular park, given that by taking a shortcut along the diagonal, a boy saved a distance equal to $\frac{1}{3}$ of the longer side of the park. | \frac{5}{12} | 0.9375 | 2,453 | 2,149 | 7,013 | |
As shown in the diagram, $E$ is the midpoint of the leg $AB$ of trapezoid $ABCD$. $DF \perp EC$, $DF=10$, and $EC=24$. Find the area of trapezoid $ABCD$. | 240 | 0.0625 | 8,192 | 8,192 | 8,192 | |
The reciprocal of the opposite number of \(-(-3)\) is \(\frac{1}{3}\). | -\frac{1}{3} | 0.3125 | 5,677.6875 | 7,025 | 5,065.272727 | |
Find the remainder when $5x^4 - 12x^3 + 3x^2 - 5x + 15$ is divided by $3x - 9$. | 108 | 0.875 | 3,821.1875 | 3,196.785714 | 8,192 | |
In a sealed box, there are three red chips and two green chips. Chips are randomly drawn from the box without replacement until either all three red chips or both green chips are drawn. What is the probability of drawing all three red chips? | $\frac{2}{5}$ | 0 | 8,192 | -1 | 8,192 | |
What is the greatest common divisor of 128, 144 and 480? | 16 | 1 | 2,341.9375 | 2,341.9375 | -1 | |
A football association stipulates that in the league, a team earns $a$ points for a win, $b$ points for a draw, and 0 points for a loss, where $a$ and $b$ are real numbers such that $a > b > 0$. If a team has 2015 possible total scores after $n$ games, find the minimum value of $n$. | 62 | 0.1875 | 7,517.6875 | 4,941.333333 | 8,112.230769 | |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $2a\cos A=c\cos B+b\cos C$.
1. Find the value of $\cos A$.
2. If $a=1$ and $\cos^{2}\frac{B}{2}+\cos^{2}\frac{C}{2}=1+\frac{\sqrt{3}}{4}$, find the length of side $c$. | \frac { \sqrt {3}}{3} | 0 | 7,223 | -1 | 7,223 | |
A truncated pyramid has a square base with a side length of 4 units, and every lateral edge is also 4 units. The side length of the top face is 2 units. What is the greatest possible distance between any two vertices of the truncated pyramid? | \sqrt{32} | 0 | 7,269.1875 | -1 | 7,269.1875 | |
Quadrilateral \(ABCD\) is inscribed in a circle with diameter \(AD\) having a length of 4. If the lengths of \(AB\) and \(BC\) are each 1, calculate the length of \(CD\). | \frac{7}{2} | 0.875 | 6,079.1875 | 5,777.357143 | 8,192 | |
Given that $\sin x \cdot \cos x = -\frac{1}{4}$ and $\frac{3\pi}{4} < x < \pi$, find the value of $\sin x + \cos x$. | -\frac{\sqrt{2}}{2} | 0 | 4,070.75 | -1 | 4,070.75 | |
For a positive integer $n$ , let $v(n)$ denote the largest integer $j$ such that $n$ is divisible by $2^j$ . Let $a$ and $b$ be chosen uniformly and independently at random from among the integers between 1 and 32, inclusive. What is the probability that $v(a) > v(b)$ ? | 341/1024 | 0.5625 | 6,841.4375 | 5,791 | 8,192 | |
Jacqueline has 2 liters of soda. Liliane has 60% more soda than Jacqueline, and Alice has 40% more soda than Jacqueline. Calculate the percentage difference between the amount of soda Liliane has compared to Alice. | 14.29\% | 0.5 | 4,048.375 | 3,689 | 4,407.75 | |
Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$ . $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\angle ZWX$ is a right angle. Given that $WX = WZ = \sqrt{13}$ and $XY = ZY$ , determine the length of $XY$ . | \sqrt{65} | 0.75 | 6,396 | 5,797.333333 | 8,192 | |
Let $z$ be a complex number such that $|z| = 2.$ Find the largest possible distance between $(3 + 4i)z^3$ and $z^5$ when plotted in the complex plane. | 72 | 0.5625 | 6,116.0625 | 4,982 | 7,574.142857 | |
For any $x \in (0, +\infty)$, the inequality $(x-a+\ln \frac{x}{a})(-2x^2+ax+10) \leq 0$ always holds. Then, the range of the real number $a$ is ______. | \sqrt{10} | 0 | 8,155.625 | -1 | 8,155.625 | |
Henry decides one morning to do a workout, and he walks $\frac{3}{4}$ of the way from his home to his gym. The gym is $2$ kilometers away from Henry's home. At that point, he changes his mind and walks $\frac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and ... | 1 \frac{1}{5} | 1. **Define the sequence of positions**: Let $A$ be the point closer to Henry’s home, and $B$ be the point closer to the gym. Define $(a_n)$ to be the position of Henry after $2n$ walks (even steps, returning towards home), and $(b_n)$ to be the position of Henry after $2n - 1$ walks (odd steps, going towards the gym).... | 0 | 7,149 | -1 | 7,149 |
Given four positive integers \(a, b, c,\) and \(d\) satisfying the equations \(a^2 = c(d + 20)\) and \(b^2 = c(d - 18)\). Find the value of \(d\). | 180 | 0.3125 | 7,939.4375 | 7,383.8 | 8,192 | |
For what values of $x$ is \[\frac{x-10x^2+25x^3}{8-x^3}\]nonnegative? Answer as an interval. | [0,2) | 0.6875 | 6,698.1875 | 6,183.363636 | 7,830.8 | |
For each integer $x$ with $1 \leq x \leq 10$, a point is randomly placed at either $(x, 1)$ or $(x,-1)$ with equal probability. What is the expected area of the convex hull of these points? Note: the convex hull of a finite set is the smallest convex polygon containing it. | \frac{1793}{128} | Let $n=10$. Given a random variable $X$, let $\mathbb{E}(X)$ denote its expected value. If all points are collinear, then the convex hull has area zero. This happens with probability $\frac{2}{2^{n}}$ (either all points are at $y=1$ or all points are at $y=-1$ ). Otherwise, the points form a trapezoid with height 2 (th... | 0 | 7,771.0625 | -1 | 7,771.0625 |
Jack walked up a hill at a speed of $(x^2-11x-22)$ miles per hour. Meanwhile, Jill walked a total distance of $(x^2-3x-54)$ miles in $(x+6)$ hours. If Jack and Jill walked at the same speed, what is that speed, in miles per hour? | 4 | 1 | 2,345.6875 | 2,345.6875 | -1 | |
Find the sum of $452_8$ and $164_8$ in base $8$. | 636_8 | 0.625 | 4,541.1875 | 2,350.7 | 8,192 | |
In the diagram, \(O\) is the center of a circle with radii \(OA=OB=7\). A quarter circle arc from \(A\) to \(B\) is removed, creating a shaded region. What is the perimeter of the shaded region? | 14 + 10.5\pi | 0 | 4,603.5625 | -1 | 4,603.5625 | |
If
\[1 \cdot 1987 + 2 \cdot 1986 + 3 \cdot 1985 + \dots + 1986 \cdot 2 + 1987 \cdot 1 = 1987 \cdot 994 \cdot x,\]compute the integer $x.$ | 663 | 0.875 | 4,955.875 | 4,493.571429 | 8,192 | |
Evaluate $|\omega^2 + 7\omega + 40|$ if $\omega = 4 + 3i$. | 15\sqrt{34} | 0.9375 | 2,548 | 2,542.4 | 2,632 | |
The graph of the power function $f(x)$ passes through the point $(3, \frac{1}{9})$, find the maximum value of the function $g(x) = (x-1)f(x)$ on the interval $[1,3]$. | \frac{1}{4} | 0.25 | 7,589.4375 | 6,647.75 | 7,903.333333 | |
Find the area of triangle $ABC$ given below:
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$1$",(B+C)/2,NE);
label("$45^\circ$",(0,0.75),E);
[/asy] | \frac{1}{4} | 0.75 | 5,339.625 | 4,892.25 | 6,681.75 | |
In the figure shown, arc $ADB$ and arc $BEC$ are semicircles, each with a radius of one unit. Point $D$, point $E$ and point $F$ are the midpoints of arc $ADB$, arc $BEC$ and arc $DFE$, respectively. If arc $DFE$ is also a semicircle, what is the area of the shaded region?
[asy]
unitsize(0.5inch);
path t=(1,1)..(2,0)-... | 2 | 0 | 7,394.8125 | -1 | 7,394.8125 | |
A rhombus has an area of 108 square units. The lengths of its diagonals have a ratio of 3 to 2. What is the length of the longest diagonal, in units? | 18 | 1 | 1,225.6875 | 1,225.6875 | -1 | |
Two symmetrical coins are flipped. What is the probability that both coins show numbers on their upper sides? | 0.25 | 0 | 4,801.4375 | -1 | 4,801.4375 | |
On a sheet of graph paper, two rectangles are outlined. The first rectangle has a vertical side shorter than the horizontal side, and for the second rectangle, the opposite is true. Find the maximum possible area of their intersection if the first rectangle contains 2015 cells and the second one contains 2016 cells. | 1302 | 0 | 8,114.6875 | -1 | 8,114.6875 | |
A and B bought the same number of sheets of stationery. A put 1 sheet of stationery into each envelope and had 40 sheets of stationery left after using all the envelopes. B put 3 sheets of stationery into each envelope and had 40 envelopes left after using all the sheets of stationery. How many sheets of stationery did... | 120 | 0.1875 | 2,335.3125 | 699.333333 | 2,712.846154 | |
Digital clocks display hours and minutes (for example, 16:15). While practicing arithmetic, Buratino finds the sum of the digits on the clock $(1+6+1+5=13)$. Write down such a time of day when the sum of the digits on the clock will be the greatest. | 19:59 | 0.125 | 8,062.875 | 8,142 | 8,051.571429 | |
The digits from 1 to 9 are each used exactly once to write three one-digit integers and three two-digit integers. The one-digit integers are equal to the length, width and height of a rectangular prism. The two-digit integers are equal to the areas of the faces of the same prism. What is the surface area of the rectang... | 198 | 0.125 | 8,036.6875 | 6,949.5 | 8,192 | |
Given two lines $l_{1}$: $x+my+6=0$, and $l_{2}$: $(m-2)x+3y+2m=0$, if the lines $l_{1}\parallel l_{2}$, then $m=$_______. | -1 | 0.375 | 6,761.4375 | 6,880.5 | 6,690 | |
Below is a portion of the graph of a function, $y=u(x)$:
[asy]
import graph; size(5.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=3.25,ymin=-3.25,ymax=3.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,... | 0 | 1 | 2,967.25 | 2,967.25 | -1 | |
How many multiples of 15 are between 15 and 305? | 20 | 0.125 | 4,685.6875 | 4,726 | 4,679.928571 | |
The bases \(AB\) and \(CD\) of the trapezoid \(ABCD\) are 41 and 24 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\). | 984 | 0.6875 | 5,307.375 | 4,744.454545 | 6,545.8 | |
Let the line $p$ be the perpendicular bisector of $A = (24, 7)$ and $B = (3, 4).$ Given that $AB$ meets $p$ at $C = (x, y),$ what is $2x - 4y$? | 5 | 1 | 2,837.9375 | 2,837.9375 | -1 | |
In a square $\mathrm{ABCD}$, point $\mathrm{E}$ is on $\mathrm{BC}$ with $\mathrm{BE} = 2$ and $\mathrm{CE} = 1$. Point $\mathrm{P}$ moves along $\mathrm{BD}$. What is the minimum value of $\mathrm{PE} + \mathrm{PC}$? | \sqrt{13} | 0.6875 | 6,791.875 | 6,155.454545 | 8,192 | |
Eleven positive integers from a list of fifteen positive integers are $3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23$. What is the largest possible value of the median of this list of fifteen positive integers? | 17 | 0 | 8,192 | -1 | 8,192 |
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