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40.3k
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5.15k
| ground_truth
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100
|
|---|---|---|---|
20,300 |
The sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can the quantity \( n^{2} d \) take under these conditions, where \( d \) is the common difference of the progression, and \( n \) is the number of its terms?
|
400
| 1.5625 |
20,301 |
Given a sequence $\{a_n\}$ satisfying $a_1=1$ and $a_{n+1}= \frac {a_n}{a_n+2}$ $(n\in\mathbb{N}^*)$, find the value of $a_{10}$.
|
\frac {1}{1023}
| 58.59375 |
20,302 |
(a) A natural number $n$ is less than 120. What is the largest remainder that the number 209 can give when divided by $n$?
(b) A natural number $n$ is less than 90. What is the largest remainder that the number 209 can give when divided by $n$?
|
69
| 32.03125 |
20,303 |
A man buys a house for $20,000 and wants to earn a $6\%$ annual return on his investment. He pays $650 a year in taxes and sets aside $15\%$ of each month's rent for repairs and upkeep. Determine the required monthly rent (in dollars) to meet his financial goals.
A) $165.25$
B) $172.50$
C) $181.38$
D) $190.75$
E) $200.00$
|
181.38
| 44.53125 |
20,304 |
What is the remainder when $8x^4 - 18x^3 + 27x^2 - 14x - 30$ is divided by $4x-12$?
|
333
| 14.84375 |
20,305 |
The radian measure of a 60° angle is $$\frac {\pi}{6}$$.
|
\frac{\pi}{3}
| 85.9375 |
20,306 |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with $a > 0, b > 0$, if the four vertices of square $ABCD$ are on the hyperbola and the midpoints of $AB$ and $CD$ are the two foci of the hyperbola, determine the eccentricity of the hyperbola.
|
\frac{1 + \sqrt{5}}{2}
| 3.90625 |
20,307 |
Given that points $P$ and $Q$ are on the circle $x^2 + (y-6)^2 = 2$ and the ellipse $\frac{x^2}{10} + y^2 = 1$, respectively, what is the maximum distance between $P$ and $Q$?
A) $5\sqrt{2}$
B) $\sqrt{46} + \sqrt{2}$
C) $7 + \sqrt{2}$
D) $6\sqrt{2}$
|
6\sqrt{2}
| 11.71875 |
20,308 |
Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$.
|
\frac{3-\sqrt{3}}{3}
| 14.0625 |
20,309 |
Given integers $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\ldots $ satisfy the following conditions: $a_{1}=0$, $a_{2}=-|a+1|$, $a_{3}=-|a_{2}+2|$, $a_{4}=-|a_{3}+3|$, and so on, then the value of $a_{2022}$ is ____.
|
-1011
| 73.4375 |
20,310 |
Compute $\sqrt{125}\cdot\sqrt{45}\cdot \sqrt{10}$.
|
75\sqrt{10}
| 63.28125 |
20,311 |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\]
|
\frac{89}{9}
| 5.46875 |
20,312 |
In a modified version of a walking game, I play by different rules. On the first move, I stand still, but for each subsequent move $n$ where $2 \le n \le 25$, I take two steps forward if $n$ is prime and three steps backward if $n$ is composite. After completing all 25 moves, I must return to my original starting point. How many steps is my return journey?
|
27
| 1.5625 |
20,313 |
There are four positive integers that are divisors of each number in the list $$20, 40, 100, 80, 180.$$ Find the sum of these four positive integers.
|
12
| 34.375 |
20,314 |
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal?
|
50
| 2.34375 |
20,315 |
Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and $\sin A=\sqrt{3}(1-\cos A)$.
$(1)$ Find $A$;
$(2)$ If $a=7$ and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of $\triangle ABC$.
|
10 \sqrt{3}
| 59.375 |
20,316 |
Find the smallest composite number that has no prime factors less than 15.
|
289
| 75 |
20,317 |
Given the planar vectors $\overrightarrow{a}, \overrightarrow{b}$, with $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a} \cdot \overrightarrow{b}=1$, let $\overrightarrow{e}$ be a unit vector in the plane. Find the maximum value of $y=\overrightarrow{a} \cdot \overrightarrow{e} + \overrightarrow{b} \cdot \overrightarrow{e}$.
|
\sqrt{7}
| 73.4375 |
20,318 |
Given the sets $A=\{x|x^2 - mx + m^2 - 19 = 0\}$, $B=\{x|x^2 - 5x + 6 = 0\}$, and $C=\{2, -4\}$. If $A \cap B \neq \emptyset$ and $A \cap C = \emptyset$, find the value of the real number $m$.
|
-2
| 70.3125 |
20,319 |
On the first day, Barry Sotter used his magic to make an object's length increase by $\frac{1}{3}$, so if the original length was $x$, it became $x + \frac{1}{3} x$. On the second day, he increased the new length by $\frac{1}{4}$; on the third day by $\frac{1}{5}$; and so on. On the $n^{\text{th}}$ day, Barry made the object's length exactly 50 times its original length. Find the value of $n$.
|
147
| 39.84375 |
20,320 |
Given the line $l_{1}: 2x + 5y = 1$ and the circle $C: x^{2} + y^{2} - 2x + 4y = 4$ with center $O_{1}$, let a moving line $l_{2}$ which is parallel to $l_{1}$ intersect the circle $C$ at points $A$ and $B$. Find the maximum value of the area $S_{\triangle ABB_{1}}$.
|
$\frac{9}{2}$
| 0 |
20,321 |
Find the largest prime divisor of $36^2 + 49^2$.
|
3697
| 98.4375 |
20,322 |
Determine the area and the circumference of a circle with the center at the point \( R(2, -1) \) and passing through the point \( S(7, 4) \). Express your answer in terms of \( \pi \).
|
10\pi \sqrt{2}
| 27.34375 |
20,323 |
Nine points are evenly spaced at intervals of one unit around a $3 \times 3$ square grid, such that each side of the square has three equally spaced points. Two of the 9 points are chosen at random. What is the probability that the two points are one unit apart?
A) $\frac{1}{3}$
B) $\frac{1}{4}$
C) $\frac{1}{5}$
D) $\frac{1}{6}$
E) $\frac{1}{7}$
|
\frac{1}{3}
| 64.0625 |
20,324 |
In Mathville, the streets are all $30$ feet wide and the blocks they enclose are all squares of side length $500$ feet. Matt runs around the block on the $500$-foot side of the street, while Mike runs on the opposite side of the street. How many more feet than Matt does Mike run for every lap around the block?
|
240
| 78.125 |
20,325 |
Given $\sqrt{2 + \frac{2}{3}} = 2\sqrt{\frac{2}{3}}, \sqrt{3 + \frac{3}{8}} = 3\sqrt{\frac{3}{8}}, \sqrt{4 + \frac{4}{15}} = 4\sqrt{\frac{4}{15}}\ldots$, if $\sqrt{6 + \frac{a}{b}} = 6\sqrt{\frac{a}{b}}$ (where $a,b$ are real numbers), please deduce $a = \_\_\_\_$, $b = \_\_\_\_$.
|
35
| 23.4375 |
20,326 |
Given a "double-single" number is a three-digit number made up of two identical digits followed by a different digit, calculate the number of double-single numbers between 100 and 1000.
|
81
| 98.4375 |
20,327 |
In the expansion of $(1-x)^{2}(2-x)^{8}$, find the coefficient of $x^{8}$.
|
145
| 64.0625 |
20,328 |
Let $x,$ $y,$ and $z$ be real numbers such that
\[x^3 + y^3 + z^3 - 3xyz = 8.\]
Find the minimum value of $x^2 + y^2 + z^2.$
|
\frac{40}{7}
| 0 |
20,329 |
On a table there are $100$ red and $k$ white buckets for which all of them are initially empty. In each move, a red and a white bucket is selected and an equal amount of water is added to both of them. After some number of moves, there is no empty bucket and for every pair of buckets that are selected together at least once during the moves, the amount of water in these buckets is the same. Find all the possible values of $k$ .
|
100
| 56.25 |
20,330 |
On a table near the sea, there are $N$ glass boxes where $N<2021$ , each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let $S$ be the sum of all values of $N$ for which Sowdha has a winning strategy and let $R$ be the sum of all values of $N$ for which Rafi has a winning strategy. What is the value of $\frac{R-S}{10}$ ?
|
101
| 62.5 |
20,331 |
Evaluate the expression $3 + 2\sqrt{3} + \frac{1}{3 + 2\sqrt{3}} + \frac{1}{2\sqrt{3} - 3}$.
|
3 + \frac{10\sqrt{3}}{3}
| 96.875 |
20,332 |
Read the following problem-solving process:<br/>The first equation: $\sqrt{1-\frac{3}{4}}=\sqrt{\frac{1}{4}}=\sqrt{(\frac{1}{2})^2}=\frac{1}{2}$.<br/>The second equation: $\sqrt{1-\frac{5}{9}}=\sqrt{\frac{4}{9}}=\sqrt{(\frac{2}{3})^2}=\frac{2}{3}$;<br/>The third equation: $\sqrt{1-\frac{7}{16}}=\sqrt{\frac{9}{16}}=\sqrt{(\frac{3}{4})^2}=\frac{3}{4}$;<br/>$\ldots $<br/>$(1)$ According to the pattern you discovered, please write down the fourth equation: ______<br/>$(2)$ According to the pattern you discovered, please write down the nth equation (n is a positive integer): ______;<br/>$(3)$ Using this pattern, calculate: $\sqrt{(1-\frac{3}{4})×(1-\frac{5}{9})×(1-\frac{7}{16})×⋯×(1-\frac{21}{121})}$;
|
\frac{1}{11}
| 35.9375 |
20,333 |
According to the Shannon formula $C=W\log_{2}(1+\frac{S}{N})$, if the bandwidth $W$ is not changed, but the signal-to-noise ratio $\frac{S}{N}$ is increased from $1000$ to $12000$, then find the approximate percentage increase in the value of $C$.
|
36\%
| 0 |
20,334 |
Expand $-(4-c)(c+2(4-c) + c^2)$. What is the sum of the coefficients of the expanded form?
|
-24
| 52.34375 |
20,335 |
Compute: \(103 \times 97\).
|
9991
| 100 |
20,336 |
A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval $[0, 1]$. This process is repeated two independent times, obtaining two real numbers $x$ and $y$. What is the probability that $|x-y| > \frac{1}{2}$?
|
$\frac{7}{16}$
| 0 |
20,337 |
Given $cos({α-\frac{π}{6}})=\frac{1}{3}$, find $sin({2α+\frac{π}{6}})$.
|
-\frac{7}{9}
| 76.5625 |
20,338 |
Given Professor Lee has ten different mathematics books on a shelf, consisting of three calculus books, four algebra books, and three statistics books, determine the number of ways to arrange the ten books on the shelf keeping all calculus books together and all statistics books together.
|
25920
| 0 |
20,339 |
The number \(abcde\) has five distinct digits, each different from zero. When this number is multiplied by 4, the result is a five-digit number \(edcba\), which is the reverse of \(abcde\). What is the value of \(a + b + c + d + e\)?
|
27
| 90.625 |
20,340 |
Given a plane $\alpha$ and two non-coincident straight lines $m$ and $n$, consider the following four propositions:
(1) If $m \parallel \alpha$ and $n \subseteq \alpha$, then $m \parallel n$.
(2) If $m \parallel \alpha$ and $n \parallel \alpha$, then $m \parallel n$.
(3) If $m \parallel n$ and $n \subseteq \alpha$, then $m \parallel \alpha$.
(4) If $m \parallel n$ and $m \parallel \alpha$, then $n \parallel \alpha$ or $n \subseteq \alpha$.
Identify which of the above propositions are correct (write the number).
|
(4)
| 0 |
20,341 |
Given the set $M=\{m\in \mathbb{Z} | x^2+mx-36=0 \text{ has integer solutions}\}$, a non-empty set $A$ satisfies the conditions:
(1) $A \subseteq M$,
(2) If $a \in A$, then $-a \in A$, the number of all such sets $A$ is.
|
31
| 32.03125 |
20,342 |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).
Given points:
\( A_{1}(1, -1, 1) \)
\( A_{2}(-2, 0, 3) \)
\( A_{3}(2, 1, -1) \)
\( A_{4}(2, -2, -4) \)
|
\frac{33}{\sqrt{101}}
| 8.59375 |
20,343 |
Given the function $f(x)=\vec{a}\cdot \vec{b}$, where $\vec{a}=(2\cos x,\sqrt{3}\sin 2x)$, $\vec{b}=(\cos x,1)$, and $x\in \mathbb{R}$.
(Ⅰ) Find the period and the intervals of monotonic increase for the function $y=f(x)$;
(Ⅱ) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $f(A)=2$, $a=\sqrt{7}$, and $\sin B=2\sin C$. Calculate the area of $\triangle ABC$.
|
\frac{7\sqrt{3}}{6}
| 75.78125 |
20,344 |
Calculate the value of $$1+\cfrac{2}{3+\cfrac{6}{7}}$$ and express it as a simplified fraction.
|
\frac{41}{27}
| 43.75 |
20,345 |
Given a geometric sequence $a$, $a(a-1)$, $a(a-1)^{2}$, $...$, where $S_{n}$ represents the sum of its first $n$ terms.
1) Find the range of real values for $a$ and the expression for $S_{n}$;
2) Does there exist a real value of $a$ such that $S_{1}$, $S_{3}$, $S_{2}$ form an arithmetic sequence? If it exists, find the value of $a$; if not, explain the reason.
|
\frac{1}{2}
| 18.75 |
20,346 |
From the $10$ numbers $0-9$, select $3$ numbers. Find:<br/>
$(1)$ How many unique three-digit numbers can be formed without repeating any digits?<br/>
$(2)$ How many unique three-digit odd numbers can be formed without repeating any digits?
|
320
| 89.0625 |
20,347 |
Determine the number of real number $a$ , such that for every $a$ , equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:
(a) $x_0$ is an even integer;
(b) $|x_0|<1000$ .
|
999
| 83.59375 |
20,348 |
Find the smallest real number \(\lambda\) such that the inequality
$$
5(ab + ac + ad + bc + bd + cd) \leq \lambda abcd + 12
$$
holds for all positive real numbers \(a, b, c, d\) that satisfy \(a + b + c + d = 4\).
|
18
| 91.40625 |
20,349 |
What is the minimum number of equilateral triangles, each of side length 1 unit, needed to completely cover an equilateral triangle of side length 15 units?
|
225
| 100 |
20,350 |
In the expression $(1+x)^{56}$, the parentheses are expanded and like terms are combined. Find the coefficients of $x^8$ and $x^{48}$.
|
\binom{56}{8}
| 0 |
20,351 |
A math teacher randomly selects 3 questions for analysis from a test paper consisting of 12 multiple-choice questions, 4 fill-in-the-blank questions, and 6 open-ended questions. The number of different ways to select questions such that at least one multiple-choice question and at least one open-ended question are selected is __________.
|
864
| 37.5 |
20,352 |
Alice has 4 sisters and 6 brothers. Given that Alice's sister Angela has S sisters and B brothers, calculate the product of S and B.
|
24
| 8.59375 |
20,353 |
Given \\(a > 0\\), \\(b > 0\\), and \\(a+4b={{(ab)}^{\\frac{3}{2}}}\\).
\\((\\)I\\()\\) Find the minimum value of \\(a^{2}+16b^{2}\\);
\\((\\)II\\()\\) Determine whether there exist \\(a\\) and \\(b\\) such that \\(a+3b=6\\), and explain the reason.
|
32
| 10.9375 |
20,354 |
Given that $||\overrightarrow{OA}||=||\overrightarrow{OB}||=2$, point $C$ is on line segment $AB$, and the minimum value of $||\overrightarrow{OC}||$ is $1$, find the minimum value of $||\overrightarrow{OA}-t\overrightarrow{OB}||$.
|
\sqrt{3}
| 35.15625 |
20,355 |
In a two-day problem-solving tournament, Alpha and Gamma both attempted questions worth a total of 600 points. Alpha scored 210 points out of 350 points on the first day, and 150 points out of 250 points on the second day. Gamma, who did not attempt 350 points on the first day, had a positive integer score on each of the two days, and Gamma's daily success ratio (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $\frac{360}{600} = 3/5$.
Find the largest possible two-day success ratio that Gamma could have achieved.
|
\frac{359}{600}
| 3.90625 |
20,356 |
$ABCDEFGH$ is a cube. Find $\sin \angle HAD$.
|
\frac{\sqrt{2}}{2}
| 32.03125 |
20,357 |
In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \sqrt{2}$, calculate the spherical distance between points $A$ and $C$.
|
\frac{\pi}{2}
| 70.3125 |
20,358 |
What is the coefficient of $x^5$ when $$2x^5 - 4x^4 + 3x^3 - x^2 + 2x - 1$$ is multiplied by $$x^3 + 3x^2 - 2x + 4$$ and the like terms are combined?
|
24
| 21.875 |
20,359 |
Let \( m \in \mathbb{N} \), and let the integer part of \( \log_2 m \) be denoted as \( f(m) \). Calculate the value of \( f(1) + f(2) + \cdots + f(1024) \).
|
8204
| 86.71875 |
20,360 |
Given the function $f(x)=2\sin (\\omega x)$, where $\\omega > 0$.
(1) When $ \\omega =1$, determine the even-odd property of the function $F(x)=f(x)+f(x+\\dfrac{\\mathrm{ }\\!\\!\\pi\\!\\!{ }}{2})$ and explain the reason.
(2) When $ \\omega =2$, the graph of the function $y=f(x)$ is translated to the left by $ \\dfrac{\\mathrm{ }\\!\\!\\pi\\!\\!{ }}{6}$ unit, and then translated upward by 1 unit to obtain the graph of the function $y=g(x)$. Find all possible values of the number of zeros of $y=g(x)$ in the interval $[a,a+10π]$ for any $a∈R$.
|
20
| 64.84375 |
20,361 |
Find the number of digit of $\sum_{n=0}^{99} 3^n$ .
You may use $\log_{10} 3=0.4771$ .
2012 Tokyo Institute of Technology entrance exam, problem 2-A
|
48
| 98.4375 |
20,362 |
On the same road, two trucks are driving in the same direction. Initially, Truck A is 4 kilometers ahead of Truck B. The speed of Truck A is 45 kilometers per hour, and the speed of Truck B is 60 kilometers per hour. How far apart are the two trucks 1 minute before Truck B catches up to Truck A, in meters?
|
250
| 43.75 |
20,363 |
In the Cartesian coordinate system $xoy$, the parametric equation of curve $C_1$ is
$$
\begin{cases}
x=2\sqrt{2}-\frac{\sqrt{2}}{2}t \\
y=\sqrt{2}+\frac{\sqrt{2}}{2}t
\end{cases}
(t \text{ is the parameter}).
$$
In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of curve $C_2$ is $\rho=4\sqrt{2}\sin \theta$.
(Ⅰ) Convert the equation of $C_2$ into a Cartesian coordinate equation;
(Ⅱ) Suppose $C_1$ and $C_2$ intersect at points $A$ and $B$, and the coordinates of point $P$ are $(\sqrt{2},2\sqrt{2})$, find $|PA|+|PB|$.
|
2\sqrt{7}
| 9.375 |
20,364 |
At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61{,}629{,}875}$ and $n = \sqrt[7]{170{,}859{,}375}$ . Given that $m$ and $n$ are both integers, compute $100m+n$ .
*Proposed by Evan Chen*
|
39515
| 99.21875 |
20,365 |
In an isosceles triangle \( \triangle ABC \), the length of the altitude to one of the equal sides is \( \sqrt{3} \) and the angle between this altitude and the base is \( 60^\circ \). Calculate the area of \( \triangle ABC \).
|
\sqrt{3}
| 67.96875 |
20,366 |
Let $ a,\ b$ be real constants. Find the minimum value of the definite integral:
$ I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.$
|
\pi - \frac{8}{\pi}
| 46.875 |
20,367 |
In city "N", there are 10 horizontal and 12 vertical streets. A pair of horizontal and a pair of vertical streets form the rectangular boundary of the city, while the rest divide it into blocks shaped like squares with a side length of 100 meters. Each block has an address consisting of two integers \((i, j)\), \(i = 1, 2, \ldots, 9\), \(j = 1, 2, \ldots, 11\), representing the numbers of the streets that bound it from below and the left. A taxi transports passengers from one block to another, observing the following rules:
1. Pickup and drop-off are carried out at any point on the block's boundary at the passenger's request.
2. It is prohibited to enter inside the block.
3. Transportation is carried out by the shortest path.
4. For every 100 meters of travel, a fare of 1 coin is charged (with distance rounded up to the nearest multiple of 100 meters in favor of the driver).
How many blocks are there in the city? What is the maximum and minimum fare the driver can request for a ride from block \((7,1)\) to block \((2,10)\) without violating the rules?
|
14
| 66.40625 |
20,368 |
The lengths of the six edges of a tetrahedron $ABCD$ are $7, 13, 18, 27, 36, 41$, and $AB = 41$. What is the length of $CD$?
|
13
| 50 |
20,369 |
Given the function $f(x) = \cos^4x + 2\sin x\cos x - \sin^4x$
(1) Determine the parity, the smallest positive period, and the intervals of monotonic increase for the function $f(x)$.
(2) When $x \in [0, \frac{\pi}{2}]$, find the maximum and minimum values of the function $f(x)$.
|
-1
| 49.21875 |
20,370 |
Given the function $f(x) = \sin(x - \varphi)$ and $|\varphi| < \frac{\pi}{2}$, and $\int_{0}^{\frac{2\pi}{3}} f(x) \, dx = 0$, find the equation of one of the axes of symmetry of the graph of function $f(x)$.
|
\frac{5\pi}{6}
| 90.625 |
20,371 |
Brand Z juice claims, "We offer 30% more juice than Brand W at a price that is 15% less." What is the ratio of the unit price of Brand Z juice to the unit price of Brand W juice? Express your answer as a common fraction.
|
\frac{17}{26}
| 68.75 |
20,372 |
A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$ , whose apex $F$ is on the leg $AC$ . Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$ ?
|
2:3
| 67.1875 |
20,373 |
Among the numbers $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is \_\_\_\_\_\_.
|
111111_{(2)}
| 99.21875 |
20,374 |
Given that \( b \) is an even number between 1 and 11 (inclusive), and \( c \) is any natural number, determine the number of quadratic equations \( x^{2} + b x + c = 0 \) that have two distinct real roots.
|
50
| 69.53125 |
20,375 |
In the parallelogram $ABCD$ , a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$ . If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$ , find the area of the quadrilateral $AFED$ .
|
250
| 2.34375 |
20,376 |
Given the data from a 2×2 contingency table calculates $k=4.073$, there is a \_\_\_\_\_\_ confidence that the two variables are related, knowing that $P(k^2 \geq 3.841) \approx 0.05$, $P(k^2 \geq 5.024) \approx 0.025$.
|
95\%
| 79.6875 |
20,377 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $\overrightarrow{m}=(\sqrt{3}b-c,\cos C)$, $\overrightarrow{n}=(a,\cos A)$. Given that $\overrightarrow{m} \parallel \overrightarrow{n}$, determine the value of $\cos A$.
|
\dfrac{\sqrt{3}}{3}
| 68.75 |
20,378 |
In the tetrahedron P-ABC, $PC \perpendicular$ plane ABC, $\angle CAB=90^\circ$, $PC=3$, $AC=4$, $AB=5$, then the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_.
|
50\pi
| 39.84375 |
20,379 |
Consider a polynomial $P(x) \in \mathbb{R}[x]$ , with degree $2023$ , such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$ . If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$ ?
|
4046
| 48.4375 |
20,380 |
Given bed A has 600 plants, bed B has 500 plants, bed C has 400 plants, beds A and B share 60 plants, beds A and C share 80 plants, beds B and C share 40 plants, and beds A, B, and C share 20 plants collectively, calculate the total number of unique plants when considering just beds A, B, and C.
|
1340
| 66.40625 |
20,381 |
Given that Adam has a triangular field ABC with AB = 5, BC = 8, and CA = 11, and he intends to separate the field into two parts by building a straight fence from A to a point D on side BC such that AD bisects ∠BAC, find the area of the part of the field ABD.
|
\frac{5 \sqrt{21}}{4}
| 91.40625 |
20,382 |
In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Calculate the area of trapezoid $EFCD$ if the height from $E$ to line $CD$ is 10 units.
|
250
| 78.90625 |
20,383 |
The Cookie Monster now encounters a different cookie, which is bounded by the equation $(x-2)^2 + (y+1)^2 = 5$. He wonders if this cookie is big enough to share. Calculate the radius of this cookie and determine the area it covers.
|
5\pi
| 82.8125 |
20,384 |
Football tickets are normally priced at $15 each. After buying 5 tickets, any additional tickets are sold at a discounted price of $12 each. If Jane has $150, what is the maximum number of tickets she can buy?
|
11
| 60.15625 |
20,385 |
Given $f(x)=x^{3}+3ax^{2}+bx+a^{2}$ has an extremum of $0$ at $x=-1$, find $a-b=\_\_\_\_\_\_\_\_$.
|
-7
| 37.5 |
20,386 |
Let $\alpha \in \left(0, \frac{\pi}{3}\right)$, satisfying $\sqrt{3}\sin\alpha + \cos\alpha = \frac{\sqrt{6}}{2}$.
$(1)$ Find the value of $\cos\left(\alpha + \frac{\pi}{6}\right)$;
$(2)$ Find the value of $\cos\left(2\alpha + \frac{7\pi}{12}\right)$.
|
\frac{\sqrt{2} - \sqrt{30}}{8}
| 53.125 |
20,387 |
There are four points that are $7$ units from the line $y=20$ and $15$ units from the point $(10,20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?
|
120
| 52.34375 |
20,388 |
The number of terms in the expansion of $(x+y+z)^4$ is 15.
|
15
| 48.4375 |
20,389 |
There is a school that offers 10 courses for students to elect. Among them, Courses A, B, and C have conflicting schedules and thus at most one of these can be chosen. The school mandates that each student must elect three courses. How many different elective schemes are available for a student?
|
98
| 68.75 |
20,390 |
Given the function $f(x)=\frac{ax+b}{{x}^{2}+4}$ attains a maximum value of $1$ at $x=-1$, find the minimum value of $f(x)$.
|
-\frac{1}{4}
| 47.65625 |
20,391 |
Given an arithmetic sequence $\{a_n\}$, the sum of its first $n$ terms is $S_n$. It is known that $S_8 \leq 6$ and $S_{11} \geq 27$. Determine the minimum value of $S_{19}$.
|
133
| 25 |
20,392 |
How many different 8-digit positive integers exist if the digits from the second to the eighth can only be 0, 1, 2, 3, or 4?
|
703125
| 0 |
20,393 |
The repeating decimal for $\frac{5}{13}$ is $0.cdc\ldots$ What is the value of the sum $c+d$?
|
11
| 88.28125 |
20,394 |
Find the value of $\dfrac{2\cos 10^\circ - \sin 20^\circ }{\sin 70^\circ }$.
|
\sqrt{3}
| 75.78125 |
20,395 |
Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ Find the prime factorization of the minimum possible value of $x$ and determine the sum of the exponents and the prime factors.
|
31
| 35.9375 |
20,396 |
According to the classification standard of the Air Pollution Index (API) for city air quality, when the air pollution index is not greater than 100, the air quality is good. The environmental monitoring department of a city randomly selected the air pollution index for 5 days from last month's air quality data, and the data obtained were 90, 110, x, y, and 150. It is known that the average of the air pollution index for these 5 days is 110.
$(1)$ If x < y, from these 5 days, select 2 days, and find the probability that the air quality is good for both of these 2 days.
$(2)$ If 90 < x < 150, find the minimum value of the variance of the air pollution index for these 5 days.
|
440
| 74.21875 |
20,397 |
Given vectors $\overrightarrow{a}=(\cos α,\sin α)$, $\overrightarrow{b}=(\cos β,\sin β)$, and $|\overrightarrow{a}- \overrightarrow{b}|= \frac {4 \sqrt {13}}{13}$.
(1) Find the value of $\cos (α-β)$;
(2) If $0 < α < \frac {π}{2}$, $- \frac {π}{2} < β < 0$, and $\sin β=- \frac {4}{5}$, find the value of $\sin α$.
|
\frac {16}{65}
| 7.8125 |
20,398 |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $\sqrt{3}\sin C - c\cos A = c$.
$(1)$ Find the value of angle $A$.
$(2)$ If $b = 2c$, point $D$ is the midpoint of side $BC$, and $AD = \sqrt{7}$, find the area of triangle $\triangle ABC$.
|
2\sqrt{3}
| 35.9375 |
20,399 |
Consider the graph of $y=f(x)$, which consists of five line segments as described below:
- From $(-5, -4)$ to $(-3, 0)$
- From $(-3, 0)$ to $(-1, -1)$
- From $(-1, -1)$ to $(1, 3)$
- From $(1, 3)$ to $(3, 2)$
- From $(3, 2)$ to $(5, 6)$
What is the sum of the $x$-coordinates of all points where $f(x) = 2.3$?
|
4.35
| 67.96875 |
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