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 |
|---|---|---|---|---|---|---|
Ms. Carr asks her students to read any $5$ of the $10$ books on a reading list. Harold randomly selects $5$ books from this list, and Betty does the same. What is the probability that there are exactly $2$ books that they both select? | \frac{25}{63} | 1. **Calculate the total number of ways Harold and Betty can each choose 5 books from 10 books:**
The number of ways to choose 5 books from 10 is given by the binomial coefficient $\binom{10}{5}$. Since both Harold and Betty are choosing 5 books independently, the total number of outcomes is:
\[
\binom{10}{5} ... | 1 | 4,807 | 4,807 | -1 |
For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have
\[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\] | \frac{1}{2n} |
To determine the largest real constant \( C_n \) such that for all positive real numbers \( a_1, a_2, \ldots, a_n \), the inequality
\[
\frac{a_1^2 + a_2^2 + \ldots + a_n^2}{n} \geq \left( \frac{a_1 + a_2 + \ldots + a_n}{n} \right)^2 + C_n \cdot (a_1 - a_n)^2
\]
holds, we start by rewriting the inequality:
\[
\frac... | 0.0625 | 8,171.0625 | 7,857 | 8,192 |
Four cars $A$, $B$, $C$, and $D$ start simultaneously from the same point on a circular track. Cars $A$ and $B$ travel clockwise, while cars $C$ and $D$ travel counterclockwise. All cars move at constant but distinct speeds. Exactly 7 minutes after the race starts, $A$ meets $C$ for the first time, and at the same mome... | 53 | 0.0625 | 7,932 | 8,192 | 7,914.666667 | |
For the system of equations \(x^{2} + x^{2} y^{2} + x^{2} y^{4} = 525\) and \(x + x y + x y^{2} = 35\), find the sum of the real y values that satisfy the equations. | \frac{5}{2} | 1 | 3,597.1875 | 3,597.1875 | -1 | |
When $\frac{1}{1001}$ is expressed as a decimal, what is the sum of the first 50 digits after the decimal point? | 216 | 0.5625 | 6,556.625 | 5,587 | 7,803.285714 | |
In the geometric sequence $\{a_{n}\}$, $a_{3}$ and $a_{7}$ are two distinct extreme points of the function $f\left(x\right)=\frac{1}{3}x^{3}+4x^{2}+9x-1$. Find $a_{5}$. | -3 | 0.6875 | 6,488.0625 | 5,713.545455 | 8,192 | |
The number $m$ is a prime number between 30 and 50. If you divide $m$ by 12, the remainder is 7. What is the value of $m$? | 43 | 0.375 | 6,837.125 | 6,344.5 | 7,132.7 | |
Given that the variables $a$ and $b$ satisfy the equation $b=-\frac{1}{2}a^2 + 3\ln{a} (a > 0)$, and that point $Q(m, n)$ lies on the line $y = 2x + \frac{1}{2}$, find the minimum value of $(a - m)^2 + (b - n)^2$. | \frac{9}{5} | 0.4375 | 6,890.8125 | 5,339.714286 | 8,097.222222 | |
Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac {1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac {m}{n}$, where $m$ and $n... | 341 | 0.6875 | 6,578.75 | 5,845.454545 | 8,192 | |
In triangle $ABC$, where $\angle A = 90^\circ$, $BC = 20$, and $\tan C = 4\cos B$. Find the length of $AB$. | 5\sqrt{15} | 0.8125 | 3,209.8125 | 2,399.923077 | 6,719.333333 | |
The sum of two numbers is $40$. If we triple the larger number and subtract four times the smaller number, the result is $10$. What is the positive difference between the two numbers? | 8.57 | 0 | 2,990.1875 | -1 | 2,990.1875 | |
On the sides \(AB\) and \(AD\) of square \(ABCD\), points \(E\) and \(F\) are marked such that \(BE : EA = AF : FD = 2022 : 2023\). Segments \(EC\) and \(FC\) intersect the diagonal \(BD\) of the square at points \(G\) and \(H\), respectively. Find the ratio \(GH : BD\). | \frac{12271519}{36814556} | 0 | 8,124.875 | -1 | 8,124.875 | |
Calculate $3(72+76+80+84+88+92+96+100+104+108)$. | 2700 | 0.875 | 2,899.625 | 2,738.571429 | 4,027 | |
Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$ .
*2021 CCA Math Bonanza Lightning Round #2.4* | 14 | 0.5625 | 7,514.6875 | 6,987.888889 | 8,192 | |
Given $\cos (\pi+\alpha)=- \frac { \sqrt {10}}{5}$, and $\alpha\in(- \frac {\pi}{2},0)$, determine the value of $\tan \alpha$. | - \frac{\sqrt{6}}{2} | 0 | 2,694.5625 | -1 | 2,694.5625 | |
Let $x,$ $y,$ and $z$ be nonnegative numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of
\[2xy \sqrt{6} + 8yz.\] | \sqrt{22} | 0.6875 | 6,946.8125 | 6,617.454545 | 7,671.4 | |
If the function $f(x) = x^2$ has a domain $D$ and its range is $\{0, 1, 2, 3, 4, 5\}$, how many such functions $f(x)$ exist? (Please answer with a number). | 243 | 0.0625 | 7,210.625 | 8,192 | 7,145.2 | |
Using systematic sampling, \\(32\\) people are selected from \\(960\\) for a questionnaire survey. They are randomly numbered from \\(1\\) to \\(960\\), and then grouped. The number drawn by simple random sampling in the first group is \\(9\\). Among the \\(32\\) people selected, those with numbers in the interval \\([... | 10 | 0.75 | 4,896.5 | 4,150.25 | 7,135.25 | |
Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning in the three events are 0.5... | 13 | 0.1875 | 6,582.5625 | 5,450.666667 | 6,843.769231 | |
In the right triangle \( \triangle ABC \), \( \angle B = 90^\circ \). Point \( P \) is on the angle bisector of \( \angle A \) within \( \triangle ABC \). Point \( M \) (distinct from \( A \) and \( B \)) is a point on side \( AB \). The lines \( AP \), \( CP \), and \( MP \) intersect sides \( BC \), \( AB \), and \( ... | 1/2 | 0 | 8,192 | -1 | 8,192 | |
In a rectangle of size $3 \times 4$, 4 points are chosen. Find the smallest number $C$ such that the distance between some two of these points does not exceed $C$. | 2.5 | 0 | 8,102.125 | -1 | 8,102.125 | |
What is the sum of the the roots of the equation $4x^3 + 5x^2 - 8x = 0$? Express your answer as a decimal to the nearest hundredth. | -1.25 | 1 | 2,308.0625 | 2,308.0625 | -1 | |
A regular hexagon PROFIT has area 1. Every minute, greedy George places the largest possible equilateral triangle that does not overlap with other already-placed triangles in the hexagon, with ties broken arbitrarily. How many triangles would George need to cover at least $90 \%$ of the hexagon's area? | 46 | It's not difficult to see that the first triangle must connect three non-adjacent vertices (e.g. POI), which covers area $\frac{1}{2}$, and leaves three 30-30-120 triangles of area $\frac{1}{6}$ each. Then, the next three triangles cover $\frac{1}{3}$ of the respective small triangle they are in, and leave six 30-30-12... | 0 | 7,720.125 | -1 | 7,720.125 |
Let $m$ be a positive integer, and let $a_0, a_1, \dots , a_m$ be a sequence of real numbers such that $a_0 = 37$, $a_1 = 72$, $a_m=0$, and $$ a_{k+1} = a_{k-1} - \frac{3}{a_k} $$for $k = 1,
2, \dots, m-1$. Find $m$. | 889 | 0.75 | 5,269.5 | 4,758.083333 | 6,803.75 | |
When each of $702$, $787$, and $855$ is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of $412$, $722$, and $815$ is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Find $m+n+r+s$. | 62 | We know that $702 = am + r, 787 = bm + r,$ and $855 = cm+r$ where $a-c$ are integers.
Subtracting the first two, the first and third, and the last two, we get $85 = (b-a)m, 153=(c-a)m,$ and $68=(c-b)m.$
We know that $b-a, c-a$ and $c-b$ must be integers, so all the numbers are divisible by $m.$
Factorizing the numbe... | 1 | 2,449.3125 | 2,449.3125 | -1 |
A pedestrian crossing signal at an intersection alternates between red and green lights, with the red light lasting for $30$ seconds. The probability that Little Ming, upon arriving at the intersection and encountering a red light, will have to wait at least $10$ seconds before the green light appears is _______. | \frac{5}{6} | 0 | 4,705.3125 | -1 | 4,705.3125 | |
Sanitation workers plan to plant 7 trees in a row on one side of a road, choosing only from plane trees and willow trees. What is the number of planting methods where no two adjacent trees are both willows? | 34 | 0.9375 | 4,416.9375 | 4,165.266667 | 8,192 | |
Given a regular decagon, calculate the number of distinct points in the interior of the decagon where two or more diagonals intersect. | 210 | 0.125 | 8,059.9375 | 7,135.5 | 8,192 | |
Two distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 6, 9\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction. | \frac{3}{5} | 0 | 4,417.875 | -1 | 4,417.875 | |
Find the number of arrangements of 4 beads (2 red, 2 green, 2 blue) in a circle such that the two red beads are not adjacent. | 11 | We divide this problem into cases based on the relative position of the two red beads: - They are adjacent. Then, there are 4 possible placements of the green and blue beads: GGBB, GBBG, GBGB, BGGB. - They are 1 bead apart. Then, there are two choices for the bead between then and 2 choices for the other bead of that c... | 0 | 8,192 | -1 | 8,192 |
Convert $1357_{10}$ to base 5. | 20412_5 | 1 | 3,416.375 | 3,416.375 | -1 | |
For the polynomial
\[ p(x) = 985 x^{2021} + 211 x^{2020} - 211, \]
let its 2021 complex roots be \( x_1, x_2, \cdots, x_{2021} \). Calculate
\[ \sum_{k=1}^{2021} \frac{1}{x_{k}^{2} + 1} = \]
| 2021 | 0 | 8,048.5 | -1 | 8,048.5 | |
The sum of two natural numbers is $17402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? | 14238 | 1. **Identify the relationship between the two numbers**: Given that one number is divisible by $10$ and removing its units digit (which is $0$) gives the other number, we can denote the smaller number as $a$ and the larger number as $10a$.
2. **Set up the equation for their sum**: The sum of the two numbers is given ... | 1 | 2,528 | 2,528 | -1 |
The angle bisectors $\mathrm{AD}$ and $\mathrm{BE}$ of the triangle $\mathrm{ABC}$ intersect at point I. It turns out that the area of triangle $\mathrm{ABI}$ is equal to the area of quadrilateral $\mathrm{CDIE}$. Find the maximum possible value of angle $\mathrm{ACB}$. | 60 | 0.0625 | 8,125.3125 | 7,125 | 8,192 | |
Let's call a natural number "remarkable" if it is the smallest among natural numbers with the same sum of digits. What is the sum of the digits of the two-thousand-and-first remarkable number? | 2001 | 0 | 8,192 | -1 | 8,192 | |
Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden(back, bottom, between). What is the total number of dots NOT visible in this view? [asy]
/* AMC8 2000 #8 Problem */
draw((0,0)--(1,0)--(1.5,0.66)--(1.5,3.66)--(.5,3.66)--(0,3)--cycle);
d... | 41 | 0.125 | 7,798.625 | 6,107 | 8,040.285714 | |
A fair 10-sided die is rolled repeatedly until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
- **A** $\frac{1}{300}$
- **B** $\frac{1}{252}$
- **C** $\frac{1}{500}$
- **D** $\frac{1}{100}$ | \frac{1}{252} | 0 | 7,983.3125 | -1 | 7,983.3125 | |
Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. The minimum height of this regular tetrahedron is: | 2 + \frac{2 \sqrt{6}}{3} | 0 | 8,002.875 | -1 | 8,002.875 | |
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$ | 901 | The smallest $S$ is $1+2+ \ldots +90 = 91 \cdot 45 = 4095$. The largest $S$ is $11+12+ \ldots +100=111\cdot 45=4995$. All numbers between $4095$ and $4995$ are possible values of S, so the number of possible values of S is $4995-4095+1=901$.
Alternatively, for ease of calculation, let set $\mathcal{B}$ be a 10-element... | 0.5625 | 6,379.6875 | 4,970.111111 | 8,192 |
The area of the shaded region $\text{BEDC}$ in parallelogram $\text{ABCD}$ is | 64 | 1. **Identify the areas to be calculated**: We need to find the area of the shaded region $\text{BEDC}$ in parallelogram $\text{ABCD}$. The area of $\text{BEDC}$ can be found by subtracting the area of triangle $\text{ABE}$ from the area of parallelogram $\text{ABCD}$:
\[
[\text{BEDC}] = [\text{ABCD}] - [\text{AB... | 0 | 8,033.8125 | -1 | 8,033.8125 |
What is the slope of the line that is tangent to a circle at point (5,5) if the center of the circle is (3,2)? Express your answer as a common fraction. | -\frac{2}{3} | 0.9375 | 1,831.9375 | 1,407.933333 | 8,192 | |
Given an ellipse $\frac{x^{2}}{8} + \frac{y^{2}}{2} = 1$, and a point $A(2,1)$ on the ellipse. The slopes of the lines $AB$ and $AC$ connecting point $A$ to two moving points $B$ and $C$ on the ellipse are $k_{1}$ and $k_{2}$, respectively, with $k_{1} + k_{2} = 0$. Determine the slope $k$ of line $BC$. | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Let $a$, $b$, $c$, and $d$ be positive integers with $a < 3b$, $b < 3c$, and $c < 4d$. Additionally, suppose $b + d = 200$. The largest possible value for $a$ is:
A) 438
B) 440
C) 445
D) 449
E) 455 | 449 | 0 | 8,104.4375 | -1 | 8,104.4375 | |
The minimum positive period of the function $f(x) = \sin \omega x + \sqrt{3}\cos \omega x + 1$ ($\omega > 0$) is $\pi$. When $x \in [m, n]$, $f(x)$ has at least 5 zeros. The minimum value of $n-m$ is \_\_\_\_\_\_. | 2\pi | 0 | 8,192 | -1 | 8,192 | |
What is the base \(2\) representation of \(125_{10}\)? | 1111101_2 | 0.3125 | 3,569.8125 | 3,319.6 | 3,683.545455 | |
Two right triangles share a side as follows: Triangle ABC and triangle ABD have AB as their common side. AB = 8 units, AC = 12 units, and BD = 8 units. There is a rectangle BCEF where point E is on line segment BD and point F is directly above E such that CF is parallel to AB. What is the area of triangle ACF? | 24 | 0 | 8,172 | -1 | 8,172 | |
As shown in the diagram, a square is divided into 4 identical rectangles, each of which has a perimeter of 20 centimeters. What is the area of this square? | 64 | 0.3125 | 6,558.375 | 5,100.4 | 7,221.090909 | |
For any integer $n$, define $\lfloor n\rfloor$ as the greatest integer less than or equal to $n$. For any positive integer $n$, let $$f(n)=\lfloor n\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n}{3}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor.$$ For how many values of $n, 1 \leq n \leq 10... | 55 | 55 Notice that, for fixed $a,\lfloor n / a\rfloor$ counts the number of integers $b \in$ $\{1,2, \ldots, n\}$ which are divisible by $a$; hence, $f(n)$ counts the number of pairs $(a, b), a, b \in$ $\{1,2, \ldots, n\}$ with $b$ divisible by $a$. For any fixed $b$, the number of such pairs is $d(b)$ (the number of divis... | 0.0625 | 8,119.125 | 7,682 | 8,148.266667 |
For how many values of \(c\) in the interval \([0, 2000]\) does the equation
\[8 \lfloor x \rfloor + 3 \lceil x \rceil = c\]
have a solution for \(x\)? | 363 | 0 | 7,217.75 | -1 | 7,217.75 | |
What is the largest number of white and black chips that can be placed on a chessboard so that on each horizontal and each vertical, the number of white chips is exactly twice the number of black chips? | 48 | 0.3125 | 7,388.375 | 5,620.4 | 8,192 | |
Let $A = (1,0)$ and $B = (5,4).$ Let $P$ be a point on the parabola $y^2 = 4x.$ Find the smallest possible value of $AP + BP.$ | 6 | 0.25 | 7,761.25 | 7,674.25 | 7,790.25 | |
Suppose that the roots of $x^3 + 2x^2 + 5x - 8 = 0$ are $p$, $q$, and $r$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $p+q$, $q+r$, and $r+p$. Find $w$. | 18 | 0.75 | 5,299.6875 | 4,649.083333 | 7,251.5 | |
In a department store, they received 10 suitcases and 10 keys separately in an envelope. Each key opens only one suitcase, and every suitcase can be matched with a corresponding key.
A worker in the department store, who received the suitcases, sighed:
- So much hassle with matching keys! I know how stubborn inanima... | 29.62 | 0 | 8,183.375 | -1 | 8,183.375 | |
If $y=x^2+px+q$, then if the least possible value of $y$ is zero $q$ is equal to: | \frac{p^2}{4} | 1. **Identify the vertex of the quadratic function**: The quadratic function given is $y = x^2 + px + q$. The vertex form of a quadratic function $y = ax^2 + bx + c$ is given by the vertex $(h, k)$ where $h = -\frac{b}{2a}$ and $k$ is the value of the function at $x = h$. Here, $a = 1$, $b = p$, and $c = q$.
2. **Calc... | 1 | 2,145.5 | 2,145.5 | -1 |
Let $A$ be a set of integers such that for each integer $m$, there exists an integer $a \in A$ and positive integer $n$ such that $a^{n} \equiv m(\bmod 100)$. What is the smallest possible value of $|A|$? | 41 | Work in $R=\mathbb{Z} / 100 \mathbb{Z} \cong \mathbb{Z} / 4 \mathbb{Z} \times \mathbb{Z} / 25 \mathbb{Z}$. Call an element $r \in R$ type $(s, t)$ if $s=\nu_{2}(r) \leq 2$ and $t=\nu_{5}(r) \leq 2$. Also, define an element $r \in R$ to be coprime if it is of type $(0,0)$, powerful if it is of types $(0,2),(2,0)$, or $(... | 0 | 8,192 | -1 | 8,192 |
What is the $100^{\mathrm{th}}$ odd positive integer, and what even integer directly follows it? | 200 | 0.8125 | 1,816.9375 | 1,541.769231 | 3,009.333333 | |
Encrypt integers by the following method: the digit of each number becomes the units digit of its product with 7, then replace each digit _a_ with $10 - _a_$. If a number is encrypted by the above method and becomes 473392, then the original number is ______. | 891134 | 0.9375 | 3,365.4375 | 3,043.666667 | 8,192 | |
The sum of $25$ consecutive even integers is $10,000$. What is the largest of these $25$ consecutive integers? | 424 | 1. **Define the sequence**: Let the smallest of the 25 consecutive even integers be $x$. Then the integers are $x, x+2, x+4, \ldots, x+48$.
2. **Formulate the sum**: The sum of these integers can be expressed as:
\[
x + (x+2) + (x+4) + \cdots + (x+48)
\]
This is an arithmetic sequence where the first term ... | 1 | 2,515.8125 | 2,515.8125 | -1 |
A gardener plans to enclose a rectangular garden with 480 feet of fencing. However, one side of the garden will be twice as long as another side. What is the maximum area of this garden? | 12800 | 1 | 5,448.75 | 5,448.75 | -1 | |
Given a right-angled triangle, one of whose acute angles is $\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which value of $\alpha$ this ratio will be the smallest. | \sqrt{2} + 1 | 0.25 | 6,691.3125 | 5,839.25 | 6,975.333333 | |
Alexio has 200 cards numbered 1-200, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 4, 5, or 7? Express your answer as a common fraction. | \frac{97}{200} | 0.9375 | 4,218.9375 | 3,954.066667 | 8,192 | |
In the diagram, $BP$ and $BQ$ trisect $\angle ABC$. $BM$ bisects $\angle PBQ$. Find the ratio of the measure of $\angle MBQ$ to the measure of $\angle ABQ$. | \frac14 | 0.75 | 2,468.5625 | 2,382.333333 | 2,727.25 | |
In the rectangular coordinate system $(xOy)$, there are two curves $C_1: x + y = 4$ and $C_2: \begin{cases} x = 1 + \cos \theta \\ y = \sin \theta \end{cases}$ (where $\theta$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar a... | \frac{1}{4}(\sqrt{2} + 1) | 0 | 7,129.625 | -1 | 7,129.625 | |
Given $\cos \left(a- \frac{\pi}{6}\right) + \sin a = \frac{4 \sqrt{3}}{5}$, find the value of $\sin \left(a+ \frac{7\pi}{6}\right)$. | -\frac{4}{5} | 0.875 | 5,725.5625 | 5,373.214286 | 8,192 | |
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? | \frac{281}{13} | 1. **Identify the triangle formed by the line and the axes**: The line $12x + 5y = 60$ intersects the x-axis and y-axis, forming a right triangle with the axes. To find the intercepts:
- **x-intercept**: Set $y = 0$ in the equation $12x + 5y = 60$:
\[
12x = 60 \implies x = \frac{60}{12} = 5
\]
- **... | 0.9375 | 4,568.375 | 4,326.8 | 8,192 |
In the Cartesian coordinate system, given the set of points $I=\{(x, y) \mid x$ and $y$ are integers, and $0 \leq x \leq 5,0 \leq y \leq 5\}$, find the number of distinct squares that can be formed with vertices from the set $I$. | 105 | 0 | 8,192 | -1 | 8,192 | |
What is the smallest integral value of $k$ such that
$2x(kx-4)-x^2+6=0$ has no real roots? | 2 | 1. **Expand and simplify the given quadratic equation**:
\[
2x(kx-4) - x^2 + 6 = 0
\]
Expanding the terms:
\[
2kx^2 - 8x - x^2 + 6 = 0
\]
Combine like terms:
\[
(2k-1)x^2 - 8x + 6 = 0
\]
2. **Condition for no real roots**:
A quadratic equation $ax^2 + bx + c = 0$ has no real roots i... | 1 | 2,485.9375 | 2,485.9375 | -1 |
Determine the fourth-largest divisor of $1,234,560,000$. | 154,320,000 | 0 | 8,070.375 | -1 | 8,070.375 | |
Given $\sin \theta + \cos \theta = \frac{1}{5}$, with $\theta \in (0,\pi)$.
$(1)$ Find the value of $\tan \theta$;
$(2)$ Find the value of $\frac{1-2\sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta}$. | -7 | 1 | 4,397.25 | 4,397.25 | -1 | |
Calculate the integer nearest to $500\sum_{n=4}^{10005}\frac{1}{n^2-9}$. | 174 | 0.0625 | 7,994.75 | 6,380 | 8,102.4 | |
A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars? | 10 | 1. **Understanding the problem**: The problem states that half-pound packages of fish are being sold for $3 per package after a 50% discount. We need to find the regular price for a full pound of fish.
2. **Setting up the equation**: Let $x$ be the regular price for a full pound of fish. Since the fish is currently be... | 0 | 1,515.6875 | -1 | 1,515.6875 |
Given points $A(2,-1,1)$, $B(1,-2,1)$, $C(0,0,-1)$, the distance from $A$ to $BC$ is ______. | \frac{\sqrt{17}}{3} | 0 | 2,696.5 | -1 | 2,696.5 | |
Compute the number of positive integers $n \leq 1000$ such that \operatorname{lcm}(n, 9)$ is a perfect square. | 43 | Suppose $n=3^{a} m$, where $3 \nmid m$. Then $$\operatorname{lcm}(n, 9)=3^{\max (a, 2)} m$$ In order for this to be a square, we require $m$ to be a square, and $a$ to either be even or 1 . This means $n$ is either a square (if $a$ is even) or of the form $3 k^{2}$ where $3 \nmid k$ (if $a=1$ ). There are 31 numbers of... | 0 | 8,192 | -1 | 8,192 |
Suppose $a, b$, and $c$ are complex numbers satisfying $$\begin{aligned} a^{2} & =b-c \\ b^{2} & =c-a, \text { and } \\ c^{2} & =a-b \end{aligned}$$ Compute all possible values of $a+b+c$. | 0, \pm i \sqrt{6} | Summing the equations gives $a^{2}+b^{2}+c^{2}=0$ and summing $a$ times the first equation and etc. gives $a^{3}+b^{3}+c^{3}=0$. Let $a+b+c=k$. Then $a^{2}+b^{2}+c^{2}=0$ means $a b+b c+c a=k^{2} / 2$, and $a^{3}+b^{3}+c^{3}=0 \Longrightarrow-3 a b c=a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)(a^{2}+b^{2}+c^{2}-a b-b c-c a)=-k^{... | 0 | 8,080.1875 | -1 | 8,080.1875 |
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other ar... | 16 | 0 | 7,479.4375 | -1 | 7,479.4375 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=2$, and $\overrightarrow{b} \perp (2\overrightarrow{a}+ \overrightarrow{b})$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{2\pi}{3} | 0 | 1,645.375 | -1 | 1,645.375 | |
How many distinct positive factors does 32 have? | 6 | 1 | 1,299.25 | 1,299.25 | -1 | |
Find the matrix $\mathbf{M}$ that doubles the first column of a matrix. In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2a & b \\ 2c & d \end{pmatrix}.\]If no such matrix $\mathbf{M}$ exists, then enter the zero matrix. | \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} | 0.625 | 6,312.5625 | 5,608 | 7,486.833333 | |
A multiple choice examination consists of $20$ questions. The scoring is $+5$ for each correct answer, $-2$ for each incorrect answer, and $0$ for each unanswered question. John's score on the examination is $48$. What is the maximum number of questions he could have answered correctly? | 12 | Let $c$ be the number of questions John answered correctly, $w$ be the number of questions he answered incorrectly, and $b$ be the number of questions he left blank. We know from the problem statement that:
1. The total number of questions is 20:
\[
c + w + b = 20
\]
2. The scoring formula given is $+5$ for ea... | 0.9375 | 4,231.4375 | 3,967.4 | 8,192 |
Let $\mathbf{m},$ $\mathbf{n},$ and $\mathbf{p}$ be unit vectors such that the angle between $\mathbf{m}$ and $\mathbf{n}$ is $\alpha,$ and the angle between $\mathbf{p}$ and $\mathbf{m} \times \mathbf{n}$ is also $\alpha.$ If $\mathbf{n} \cdot (\mathbf{p} \times \mathbf{m}) = \frac{1}{4},$ find the smallest possible ... | 30^\circ | 0 | 5,957 | -1 | 5,957 | |
Observe that $7 = 5 \times 1 + 2$, $12 = 5 \times 2 + 2$, and $17 = 5 \times 3 + 2$. Here, 7, 12, and 17 are called "3 consecutive numbers that leave a remainder of 2 when divided by 5." If the sum of 3 consecutive numbers that leave a remainder of 2 when divided by 5 is equal to 336, find the smallest of these numbers... | 107 | 1 | 3,645.5625 | 3,645.5625 | -1 | |
In the Cartesian coordinate system, A and B are points moving on the x-axis and y-axis, respectively. If the circle C with AB as its diameter is tangent to the line $3x+y-4=0$, then the minimum area of circle C is \_\_\_\_\_\_. | \frac {2}{5}\pi | 0 | 8,164.6875 | -1 | 8,164.6875 | |
Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=3$, and $DC=4$. What is the area of $\triangle ABC$? | $7\sqrt{3}$ | 1. **Identify the relationship between segments in the triangle**: Given that $ABC$ is a right triangle with the right angle at $B$, and $D$ is the foot of the altitude from $B$ to $AC$, we know that $AD = 3$ and $DC = 4$.
2. **Use the geometric property of right triangles**: In a right triangle, the altitude from th... | 0 | 3,602 | -1 | 3,602 |
Evaluate $$\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{29}\rceil$$Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$ | 112 | 0.1875 | 6,168.5625 | 5,272.333333 | 6,375.384615 | |
The high-speed train "Sapsan," approaching a railway station at a speed of \( v = 216 \) km/h, emits a warning sound signal lasting \( \Delta t = 5 \) seconds when it is half a kilometer away from the station. What will be the duration of the signal \( \Delta t_{1} \) from the perspective of passengers standing on the ... | 4.12 | 0.0625 | 8,186.0625 | 8,097 | 8,192 | |
What is the sum of the 2023 fractions of the form $\frac{2}{n(n+3)}$ for $n$ as the positive integers from 1 through 2023? Express your answer as a decimal to the nearest thousandth. | 1.222 | 0 | 8,089.375 | -1 | 8,089.375 | |
Alexio now has 100 cards numbered from 1 to 100. He again randomly selects one card from the box. What is the probability that the number on the chosen card is a multiple of 3, 5, or 7? Express your answer as a common fraction. | \frac{11}{20} | 0.8125 | 3,606.3125 | 3,163.307692 | 5,526 | |
To actively create a "demonstration school for the prevention and control of myopia in children and adolescents in the city" and cultivate students' good eye habits, a certain school conducted a competition on correct eye knowledge this semester. $20$ student answer sheets were randomly selected and their scores (denot... | 1755 | 0.3125 | 6,096.9375 | 4,994.2 | 6,598.181818 | |
In a convex polygon with 1992 sides, the minimum number of interior angles that are not acute is: | 1989 | 0.25 | 7,659.4375 | 6,061.75 | 8,192 | |
If $y=kx^{\frac{1}{4}}$ and $y=3\sqrt{2}$ at $x=81$, what is the value of $y$ at $x=4$? | 2 | 1 | 2,729.1875 | 2,729.1875 | -1 | |
Given that the sum of the coefficients of the expansion of $(\frac{3}{x}-\sqrt{x})^n$ is $512$. Find:<br/>
$(1)$ The coefficient of the term containing $x^{3}$ in the expansion;<br/>
$(2)$ The constant term in the expansion of $(1+\frac{1}{x})(2x-1)^n$. | 17 | 0.6875 | 5,371 | 4,567.363636 | 7,139 | |
Let $x, y$, and $N$ be real numbers, with $y$ nonzero, such that the sets $\left\{(x+y)^{2},(x-y)^{2}, x y, x / y\right\}$ and $\{4,12.8,28.8, N\}$ are equal. Compute the sum of the possible values of $N$. | 85.2 | First, suppose that $x$ and $y$ were of different signs. Then $x y<0$ and $x / y<0$, but the set has at most one negative value, a contradiction. Hence, $x$ and $y$ have the same sign; without loss of generality, we say $x$ and $y$ are both positive. Let $(s, d):=(x+y, x-y)$. Then the set given is equal to $\left\{s^{2... | 0 | 8,192 | -1 | 8,192 |
Five students, $A$, $B$, $C$, $D$, and $E$, entered the final of a school skills competition and the rankings from first to fifth were determined (with no ties). It is known that students $A$ and $B$ are neither first nor last. Calculate the number of different arrangements of the final rankings for these 5 students. | 36 | 0.625 | 5,793.0625 | 4,353.7 | 8,192 | |
Given the positive numbers \( a, b, c, x, y, z \) that satisfy the equations \( cy + bz = a \), \( az + cx = b \), and \( bx + ay = c \), find the minimum value of the function
\[
f(x, y, z) = \frac{x^{2}}{1+x} + \frac{y^{2}}{1+y} + \frac{z^{2}}{1+z}.
\] | 1/2 | 0 | 8,192 | -1 | 8,192 | |
Given a sequence of positive terms $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, it satisfies the equation $2S_n = a_n^2 + a_n$ for all natural numbers $n$. Define a new sequence $\{c_n\}$ where $c_n = (-1)^n \frac{2a_n + 1}{2S_n}$. Find the sum of the first 2016 terms of the sequence $\{c_n\}$. | - \frac{2016}{2017} | 0.375 | 7,330.25 | 5,894 | 8,192 | |
Determine the coefficient of the $x^5$ term in the expansion of $(x+1)(x^2-x-2)^3$. | -6 | 0.4375 | 7,637.4375 | 6,924.428571 | 8,192 | |
Given triangle $ABC$, $\overrightarrow{CA}•\overrightarrow{CB}=1$, the area of the triangle is $S=\frac{1}{2}$,<br/>$(1)$ Find the value of angle $C$;<br/>$(2)$ If $\sin A\cos A=\frac{{\sqrt{3}}}{4}$, $a=2$, find $c$. | \frac{2\sqrt{6}}{3} | 0 | 7,776.8125 | -1 | 7,776.8125 | |
In $\triangle ABC$, we have $AC = BC = 10$, and $AB = 8$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 12$. What is $BD$? | 2\sqrt{15} | 0 | 8,002.375 | -1 | 8,002.375 | |
Two random points are chosen on a segment and the segment is divided at each of these two points. Of the three segments obtained, find the probability that the largest segment is more than three times longer than the smallest segment. | \frac{27}{35} | We interpret the problem with geometric probability. Let the three segments have lengths $x, y, 1-x-y$ and assume WLOG that $x \geq y \geq 1-x-y$. The every possible $(x, y)$ can be found in the triangle determined by the points $\left(\frac{1}{3}, \frac{1}{3}\right),\left(\frac{1}{2}, \frac{1}{2}\right),(1,0)$ in $\ma... | 0 | 8,192 | -1 | 8,192 |
Given \(x \geqslant 1\), the minimum value of the function \(y=f(x)= \frac {4x^{2}-2x+16}{2x-1}\) is \_\_\_\_\_\_, and the corresponding value of \(x\) is \_\_\_\_\_\_. | \frac {5}{2} | 0.875 | 3,591.6875 | 3,417.428571 | 4,811.5 |
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