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 |
|---|---|---|---|---|---|---|
Five cards labeled $1,3,5,7,9$ are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected value ... | 50308 | For a given card, let $p(n)$ denote the probability that it is in its original position after $n$ swaps. Then $p(n+1)=p(n) \cdot \frac{3}{5}+(1-p(n)) \cdot \frac{1}{10}$, by casework on whether the card is in the correct position or not after $n$ swaps. In particular, $p(0)=1, p(1)=3 / 5, p(2)=2 / 5$, and $p(3)=3 / 10$... | 0 | 7,666.5625 | -1 | 7,666.5625 |
Each vertex of a regular dodecagon ($12$-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle. | 928 | Note that the condition is equivalent to stating that there are no 2 pairs of oppositely spaced vertices with the same color.
Case 1: There are no pairs. This yields $2$ options for each vertices 1-6, and the remaining vertices 7-12 are set, yielding $2^6=64$ cases.
Case 2: There is one pair. Again start with 2 optio... | 0 | 8,192 | -1 | 8,192 |
Let $A B C D$ be a quadrilateral with $A=(3,4), B=(9,-40), C=(-5,-12), D=(-7,24)$. Let $P$ be a point in the plane (not necessarily inside the quadrilateral). Find the minimum possible value of $A P+B P+C P+D P$. | 16 \sqrt{17}+8 \sqrt{5} | By the triangle inequality, $A P+C P \geq A C$ and $B P+D P \geq B D$. So $P$ should be on $A C$ and $B D$; i.e. it should be the intersection of the two diagonals. Then $A P+B P+C P+D P=A C+B D$, which is easily computed to be $16 \sqrt{17}+8 \sqrt{5}$ by the Pythagorean theorem. Note that we require the intersection ... | 0 | 8,192 | -1 | 8,192 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -\sqrt{3}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{5\pi}{6} | 0.6875 | 1,794.0625 | 1,956.727273 | 1,436.2 | |
In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE=8$, and $AF=4$. Find the area of $ABCD$. | 192\sqrt{3}-96 | 0.0625 | 7,688.5 | 7,354 | 7,710.8 | |
There are 21 nonzero numbers. For each pair of these numbers, their sum and product are calculated. It turns out that half of all the sums are positive and half are negative. What is the maximum possible number of positive products? | 120 | 0.0625 | 8,127.1875 | 7,155 | 8,192 | |
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
(G. Galperin) | 1990 | 0.8125 | 5,715.4375 | 5,143.923077 | 8,192 | |
In three-dimensional space, find the number of lattice points that have a distance of 5 from the origin.
Note: A point is a lattice point if all its coordinates are integers. | 54 | 0 | 8,071.5625 | -1 | 8,071.5625 | |
Let $n^{}_{}$ be the smallest positive integer that is a multiple of $75_{}^{}$ and has exactly $75_{}^{}$ positive integral divisors, including $1_{}^{}$ and itself. Find $\frac{n}{75}$. | 432 | The prime factorization of $75 = 3^15^2 = (2+1)(4+1)(4+1)$. For $n$ to have exactly $75$ integral divisors, we need to have $n = p_1^{e_1-1}p_2^{e_2-1}\cdots$ such that $e_1e_2 \cdots = 75$. Since $75|n$, two of the prime factors must be $3$ and $5$. To minimize $n$, we can introduce a third prime factor, $2$. Also to ... | 0.375 | 7,814.4375 | 7,185.166667 | 8,192 |
A regular 2015-gon \( A_{1} A_{2} \cdots A_{2015} \) is inscribed in a unit circle \( O \). What is the probability that for any two distinct vertices \( A_{i}, A_{j} \), the magnitude \( \left|\overrightarrow{O A_{i}}+\overrightarrow{O A_{j}}\right| \geqslant 1 \) is true? | 671/1007 | 0 | 7,894.5 | -1 | 7,894.5 | |
The math scores of a high school's college entrance examination, denoted as $\xi$, approximately follow a normal distribution $N(100, 5^2)$. Given that $P(\xi < 110) = 0.98$, find the value of $P(90 < \xi < 100)$. | 0.48 | 0.5625 | 5,084.625 | 4,058 | 6,404.571429 | |
John's flight departed from Chicago at 3:15 PM and landed in Denver at 4:50 PM. Considering that Denver is one hour behind Chicago, if his flight took $h$ hours and $m$ minutes, with $0 < m < 60$, calculate the value of $h + m$. | 36 | 0.25 | 665.25 | 514.25 | 715.583333 | |
What is the largest real number $\theta$ less than $\pi$ (i.e. $\theta<\pi$ ) such that $\prod_{k=0}^{10} \cos \left(2^{k} \theta\right) \neq 0$ and $\prod_{k=0}^{10}\left(1+\frac{1}{\cos \left(2^{k} \theta\right)}\right)=1 ? | \frac{2046 \pi}{2047} | For equality to hold, note that $\theta$ cannot be an integer multiple of $\pi$ (or else $\sin =0$ and $\cos = \pm 1$ ). Let $z=e^{i \theta / 2} \neq \pm 1$. Then in terms of complex numbers, we want $\prod_{k=0}^{10}\left(1+\frac{2}{z^{2^{k+1}}+z^{-2^{k+1}}}\right)=\prod_{k=0}^{10} \frac{\left(z^{2^{k}}+z^{-2^{k}}\rig... | 0 | 8,192 | -1 | 8,192 |
Let $f$ be a function from the nonnegative integers to the positive reals such that $f(x+y)=f(x) \cdot f(y)$ holds for all nonnegative integers $x$ and $y$. If $f(19)=524288 k$, find $f(4)$ in terms of $k$. | 16 k^{4 / 19} | The given condition implies $f(m n)=f(m)^{n}$, so $$f(4)^{19}=f(4 \cdot 19)=f(19 \cdot 4)=f(19)^{4}$$ and it follows that $f(4)=16 k^{4 / 19}$. | 0.875 | 3,028.0625 | 3,072.857143 | 2,714.5 |
Jerry and Neil have a 3-sided die that rolls the numbers 1, 2, and 3, each with probability $\frac{1}{3}$. Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is 3. | \frac{11}{18} | If Jerry rolls $k$, then there is a $\frac{1}{4-k}$ probability that Neil's number is 3, since Neil has an equal chance of rolling any of the $4-k$ integers not less than $k$. Thus, the answer is $$\frac{1}{3}\left(1+\frac{1}{2}+\frac{1}{3}\right)=\frac{11}{18}$$. | 0.4375 | 6,972.375 | 5,713.857143 | 7,951.222222 |
Two people are flipping a coin: one flipped it 10 times, and the other flipped it 11 times. Find the probability that the second person got heads more times than the first person. | \frac{1}{2} | 0 | 7,939.6875 | -1 | 7,939.6875 | |
Let \( S = \{ 1, 2, \cdots, 2005 \} \). Find the smallest number \( n \) such that in any subset of \( n \) pairwise coprime numbers from \( S \), there is at least one prime number. | 16 | 0.0625 | 7,747.8125 | 6,199 | 7,851.066667 | |
A pedestrian traffic light allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for a pedestrian who approaches the intersection. | 40 | 0 | 5,670.1875 | -1 | 5,670.1875 | |
Simplify $\frac{\sin 7^{\circ}+\cos 15^{\circ} \cdot \sin 8^{\circ}}{\cos 7^{\circ}-\sin 15^{\circ} \cdot \sin 8^{\circ}}$. The value equals ( ). | $2-\sqrt{3}$ | 0 | 5,251.0625 | -1 | 5,251.0625 | |
Given a hyperbola with the equation $\frac{y^2}{4} - \frac{x^2}{a} = 1$ and asymptote equations $y = \pm\frac{2\sqrt{3}}{3}x$, find the eccentricity of this hyperbola. | \frac{\sqrt{7}}{2} | 0 | 3,177.0625 | -1 | 3,177.0625 | |
Three coplanar circles intersect as shown. What is the maximum number of points on the circles that a line passing through all three circles can touch?
[asy]import graph;
draw(Circle((-9,9),15));
draw(Circle((0,-9),15));
draw(Circle((9,9),15));
[/asy] | 6 | 0.75 | 5,947.25 | 5,297.083333 | 7,897.75 | |
Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$ .
Find the greatest value of $M = abc$ | 6171 | 0 | 8,192 | -1 | 8,192 | |
Given $M$ be the second smallest positive integer that is divisible by every positive integer less than 9, find the sum of the digits of $M$. | 15 | 0.9375 | 2,987.25 | 2,640.266667 | 8,192 | |
Chess piece called *skew knight*, if placed on the black square, attacks all the gray squares.
What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other?
*Proposed by Arsenii Nikola... | 32 | 0.6875 | 6,957.1875 | 6,395.909091 | 8,192 | |
A straight line joins the points $(-1,1)$ and $(3,9)$. Its $x$-intercept is: | -\frac{3}{2} | 1. **Identify the coordinates of the given points and calculate the slope ($m$) of the line.**
The points given are $(-1,1)$ and $(3,9)$. The slope formula is:
\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 1}{3 - (-1)} = \frac{8}{4} = 2.
\]
2. **Write the equation of the line u... | 0.8125 | 2,713.75 | 2,706.153846 | 2,746.666667 |
Descartes's Blackjack: How many integer lattice points (points of the form $(m, n)$ for integers $m$ and $n$) lie inside or on the boundary of the disk of radius 2009 centered at the origin? | 12679605 | The number of lattice points inside or on the boundary of a circle with radius $r$ centered at the origin can be approximated using the formula $\pi r^2 + \text{error term}$. For a circle with radius 2009, this results in approximately 12679605 lattice points. | 0 | 8,192 | -1 | 8,192 |
Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters? | 26 | 1 | 1,922.4375 | 1,922.4375 | -1 | |
A class has a group of 7 people, and now 3 of them are chosen to swap seats with each other, while the remaining 4 people's seats remain unchanged. Calculate the number of different rearrangement plans. | 70 | 0.875 | 4,166.0625 | 3,980.785714 | 5,463 | |
At the end of the year, the Math Club decided to hold an election for which 5 equal officer positions were available. However, 16 candidates were nominated, of whom 7 were past officers. Of all possible elections of the officers, how many will have at least 1 of the past officers? | 4242 | 0.8125 | 3,448.9375 | 3,064.230769 | 5,116 | |
Evaluate the expression $1 - \frac{1}{1 + \sqrt{5}} + \frac{1}{1 - \sqrt{5}}$. | 1 - \frac{\sqrt{5}}{2} | 0 | 5,734.5625 | -1 | 5,734.5625 | |
Four distinct natural numbers, one of which is an even prime number, have the following properties:
- The sum of any two numbers is a multiple of 2.
- The sum of any three numbers is a multiple of 3.
- The sum of all four numbers is a multiple of 4.
Find the smallest possible sum of these four numbers. | 44 | 0.3125 | 7,728.0625 | 6,707.4 | 8,192 | |
From a deck of cards marked with 1, 2, 3, and 4, two cards are drawn consecutively. The probability of drawing the card with the number 4 on the first draw, the probability of not drawing it on the first draw but drawing it on the second, and the probability of drawing the number 4 at any point during the drawing proce... | \frac{1}{2} | 0.9375 | 3,953.3125 | 3,670.733333 | 8,192 | |
Consider a $2 \times 2$ grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of the numbers in each column. If the sum of the eight numbers he writes down is 2015, what is the minimum po... | 88 | Let the four numbers be $a, b, c, d$, so that the other four numbers are $a b, a d, b c, b d$. The sum of these eight numbers is $a+b+c+d+a b+a d+b c+b d=(a+c)+(b+d)+(a+c)(b+d)=2015$, and so $(a+c+1)(b+d+1)=2016$. Since we seek to minimize $a+b+c+d$, we need to find the two factors of 2016 that are closest to each othe... | 0.4375 | 7,137 | 5,780.571429 | 8,192 |
Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides $5$ times as fast as Bella walks. The distance between their houses is $2$ miles, which is $10,560$ feet, and Bella covers $2 \f... | 704 | 1. **Understanding the problem**: Bella and Ella start moving towards each other from their respective houses, which are 2 miles apart. Bella walks and Ella rides a bicycle, with Ella's speed being 5 times Bella's speed. We need to find how many steps Bella takes before they meet.
2. **Convert the distance from miles ... | 1 | 1,916.5625 | 1,916.5625 | -1 |
Rationalize the denominator of $\displaystyle\frac{21}{\sqrt{21}}$. | \!\sqrt{21} | 1 | 1,707.1875 | 1,707.1875 | -1 | |
Ali chooses one of the stones from a group of $2005$ stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two ... | 11 | 0.0625 | 7,791.9375 | 8,192 | 7,765.266667 | |
If $P = \sqrt{1988 \cdot 1989 \cdot 1990 \cdot 1991 + 1} + \left(-1989^{2}\right)$, then the value of $P$ is | 1988 | 0.6875 | 5,194.9375 | 4,508.909091 | 6,704.2 | |
A mini soccer team has 12 members. We want to choose a starting lineup of 5 players, which includes one goalkeeper and four outfield players (order of outfield players does not matter). In how many ways can we choose this starting lineup? | 3960 | 0.9375 | 2,306.9375 | 1,914.6 | 8,192 | |
Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left focus $F$ and a chord perpendicular to the major axis of length $6\sqrt{2}$, a line passing through point $P(2,1)$ with slope $-1$ intersects $C$ at points $A$ and $B$, where $P$ is the midpoint of $AB$. Find the maximum distance fr... | 6\sqrt{2} + 6 | 0 | 7,836.1875 | -1 | 7,836.1875 | |
If there exists a line $l$ that is a tangent to the curve $y=x^{2}$ and also a tangent to the curve $y=a\ln x$, then the maximum value of the real number $a$ is ____. | 2e | 0.875 | 4,454.8125 | 3,920.928571 | 8,192 | |
Lucy started with a bag of 180 oranges. She sold $30\%$ of them to Max. From the remaining, she then sold $20\%$ to Maya. Of the oranges left, she donated 10 to a local charity. Find the number of oranges Lucy had left. | 91 | 0.3125 | 5,256 | 1,702.4 | 6,871.272727 | |
A certain school arranges $5$ senior high school teachers to visit $3$ schools for exchange and learning. One school has $1$ teacher, one school has $2$ teachers, and one school has $2$ teachers. The total number of different arrangements is _______. | 90 | 0.0625 | 5,280.9375 | 7,492 | 5,133.533333 | |
A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ vertices equals 2009. The midpoints of the sides of $P_1$ form a second $100$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $100$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of ... | 2009 | 0.9375 | 4,396.375 | 4,143.333333 | 8,192 | |
For a nonnegative integer $n$, let $r_9(n)$ stand for the remainder left when $n$ is divided by $9.$ For example, $r_9(25)=7.$
What is the $22^{\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_9(5n)\le 4~?$$(Note that the first entry in this list is $0$.) | 38 | 0.625 | 6,874.375 | 6,083.8 | 8,192 | |
A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 5$ passes through no lattice point with $0 < x \leq 150$ for all $m$ such that $0 < m < b$. What is the maximum possible value of $b$?
**A)** $\frac{1}{150}$
**B)** $\frac{1}{151}$
**C)** $\fra... | \frac{1}{151} | 0 | 7,977.1875 | -1 | 7,977.1875 | |
Given a moving circle $C$ that passes through points $A(4,0)$ and $B(0,-2)$, and intersects with the line passing through point $M(1,-2)$ at points $E$ and $F$. Find the minimum value of $|EF|$ when the area of circle $C$ is at its minimum. | 2\sqrt{3} | 0.4375 | 7,292.4375 | 6,135.857143 | 8,192 | |
On the base \( AC \) of an isosceles triangle \( ABC \), a circle is constructed using it as a diameter. This circle intersects side \( BC \) at point \( N \) such that \( BN: NC = 7: 2 \). Find the ratio of the lengths of segments \( AN \) and \( BC \). | \frac{4 \sqrt{2}}{9} | 0 | 6,852.125 | -1 | 6,852.125 | |
Let $p,$ $q,$ $r,$ and $s$ be the roots of \[x^4 + 6x^3 + 11x^2 + 6x + 3 = 0.\] Find the value of \[\frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}.\] | \frac{11}{3} | 0.8125 | 5,402.1875 | 4,758.384615 | 8,192 | |
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate t... | 1925 | To solve this problem, we need to determine the smallest number of coins in the chest initially, such that each pirate receives a whole number of coins when taking their share according to the given rule. The rule states that the $k^{\text{th}}$ pirate takes $\frac{k}{12}$ of the remaining coins.
Let $x$ be the initia... | 0 | 8,192 | -1 | 8,192 |
A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles than the area of the small semicircles? Express your answer to the nearest whole number. | 178\% | 0.9375 | 3,524.875 | 3,542.466667 | 3,261 | |
Paul fills in a $7 \times 7$ grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same thing again (to obtain two different random arrangements of the numbers in the grid). What is the expected number of pairs of numbers that occur in either the same row as each other or the ... | 147 / 2 | Each of the $\binom{49}{2}$ pairs of numbers has a probability of $\frac{14 \cdot\binom{7}{2}}{\binom{49}{2}}=1 / 4$ of being in the same row or column in one of the arrangements, so the expected number that are in the same row or column in both arrangements is $$\binom{49}{2} \cdot(1 / 4)^{2}=\frac{147}{2}$$ | 0.0625 | 7,042.9375 | 6,151 | 7,102.4 |
Six positive numbers, not exceeding 3, satisfy the equations \(a + b + c + d = 6\) and \(e + f = 2\). What is the minimum value of the expression
$$
\left(\sqrt{a^{2}+4}+\sqrt{b^{2}+e^{2}}+\sqrt{c^{2}+f^{2}}+\sqrt{d^{2}+4}\right)^{2}
$$ | 72 | 0.1875 | 7,698.3125 | 6,325 | 8,015.230769 | |
If $f(x) = (x-1)^3 + 1$, calculate the value of $f(-5) + f(-4) + \ldots + f(0) + \ldots + f(7)$. | 13 | 0.875 | 4,741.75 | 4,248.857143 | 8,192 | |
If $a$, $b$, and $c$ are positive numbers such that $ab = 24\sqrt{3}$, $ac = 30\sqrt{3}$, and $bc = 40\sqrt{3}$, find the value of $abc$. | 120\sqrt{6} | 0 | 8,192 | -1 | 8,192 | |
Two distinct integers $x$ and $y$ are factors of 48. One of these integers must be even. If $x\cdot y$ is not a factor of 48, what is the smallest possible value of $x\cdot y$? | 32 | 0.0625 | 7,901.5 | 5,239 | 8,079 | |
A *substring* of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also prime.
| 576 | 0.125 | 7,809.1875 | 7,319 | 7,879.214286 | |
Given that the sequence starts with 2 and alternates by adding 2 between consecutive terms, find the 30th term of this arithmetic sequence. | 60 | 1 | 1,483.5 | 1,483.5 | -1 | |
If $f(x)=f(2-x)$ for all $x$, then what line is necessarily an axis of symmetry of the graph of $y=f(x)$? (Give the simplest equation of this line.) | x=1 | 1 | 1,750.6875 | 1,750.6875 | -1 | |
Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$. | 131 | Continuing from Solution 1, we have $N=\frac{39}{50}k$ and $N=\frac{29}{25}m$. It follows that $k=\frac{50}{39}N$ and $m=\frac{25}{29}N$. Both $m$ and $k$ have to be integers, so, in order for that to be true, $N$ has to cancel the denominators of both $\frac{50}{39}$ and $\frac{25}{29}$. In other words, $N$ is a multi... | 1 | 4,154.9375 | 4,154.9375 | -1 |
Given a deck consisting of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning set. | \frac{1}{14} | 0.75 | 5,256.8125 | 5,410.416667 | 4,796 | |
An ordered list of four numbers is called a quadruple. A quadruple $(p, q, r, s)$ of integers with $p, q, r, s \geq 0$ is chosen at random such that $2 p+q+r+s=4$. What is the probability that $p+q+r+s=3$? | \frac{3}{11} | First, we count the number of quadruples $(p, q, r, s)$ of non-negative integer solutions to the equation $2 p+q+r+s=4$. Then, we determine which of these satisfies $p+q+r+s=3$. This will allow us to calculate the desired probability. Since each of $p, q, r$, and $s$ is a non-negative integer and $2 p+q+r+s=4$, then th... | 0.875 | 3,131.375 | 3,090.928571 | 3,414.5 |
In parallelogram $ABCD$, point $M$ is on $\overline{AB}$ so that $\frac {AM}{AB} = \frac {17}{1000}$ and point $N$ is on $\overline{AD}$ so that $\frac {AN}{AD} = \frac {17}{2009}$. Let $P$ be the point of intersection of $\overline{AC}$ and $\overline{MN}$. Find $\frac {AC}{AP}$.
| 177 | 0.9375 | 4,729.6875 | 4,498.866667 | 8,192 | |
Sophie has written three tests. Her marks were $73\%$, $82\%$, and $85\%$. She still has two tests to write. All tests are equally weighted. Her goal is an average of $80\%$ or higher. With which of the following pairs of marks on the remaining tests will Sophie not reach her goal: $79\%$ and $82\%$, $70\%$ and $91\%$,... | 73\% and 83\% | For Sophie's average over 5 tests to be $80\%$, the sum of her marks on the 5 tests must be $5 \times 80\% = 400\%$. After the first 3 tests, the sum of her marks is $73\% + 82\% + 85\% = 240\%$. Therefore, she will reach her goal as long as the sum of her marks on the two remaining tests is at least $400\% - 240\% = 1... | 0 | 2,866.625 | -1 | 2,866.625 |
A writer composed a series of essays totaling 60,000 words over a period of 150 hours. However, during the first 50 hours, she was exceptionally productive and wrote half of the total words. Calculate the average words per hour for the entire 150 hours and separately for the first 50 hours. | 600 | 0.6875 | 538.9375 | 486.818182 | 653.6 | |
When Person A has traveled 100 meters, Person B has traveled 50 meters. When Person A reaches point $B$, Person B is still 100 meters away from $B$. Person A immediately turns around and heads back towards $A$, and they meet 60 meters from point $B$. What is the distance between points $A$ and $B$ in meters? | 300 | 0 | 7,628.125 | -1 | 7,628.125 | |
\( p(x, y, z) \) is a polynomial with real coefficients such that:
1. \( p(tx, ty, tz) = t^2 f(y - x, z - x) \) for all real \( x, y, z, t \) (and some function \( f \));
2. \( p(1, 0, 0) = 4 \), \( p(0, 1, 0) = 5 \), and \( p(0, 0, 1) = 6 \);
3. \( p(\alpha, \beta, \gamma) = 0 \) for some complex numbers \( \alpha, \b... | \frac{5 \sqrt{30}}{3} | 0 | 8,192 | -1 | 8,192 | |
Compute the sum:
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | 15 | 0.5 | 7,185.6875 | 6,179.375 | 8,192 | |
A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,2), (2,2), and (2,0). What is the probability that $x+y < 3$? | \dfrac{7}{8} | 0.75 | 5,397.6875 | 4,466.25 | 8,192 | |
Calculate: $(-2)^{2}+\sqrt{(-3)^{2}}-\sqrt[3]{27}+|\sqrt{3}-2|$. | 6 - \sqrt{3} | 0.3125 | 574.4375 | 633.4 | 547.636364 | |
In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle? | 50 | 0.8125 | 4,709.75 | 4,257.384615 | 6,670 | |
Mia is researching a yeast population. There are 50 yeast cells present at 10:00 a.m. and the population triples every 5 minutes. Assuming none of the yeast cells die, how many yeast cells are present at 10:18 a.m. the same day? | 1350 | 0.6875 | 5,193 | 4,112.090909 | 7,571 | |
A stock investment increased by $15\%$ in the first year. At the start of the next year, by what percent must the stock now decrease to return to its original price at the beginning of the first year? | 13.04\% | 0.6875 | 5,851.375 | 5,575 | 6,459.4 | |
Given that $[x]$ represents the greatest integer less than or equal to $x$, if
$$
[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104
$$
then the smallest value of $x$ is ( ). | 11.5 | 0 | 8,192 | -1 | 8,192 | |
The line $y = 2$ intersects the graph of $y = 3x^2 + 2x - 5$ at the points $C$ and $D$. Determine the distance between $C$ and $D$ and express this distance in the form $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are coprime positive integers. | \frac{2\sqrt{22}}{3} | 0 | 3,852.5 | -1 | 3,852.5 | |
Let $f(x) = Ax + B$ and $g(x) = Bx + A$, where $A \neq B$. If $f(g(x)) - g(f(x)) = B - A$, what is $A + B$? | 0 | 0.9375 | 2,298.625 | 1,905.733333 | 8,192 | |
Find $1-0.\overline{3}.$ | \frac{2}{3} | 0.875 | 2,483 | 2,191.285714 | 4,525 | |
Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a heart and the second card is a 10? | \frac{1}{52} | 0.75 | 6,284.0625 | 5,921.916667 | 7,370.5 | |
The matrices
\[\begin{pmatrix} 3 & -8 \\ a & 11 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 11 & b \\ 4 & 3 \end{pmatrix}\]are inverses. Enter the ordered pair $(a,b).$ | (-4,8) | 1 | 1,669.375 | 1,669.375 | -1 | |
Given that \(\alpha, \beta \in \left(\frac{3\pi}{4}, \pi \right)\), \(\cos (\alpha + \beta) = \frac{4}{5}\), and \(\sin \left(\alpha - \frac{\pi}{4}\right) = \frac{12}{13}\), find \(\cos \left(\beta + \frac{\pi}{4}\right)\). | -\frac{56}{65} | 0.4375 | 6,015.6875 | 3,959.571429 | 7,614.888889 | |
Find $537_{8} - 261_{8}$. Verify by adding the result to $261_{8}$ and checking if it matches $537_{8}$. | 256_8 | 0.25 | 1,893.1875 | 1,530.75 | 2,014 | |
Given the function $f(x) = x^2 + (a+8)x + a^2 + a - 12$ ($a < 0$), and $f(a^2 - 4) = f(2a - 8)$, determine the minimum value of $\frac {f(n)-4a}{n+1}$ for $n \in \mathbb{N}^{+}$. | \frac {37}{4} | 0.8125 | 5,707.5 | 5,435.846154 | 6,884.666667 | |
What is the value of $\frac{(2112-2021)^2}{169}$? | 49 | 1. **Calculate the difference in the numerator**:
\[
2112 - 2021 = 91
\]
This is the exact value, not an approximation.
2. **Square the difference**:
\[
(2112 - 2021)^2 = 91^2 = 8281
\]
3. **Divide by the denominator**:
\[
\frac{8281}{169}
\]
To simplify this, we can either perform t... | 1 | 1,727.9375 | 1,727.9375 | -1 |
Given that the graph of the power function $y=(m^{2}-2m-2)x^{m^{2}+4m}$ is symmetric with respect to the origin and does not intersect the $x$-axis or $y$-axis, determine the value of the integer $m$. | -1 | 0.125 | 8,158.375 | 8,192 | 8,153.571429 | |
What is the sum of all the odd integers between $300$ and $500$? | 40000 | 0.75 | 4,363.25 | 3,087 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, with $c=5$ and $b(2\sin B+\sin A)+(2a+b)\sin A=2c\sin C$.
(1) Find the value of $C$.
(2) If $\cos A= \frac {4}{5}$, find the value of $b$. | 4- \sqrt {3} | 0 | 5,428.8125 | -1 | 5,428.8125 | |
Given that the base of the triangular pyramid \( S-ABC \) is an isosceles right triangle with \( AB \) as the hypotenuse, and \( SA = SB = SC = 2 \), and \( AB = 2 \), if points \( S \), \( A \), \( B \), and \( C \) are all on the surface of a sphere centered at \( O \), then the distance from point \( O \) to the pla... | \frac{\sqrt{3}}{3} | 0 | 6,075.1875 | -1 | 6,075.1875 | |
The first term of a sequence is 1. Each subsequent term is 4 times the square root of the sum of all preceding terms plus 4. What is the sum of the first 1971 terms in the sequence? | 15531481 | 0.5625 | 5,742.3125 | 3,837 | 8,192 | |
There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denominators to create 100 new numbers.
Find th... | 99 |
To solve this problem, we aim to find the smallest positive integer \( n \) such that after \( n \) steps, the 100 numbers on the board are all pairwise coprime regardless of their initial values.
### Key Observations
1. **Irreducible Fractions**: At each step, Alex forms 50 fractions out of the 100 numbers. Each fr... | 0 | 8,192 | -1 | 8,192 |
Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both). | 65 | 0 | 8,192 | -1 | 8,192 | |
Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume tha... | 120 | Let $e$ and $b$ be the speeds of the escalator and Bob, respectively.
When Al was on his way down, he took $150$ steps with a speed of $3b-e$ per step. When Bob was on his way up, he took $75$ steps with a speed of $b+e$ per step. Since Al and Bob were walking the same distance, we have \[150(3b-e)=75(b+e)\] Solving g... | 0.125 | 7,443.75 | 6,597 | 7,564.714286 |
In a convex 10-gon \(A_{1} A_{2} \ldots A_{10}\), all sides and all diagonals connecting vertices skipping one (i.e., \(A_{1} A_{3}, A_{2} A_{4},\) etc.) are drawn, except for the side \(A_{1} A_{10}\) and the diagonals \(A_{1} A_{9}\), \(A_{2} A_{10}\).
A path from \(A_{1}\) to \(A_{10}\) is defined as a non-self-int... | 55 | 0 | 8,192 | -1 | 8,192 | |
Find $\left \lceil \frac{12}{7} \cdot \frac{-29}{3}\right\rceil - \left\lfloor \frac{12}{7} \cdot \left \lfloor \frac{-29}{3}\right \rfloor \right \rfloor$. | 2 | 1 | 3,328.875 | 3,328.875 | -1 | |
An integer, whose decimal representation reads the same left to right and right to left, is called symmetrical. For example, the number 513151315 is symmetrical, while 513152315 is not. How many nine-digit symmetrical numbers exist such that adding the number 11000 to them leaves them symmetrical? | 8100 | 0 | 8,192 | -1 | 8,192 | |
Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy? | 406 | 0.0625 | 7,222.375 | 2,185 | 7,558.2 | |
Is it possible to append two digits to the right of the number 277 so that the resulting number is divisible by any number from 2 to 12? | 27720 | 0.25 | 5,910.1875 | 4,614 | 6,342.25 | |
Let
\[f(x) = (\arccos x)^3 + (\arcsin x)^3.\]Find the range of $f(x).$ All functions are in radians. | \left[ \frac{\pi^3}{32}, \frac{7 \pi^3}{8} \right] | 0.4375 | 5,856.75 | 5,830.857143 | 5,876.888889 | |
Let \(CD\) be a chord of a circle \(\Gamma_{1}\) and \(AB\) a diameter of \(\Gamma_{1}\) perpendicular to \(CD\) at \(N\) with \(AN > NB\). A circle \(\Gamma_{2}\) centered at \(C\) with radius \(CN\) intersects \(\Gamma_{1}\) at points \(P\) and \(Q\), and the segments \(PQ\) and \(CD\) intersect at \(M\). Given that ... | 78 | 0.125 | 7,808.8125 | 5,126.5 | 8,192 | |
Given that the ratio of $3x - 4$ to $y + 15$ is constant, and $y = 3$ when $x = 2$, then, when $y = 12$, $x$ equals: | \frac{7}{3} | 1. **Identify the constant ratio**: Given that the ratio of $3x - 4$ to $y + 15$ is constant, we can express this relationship as:
\[
\frac{3x - 4}{y + 15} = k
\]
where $k$ is the constant ratio.
2. **Use the given condition to find $k$**: We know that when $x = 2$ and $y = 3$, the ratio holds. Plugging th... | 1 | 1,985.125 | 1,985.125 | -1 |
The smaller root of the equation $\left(x- \frac{3}{4} \right) \left(x - \frac{3}{4} \right) + \left(x - \frac{3}{4} \right) \left(x - \frac{1}{2} \right) = 0$ is: | \frac{5}{8} | 1. **Identify the form of the equation**: The given equation is:
\[
\left(x-\frac{3}{4}\right)\left(x-\frac{3}{4}\right)+\left(x-\frac{3}{4}\right)\left(x-\frac{1}{2}\right) = 0
\]
This can be rewritten using $a = x - \frac{3}{4}$ and $b = x - \frac{1}{2}$, so the equation becomes:
\[
a^2 + ab = 0
... | 1 | 2,665 | 2,665 | -1 |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2016} > 0$, $S_{2017} < 0$. For any positive integer $n$, we have $|a_n| \geqslant |a_k|$. Determine the value of $k$. | 1009 | 0.25 | 7,264 | 6,463.5 | 7,530.833333 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.