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40.3k
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stringlengths 10
5.15k
| ground_truth
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float64 0
100
|
|---|---|---|---|
17,600 |
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ satisfy $\overrightarrow{a}=(1, \sqrt {3}), |\overrightarrow{b}|=1, |\overrightarrow{a}+ \overrightarrow{b}|= \sqrt {3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______.
|
\frac{2\pi}{3}
| 98.4375 |
17,601 |
Jeff decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 1/3 chance of giving him a positive answer. If he asks it 7 questions, what is the probability that it gives him exactly 3 positive answers?
|
\frac{560}{2187}
| 3.90625 |
17,602 |
Circles \(P\), \(Q\), and \(R\) are externally tangent to each other and internally tangent to circle \(S\). Circles \(Q\) and \(R\) are congruent. Circle \(P\) has radius 2 and passes through the center of \(S\). What is the radius of circle \(Q\)?
|
\frac{16}{9}
| 1.5625 |
17,603 |
Given a pair of concentric circles, chords $AB,BC,CD,\dots$ of the outer circle are drawn such that they all touch the inner circle. If $\angle ABC = 75^{\circ}$ , how many chords can be drawn before returning to the starting point ?
|
24
| 78.90625 |
17,604 |
Given $a$, $b$, $c > 0$ and $$a(a+b+c)+bc=4-2 \sqrt {3}$$, calculate the minimum value of $2a+b+c$.
|
2\sqrt{3}-2
| 39.84375 |
17,605 |
Convert $5214_8$ to a base 10 integer.
|
2700
| 67.1875 |
17,606 |
Given a sector with a central angle of $\alpha$ and a radius of $r$.
$(1)$ If $\alpha = 60^{\circ}$ and $r = 3$, find the arc length of the sector.
$(2)$ If the perimeter of the sector is $16$, at what angle $\alpha$ will the area of the sector be maximized? Also, find the maximum area.
|
16
| 86.71875 |
17,607 |
A ferry boat begins shuttling tourists to an island every hour starting at 10 AM, with its last trip starting at 4 PM. On the 10 AM trip, there were 100 tourists on the ferry, and on each successive trip, the number of tourists decreased by 2 from the previous trip. Calculate the total number of tourists transported to the island that day.
|
658
| 23.4375 |
17,608 |
Given $M$ be the second smallest positive integer that is divisible by every positive integer less than 9, find the sum of the digits of $M$.
|
15
| 72.65625 |
17,609 |
Given the general term formula of the sequence $\{a\_n\}$, where $a\_n=n\cos \frac {nΟ}{2}$, and the sum of the first $n$ terms is represented by $S\_n$, find the value of $S\_{2016}$.
|
1008
| 60.9375 |
17,610 |
In how many ways is it possible to arrange the digits of 11250 to get a five-digit multiple of 5?
|
21
| 42.1875 |
17,611 |
An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. Find the smallest abundant number that is not a multiple of 10.
|
12
| 0 |
17,612 |
In an isosceles triangle \(ABC\), the base \(AC\) is equal to 1, and the angle \(\angle ABC\) is \(2 \arctan \frac{1}{2}\). Point \(D\) lies on the side \(BC\) such that the area of triangle \(ABC\) is four times the area of triangle \(ADC\). Find the distance from point \(D\) to the line \(AB\) and the radius of the circle circumscribed around triangle \(ADC\).
|
\frac{\sqrt{265}}{32}
| 0 |
17,613 |
In a district of Shanghai, the government convened the heads of 5 companies for an annual experience exchange meeting. Among them, Company A had 2 representatives attending, while the other 4 companies each had 1 representative attending. If 3 representatives are to be selected to speak at the meeting, the number of possible situations where these 3 representatives come from 3 different companies is ____.
|
16
| 60.9375 |
17,614 |
An exchange point conducts two types of operations:
1) Give 2 euros - receive 3 dollars and a candy as a gift;
2) Give 5 dollars - receive 3 euros and a candy as a gift.
When the wealthy Buratino came to the exchange point, he had only dollars. When he left, he had fewer dollars, he did not acquire any euros, but he received 50 candies. How many dollars did this "gift" cost Buratino?
|
10
| 5.46875 |
17,615 |
Given the function $f(x) = (2-a)(x-1) - 2\ln x$
(1) When $a=1$, find the intervals of monotonicity for $f(x)$.
(2) If the function $f(x)$ has no zeros in the interval $\left(0, \frac{1}{2}\right)$, find the minimum value of $a$.
|
2 - 4\ln 2
| 64.84375 |
17,616 |
A survey of participants was conducted at the Olympiad. $50\%$ of the participants liked the first round, $60\%$ of the participants liked the second round, $70\%$ of the participants liked the opening of the Olympiad. It is known that each participant liked either one option or all three. Determine the percentage of participants who rated all three events positively.
|
40
| 64.0625 |
17,617 |
Two numbers are independently selected from the set of positive integers less than or equal to 6. Exactly one of the numbers must be even. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction.
|
\frac{2}{3}
| 38.28125 |
17,618 |
Let \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} (n>2) \) be sequences of arbitrary real numbers satisfying the following conditions:
1. \( x_{1}+x_{2}+\cdots+x_{n}=0 \);
2. \( \left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=1 \);
3. \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \).
To ensure that the inequality \( \mid a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n} \mid \leqslant A\left(a_{1}-a_{n}\right) \) holds, find the minimum value of \( A \).
|
\frac{1}{2}
| 79.6875 |
17,619 |
Find the least odd prime factor of $2047^4 + 1$.
|
41
| 2.34375 |
17,620 |
Given that $\log_{10}2 \approx 0.30103$ , ο¬nd the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$ .
|
30
| 92.96875 |
17,621 |
Let $ABCD$ be a trapezoid with $AB\parallel DC$ . Let $M$ be the midpoint of $CD$ . If $AD\perp CD, AC\perp BM,$ and $BC\perp BD$ , find $\frac{AB}{CD}$ .
[i]Proposed by Nathan Ramesh
|
\frac{\sqrt{2}}{2}
| 44.53125 |
17,622 |
Given vectors $\overrightarrow{m}=(\cos \frac {x}{2},-1)$ and $\overrightarrow{n}=( \sqrt {3}\sin \frac {x}{2},\cos ^{2} \frac {x}{2})$, and the function $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}+1$.
(I) If $x \in [\frac{\pi}{2}, \pi]$, find the minimum value of $f(x)$ and the corresponding value of $x$.
(II) If $x \in [0, \frac{\pi}{2}]$ and $f(x)= \frac{11}{10}$, find the value of $\sin x$.
|
\frac{3\sqrt{3}+4}{10}
| 52.34375 |
17,623 |
Given that the line passing through the focus of the parabola $y^2=x$ intersects the parabola at points A and B, and O is the origin of the coordinates, calculate $\overrightarrow {OA}\cdot \overrightarrow {OB}$.
|
-\frac{3}{16}
| 44.53125 |
17,624 |
Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\widehat{AOD})=60^\circ$ and the altitude of the trapezoid is $10$ , what is the area of the trapezoid?
|
100\sqrt{3}
| 20.3125 |
17,625 |
There is a pile of eggs. Joan counted the eggs, but her count was way off by $1$ in the $1$ 's place. Tom counted in the eggs, but his count was off by $1$ in the $10$ 's place. Raoul counted the eggs, but his count was off by $1$ in the $100$ 's place. Sasha, Jose, Peter, and Morris all counted the eggs and got the correct count. When these seven people added their counts together, the sum was $3162$ . How many eggs were in the pile?
|
439
| 1.5625 |
17,626 |
The line $y=kx$ intersects the graph of the function $y=\tan x$ ($-\frac{Ο}{2}οΌxοΌ\frac{Ο}{2}$) at points $M$ and $N$ (not coinciding with the origin $O$). The coordinates of point $A$ are $(-\frac{Ο}{2},0)$. Find $(\overrightarrow{AM}+\overrightarrow{AN})\cdot\overrightarrow{AO}$.
|
\frac{\pi^2}{2}
| 81.25 |
17,627 |
Given the function $f(x)=\sin \omega x (\omega > 0)$, translate the graph of this function to the left by $\dfrac{\pi}{4\omega}$ units to obtain the graph of the function $g(x)$. If the graph of $g(x)$ is symmetric about the line $x=\omega$ and is monotonically increasing in the interval $(-\omega,\omega)$, determine the value of $\omega$.
|
\dfrac{\sqrt{\pi}}{2}
| 57.8125 |
17,628 |
Simplify the expression:
\[
\frac{1}{\dfrac{3}{\sqrt{5}+2} - \dfrac{4}{\sqrt{7}+2}}.
\]
|
\frac{3(9\sqrt{5} + 4\sqrt{7} + 10)}{(9\sqrt{5} - 4\sqrt{7} - 10)(9\sqrt{5} + 4\sqrt{7} + 10)}
| 0 |
17,629 |
For positive integers $n,$ let $s(n)$ be the sum of the digits of $n.$ Over all four-digit positive integers $n,$ which value of $n$ maximizes the ratio $\frac{s(n)}{n}$ ?
*Proposed by Michael Tang*
|
1099
| 85.9375 |
17,630 |
Given a positive integer \( n \geq 3 \), for an \( n \)-element real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\), if every permutation \( y_{1}, y_{2}, \cdots, y_{n} \) of it satisfies \(\sum_{i=1}^{n-1} y_{i} y_{i+1} \geq -1\), then the real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) is called "glowing". Find the largest constant \( C = C(n) \) such that for every glowing \( n \)-element real array, \(\sum_{1 \leq i < j \leq n} x_{i} x_{j} \geq C \).
|
-1
| 2.34375 |
17,631 |
What is the sum of every third odd integer between $200$ and $500$?
|
17400
| 48.4375 |
17,632 |
What is the smallest positive number that is both prime and a palindrome, and is exactly $8$ less than a perfect square?
|
17
| 19.53125 |
17,633 |
Five volunteers and two elderly people need to line up in a row, with the two elderly people next to each other but not at the ends. How many different ways can they arrange themselves?
|
960
| 44.53125 |
17,634 |
Suppose \( S = \{1, 2, \cdots, 2005\} \). If any subset of \( S \) containing \( n \) pairwise coprime numbers always includes at least one prime number, find the minimum value of \( n \).
|
16
| 3.125 |
17,635 |
There are seven students taking a graduation photo in a row. Student A must stand in the middle, and students B and C must stand together. How many different arrangements are possible?
|
192
| 2.34375 |
17,636 |
For any positive integer $n$ , let $a_n=\sum_{k=1}^{\infty}[\frac{n+2^{k-1}}{2^k}]$ , where $[x]$ is the largest integer that is equal or less than $x$ . Determine the value of $a_{2015}$ .
|
2015
| 92.1875 |
17,637 |
Given two unit vectors \\( \overrightarrow{a} \\) and \\( \overrightarrow{b} \\). If \\( |3 \overrightarrow{a} - 2 \overrightarrow{b}| = 3 \\), find the value of \\( |3 \overrightarrow{a} + \overrightarrow{b}| \\).
|
2 \sqrt {3}
| 0 |
17,638 |
A grocer arranges a display of cans where the top row has three cans, and each lower row has two more cans than the row above it. The total number of cans in the display is 169. How many rows does the display contain?
|
12
| 21.875 |
17,639 |
Given real numbers \(a_{1}, a_{2}, \cdots, a_{n}\) which are all greater than 0 (where \(n\) is a natural number no less than 4) and satisfy the equation \(a_{1} + a_{2} + \cdots + a_{n} = 1\), find the maximum value of the sum \(S = \sum_{k=1}^{n} \frac{a_{k}^{2}}{a_{k} + a_{k+1} + a_{k+2}} \quad\text{with}\ (a_{n+1} = a_{1}, a_{n+2} = a_{2})\).
|
\frac{1}{3}
| 98.4375 |
17,640 |
Numbers from 1 to 100 are written in a vertical row in ascending order. Fraction bars of different sizes are inserted between them. The calculation starts with the smallest fraction bar and ends with the largest one, for example, $\frac{1}{\frac{5}{3}}=\frac{15}{4}$. What is the greatest possible value that the resulting fraction can have?
|
100
| 13.28125 |
17,641 |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 20.\]
Find the largest possible value of $d.$
|
\frac{5 + \sqrt{105}}{2}
| 28.125 |
17,642 |
How many license plates can be made if they consist of 3 letters followed by 2 digits, where the first digit must be odd, the second digit must be even, and the first letter must be a vowel?
|
84,500
| 0 |
17,643 |
The moisture content of freshly cut grass is $70\%$, while the moisture content of hay is $16\%. How much grass needs to be cut to obtain 1 ton of hay?
|
2800
| 35.15625 |
17,644 |
Given a sequence $\{b_n\}$ with 8 terms satisfying $b_1=2014$, $b_8=2015$, and $b_{n+1}-b_n \in \{-1, \frac{1}{3}, 1\}$ (where $n=1,2,\ldots,7$), determine the total number of such sequences $\{b_n\}$.
|
252
| 41.40625 |
17,645 |
Mateo receives $20 every hour for one week, and Sydney receives $400 every day for one week. Calculate the difference between the total amounts of money that Mateo and Sydney receive over the one week period.
|
560
| 82.03125 |
17,646 |
In the diagram, the largest circle has a radius of 10 meters. Seven congruent smaller circles are symmetrically aligned in such a way that in an east-to-west and north-to-south orientation, the diameter of four smaller circles equals the diameter of the largest circle. What is the radius in meters of one of the seven smaller circles?
|
2.5
| 71.09375 |
17,647 |
The diagram shows two 10 by 14 rectangles which are edge-to-edge and share a common vertex. It also shows the center \( O \) of one rectangle and the midpoint \( M \) of one edge of the other. What is the distance \( OM \)?
A) 12
B) 15
C) 18
D) 21
E) 24
|
15
| 33.59375 |
17,648 |
Two dice are thrown one after the other, and the numbers obtained are denoted as $a$ and $b$.
(β
) Find the probability that $a^2 + b^2 = 25$;
(β
‘) Given that the lengths of three line segments are $a$, $b$, and $5$, find the probability that these three line segments can form an isosceles triangle.
|
\dfrac{7}{18}
| 11.71875 |
17,649 |
Given that acute angles $\alpha$ and $\beta$ satisfy $\alpha+2\beta=\frac{2\pi}{3}$ and $\tan\frac{\alpha}{2}\tan\beta=2-\sqrt{3}$, find the value of $\alpha +\beta$.
|
\frac{5\pi}{12}
| 30.46875 |
17,650 |
Given that $F$ is the right focus of the ellipse $C:\frac{x^2}{4}+\frac{y^2}{3}=1$, $P$ is a point on the ellipse $C$, and $A(1,2\sqrt{2})$, find the maximum value of $|PA|+|PF|$.
|
4 + 2\sqrt{3}
| 57.03125 |
17,651 |
How many 5-digit positive numbers contain only odd numbers and have at least one pair of consecutive digits whose sum is 10?
|
1845
| 0 |
17,652 |
Math City plans to add more streets and now has 10 streets, but two pairs of these streets are parallel to each other. No other streets are parallel, and no street is parallel to more than one other street. What is the greatest number of police officers needed at intersections, assuming that each intersection has exactly one police officer stationed?
|
43
| 5.46875 |
17,653 |
What is the product of the numerator and the denominator when $0.\overline{012}$ is expressed as a fraction in lowest terms?
|
1332
| 93.75 |
17,654 |
Two circles have centers at $(3,5)$ and $(20,15)$. Both circles are tangent to the x-axis. Determine the distance between the closest points of the two circles.
|
\sqrt{389} - 20
| 56.25 |
17,655 |
What is the smallest positive integer $n$ such that all the roots of $z^6 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity?
|
18
| 93.75 |
17,656 |
In the arithmetic sequence $\{a_n\}$, it is known that $a_1=10$, and the sum of the first $n$ terms is $S_n$. If $S_9=S_{12}$, find the maximum value of $S_n$ and the corresponding value of $n$.
|
55
| 29.6875 |
17,657 |
In triangle \\(ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, and it is given that \\(A < B < C\\) and \\(C = 2A\\).
\\((1)\\) If \\(c = \sqrt{3}a\\), find the measure of angle \\(A\\).
\\((2)\\) If \\(a\\), \\(b\\), and \\(c\\) are three consecutive positive integers, find the area of \\(\triangle ABC\\).
|
\dfrac{15\sqrt{7}}{4}
| 58.59375 |
17,658 |
How many ways are there to put 6 balls into 4 boxes if the balls are indistinguishable, the boxes are distinguishable, and each box must contain at least one ball?
|
10
| 98.4375 |
17,659 |
The line $l: x - 2y + 2 = 0$ passes through the left focus F<sub>1</sub> and a vertex B of an ellipse. Find the eccentricity of the ellipse.
|
\frac{2\sqrt{5}}{5}
| 95.3125 |
17,660 |
Rounded to 3 decimal places, what is $\frac{8}{11}$?
|
0.727
| 58.59375 |
17,661 |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. If $$a=b\cos C+\frac{\sqrt{3}}{3}c\sin B$$.
(1) Find the value of angle B.
(2) If the area of triangle ABC is S=$$5\sqrt{3}$$, and a=5, find the value of b.
|
\sqrt{21}
| 48.4375 |
17,662 |
Given Chelsea leads by 60 points halfway through a 120-shot archery tournament, scores at least 5 points per shot, and scores at least 10 points for each of her next n shots, determine the minimum number of shots, n, she must get as bullseyes to guarantee her victory.
|
49
| 16.40625 |
17,663 |
On the ellipse $\frac {x^{2}}{3}+ \frac {y^{2}}{2}=1$, the distance from a point P to the left focus is $\frac { \sqrt {3}}{2}$. Find the distance from P to the right directrix.
|
\frac{9}{2}
| 42.1875 |
17,664 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The perimeter of $\triangle ABC$ is $\sqrt {2}+1$, and $\sin A + \sin B = \sqrt {2} \sin C$.
(I) Find the length of side $c$.
(II) If the area of $\triangle ABC$ is $\frac {1}{5} \sin C$, find the value of $\cos C$.
|
\frac{1}{4}
| 86.71875 |
17,665 |
Add 76.893 to 34.2176 and round to the nearest tenth.
|
111.1
| 93.75 |
17,666 |
Find the product of the greatest common divisor (gcd) and the least common multiple (lcm) of 225 and 252.
|
56700
| 99.21875 |
17,667 |
Jennifer plans a profit of 20% on the selling price of an item, and her expenses are 10% of the selling price. There is also a sales tax of 5% on the selling price of the item. The item sells for $10.00. Calculate the rate of markup on cost of this item.
|
53.85\%
| 54.6875 |
17,668 |
On the radius \( AO \) of a circle centered at \( O \), a point \( M \) is chosen. On one side of \( AO \), points \( B \) and \( C \) are chosen on the circle such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \) if the radius of the circle is 10 and \( \cos \alpha = \frac{4}{5} \).
|
16
| 36.71875 |
17,669 |
Given that $0 < \alpha < \pi$ and $\cos{\alpha} = -\frac{3}{5}$, find the value of $\tan{\alpha}$ and the value of $\cos{2\alpha} - \cos{(\frac{\pi}{2} + \alpha)}$.
|
\frac{13}{25}
| 70.3125 |
17,670 |
In the "Black White Pair" game, a common game among children often used to determine who goes first, participants (three or more) reveal their hands simultaneously, using the palm (white) or the back of the hand (black) to decide the winner. If one person shows a gesture different from everyone else's, that person wins; in all other cases, there is no winner. Now, A, B, and C are playing the "Black White Pair" game together. Assuming A, B, and C each randomly show "palm (white)" or "back of the hand (black)" with equal probability, the probability of A winning in one round of the game is _______.
|
\frac{1}{4}
| 66.40625 |
17,671 |
In a certain school, 3 teachers are chosen from a group of 6 to give support teaching in 3 remote areas, with each area receiving one teacher. There are restrictions such that teacher A and teacher B cannot go together, and teacher A can only go with teacher C or not go at all. How many different dispatch plans are there?
|
42
| 8.59375 |
17,672 |
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.
|
309
| 0 |
17,673 |
Calculate \(3^3 \cdot 4^3\).
|
1728
| 99.21875 |
17,674 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $c\sin A= \sqrt {3}a\cos C$ and $(a-c)(a+c)=b(b-c)$, consider the function $f(x)=2\sin x\cos ( \frac {Ο}{2}-x)- \sqrt {3}\sin (Ο+x)\cos x+\sin ( \frac {Ο}{2}+x)\cos x$.
(1) Find the period and the equation of the axis of symmetry of the function $y=f(x)$.
(2) Find the value of $f(B)$.
|
\frac {5}{2}
| 71.875 |
17,675 |
In how many ways can $100$ be written as the sum of three positive integers $x, y$ , and $z$ satisfying $x < y < z$ ?
|
784
| 99.21875 |
17,676 |
Given a point $P$ on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with foci $F_{1}$, $F_{2}$, find the eccentricity of the ellipse given that $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}=0$ and $\tan \angle PF_{1}F_{2}= \frac{1}{2}$.
|
\frac{\sqrt{5}}{3}
| 30.46875 |
17,677 |
What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6?
|
144
| 2.34375 |
17,678 |
In a bag containing 12 green marbles and 8 purple marbles, Phil draws a marble at random, records its color, replaces it, and repeats this process until he has drawn 10 marbles. What is the probability that exactly five of the marbles he draws are green? Express your answer as a decimal rounded to the nearest thousandth.
|
0.201
| 90.625 |
17,679 |
A regular hexagon is inscribed in another regular hexagon such that each vertex of the inscribed hexagon divides a side of the original hexagon into two parts in the ratio 2:1. Find the ratio of the area of the inscribed hexagon to the area of the larger hexagon.
|
7/9
| 0 |
17,680 |
Given (1+i)x=1+yi, find |x+yi|.
|
\sqrt{2}
| 78.125 |
17,681 |
In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.2 times the probability of earning a B. Assuming that all grades are A, B, or C, how many B's will there be in a mathematics class of 40 students?
|
14
| 93.75 |
17,682 |
Given that $O$ is any point in space, and $A$, $B$, $C$, $D$ are four points such that no three of them are collinear, but they are coplanar, and $\overrightarrow{OA}=2x\cdot \overrightarrow{BO}+3y\cdot \overrightarrow{CO}+4z\cdot \overrightarrow{DO}$, find the value of $2x+3y+4z$.
|
-1
| 54.6875 |
17,683 |
Solve the congruence $15x + 3 \equiv 9 \pmod{21}$ for $x$, and express the solution as $x \equiv a \pmod{m}$, where $a < m$ and find $a + m$.
|
13
| 56.25 |
17,684 |
Through the focus of the parabola $y^{2}=4x$, two chords $AB$ and $CD$ are drawn perpendicular to each other. Calculate $\left( \frac{1}{|AB|}+ \frac{1}{|CD|} \right)$.
|
\frac{1}{4}
| 80.46875 |
17,685 |
Given the sum of the first n terms of two arithmetic sequences, ${a_n}$ and ${b_n}$, are $S_n$ and $T_n$ respectively, and the ratio $\frac{S_n}{T_n} = \frac{2n+1}{3n+2}$, calculate the value of $\frac{a_3 + a_{11} + a_{19}}{b_7 + b_{15}}$.
|
\frac{129}{130}
| 27.34375 |
17,686 |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?
|
115
| 26.5625 |
17,687 |
Given $f(x)= \begin{cases} \sin \frac{\pi}{3}x, & x\leqslant 2011, \\ f(x-4), & x > 2011, \end{cases}$ find $f(2012)$.
|
-\frac{\sqrt{3}}{2}
| 66.40625 |
17,688 |
Given the function $f(x)=4\cos (3x+Ο)(|Ο| < \dfrac{Ο}{2})$, its graph is symmetric about the line $x=\dfrac{11Ο}{12}$. When $x\_1$, $x\_2β(β\dfrac{7Ο}{12},β\dfrac{Ο}{12})$, $x\_1β x\_2$, and $f(x\_1)=f(x\_2)$, determine the value of $f(x\_1+x\_2)$.
|
2\sqrt{2}
| 64.0625 |
17,689 |
What is the least possible value of the expression (x+1)(x+2)(x+3)(x+4) + 2021 where x is a real number?
|
2020
| 74.21875 |
17,690 |
Find the smallest integer satisfying the following conditions:
$\bullet$ I. The sum of the squares of its digits is $85$.
$\bullet$ II. Each digit is larger than the one on its left.
What is the product of the digits of this integer?
|
18
| 0.78125 |
17,691 |
For the Shanghai World Expo, 20 volunteers were recruited, with each volunteer assigned a unique number from 1 to 20. If four individuals are to be selected randomly from this group and divided into two teams according to their numbers, with the smaller numbers in one team and the larger numbers in another, what is the total number of ways to ensure that both volunteers number 5 and number 14 are selected and placed on the same team?
|
21
| 7.8125 |
17,692 |
In the Cartesian coordinate system $xOy$, the equation of curve $C$ is $x^{2}+y^{2}=16$, and the parameter equation of the line is $\left\{\begin{array}{l}x=1+t\cos \theta \\y=2+t\sin \theta \end{array}\right.$ (where $t$ is the parameter). If the line $l$ intersects curve $C$ at points $M$ and $N$, and the midpoint of line segment $MN$ is $(1,2)$, calculate the slope of the line.
|
-\frac{1}{2}
| 89.84375 |
17,693 |
A student types the following pattern on a computer (where 'γ' represents an empty circle and 'β' represents a solid circle): γβγγβγγγβγγγγβ... If this pattern of circles continues, what is the number of solid circles among the first 2019 circles?
|
62
| 44.53125 |
17,694 |
Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$ . Find the number of Juan.
|
532
| 16.40625 |
17,695 |
Given \\(a > 0\\), the function \\(f(x)= \frac {1}{3}x^{3}+ \frac {1-a}{2}x^{2}-ax-a\\).
\\((1)\\) Discuss the monotonicity of \\(f(x)\\);
\\((2)\\) When \\(a=1\\), let the function \\(g(t)\\) represent the difference between the maximum and minimum values of \\(f(x)\\) on the interval \\([t,t+3]\\). Find the minimum value of \\(g(t)\\) on the interval \\([-3,-1]\\).
|
\frac {4}{3}
| 14.0625 |
17,696 |
In a triangle $ABC$ , let $I$ denote the incenter. Let the lines $AI,BI$ and $CI$ intersect the incircle at $P,Q$ and $R$ , respectively. If $\angle BAC = 40^o$ , what is the value of $\angle QPR$ in degrees ?
|
20
| 31.25 |
17,697 |
Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction.
|
\frac{1}{12}
| 89.84375 |
17,698 |
Jeff wants to calculate the product $0.52 \times 7.35$ using a calculator. However, he mistakenly inputs the numbers as $52 \times 735$ without the decimal points. The calculator then shows a product of $38220$. What would be the correct product if Jeff had correctly entered the decimal points?
A) $0.3822$
B) $38.22$
C) $3.822$
D) $0.03822$
E) $382.2$
|
3.822
| 13.28125 |
17,699 |
On a clock, there are two instants between $12$ noon and $1 \,\mathrm{PM}$ , when the hour hand and the minute hannd are at right angles. The difference *in minutes* between these two instants is written as $a + \dfrac{b}{c}$ , where $a, b, c$ are positive integers, with $b < c$ and $b/c$ in the reduced form. What is the value of $a+b+c$ ?
|
51
| 75.78125 |
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