Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
19,300 |
How many positive integers less than $201$ are multiples of either $8$ or $11$, but not both at once?
|
39
| 4.6875 |
19,301 |
Given the sets \( A = \{2, 0, 1, 7\} \) and \( B = \{ x \mid x^2 - 2 \in A, \, x - 2 \notin A \} \), the product of all elements in set \( B \) is:
|
36
| 23.4375 |
19,302 |
Given that the function $f(x+2)$ is an odd function and it satisfies $f(6-x)=f(x)$, and $f(3)=2$, determine the value of $f(2008)+f(2009)$.
|
-2
| 19.53125 |
19,303 |
A standard deck of 52 cards is randomly arranged. What is the probability that the top three cards are $\spadesuit$, $\heartsuit$, and $\spadesuit$ in that sequence?
|
\dfrac{78}{5100}
| 0 |
19,304 |
How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips.
|
14
| 72.65625 |
19,305 |
Let \( a \) be an integer such that \( |a| \leq 2005 \). Find the number of values of \( a \) for which the system of equations
\[
\begin{cases}
x^2 = y + a, \\
y^2 = x + a
\end{cases}
\]
has integer solutions.
|
90
| 32.8125 |
19,306 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Angles $A$, $B$, $C$ form an arithmetic sequence, $c - a = 1$, and $b = \sqrt{7}$.
(I) Find the area $S$ of $\triangle ABC$.
(II) Find the value of $\sin\left(2C + \frac{\pi}{4}\right)$.
|
\frac{3\sqrt{6} - 13\sqrt{2}}{28}
| 32.03125 |
19,307 |
There are five positive integers that are common divisors of each number in the list $$36, 72, -24, 120, 96.$$ Find the sum of these five positive integers.
|
16
| 31.25 |
19,308 |
Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG=5$. If $EFGH$ has three sides with distinct integer lengths and $FG = 1$, then what is the area of $EFGH$? Express your answer in simplest radical form.
|
\sqrt{6} + 6
| 0.78125 |
19,309 |
A piece of iron wire with a length of $80cm$ is randomly cut into three segments. Calculate the probability that each segment has a length of no less than $20cm$.
|
\frac{1}{16}
| 10.9375 |
19,310 |
Given a cubic function $f(x) = ax^3 + bx^2 + cx + d$ ($a \neq 0$), the definition of an inflection point is provided: Let $f'(x)$ be the derivative of $y = f(x)$, and let $f''(x)$ be the derivative of $f'(x)$. If the equation $f''(x) = 0$ has a real number solution $x_0$, then the point $(x_0, f(x_0))$ is called an "inflection point" of the function $y = f(x)$. A student discovered that every cubic function has an "inflection point" and a center of symmetry, and that the "inflection point" is the center of symmetry. Let the function be $$g(x) = \frac{1}{3}x^3 - \frac{1}{2}x^2 + 3x - \frac{5}{12}$$. Find the value of $$g(\frac{1}{2018}) + g(\frac{2}{2018}) + \ldots + g(\frac{2017}{2018})$$.
|
2017
| 92.96875 |
19,311 |
Find the remainder when $2^{2^{2^2}}$ is divided by 500.
|
36
| 41.40625 |
19,312 |
How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not?
|
64
| 68.75 |
19,313 |
Three distinct integers are chosen uniformly at random from the set $$ \{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}. $$ Compute the probability that their arithmetic mean is an integer.
|
7/20
| 87.5 |
19,314 |
A store received a cistern of milk. The seller has a balance scale without weights (you can place cans on the pans of the scale) and three identical cans, two of which are empty and the third contains 1 liter of milk. How can you measure exactly 85 liters of milk into one can using no more than eight weighings?
|
85
| 43.75 |
19,315 |
Given $y=f(x)+x^2$ is an odd function, and $f(1)=1$, then $f(-1)=$?
|
-3
| 100 |
19,316 |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, $f(x) = \begin{cases} \log_{2}(x+1) & \text{for } x \geq 0 \\ g(x) & \text{for } x < 0 \\ \end{cases}$. Find the value of $g\left(f(-7)\right)$.
|
-2
| 92.96875 |
19,317 |
There are 6 people including A, B, and C standing in a row for a photo, where A cannot stand at either end, and B and C must stand next to each other. How many such arrangements are there?
|
144
| 12.5 |
19,318 |
There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$ . $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$ . Call $X$ the intersection of $AF$ and $DE$ . What is the area of pentagon $BCFXE$ ?
Proposed by Minseok Eli Park (wolfpack)
|
47
| 6.25 |
19,319 |
Simplify first, then evaluate: $\frac{1}{{{x^2}+2x+1}}\cdot (1+\frac{3}{x-1})\div \frac{x+2}{{{x^2}-1}$, where $x=2\sqrt{5}-1$.
|
\frac{\sqrt{5}}{10}
| 89.84375 |
19,320 |
Given that $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$ are all unit vectors, and satisfy $\frac{1}{2}\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}=\overrightarrow{0}$, calculate the value of $\overrightarrow{AB} \cdot \overrightarrow{AC}$.
|
\frac{5}{8}
| 60.15625 |
19,321 |
Given that A and B are any two points on the line l, and O is a point outside of l. If there is a point C on l that satisfies the equation $\overrightarrow {OC}= \overrightarrow {OA}cosθ+ \overrightarrow {OB}cos^{2}θ$, find the value of $sin^{2}θ+sin^{4}θ+sin^{6}θ$.
|
\sqrt {5}-1
| 0 |
19,322 |
Given $\overrightarrow{a}=(\sin \pi x,1)$, $\overrightarrow{b}=( \sqrt {3},\cos \pi x)$, and $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$:
(I) If $x\in[0,2]$, find the interval(s) where $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$ is monotonically increasing.
(II) Let $P$ be the coordinates of the first highest point and $Q$ be the coordinates of the first lowest point on the graph of $y=f(x)$ to the right of the $y$-axis. Calculate the cosine value of $\angle POQ$.
|
-\frac{16\sqrt{481}}{481}
| 28.125 |
19,323 |
Throw a fair die, and let event $A$ be that the number facing up is even, and event $B$ be that the number facing up is greater than $2$ and less than or equal to $5$. Then, the probability of the complement of event $B$ is ____, and the probability of event $A \cup B$ is $P(A \cup B) = $ ____.
|
\dfrac{5}{6}
| 92.96875 |
19,324 |
The image shows a 3x3 grid where each cell contains one of the following characters: 华, 罗, 庚, 杯, 数, 学, 精, 英, and 赛. Each character represents a different number from 1 to 9, and these numbers satisfy the following conditions:
1. The sum of the four numbers in each "田" (four cells in a square) is equal.
2. 华 $\times$ 华 $=$ 英 $\times$ 英 + 赛 $\times$ 赛.
3. 数 > 学
According to the above conditions, find the product of the numbers represented by 华, 杯, and 赛.
|
120
| 21.09375 |
19,325 |
How many different four-digit numbers, divisible by 4, can be composed of the digits 1, 2, 3, and 4,
a) if each digit can occur only once?
b) if each digit can occur multiple times?
|
64
| 67.1875 |
19,326 |
Simplify first, then evaluate: $-2(-x^2y+xy^2)-[-3x^2y^2+3x^2y+(3x^2y^2-3xy^2)]$, where $x=-1$, $y=2$.
|
-6
| 60.15625 |
19,327 |
In trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$, $AD = AB = 5$, and $DC = 10$. Additionally, $E$ is on $\overline{DC}$ such that $\overline{BE}$ is parallel to $\overline{AD}$. Find the area of the parallelogram formed by $\overline{BE}$.
|
25
| 75.78125 |
19,328 |
Let $D$ be the circle with the equation $x^2 + 8x + 20y + 89 = -y^2 - 6x$. Find the value of $c + d + s$ where $(c, d)$ is the center of $D$ and $s$ is its radius.
|
-17 + 2\sqrt{15}
| 67.1875 |
19,329 |
What is the greatest four-digit number which is a multiple of 17?
|
9996
| 0 |
19,330 |
Given that the polar coordinate equation of circle $C$ is $ρ=2\cos θ$, the parametric equation of line $l$ is $\begin{cases}x= \frac{1}{2}+ \frac{ \sqrt{3}}{2}t \\ y= \frac{1}{2}+ \frac{1}{2}t\end{cases} (t\text{ is a parameter})$, and the polar coordinates of point $A$ are $(\frac{ \sqrt{2}}{2} ,\frac{π}{4} )$, let line $l$ intersect with circle $C$ at points $P$ and $Q$.
(1) Write the Cartesian coordinate equation of circle $C$;
(2) Find the value of $|AP|⋅|AQ|$.
By Vieta's theorem, we get $t_{1}⋅t_{2}=- \frac{1}{2} < 0$, and according to the geometric meaning of the parameters, we have $|AP|⋅|AQ|=|t_{1}⋅t_{2}|= \frac{1}{2}$.
|
\frac{1}{2}
| 94.53125 |
19,331 |
Given vectors $\overrightarrow{a} = (5\sqrt{3}\cos x, \cos x)$ and $\overrightarrow{b} = (\sin x, 2\cos x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + |\overrightarrow{b}|^2 + \frac{3}{2}$.
(I) Find the range of $f(x)$ when $x \in [\frac{\pi}{6}, \frac{\pi}{2}]$.
(II) If $f(x) = 8$ when $x \in [\frac{\pi}{6}, \frac{\pi}{2}]$, find the value of $f(x - \frac{\pi}{12})$.
|
\frac{3\sqrt{3}}{2} + 7
| 0.78125 |
19,332 |
In triangle $\triangle ABC$, $\overrightarrow{BC}=\sqrt{3}\overrightarrow{BD}$, $AD\bot AB$, $|{\overrightarrow{AD}}|=1$, then $\overrightarrow{AC}•\overrightarrow{AD}=\_\_\_\_\_\_$.
|
\sqrt{3}
| 33.59375 |
19,333 |
Given that positive real numbers a and b satisfy $a^{2}+2ab+4b^{2}=6$, calculate the maximum value of a+2b.
|
2\sqrt{2}
| 78.90625 |
19,334 |
Select two distinct numbers a, b from the set {0,1,2,3,4,5,6} to form a complex number a+bi, and determine the total number of such complex numbers with imaginary parts.
|
36
| 39.84375 |
19,335 |
An investor placed \$12,000 in a one-year fixed deposit that yielded a simple annual interest rate of 8%. After one year, the total amount was reinvested in another one-year deposit. At the end of the second year, the total amount was \$13,500. If the annual interest rate of the second deposit is \( s\% \), what is \( s \)?
|
4.17\%
| 27.34375 |
19,336 |
A fair six-sided die is rolled twice, and the resulting numbers are denoted as $a$ and $b$.
(1) Find the probability that $a^2 + b^2 = 25$.
(2) Given three line segments with lengths $a$, $b$, and $5$, find the probability that they can form an isosceles triangle (including equilateral triangles).
|
\frac{7}{18}
| 10.9375 |
19,337 |
My grandpa has 12 pieces of art, including 4 prints by Escher and 3 by Picasso. What is the probability that all four Escher prints and all three Picasso prints will be placed consecutively?
|
\dfrac{1}{660}
| 0 |
19,338 |
Sam places a total of 30 yellow Easter eggs in several purple baskets and a total of 45 pink Easter eggs in some orange baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Sam put in each basket?
|
15
| 63.28125 |
19,339 |
Given a sequence $\{a_n\}$ satisfies $a_n + (-1)^{n+1}a_{n+1} = 2n - 1$, find the sum of the first 40 terms, $S_{40}$.
|
780
| 13.28125 |
19,340 |
A student's written work has a two-grade evaluation system; i.e., the work will either pass if it is done well, or fail if it is done poorly. The works are first checked by a neural network that gives incorrect answers in 10% of cases, and then all works deemed failed are rechecked manually by experts who do not make mistakes. The neural network can both classify good work as failed and vice versa – classify bad work as passed. It is known that among all the submitted works, 20% are actually bad. What is the minimum percentage of bad work that can be among those rechecked by experts after the selection by the neural network? In your answer, indicate the whole number part.
|
69
| 36.71875 |
19,341 |
Jenny wants to create all the six-letter words where the first two letters are the same as the last two letters. How many combinations of letters satisfy this property?
|
17576
| 14.0625 |
19,342 |
Given $$∫_{ 0 }^{ 2 }(\cos \frac {π}{4}x+ \sqrt {4-x^{2}})dx$$, evaluate the definite integral.
|
\pi+\frac{4}{\pi}
| 7.03125 |
19,343 |
Compute the sum:
\[\sum_{1 \le a < b < c} \frac{1}{3^a 5^b 7^c}.\]
(The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \le a < b < c.$)
|
\frac{1}{21216}
| 49.21875 |
19,344 |
How many rational terms are in the expansion of
a) $(\sqrt{2}+\sqrt[4]{3})^{100}$
b) $(\sqrt{2}+\sqrt[3]{3})^{300}$?
|
51
| 62.5 |
19,345 |
Find the least real number $k$ with the following property: if the real numbers $x$ , $y$ , and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]
|
\frac{16}{9}
| 34.375 |
19,346 |
Given $\tan \alpha = \frac{1}{3}$, calculate the value of $\frac{\cos^2 \alpha - 2\sin^2 \alpha}{\cos^2 \alpha}$.
|
\frac{7}{9}
| 96.09375 |
19,347 |
What is the least common multiple of the numbers 1056 and 792, and then add 100 to your result?
|
3268
| 18.75 |
19,348 |
Given six test scores have a mean of $85$, a median of $86$, and a mode of $88$. Determine the sum of the two lowest test scores.
|
162
| 85.9375 |
19,349 |
Given \\(|a|=1\\), \\(|b|= \sqrt{2}\\), and \\(a \perp (a-b)\\), the angle between vector \\(a\\) and vector \\(b\\) is ______.
|
\frac{\pi}{4}
| 96.09375 |
19,350 |
Given that in a class of 36 students, more than half purchased notebooks from a store where each notebook had the same price in cents greater than the number of notebooks bought by each student, and the total cost for the notebooks was 990 cents, calculate the price of each notebook in cents.
|
15
| 7.8125 |
19,351 |
A card is chosen at random from a standard deck of 54 cards, including 2 jokers, and then it is replaced, and another card is chosen. What is the probability that at least one of the cards is a diamond, an ace, or a face card?
|
\frac{533}{729}
| 10.15625 |
19,352 |
In $\triangle ABC$, it is known that the internal angle $A= \frac{\pi}{3}$, side $BC=2\sqrt{3}$. Let internal angle $B=x$, and the area be $y$.
(1) If $x=\frac{\pi}{4}$, find the length of side $AC$;
(2) Find the maximum value of $y$.
|
3\sqrt{3}
| 50.78125 |
19,353 |
Let $\triangle ABC$ be an acute isosceles triangle with circumcircle $\omega$. The tangents to $\omega$ at vertices $B$ and $C$ intersect at point $T$. Let $Z$ be the projection of $T$ onto $BC$. Assume $BT = CT = 20$, $BC = 24$, and $TZ^2 + 2BZ \cdot CZ = 478$. Find $BZ \cdot CZ$.
|
144
| 18.75 |
19,354 |
In a city, from 7:00 to 8:00, is a peak traffic period, during which all vehicles travel at half their normal speed. Every morning at 6:50, two people, A and B, start from points A and B respectively and travel towards each other. They meet at a point 24 kilometers from point A. If person A departs 20 minutes later, they meet exactly at the midpoint of the route between A and B. If person B departs 20 minutes earlier, they meet at a point 20 kilometers from point A. What is the distance between points A and B in kilometers?
|
48
| 11.71875 |
19,355 |
Calculate \( \left[6 \frac{3}{5}-\left(8.5-\frac{1}{3}\right) \div 3.5\right] \times\left(2 \frac{5}{18}+\frac{11}{12}\right) = \)
|
\frac{368}{27}
| 71.875 |
19,356 |
A construction company in Changsha has a total of 50 slag transport vehicles, including 20 Type A and 30 Type B vehicles. Now all these slag transport vehicles are allocated to the construction of the Changsha-Zhuzhou-Xiangtan intercity light rail, with 30 vehicles sent to Site A and 20 vehicles sent to Site B. The daily rental prices agreed upon by the two sites and the construction company are as follows:
| | Type A Slag Truck Rental | Type B Slag Truck Rental |
|------------|--------------------------|--------------------------|
| Site A | $1800 per vehicle | $1600 per vehicle |
| Site B | $1600 per vehicle | $1200 per vehicle |
$(1)$ If $x$ Type A slag transport vehicles are sent to Site A, the rental income obtained by the construction company for these 50 slag transport vehicles in one day is $y (in yuan)$. Find the analytical expression of $y$ in terms of $x$.
$(2)$ If the total rental income of these 50 slag transport vehicles in one day is not less than $79600 yuan$, determine how many allocation schemes exist and list all the schemes.
$(3)$ Under the condition of $(2)$, which allocation scheme will result in the construction company earning the highest daily rental income? What is the maximum rental income? Please explain the reason.
|
80000
| 70.3125 |
19,357 |
Let \\((x^{2}+1)(2x+1)^{9}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\ldots+a_{n}(x+2)^{n}\\), then \\(a_{0}+a_{1}+a_{2}+\ldots+a_{n}=\\) \_\_\_\_\_\_.
|
-2
| 78.90625 |
19,358 |
You walk for 90 minutes at a rate of 3 mph, then rest for 15 minutes, and then cycle for 45 minutes at a rate of 20 kph. Calculate the total distance traveled in 2 hours and 30 minutes.
|
13.82
| 49.21875 |
19,359 |
When $\sqrt[4]{2^6 \cdot 3^5 \cdot 5^2}$ is fully simplified, the result is $x\sqrt[4]{y}$, where $x$ and $y$ are positive integers. What is $x+y$?
|
306
| 66.40625 |
19,360 |
Let S<sub>n</sub> represent the sum of the first n terms of the arithmetic sequence {a<sub>n</sub>}. If S<sub>5</sub> = 2S<sub>4</sub> and a<sub>2</sub> + a<sub>4</sub> = 8, find the value of a<sub>5</sub>.
|
10
| 89.84375 |
19,361 |
The diagram shows a square \(PQRS\) with sides of length 2. The point \(T\) is the midpoint of \(RS\), and \(U\) lies on \(QR\) so that \(\angle SPT = \angle TPU\). What is the length of \(UR\)?
|
1/2
| 7.8125 |
19,362 |
Express 2.175 billion yuan in scientific notation.
|
2.175 \times 10^9
| 92.1875 |
19,363 |
How many different five-letter words can be formed such that they start and end with the same letter, and the middle letter is always 'A'?
|
17576
| 67.1875 |
19,364 |
A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?
|
\frac{6}{2 + 3\sqrt{2}}
| 0 |
19,365 |
The inhabitants of the Isle of Concatenate use an extended alphabet of 25 letters (A through Y). Each word in their language has a maximum length of 5 letters, and every word must include the letter A at least once. How many such words are possible?
|
1863701
| 0 |
19,366 |
A cube with side length $2$ is sliced by a plane that passes through a vertex $A$, the midpoint $M$ of an adjacent edge, and the midpoint $P$ of the face diagonal of the top face, not containing vertex $A$. Find the area of the triangle $AMP$.
|
\frac{\sqrt{5}}{2}
| 23.4375 |
19,367 |
A company plans to promote the same car in two locations, A and B. It is known that the relationship between the sales profit (unit: ten thousand yuan) and the sales volume (unit: cars) in the two locations is $y_1=5.06t-0.15t^2$ and $y_2=2t$, respectively, where $t$ is the sales volume ($t\in\mathbb{N}$). The company plans to sell a total of 15 cars in these two locations.
(1) Let the sales volume in location A be $x$, try to write the function relationship between the total profit $y$ and $x$;
(2) Find the maximum profit the company can obtain.
|
45.6
| 71.875 |
19,368 |
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
|
21
| 1.5625 |
19,369 |
In a circle with center $O$, the measure of $\angle TIQ$ is $45^\circ$ and the radius $OT$ is 12 cm. Find the number of centimeters in the length of arc $TQ$. Express your answer in terms of $\pi$.
|
6\pi
| 5.46875 |
19,370 |
Given three forces in space, $\overrightarrow {F_{1}}$, $\overrightarrow {F_{2}}$, and $\overrightarrow {F_{3}}$, each with a magnitude of 2, and the angle between any two of them is 60°, the magnitude of their resultant force $\overrightarrow {F}$ is ______.
|
2 \sqrt {6}
| 0 |
19,371 |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 630 = 0$ has integral solutions?
|
160
| 22.65625 |
19,372 |
Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence?
|
192
| 5.46875 |
19,373 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\frac {2a+b}{c}= \frac {\cos (A+C)}{\cos C}$.
(I) Find the magnitude of angle $C$,
(II) If $c=2$, find the maximum area of $\triangle ABC$.
|
\frac { \sqrt {3}}{3}
| 0 |
19,374 |
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}
| 3.90625 |
19,375 |
How many numbers are in the list $165, 159, 153, \ldots, 30, 24?$
|
24
| 18.75 |
19,376 |
Six students with distinct heights take a group photo, the photographer arranges them into two rows with three people each. What is the probability that every student in the back row is taller than the students in the front row?
|
\frac{1}{20}
| 60.15625 |
19,377 |
Suppose $\cos Q = 0.6$ in a right triangle $PQR$ where $PQ$ measures 15 units. What is the length of $QR$?
[asy]
pair P,Q,R;
P = (0,0);
Q = (7.5,0);
R = (0,7.5*tan(acos(0.6)));
draw(P--Q--R--P);
draw(rightanglemark(Q,P,R,20));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$15$",Q/2,S);
[/asy]
|
25
| 8.59375 |
19,378 |
Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$ , where $n$ is an integer that is not a multiple of $3$ . What is the largest integer that is a divisor of every number in $S$ ?
|
360
| 3.125 |
19,379 |
Randomly assign numbers 1 to 400 to 400 students. Then decide to use systematic sampling to draw a sample of size 20 from these 400 students. By order of their numbers, evenly divide them into 20 groups (1-20, 21-40, ..., 381-400). If the number drawn from the first group is 11 by lottery, the number drawn from the third group is __________.
|
51
| 67.1875 |
19,380 |
A particle is placed on the curve $y = x^3 - 3x^2 - x + 3$ at a point $P$ whose $y$-coordinate is $5$. It is allowed to roll along the curve until it reaches the nearest point $Q$ whose $y$-coordinate is $-2$. Compute the horizontal distance traveled by the particle.
A) $|\sqrt{6} - \sqrt{3}|$
B) $\sqrt{3}$
C) $\sqrt{6}$
D) $|1 - \sqrt{3}|$
|
|\sqrt{6} - \sqrt{3}|
| 21.875 |
19,381 |
Given the equations $\left(625\right)^{0.24}$ and $\left(625\right)^{0.06}$, find the value of their product.
|
5^{6/5}
| 0.78125 |
19,382 |
We place planes through each edge and the midpoint of the edge opposite to it in a tetrahedron. Into how many parts do these planes divide the tetrahedron, and what are the volumes of these parts?
|
24
| 36.71875 |
19,383 |
Calculate the sum of all integers greater than 4 and less than 21.
|
200
| 53.90625 |
19,384 |
The value of \(1 + 0.01 + 0.0001\) is:
|
1.0101
| 92.96875 |
19,385 |
Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, with $g(x) \neq 0$, $f(x)g'(x) > f'(x)g(x)$, and $f(x) = a^{x}g(x)$ ($a > 0$ and $a \neq 1$), $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}$. For the finite sequence $\frac{f(n)}{g(n)} = (n = 1, 2, \ldots, 0)$, find the probability that the sum of the first $k$ terms is greater than $\frac{15}{16}$ for any positive integer $k$ ($1 \leq k \leq 10$).
|
\frac{3}{5}
| 35.9375 |
19,386 |
Given the graph of the power function $y=f(x)$ passes through the point $\left( \frac{1}{2}, \frac{\sqrt{2}}{2} \right)$, then $\lg [f(2)]+\lg [f(5)]=$ \_\_\_\_\_\_ .
|
\frac{1}{2}
| 88.28125 |
19,387 |
Given that the sequence $\{a_n\}$ is an arithmetic sequence with a common difference greater than $0$, and it satisfies $a_1+a_5=4$, $a_2a_4=-5$, calculate the sum of the first $10$ terms of the sequence $\{a_n\}$.
|
95
| 90.625 |
19,388 |
How many positive divisors of 50! are either prime or the product of exactly two distinct primes?
|
120
| 17.96875 |
19,389 |
What is the least positive integer $m$ such that the following is true?
*Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$* \[\mathrm a. ~ 10\qquad \mathrm b.~11\qquad \mathrm c. ~12 \qquad \mathrm d. ~13 \qquad \mathrm e. ~1415\]
|
12
| 60.9375 |
19,390 |
Compute $\frac{x^8 + 16x^4 + 64 + 4x^2}{x^4 + 8}$ when $x = 3$.
|
89 + \frac{36}{89}
| 0.78125 |
19,391 |
There are two identical cups, A and B. Cup A is half-filled with pure water, and cup B is fully filled with a 50% alcohol solution. First, half of the alcohol solution from cup B is poured into cup A and mixed thoroughly. Then, half of the alcohol solution in cup A is poured back into cup B. How much of the solution in cup B is alcohol at this point?
|
3/7
| 6.25 |
19,392 |
From the $7$ integers from $2$ to $8$, randomly select $2$ different numbers, and calculate the probability that these $2$ numbers are coprime.
|
\frac{2}{3}
| 41.40625 |
19,393 |
Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: "The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles. If you reach 50 points, the machine gives out a prize. If you get a number greater than 50, all the points are lost." What is the minimum amount of rubles Vovochka needs to get the prize? Answer: 11 rubles.
|
11
| 48.4375 |
19,394 |
Given the complex number z = $$\frac{a^2i}{2-i} + \frac{1-2ai}{5}$$ (where a ∈ R, i is the imaginary unit), find the value(s) of a if z is a purely imaginary number.
|
-1
| 7.8125 |
19,395 |
Let $p$, $q$, $r$, $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s$, and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q$. Additionally, $p + q + r + s = 201$. Find the value of $pq + rs$.
|
-\frac{28743}{12}
| 0 |
19,396 |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its left focus is $F$, left vertex is $A$, and point $B$ is a point on the ellipse in the first quadrant. The line $OB$ intersects the ellipse at another point $C$. If the line $BF$ bisects the line segment $AC$, find the eccentricity of the ellipse.
|
\frac{1}{3}
| 62.5 |
19,397 |
In $\triangle ABC$, if $a^{2} + b^{2} - 6c^{2}$, then the value of $(\cot A + \cot B) \tan C$ is equal to:
|
$\frac{2}{5}$
| 0 |
19,398 |
Triangle $PQR$ has vertices $P(0, 10)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ bisects the area of $\triangle PQR$. Find the sum of the slope and $y$-intercept of this line.
|
\frac{-20}{3}
| 0 |
19,399 |
Find a positive integer that is divisible by 18 and has a square root between 26 and 26.2.
|
684
| 92.96875 |
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