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 |
|---|---|---|---|---|---|---|
Given a mall with four categories of food: grains, vegetable oils, animal products, and fruits and vegetables, with 40, 10, 20, and 20 varieties, respectively, calculate the total sample size if 6 types of animal products are sampled. | 27 | 0.75 | 4,358.9375 | 3,960.25 | 5,555 | |
In an experimental field, the number of fruits grown on a single plant of a certain crop, denoted as $x$, follows a normal distribution $N(90, \sigma ^{2})$, and $P(x < 70) = 0.2$. Ten plants are randomly selected from the field, and the number of plants with fruit numbers in the range $[90, 110]$ is denoted as the ran... | 2.1 | 0.4375 | 6,006.9375 | 3,740.428571 | 7,769.777778 | |
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\sin x)^{2}}
$$ | \frac{1}{3} | 0.125 | 8,013.0625 | 7,389 | 8,102.214286 | |
What is the sum of all the odd integers between $200$ and $600$? | 80000 | 0.9375 | 5,096.3125 | 4,889.933333 | 8,192 | |
I have a bag with $6$ marbles numbered from $1$ to $6.$ Mathew has a bag with $12$ marbles numbered from $1$ to $12.$ Mathew chooses one marble from his bag and I choose two from mine. In how many ways can we choose the marbles (where the order of my choices does matter) such that the sum of the numbers on my marbles e... | 30 | 0.375 | 6,925.1875 | 5,475 | 7,795.3 | |
Two spinners are divided into fifths and sixths, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? The numbers on the first spinner are 3, 5, 6, 7, and 9. The numbers on the second spinner are 2, 4, 6, 8, 9, and 11. | \frac{11}{15} | 0.6875 | 4,439.4375 | 2,733.727273 | 8,192 | |
In rectangle $ABCD$, $AB = 3$ and $BC = 9$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form.
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size(200);
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draw((0,0)--(9,0)--(9,3)--(0,3)--(0,0)--c... | \sqrt{10} | 0.8125 | 5,448.1875 | 4,815 | 8,192 | |
Given the hyperbola $\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{16}=1$ with a point $P$ on its right branch. The difference in distances from $P$ to the left and right foci is $6$, and the distance from $P$ to the left directrix is $\dfrac {34}{5}$. Calculate the distance from $P$ to the right focus. | \dfrac {16}{3} | 0.8125 | 3,420.4375 | 2,735.153846 | 6,390 | |
Each of the integers 226 and 318 has digits whose product is 24. How many three-digit positive integers have digits whose product is 24? | 21 | 0.4375 | 7,193.25 | 5,909.142857 | 8,192 | |
Let $a, b, c$ , and $d$ be real numbers such that $a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018$ . Evaluate $3b + 8c + 24d + 37a$ . | 1215 | 0.375 | 7,257 | 5,698.666667 | 8,192 | |
The number of students in James' graduating class is greater than 100 but fewer than 200 and is 1 less than a multiple of 4, 2 less than a multiple of 5, and 3 less than a multiple of 6. How many students are in James' graduating class? | 183 | 0 | 8,188.0625 | -1 | 8,188.0625 | |
In triangle $ABC,$ $\angle A = 45^\circ,$ $\angle B = 75^\circ,$ and $AC = 6.$ Find $BC$. | 6\sqrt{3} - 6 | 0.125 | 6,320.0625 | 4,257 | 6,614.785714 | |
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ | 117 | 1. **Identify the prime factorization of 100,000**:
The prime factorization of $100,000$ is $100,000 = 10^5 = (2 \cdot 5)^5 = 2^5 \cdot 5^5$.
2. **Determine the form of elements in set $S$**:
The elements of $S$, the set of all positive integer divisors of $100,000$, are of the form $2^a5^b$ where $0 \leq a, b ... | 0 | 8,192 | -1 | 8,192 |
Given $A=\{x|ax^{2}+bx+c\leqslant 0\left(a \lt b\right)\}$ has one and only one element, then the minimum value of $M=\frac{{a+3b+4c}}{{b-a}}$ is ______. | 2\sqrt{5} + 5 | 0 | 5,835.4375 | -1 | 5,835.4375 | |
Five points are chosen uniformly at random on a segment of length 1. What is the expected distance between the closest pair of points? | \frac{1}{24} | Choose five points arbitrarily at $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ in increasing order. Then the intervals $\left(a_{2}-x, a_{2}\right),\left(a_{3}-x, a_{3}\right),\left(a_{4}-x, a_{4}\right),\left(a_{5}-x, a_{5}\right)$ must all be unoccupied. The probability that this happens is the same as doing the process in re... | 0 | 8,192 | -1 | 8,192 |
The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$?
| 799 | 0.625 | 6,082.9375 | 5,059.3 | 7,789 | |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2023,0),(2023,2024),$ and $(0,2024)$. What is the probability that $x > 9y$? Express your answer as a common fraction. | \frac{2023}{36432} | 0.25 | 7,756.0625 | 6,448.25 | 8,192 | |
Given a line segment $\overline{AB}=10$ cm, a point $C$ is placed on $\overline{AB}$ such that $\overline{AC} = 6$ cm and $\overline{CB} = 4$ cm. Three semi-circles are drawn with diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, external to the segment. If a line $\overline{CD}$ is drawn perpendicular t... | \frac{3}{2} | 0.1875 | 6,344.375 | 4,118 | 6,858.153846 | |
Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy? | 36 | 1. **Assume the cost of each balloon**: Let's assume each balloon costs $3$ dollars. This assumption simplifies calculations and does not affect the generality of the problem since we are interested in the number of balloons, not the actual cost.
2. **Calculate total money Orvin has**: If Orvin has enough money to buy... | 0.9375 | 6,094.5625 | 5,954.733333 | 8,192 |
Let $N$ denote the number of subsets of $\{1,2,3, \ldots, 100\}$ that contain more prime numbers than multiples of 4. Compute the largest integer $k$ such that $2^{k}$ divides $N$. | \[
52
\] | Let $S$ denote a subset with the said property. Note that there are 25 multiples of 4 and 25 primes in the set $\{1,2,3, \ldots, 100\}$, with no overlap between the two. Let $T$ denote the subset of 50 numbers that are neither prime nor a multiple of 4, and let $U$ denote the 50 other numbers. Elements of $T$ can be ar... | 0 | 8,019 | -1 | 8,019 |
If the sum of the digits of a positive integer $a$ equals 6, then $a$ is called a "good number" (for example, 6, 24, 2013 are all "good numbers"). List all "good numbers" in ascending order as $a_1$, $a_2$, $a_3$, …, if $a_n = 2013$, then find the value of $n$. | 51 | 0 | 7,909.375 | -1 | 7,909.375 | |
The diagram shows the two squares \( BCDE \) and \( FGHI \) inside the triangle \( ABJ \), where \( E \) is the midpoint of \( AB \) and \( C \) is the midpoint of \( FG \). What is the ratio of the area of the square \( BCDE \) to the area of the triangle \( ABJ \)? | 1/3 | 0 | 8,081 | -1 | 8,081 | |
Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $... | 620 | Let $r$ be the time Rudolph takes disregarding breaks and $\frac{4}{3}r$ be the time Jennifer takes disregarding breaks. We have the equation \[r+5\left(49\right)=\frac{4}{3}r+5\left(24\right)\] \[125=\frac13r\] \[r=375.\] Thus, the total time they take is $375 + 5(49) = \boxed{620}$ minutes. | 0.375 | 5,152.375 | 4,220.666667 | 5,711.4 |
Let $ABC$ be a triangle with $AB=13$ , $BC=14$ , and $CA=15$ . Points $P$ , $Q$ , and $R$ are chosen on segments $BC$ , $CA$ , and $AB$ , respectively, such that triangles $AQR$ , $BPR$ , $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $PQR$ . What is the perimeter of $PQR$... | 30 | 0.25 | 7,743.6875 | 6,398.75 | 8,192 | |
Solve for $x$: $$\log_2 \frac{3x+9}{5x-3} +\log_2\frac{5x-3}{x-2}=2$$ | 17 | 1 | 1,942.6875 | 1,942.6875 | -1 | |
To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains $60$ pills, then the supply of medicine would last approximately | 8\text{ months} | 1. **Determine the rate of consumption**: Jill's grandmother takes one half of a pill every other day. This means that in two days, she consumes one half of a pill.
2. **Calculate the number of days per pill**: Since she takes half a pill every two days, she will take a full pill every four days (since half a pill for... | 0 | 5,101.0625 | -1 | 5,101.0625 |
In the diagram, \( Z \) lies on \( XY \) and the three circles have diameters \( XZ \), \( ZY \), and \( XY \). If \( XZ = 12 \) and \( ZY = 8 \), calculate the ratio of the area of the shaded region to the area of the unshaded region. | \frac{12}{13} | 0.4375 | 5,547.75 | 4,584.857143 | 6,296.666667 | |
Given that $2ax^3 - 3bx + 8 = 18$ when $x = -1$, determine the value of $9b - 6a + 2$. | 32 | 1 | 1,530.9375 | 1,530.9375 | -1 | |
Given points P(-2,-3) and Q(5,3) in the xy-plane; point R(x,m) is such that x=2 and PR+RQ is a minimum. Find m. | \frac{3}{7} | 0.25 | 7,215.8125 | 5,664.5 | 7,732.916667 | |
When young fishermen were asked how many fish each of them caught, the first one replied, "I caught half the number of fish that my friend caught, plus 10 fish." The second one said, "And I caught as many as my friend, plus 20 fish." How many fish did the fishermen catch? | 100 | 0.0625 | 1,881.125 | 4,663 | 1,695.666667 | |
The numbers 2, 4, 6, and 8 are a set of four consecutive even numbers. Suppose the sum of five consecutive even numbers is 320. What is the smallest of the five numbers? | 60 | 1 | 2,378.625 | 2,378.625 | -1 | |
On a backpacking trip with 10 people, in how many ways can I choose 2 cooks and 1 medical helper if any of the 10 people may fulfill these roles? | 360 | 0.1875 | 6,274.3125 | 4,929.666667 | 6,584.615385 | |
When $x=$____, the expressions $\frac{x-1}{2}$ and $\frac{x-2}{3}$ are opposite in sign. | \frac{7}{5} | 0.5625 | 631.5 | 550.333333 | 735.857143 | |
Given an ellipse, if its two foci and the two vertices on its minor axis form a square, calculate its eccentricity. | \dfrac{\sqrt{2}}{2} | 0 | 5,891.3125 | -1 | 5,891.3125 | |
In how many ways can a committee of three people be formed if the members are to be chosen from four married couples? | 32 | 0.0625 | 5,715.125 | 4,222 | 5,814.666667 | |
Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \( L \) cm. What is the value of \( L \)? | 13 | 0.0625 | 7,007.25 | 8,166 | 6,930 | |
A graduating high school class has $45$ people. Each student writes a graduation message to every other student, with each pair writing only one message between them. How many graduation messages are written in total? (Answer with a number) | 1980 | 0.6875 | 339.3125 | 311.454545 | 400.6 | |
A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$. Let $f$ be a function such that:
$f(n)=0$, if n is perfect
$f(n)=0$, if the last digit of n is 4
$f(a.b)=f(a)+f(b)$
Find $f(1998)$ | 0 |
To determine \( f(1998) \), we start by analyzing the given function \( f \) and the properties it holds.
1. **Perfect Number Property**:
If \( n \) is a perfect number, then \( f(n) = 0 \).
2. **Ending with Digit 4 Property**:
If the last digit of \( n \) is 4, then \( f(n) = 0 \).
3. **Multiplicative Prop... | 0.5625 | 7,349.6875 | 6,694.555556 | 8,192 |
How many of the 2401 smallest positive integers written in base 7 use 3 or 6 (or both) as a digit? | 1776 | 0.0625 | 7,858.125 | 4,362 | 8,091.2 | |
If a restaurant offers 15 different dishes, and Yann and Camille each decide to order either one or two different dishes, how many different combinations of meals can they order? Assume that the dishes can be repeated but the order in which each person orders the dishes matters. | 57600 | 0.25 | 4,791.1875 | 5,106.5 | 4,686.083333 | |
In the Cartesian coordinate system, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=-\sqrt{3}t}\\{y=t}\end{array}}\right.$ ($t$ is the parameter), and the parametric equations of curve $C_{2}$ are $\left\{{\begin{array}{l}{x=4\cos\theta}\\{y=4\sin\theta}\end{array}}\right.$ ($\theta$ is the pa... | 30 | 0.6875 | 7,642.5625 | 7,392.818182 | 8,192 | |
What is the digit in the thousandths place of the decimal equivalent of $\frac{3}{16}$? | 7 | 1 | 1,621.75 | 1,621.75 | -1 | |
A finite set $S$ of points in the coordinate plane is called [i]overdetermined[/i] if $|S|\ge 2$ and there exists a nonzero polynomial $P(t)$, with real coefficients and of degree at most $|S|-2$, satisfying $P(x)=y$ for every point $(x,y)\in S$.
For each integer $n\ge 2$, find the largest integer $k$ (in terms of $... | 2^{n-1} - n |
Given a finite set \( S \) of points in the coordinate plane, a set \( S \) is called \textit{overdetermined} if \( |S| \ge 2 \) and there exists a nonzero polynomial \( P(t) \) with real coefficients of degree at most \( |S| - 2 \), such that \( P(x) = y \) for every point \( (x, y) \in S \).
For each integer \( n \... | 0 | 8,192 | -1 | 8,192 |
Given that $x > 0$, $y > 0$ and $x + y = 4$, find the minimum value of $$\frac{x^2}{x + 1} + \frac{y^2}{y + 2}$$. | \frac{16}{7} | 0.625 | 6,823.625 | 6,002.6 | 8,192 | |
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE$? | 170 | 1. **Identify the given information and setup**: We are given a square $ABCD$ and a point $E$ such that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ such that $DE = DF$. We need to find $\angle AFE$.
2. **Extend $\overline{AD}$ to a point $G$**: By extending $\overline{AD}$ to a point $G$ such that $\ov... | 0.375 | 7,781 | 7,096 | 8,192 |
A frog sits at the point $(2, 3)$ on a grid within a larger square bounded by points $(0,0), (0,6), (6,6)$, and $(6,0)$. Each jump the frog makes is parallel to one of the coordinate axes and has a length $1$. The direction of each jump (up, down, left, right) is not necessarily chosen with equal probability. Instead, ... | \frac{5}{8} | 0 | 8,152.8125 | -1 | 8,152.8125 | |
Four vehicles were traveling on the highway at constant speeds: a car, a motorcycle, a scooter, and a bicycle. The car passed the scooter at 12:00, encountered the bicyclist at 14:00, and met the motorcyclist at 16:00. The motorcyclist met the scooter at 17:00 and caught up with the bicyclist at 18:00.
At what time d... | 15:20 | 0 | 8,192 | -1 | 8,192 | |
What is the remainder when the product $1734\times 5389 \times 80,\!607$ is divided by 10? | 2 | 0.9375 | 2,655.3125 | 2,286.2 | 8,192 | |
$ABCD$ is a trapezoid with the measure of base $\overline{AB}$ twice the measure of the base $\overline{CD}$. Point $E$ is the point of intersection of the diagonals. The measure of diagonal $\overline{AC}$ is 11. Find the length of segment $\overline{EC}$. Express your answer as a common fraction.
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p... | \dfrac{11}{3} | 1 | 2,374.5 | 2,374.5 | -1 | |
A standard six-sided die is rolled, and $P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $P$? | 12 | 0.875 | 5,485.1875 | 5,098.5 | 8,192 | |
The inscribed circle of triangle $DEF$ is tangent to $\overline{DE}$ at point $P$ and its radius is $13$. Given that $DP = 17$ and $PE = 31$, and the tangent from vertex $F$ to the circle is $20$, determine the perimeter of triangle $DEF$. | 136 | 0.25 | 7,042.875 | 3,595.5 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The vectors $m=(\cos (A-B),\sin (A-B))$, $n=(\cos B,-\sin B)$, and $m\cdot n=-\frac{3}{5}$.
(1) Find the value of $\sin A$.
(2) If $a=4\sqrt{2}$, $b=5$, find the measure of angle $B$ and the projection of vector $\overrighta... | \frac{\sqrt{2}}{2} | 0 | 4,782.3125 | -1 | 4,782.3125 | |
Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | 27 | 0.0625 | 7,528.875 | 6,883 | 7,571.933333 | |
In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10$. Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCF... | 148 | By the Pythagorean Theorem, $BC=35$. By the Angle Bisector Theorem $BD = 60/7$ and $DC = 185/7$. We can find the coordinates of F, and use that the find the equation of line EF. Then, we can find the coordinates of G. Triangle $ADC = 1110/7$ and we can find the area triangle $AGF$ with the shoelace theorem, so subtract... | 0 | 8,101 | -1 | 8,101 |
How many distinct four-digit positive integers are there such that the product of their digits equals 18? | 36 | 0.25 | 7,796.875 | 6,611.5 | 8,192 | |
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12},$ find all possible values for $x.$
(Enter your answer as a comma-separated list.) | 4, 6 | 0.3125 | 2,362.6875 | 2,759.4 | 2,182.363636 | |
For what value of $n$ is $3^3-5=4^2+n$? | 6 | 1 | 1,098.6875 | 1,098.6875 | -1 | |
What is $4\cdot 6+8\cdot 3-28\div 2$? | 34 | 1 | 1,626.375 | 1,626.375 | -1 | |
A building contractor needs to pay his $108$ workers $\$ 200 $ each. He is carrying $ 122 $ one hundred dollar bills and $ 188 $ fifty dollar bills. Only $ 45 $ workers get paid with two $ \ $100$ bills. Find the number of workers who get paid with four $\$ 50$ bills. | 31 | 0.5 | 4,788.8125 | 4,265.75 | 5,311.875 | |
In a redesign of his company's logo, Wei decided to use a larger square and more circles. Each circle is still tangent to two sides of the square and its adjacent circles, but now there are nine circles arranged in a 3x3 grid instead of a 2x2 grid. If each side of the new square measures 36 inches, calculate the total ... | 1296 - 324\pi | 0.1875 | 5,780.75 | 6,685 | 5,572.076923 | |
Determine the numerical value of $k$ such that
\[\frac{12}{x + z} = \frac{k}{z - y} = \frac{5}{y - x}.\] | 17 | 0.0625 | 8,007.0625 | 8,192 | 7,994.733333 | |
The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is | 80 | 1. **Identify the pattern in the sequence of odd and even integers:**
- The $n$-th odd integer can be represented as $2n - 1$.
- The $n$-th even integer can be represented as $2n$.
2. **Calculate the difference between the $n$-th even and $n$-th odd integer:**
- The difference between the $n$-th even and $n$-... | 1 | 1,464 | 1,464 | -1 |
Twelve delegates, four each from three different countries, randomly select chairs at a round table that seats twelve people. Calculate the probability that each delegate sits next to at least one delegate from another country, and express this probability as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively p... | 33 | 0 | 8,192 | -1 | 8,192 | |
If $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, then the minimum value of $$\frac {4}{A}+ \frac {1}{B+C}$$ is \_\_\_\_\_\_. | \frac {9}{\pi} | 0.875 | 3,941.875 | 3,520.714286 | 6,890 | |
I flip a fair coin once and roll a regular six-sided die. What is the probability the coin will show heads and the die will show a 2? | \dfrac{1}{12} | 1 | 1,639.75 | 1,639.75 | -1 | |
A, B, and C are guessing a two-digit number.
A says: The number has an even number of factors, and it is greater than 50.
B says: The number is odd, and it is greater than 60.
C says: The number is even, and it is greater than 70.
If each of them is only half correct, what is the number? | 64 | 0.125 | 7,934.5625 | 6,132.5 | 8,192 | |
Alice cycled 240 miles in 4 hours, 30 minutes. Then, she cycled another 300 miles in 5 hours, 15 minutes. What was Alice's average speed in miles per hour for her entire journey? | 55.38 | 0.125 | 1,605.5625 | 3,222 | 1,374.642857 | |
Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have? | 820 | We can start how we did above in solution 4 to get $\frac{9}{4} * \sqrt{(81x^2-1)(81-x^2)}$. Then, we can notice the inside is a quadratic in terms of $x^2$, which is $-81(x^2)^2+6562x^2-81$. This is maximized when $x^2 = \frac{3281}{81}$.If we plug it into the equation, we get $\frac{9}{4} *\frac{9}{4}*\frac{3280}{9} ... | 0.125 | 7,708.9375 | 6,110.5 | 7,937.285714 |
A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? | \frac{1}{3} |
We are given that the probability that a ball is tossed into bin $k$ is $2^{-k}$ for $k = 1, 2, 3, \ldots$. We need to find the probability that the red ball is tossed into a higher-numbered bin than the green ball.
#### Step-by-step Analysis:
1. **Symmetry Argument**:
- The problem is symmetric with respect to t... | 0.9375 | 4,230.625 | 3,966.533333 | 8,192 |
Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes. | 132 | Suppose $p=11$; then $m^2-m+11=11qrs$. Reducing modulo 11, we get $m\equiv 1,0 \pmod{11}$ so $k(11k\pm 1)+1 = qrs$.
Suppose $q=11$. Then we must have $11k^2\pm k + 1 = 11rs$, which leads to $k\equiv \mp 1 \pmod{11}$, i.e., $k\in \{1,10,12,21,23,\ldots\}$.
$k=1$ leads to $rs=1$ (impossible)! Then $k=10$ leads to $rs=10... | 0 | 8,192 | -1 | 8,192 |
Let square $WXYZ$ have sides of length $8$. An equilateral triangle is drawn such that no point of the triangle lies outside $WXYZ$. Determine the maximum possible area of such a triangle. | 16\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Find the sum of $453_6$, $436_6$, and $42_6$ in base 6. | 1415_6 | 0.625 | 7,042.375 | 6,352.6 | 8,192 | |
How many positive integers less than or equal to 240 can be expressed as a sum of distinct factorials? Consider 0 ! and 1 ! to be distinct. | 39 | Note that $1=0$ !, $2=0$ ! +1 !, $3=0$ ! +2 !, and $4=0!+1$ ! +2 !. These are the only numbers less than 6 that can be written as the sum of factorials. The only other factorials less than 240 are $3!=6,4!=24$, and $5!=120$. So a positive integer less than or equal to 240 can only contain 3 !, 4 !, 5 !, and/or one of $... | 0 | 8,192 | -1 | 8,192 |
Given a geometric progression of five terms, each a positive integer less than $100$. The sum of the five terms is $211$. If $S$ is the sum of those terms in the progression which are squares of integers, then $S$ is: | 133 | 1. **Identify the terms of the geometric progression**: Let the first term be $a$ and the common ratio be $r$. The terms of the progression are $a, ar, ar^2, ar^3, ar^4$. The sum of these terms is given by:
\[
a + ar + ar^2 + ar^3 + ar^4 = a(1 + r + r^2 + r^3 + r^4) = 211
\]
2. **Constraint on $r$ and $a$**: ... | 1 | 4,772.75 | 4,772.75 | -1 |
$A B C D$ is a cyclic quadrilateral in which $A B=3, B C=5, C D=6$, and $A D=10 . M, I$, and $T$ are the feet of the perpendiculars from $D$ to lines $A B, A C$, and $B C$ respectively. Determine the value of $M I / I T$. | \frac{25}{9} | Quadrilaterals $A M I D$ and $D I C T$ are cyclic, having right angles $\angle A M D, \angle A I D$, and $\angle C I D, \angle C T D$ respectively. We see that $M, I$, and $T$ are collinear. For, $m \angle M I D=\pi-m \angle D A M=$ $\pi-m \angle D A B=m \angle B C D=\pi-m \angle D C T=\pi-m \angle D I T$. Therefore, M... | 0 | 8,189.625 | -1 | 8,189.625 |
Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5$, no collection of $k$ pairs made by the child contains the shoes from exactly... | 28 | 0.5625 | 6,212 | 5,066.222222 | 7,685.142857 | |
If $(x+2)(x-3)=14$, find the sum of the possible values of $x$. | 1 | 1 | 1,362.1875 | 1,362.1875 | -1 | |
Given the ellipse $C:\frac{{x}^{2}}{4}+\frac{{y}^{2}}{3}=1$, let ${F}_{1}$ and ${F}_{2}$ be its left and right foci, respectively. A line $l$ passing through point ${F}_{2}$ with a slope of $1$ intersects ellipse $C$ at two distinct points $M$ and $N$. Calculate the area of triangle $MN{F}_{1}$. | \frac{12\sqrt{2}}{7} | 0 | 6,665.8125 | -1 | 6,665.8125 | |
Xiao Hong asked Da Bai: "Please help me calculate the result of $999 \quad 9 \times 999 \quad 9$ and determine how many zeros appear in it."
2019 nines times 2019 nines
Da Bai quickly wrote a program to compute it. Xiao Hong laughed and said: "You don't need to compute the exact result to know how many zeros there are.... | 2018 | 0.75 | 5,518.4375 | 5,070 | 6,863.75 | |
A circle with a radius of 2 passes through the midpoints of three sides of triangle \(ABC\), where the angles at vertices \(A\) and \(B\) are \(30^{\circ}\) and \(45^{\circ}\), respectively.
Find the height drawn from vertex \(A\). | 2 + 2\sqrt{3} | 0.625 | 5,846.25 | 4,556.7 | 7,995.5 | |
In the isosceles trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are related by the equation \(AD = (1 + \sqrt{15}) BC\). A circle is constructed with its center at point \(C\) and radius \(\frac{2}{3} BC\), which cuts a chord \(EF\) on the base \(AD\) of length \(\frac{\sqrt{7}}{3} BC\). In what ratio does the circle ... | 2:1 | 0 | 8,158.375 | -1 | 8,158.375 | |
Find the matrix $\mathbf{M}$ if it satisfies $\mathbf{M} \mathbf{i} = \begin{pmatrix} 2 \\ 3 \\ -8 \end{pmatrix},$ $\mathbf{M} \mathbf{j} = \begin{pmatrix} 0 \\ 5 \\ -2 \end{pmatrix},$ and $\mathbf{M} \mathbf{k} = \begin{pmatrix} 7 \\ -1 \\ 4 \end{pmatrix}.$ | \begin{pmatrix} 2 & 0 & 7 \\ 3 & 5 & -1 \\ -8 & -2 & 4 \end{pmatrix} | 0.9375 | 1,417.25 | 1,418.666667 | 1,396 | |
Alice starts to make a list, in increasing order, of the positive integers that have a first digit of 2. She writes $2, 20, 21, 22, \ldots$ but by the 1000th digit she (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits she wrote (th... | 216 | 0.375 | 7,629.25 | 7,008.5 | 8,001.7 | |
Find all three-digit numbers $\overline{\Pi B \Gamma}$, consisting of distinct digits $\Pi, B$, and $\Gamma$, for which the following equality holds: $\overline{\Pi B \Gamma} = (\Pi + B + \Gamma) \times (\Pi + B + \Gamma + 1)$. | 156 | 0.5625 | 7,497.75 | 6,957.777778 | 8,192 | |
Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three ... | 47 | 0.5 | 6,092.3125 | 4,801.625 | 7,383 | |
Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET. | 540 | 0.625 | 5,839.9375 | 4,954 | 7,316.5 | |
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4, 5, x$. The sum of the possible values of x equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$? | 36 | 1. **Identify the angles in the triangle**: Since the angles are in arithmetic progression and their sum is $180^\circ$, let the angles be $\alpha - d$, $\alpha$, and $\alpha + d$. Solving $\alpha - d + \alpha + \alpha + d = 180^\circ$ gives $3\alpha = 180^\circ$, so $\alpha = 60^\circ$. Thus, the angles are $60^\circ ... | 0 | 8,192 | -1 | 8,192 |
Let \[f(x) =
\begin{cases}
9x+16 &\text{if }x<2, \\
2x-14&\text{if }x\ge2.
\end{cases}
\]If $f(x)=-2$, find the sum of all possible values of $x$. | 4 | 1 | 1,151.125 | 1,151.125 | -1 | |
How many positive integer divisors of $2004^{2004}$ are divisible by exactly 2004 positive integers?
| 54 | 0.6875 | 5,994.875 | 4,996.181818 | 8,192 | |
Given that \( n \) is a positive integer, \( P \) is a prime number, and \( pn \) has exactly 8 positive divisors, arrange them in ascending order as \( 1=d_{1}<d_{2}< \cdots <d_{8}=pn \). Additionally, let \( d_{17p-d_{3}}=\left(d_{1}+d_{2}+d_{3}\right)\left(d_{3}+d_{4}+13p\right) \). Find \( n \). | 2021 | 0 | 8,192 | -1 | 8,192 | |
Find the sum of all positive integral values of $n$ for which $\frac{n+6}{n}$ is an integer. | 12 | 1 | 2,036.375 | 2,036.375 | -1 | |
Given the function $f(x)=\sin^{2}x+\sqrt{3}\sin x\sin (x+\frac{\pi}{2})$.
1. Find the value of $f(\frac{\pi}{12})$;
2. Find the maximum and minimum values of the function $f(x)$ when $x\in[0,\frac{\pi}{2}]$. | \frac{3}{2} | 0 | 5,742.6875 | -1 | 5,742.6875 | |
Given the enclosure dimensions are 15 feet long, 8 feet wide, and 7 feet tall, with each wall and floor being 1 foot thick, determine the total number of one-foot cubical blocks used to create the enclosure. | 372 | 0.125 | 758.5625 | 674.5 | 770.571429 | |
The perpendicular bisectors of the sides of triangle $ABC$ meet its circumcircle at points $A',$ $B',$ and $C',$ as shown. If the perimeter of triangle $ABC$ is 35 and the radius of the circumcircle is 8, then find the area of hexagon $AB'CA'BC'.$
[asy]
unitsize(2 cm);
pair A, B, C, Ap, Bp, Cp, O;
O = (0,0);
A = di... | 140 | 0 | 8,192 | -1 | 8,192 | |
The energy stored by any pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Three identical point charges start at the vertices of an equilateral triangle, and this configuration stores 15 Joules of energy. How much more energy, in Joules, would ... | 10 | 0.625 | 5,040.9375 | 3,716.5 | 7,248.333333 | |
If $\log_2 x^2 + \log_{1/2} x = 5,$ compute $x.$ | 32 | 0.9375 | 2,508.8125 | 2,129.933333 | 8,192 | |
\(A, B, C, D\) are consecutive vertices of a parallelogram. Points \(E, F, P, H\) lie on sides \(AB\), \(BC\), \(CD\), and \(AD\) respectively. Segment \(AE\) is \(\frac{1}{3}\) of side \(AB\), segment \(BF\) is \(\frac{1}{3}\) of side \(BC\), and points \(P\) and \(H\) bisect the sides they lie on. Find the ratio of t... | 37/72 | 0.4375 | 7,237 | 6,030.428571 | 8,175.444444 | |
Given the discrete random variable $X$ follows a two-point distribution, and $P\left(X=1\right)=p$, $D(X)=\frac{2}{9}$, determine the value of $p$. | \frac{2}{3} | 0.5625 | 6,270.6875 | 5,239.444444 | 7,596.571429 | |
The teacher asks Bill to calculate $a-b-c$, but Bill mistakenly calculates $a-(b-c)$ and gets an answer of 11. If the correct answer was 3, what is the value of $a-b$? | 7 | 1 | 1,306.5625 | 1,306.5625 | -1 | |
There are $n\geq 3$ cities in a country and between any two cities $A$ and $B$ , there is either a one way road from $A$ to $B$ , or a one way road from $B$ to $A$ (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define $d... | 3/2 | 0.0625 | 8,108 | 7,012 | 8,181.066667 |
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