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30,500 | Given that the sequence {a<sub>n</sub>} is an arithmetic sequence, a<sub>1</sub> < 0, a<sub>8</sub> + a<sub>9</sub> > 0, a<sub>8</sub> • a<sub>9</sub> < 0. Find the smallest value of n for which S<sub>n</sub> > 0. | 16 | 25 |
30,501 | Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.294 | 0 |
30,502 | In triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and the vectors $\overrightarrow{m}=({\cos C, \cos({\frac{\pi}{2}-B})})$, $\overrightarrow{n}=({\cos({-4\pi+B}), -\sin C})$, and $\overrightarrow{m} \cdot \overrightarrow{n}=-\frac{\sqrt{2}}{2}$. <br/>$(1)$ Find the measure of angle $A$; <br/>$(2)$ If the altitude on side $AC$ is $2$, $a=3$, find the perimeter of $\triangle ABC$. | 5 + 2\sqrt{2} + \sqrt{5} | 2.34375 |
30,503 | Find the smallest value x such that, given any point inside an equilateral triangle of side 1, we can always choose two points on the sides of the triangle, collinear with the given point and a distance x apart. | \frac{2}{3} | 0.78125 |
30,504 | Given the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$, its foci are equal to the minor axis length of the ellipse $Ω:x^{2}+ \frac{y^{2}}{4}=1$, and the major axis lengths of C and Ω are equal.
(1) Find the equation of ellipse C;
(2) Let $F_1$, $F_2$ be the left and right foci of ellipse C, respectively. A line l that does not pass through $F_1$ intersects ellipse C at two distinct points A and B. If the slopes of lines $AF_1$ and $BF_1$ form an arithmetic sequence, find the maximum area of △AOB. | \sqrt{3} | 24.21875 |
30,505 | The eight digits in Jane's phone number and the four digits in her office number have the same sum. The digits in her office number are distinct, and her phone number is 539-8271. What is the largest possible value of Jane's office number? | 9876 | 10.9375 |
30,506 | If four departments A, B, C, and D select from six tourist destinations, calculate the total number of ways in which at least three departments have different destinations. | 1080 | 1.5625 |
30,507 | 1. On a semicircle with AB as the diameter, besides points A and B, there are 6 other points. Since there are also 4 other points on AB, making a total of 12 points, how many quadrilaterals can be formed with these 12 points as vertices?
2. On one side of angle A, there are five points (excluding A), and on the other side, there are four points (excluding A). With these ten points (including A), how many triangles can be formed?
3. Suppose there are 3 equally spaced parallel lines intersecting with another set of 4 equally spaced parallel lines. How many triangles can be formed with these intersection points as vertices? | 200 | 0 |
30,508 | A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.<br/>$(1)$ Find the probability of drawing exactly one red ball.<br/>$(2)$ If the random variable $X$ represents the number of red balls drawn, find the distribution of the random variable $X$. | \frac{3}{10} | 6.25 |
30,509 | Choose any $2$ numbers from $-5$, $-3$, $-1$, $2$, and $4$. Let the maximum product obtained be denoted as $a$, and the minimum quotient obtained be denoted as $b$. Then the value of $\frac{a}{b}$ is ______. | -\frac{15}{4} | 2.34375 |
30,510 | 1. Given that the line $l$ passing through the point $M(-3,0)$ is intercepted by the circle $x^{2}+(y+2)^{2}=25$ to form a chord of length $8$, what is the equation of line $l$?
2. Circles $C_{1}$: $x^{2}+y^{2}+2x+8y-8=0$ and $C_{2}$: $x^{2}+y^{2}-4x-4y-2=0$ intersect. What is the length of their common chord?
3. What is the equation of a circle passing through points $A(0,2)$ and $B(-2,2)$, with its center lying on the line $x-y-2=0$?
4. Given the circle with the equation $(x-2)^{2}+y^{2}=1$, a line passing through the external point $P(3,4)$ intersects the circle at points $A$ and $B$. What is the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$? | 16 | 71.875 |
30,511 | In the encoded equality $A B + A B + A B + A B + A B + A B + A B + A B + A B = A A B$, digits are replaced with letters: the same digits with the same letter, and different digits with different letters. Find all possible decipherings. (I. Rubanov) | 25 | 42.96875 |
30,512 | If \(1 + 1.1 + 1.11 + \square = 4.44\), what number should be put in the box to make the equation true? | 1.23 | 100 |
30,513 | Given M be the greatest five-digit number whose digits have a product of 36, determine the sum of the digits of M. | 15 | 7.03125 |
30,514 | Given that the asymptotic line of the hyperbola $\frac{x^2}{a}+y^2=1$ has a slope of $\frac{5π}{6}$, determine the value of $a$. | -3 | 2.34375 |
30,515 | Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}|=5$, $|\overrightarrow {a}- \overrightarrow {b}|=6$, and $|\overrightarrow {a}+ \overrightarrow {b}|=4$, find the projection of vector $\overrightarrow {b}$ on vector $\overrightarrow {a}$. | -1 | 14.84375 |
30,516 | Find the three-digit integer in the decimal system that satisfies the following properties:
1. When the digits in the tens and units places are swapped, the resulting number can be represented in the octal system as the original number.
2. When the digits in the hundreds and tens places are swapped, the resulting number is 16 less than the original number when read in the hexadecimal system.
3. When the digits in the hundreds and units places are swapped, the resulting number is 18 more than the original number when read in the quaternary system. | 139 | 0 |
30,517 | Given vectors $\overrightarrow{a}=(\cos x,\sin x)$ and $\overrightarrow{b}=(3,-\sqrt{3})$, with $x\in[0,\pi]$.
$(1)$ If $\overrightarrow{a}\parallel\overrightarrow{b}$, find the value of $x$; $(2)$ Let $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$, find the maximum and minimum values of $f(x)$ and the corresponding values of $x$. | -2\sqrt{3} | 8.59375 |
30,518 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5 \times 5$ square array of dots? | 100 | 21.875 |
30,519 | The union of sets \( A \) and \( B \) is \( A \cup B = \left\{a_{1}, a_{2}, a_{3}\right\} \). When \( A \neq B \), the pairs \((A, B)\) and \((B, A)\) are considered different. How many such pairs \((A, B)\) are there? | 27 | 12.5 |
30,520 | If $y=f(x)$ has an inverse function $y=f^{-1}(x)$, and $y=f(x+2)$ and $y=f^{-1}(x-1)$ are inverse functions of each other, then $f^{-1}(2007)-f^{-1}(1)=$ . | 4012 | 36.71875 |
30,521 | The smaller square has an area of 16 and the grey triangle has an area of 1. What is the area of the larger square?
A) 17
B) 18
C) 19
D) 20
E) 21 | 18 | 6.25 |
30,522 | Solve the equation: $4x^2 - (x^2 - 2x + 1) = 0$. | -1 | 0 |
30,523 | Consider a sequence of complex numbers $\left\{z_{n}\right\}$ defined as "interesting" if $\left|z_{1}\right|=1$ and for every positive integer $n$, the following holds:
$$
4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0.
$$
Find the largest constant $C$ such that for any "interesting" sequence $\left\{z_{n}\right\}$ and any positive integer $m$, the inequality below holds:
$$
\left|z_{1}+z_{2}+\cdots+z_{m}\right| \geqslant C.
$$ | \frac{\sqrt{3}}{3} | 0.78125 |
30,524 | The product of the first three terms of a geometric sequence is 2, the product of the last three terms is 4, and the product of all terms is 64. Find the number of terms in the sequence. | 12 | 36.71875 |
30,525 | Find the $y$-intercepts of the following system of equations:
1. $2x - 3y = 6$
2. $x + 4y = -8$ | -2 | 80.46875 |
30,526 | **Problem Statement**: Let $r$ and $k$ be integers such that $-5 < r < 8$ and $0 < k < 10$. What is the probability that the division $r \div k$ results in an integer value? Express your answer as a common fraction. | \frac{33}{108} | 0 |
30,527 | A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 such that its bases are parallel to the base of the hemisphere and the top of the cylinder touches the top of the hemisphere. What is the height of the cylinder? | 2\sqrt{10} | 14.84375 |
30,528 | An entrepreneur took out a discounted loan of 12 million HUF with a fixed annual interest rate of 8%. What will be the debt after 10 years if they can repay 1.2 million HUF annually? | 8523225 | 0 |
30,529 | Given three positive numbers \( a, b, \mathrm{and} c \) satisfying \( a \leq b+c \leq 3a \) and \( 3b^2 \leq a(a+c) \leq 5b^2 \), what is the minimum value of \(\frac{b-2c}{a}\)? | -\frac{18}{5} | 0.78125 |
30,530 | Five people can paint a house in 10 hours. How many hours would it take four people to paint the same house and mow the lawn if mowing the lawn takes an additional 3 hours per person, assuming that each person works at the same rate for painting and different rate for mowing? | 15.5 | 10.9375 |
30,531 | On the trip home from the meeting where this AMC $ 10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $ bbcac$, where $ 0 \le a < b < c \le 9$, and $ b$ was the average of $ a$ and $ c$. How many different five-digit numbers satisfy all these properties? | 20 | 91.40625 |
30,532 | Given a $5\times 5$ chess board, how many ways can you place five distinct pawns on the board such that each column and each row contains exactly one pawn and no two pawns are positioned as if they were "attacking" each other in the manner of queens in chess? | 1200 | 0.78125 |
30,533 | Xiao Ming's family has three hens. The first hen lays one egg every day, the second hen lays one egg every two days, and the third hen lays one egg every three days. Given that all three hens laid eggs on January 1st, how many eggs did these three hens lay in total in the 31 days of January? | 56 | 39.84375 |
30,534 | Given the function $f(x)=2\sin(2x-\frac{\pi}{3})-1$, find the probability that a real number $a$ randomly selected from the interval $\left[0,\frac{\pi}{2}\right]$ satisfies $f(a) > 0$. | \frac{1}{2} | 79.6875 |
30,535 | Given the parabola $y^{2}=2x$ with focus $F$, a line passing through $F$ intersects the parabola at points $A$ and $B$. If $|AB|= \frac{25}{12}$ and $|AF| < |BF|$, determine the value of $|AF|$. | \frac{5}{6} | 24.21875 |
30,536 | Let us call a number \( \mathrm{X} \) "50-podpyirayushchim" if for any 50 real numbers \( a_{1}, \ldots, a_{50} \) whose sum is an integer, there exists at least one \( a_i \) such that \( \left|a_{i}-\frac{1}{2}\right| \geq X \).
Find the greatest 50-podpyirayushchee \( X \), rounded to the nearest hundredth according to standard mathematical rules. | 0.01 | 39.84375 |
30,537 | Consider the ellipse (C) with the equation \\(\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)\\) and an eccentricity of \\(\frac{\sqrt{3}}{3}\\). A line passing through the right focus (F2(c,0)) perpendicular to the x-axis intersects the ellipse at points A and B, such that |AB|=\\(\frac{4\sqrt{3}}{3}\\). Additionally, a line (l) passing through the left focus (F1(-c,0)) intersects the ellipse at point M.
1. Find the equation of the ellipse (C).
2. Given that points A and B on the ellipse (C) are symmetric with respect to line (l), find the maximum area of triangle AOB. | \frac{\sqrt{6}}{2} | 3.125 |
30,538 | Given $m$ and $n∈\{lg2+lg5,lo{g}_{4}3,{(\frac{1}{3})}^{-\frac{3}{5}},tan1\}$, the probability that the function $f\left(x\right)=x^{2}+2mx+n^{2}$ has two distinct zeros is ______. | \frac{3}{8} | 10.15625 |
30,539 | Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 2020\}$. What is the probability that $abc + ab + a$ is divisible by $4$?
A) $\frac{1}{4}$
B) $\frac{1}{32}$
C) $\frac{8}{32}$
D) $\frac{9}{32}$
E) $\frac{1}{16}$ | \frac{9}{32} | 47.65625 |
30,540 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\frac {(a+b)^{2}-c^{2}}{3ab}=1$.
$(1)$ Find $\angle C$;
$(2)$ If $c= \sqrt {3}$ and $b= \sqrt {2}$, find $\angle B$ and the area of $\triangle ABC$. | \frac {3+ \sqrt {3}}{4} | 0 |
30,541 | Angle ABC is a right angle. The diagram shows four quadrilaterals, where three are squares on each side of triangle ABC, and one square is on the hypotenuse. The sum of the areas of all four squares is 500 square centimeters. What is the number of square centimeters in the area of the largest square? | \frac{500}{3} | 42.1875 |
30,542 | Person A and Person B decided to go to a restaurant. Due to high demand, Person A arrived first and took a waiting number, while waiting for Person B. After a while, Person B arrived but did not see Person A, so he also took a waiting number. While waiting, Person B saw Person A, and they compared their waiting numbers. They found that the digits of these two numbers are two-digit numbers in reverse order, and the sum of the digits of both numbers is 8. Additionally, Person B's number is 18 greater than Person A's. What is Person A's number? $\qquad$ | 35 | 64.84375 |
30,543 | In the Cartesian coordinate system, there are points $P_{1}, P_{2}, \ldots, P_{n-1}, P_{n}, \ldots (n \in \mathbb{N}^{*})$. Let the coordinates of point $P_{n}$ be $(n, a_{n})$, where $a_{n}= \frac {2}{n} (n \in \mathbb{N}^{*})$. The line passing through points $P_{n}$ and $P_{n+1}$ forms a triangle with the coordinate axes, and the area of this triangle is $b_{n}$. Let $S_{n}$ represent the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Then, $S_{5}=$ \_\_\_\_\_\_. | \frac {125}{6} | 44.53125 |
30,544 | Determine the distance in feet between the 5th red light and the 23rd red light, where the lights are hung on a string 8 inches apart in the pattern of 3 red lights followed by 4 green lights. Recall that 1 foot is equal to 12 inches. | 28 | 4.6875 |
30,545 | Convert $6351_8$ to base 7. | 12431_7 | 11.71875 |
30,546 | Given that a smaller circle is entirely inside a larger circle, such that the larger circle has a radius $R = 2$, and the areas of the two circles form an arithmetic progression, with the largest area being that of the larger circle, find the radius of the smaller circle. | \sqrt{2} | 53.125 |
30,547 | Suppose convex hexagon $ \text{HEXAGN}$ has $ 120^\circ$ -rotational symmetry about a point $ P$ —that is, if you rotate it $ 120^\circ$ about $ P$ , it doesn't change. If $ PX\equal{}1$ , find the area of triangle $ \triangle{GHX}$ . | \frac{\sqrt{3}}{4} | 89.0625 |
30,548 | Let \(\theta\) be an angle in the second quadrant, and if \(\tan (\theta+ \frac {\pi}{3})= \frac {1}{2}\), calculate the value of \(\sin \theta+ \sqrt {3}\cos \theta\). | - \frac {2 \sqrt {5}}{5} | 0 |
30,549 | Let $T = \{ 1, 2, 3, \dots, 14, 15 \}$ . Say that a subset $S$ of $T$ is *handy* if the sum of all the elements of $S$ is a multiple of $5$ . For example, the empty set is handy (because its sum is 0) and $T$ itself is handy (because its sum is 120). Compute the number of handy subsets of $T$ . | 6560 | 12.5 |
30,550 | To investigate a non-luminous black planet in distant space, Xiao Feitian drives a high-speed spaceship equipped with a powerful light, traveling straight towards the black planet at a speed of 100,000 km/s. When Xiao Feitian had just been traveling for 100 seconds, the spaceship instruments received light reflected back from the black planet. If the speed of light is 300,000 km/s, what is the distance from Xiao Feitian's starting point to the black planet in 10,000 kilometers? | 2000 | 0.78125 |
30,551 | Given that $n \in \mathbb{N}^*$, the coefficient of the second term in the expansion of $(x+2)^n$ is $\frac{1}{5}$ of the coefficient of the third term.
(1) Find the value of $n$;
(2) Find the term with the maximum binomial coefficient in the expansion;
(3) If $(x+2)^n = a\_0 + a\_1(x+1) + a\_2(x+1)^2 + \dots + a\_n(x+1)^n$, find the value of $a\_0 + a\_1 + \dots + a\_n$. | 64 | 82.8125 |
30,552 |
Calculate the volume of a tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$, and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(-2, 0, -4)$
$A_{2}(-1, 7, 1)$
$A_{3}(4, -8, -4)$
$A_{4}(1, -4, 6)$ | 5\sqrt{2} | 16.40625 |
30,553 | In $\triangle ABC$, $2\sin ^{2} \frac{A}{2}= \sqrt{3}\sin A$, $\sin (B-C)=2\cos B\sin C$, find the value of $\frac{AC}{AB}$ . | \frac{1+\sqrt{13}}{2} | 3.125 |
30,554 | For a finite sequence \( B = (b_1, b_2, \dots, b_{50}) \) of numbers, the Cesaro sum is defined as
\[
\frac{S_1 + \cdots + S_{50}}{50},
\]
where \( S_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 50 \).
If the Cesaro sum of the 50-term sequence \( (b_1, \dots, b_{50}) \) is 500, what is the Cesaro sum of the 51-term sequence \( (2, b_1, \dots, b_{50}) \)? | 492 | 13.28125 |
30,555 | Eight students participate in a pie-eating contest. The graph shows the number of pies eaten by each participating student. Sarah ate the most pies and Tom ate the fewest. Calculate how many more pies than Tom did Sarah eat and find the average number of pies eaten by all the students.
[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[10];
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yaxis[i] = (0,i);
draw(yaxis[i]--yaxis[i]+(17,0));
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draw((0,0)--(0,9));
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fill((1,0)--(1,3)--(2,3)--(2,0)--cycle,grey);
fill((3,0)--(3,4)--(4,4)--(4,0)--cycle,grey);
fill((5,0)--(5,8)--(6,8)--(6,0)--cycle,grey);
fill((7,0)--(7,5)--(8,5)--(8,0)--cycle,grey);
fill((9,0)--(9,7)--(10,7)--(10,0)--cycle,grey);
fill((11,0)--(11,6)--(12,6)--(12,0)--cycle,grey);
fill((13,0)--(13,2)--(14,2)--(14,0)--cycle,grey);
fill((15,0)--(15,1)--(16,1)--(16,0)--cycle,grey);
label("0",yaxis[0],W);
label("1",yaxis[1],W);
label("2",yaxis[2],W);
label("3",yaxis[3],W);
label("4",yaxis[4],W);
label("5",yaxis[5],W);
label("6",yaxis[6],W);
label("7",yaxis[7],W);
label("8",yaxis[8],W);
label("9",yaxis[9],W);
label("Students/Participants",(8.5,0),S);
label("Results of a Pie-Eating Contest",(8.5,9),N);
label(rotate(90)*"$\#$ of Pies Eaten",(-1,4.5),W);
[/asy] | 4.5 | 67.96875 |
30,556 | In an isosceles trapezoid, the longer base \(AB\) is 24 units, the shorter base \(CD\) is 12 units, and each of the non-parallel sides has a length of 13 units. What is the length of the diagonal \(AC\)? | \sqrt{457} | 29.6875 |
30,557 | Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, if $\cos B= \frac {4}{5}$, $a=5$, and the area of $\triangle ABC$ is $12$, find the value of $\frac {a+c}{\sin A+\sin C}$. | \frac {25}{3} | 86.71875 |
30,558 | In how many ways can one choose distinct numbers a and b from {1, 2, 3, ..., 2005} such that a + b is a multiple of 5? | 401802 | 7.8125 |
30,559 | Given that \(11 \cdot 14 n\) is a non-negative integer and \(f\) is defined by \(f(0)=0\), \(f(1)=1\), and \(f(n)=f\left(\left\lfloor \frac{n}{2} \right\rfloor \right)+n-2\left\lfloor \frac{n}{2} \right\rfloor\), find the maximum value of \(f(n)\) for \(0 \leq n \leq 1991\). Here, \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\). | 10 | 100 |
30,560 | An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$? | 48 | 0.78125 |
30,561 | The prime factorization of 1386 is $2 \times 3 \times 3 \times 7 \times 11$. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $xy = 1386$, and both $x$ and $y$ are even? | 12 | 11.71875 |
30,562 | Given the function $f(x)=\sin(2x+\frac{\pi}{6})+\sin(2x-\frac{\pi}{6})+\cos{2x}+a$ (where $a \in \mathbb{R}$ and $a$ is a constant):
(1) Find the smallest positive period of the function and the intervals of monotonic increase.
(2) If the minimum value of $f(x)$ for $x \in \left[0, \frac{\pi}{2}\right]$ is $-2$, determine the value of $a$. | -1 | 21.875 |
30,563 | Given the function $f(x)= \sqrt {3}\sin x\cdot\cos x- \frac {1}{2}\cos 2x$ $(x\in\mathbb{R})$.
$(1)$ Find the minimum value and the smallest positive period of the function $f(x)$.
$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $f(C)=1$, $B=30^{\circ}$, $c=2 \sqrt {3}$. Find the area of $\triangle ABC$. | 2 \sqrt {3} | 0 |
30,564 | In an isosceles triangle \(ABC\) with base \(AC\) equal to 37, the exterior angle at vertex \(B\) is \(60^\circ\). Find the distance from vertex \(C\) to line \(AB\). | 18.5 | 8.59375 |
30,565 | Given that \( p \) and \( q \) are positive integers such that \( p + q > 2017 \), \( 0 < p < q \leq 2017 \), and \((p, q) = 1\), find the sum of all fractions of the form \(\frac{1}{pq}\). | 1/2 | 94.53125 |
30,566 | (1) Convert the following angles between degrees and radians: ① -15°; ② $\frac {7\pi }{12}$
(2) Given that the terminal side of angle $\alpha$ passes through point P(2sin30°, -2cos30°), find the sine, cosine, and tangent values of angle $\alpha$. | -\sqrt {3} | 0 |
30,567 | Frederik wants to make a special seven-digit password. Each digit of his password occurs exactly as many times as its digit value. The digits with equal values always occur consecutively, e.g., 4444333 or 1666666. How many possible passwords can he make?
A) 6
B) 7
C) 10
D) 12
E) 13 | 13 | 32.8125 |
30,568 | The average age of 40 sixth-graders is 12. The average age of 30 of their teachers is 45. What is the average age of all these sixth-graders and their teachers? | 26.14 | 12.5 |
30,569 | Given the equation $x^3 - 12x^2 + 27x - 18 = 0$ with roots $a$, $b$, $c$, find the value of $\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}$. | \frac{13}{24} | 15.625 |
30,570 | Given a regular quadrilateral pyramid $S-ABCD$ with lateral edge length of $4$ and $\angle ASB = 30^\circ$, points $E$, $F$, and $G$ are taken on lateral edges $SB$, $SC$, and $SD$ respectively. Find the minimum value of the perimeter of the spatial quadrilateral $AEFG$. | 4\sqrt{3} | 2.34375 |
30,571 | Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $ , we have $ a_{pk+1}=pa_k-3a_p+13 $ .Determine all possible values of $ a_{2013} $ . | 2016 | 15.625 |
30,572 | Given that our number system has a base of eight, determine the fifteenth number in the sequence. | 17 | 69.53125 |
30,573 | How many natural numbers between 200 and 400 are divisible by 8? | 24 | 0.78125 |
30,574 | Given vectors $\overrightarrow {a}$=(2,6) and $\overrightarrow {b}$=(m,-1), find the value of m when $\overrightarrow {a}$ is perpendicular to $\overrightarrow {b}$ and when $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$. | -\frac{1}{3} | 93.75 |
30,575 | Given that the terminal side of angle $\alpha$ passes through the point $(-3, 4)$, then $\cos\alpha=$ _______; $\cos2\alpha=$ _______. | -\frac{7}{25} | 98.4375 |
30,576 | The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is denoted as $S_n$. Given that $a_1-a_5-a_10-a_15+a_19=2$, find the value of $S_{19}$. | -38 | 53.90625 |
30,577 | 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 \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(-2, -1, -1) \)
\( A_{2}(0, 3, 2) \)
\( A_{3}(3, 1, -4) \)
\( A_{4}(-4, 7, 3) \) | \frac{140}{\sqrt{1021}} | 7.8125 |
30,578 | There are 11 of the number 1, 22 of the number 2, 33 of the number 3, and 44 of the number 4 on the blackboard. The following operation is performed: each time, three different numbers are erased, and the fourth number, which is not erased, is written 2 extra times. For example, if 1 of 1, 1 of 2, and 1 of 3 are erased, then 2 more of 4 are written. After several operations, there are only 3 numbers left on the blackboard, and no further operations can be performed. What is the product of the last three remaining numbers? | 12 | 7.8125 |
30,579 | Let $ABCD$ be a cyclic quadrilateral with sides $AB$, $BC$, $CD$, and $DA$. The side lengths are distinct integers less than $10$ and satisfy $BC + CD = AB + DA$. Find the largest possible value of the diagonal $BD$.
A) $\sqrt{93}$
B) $\sqrt{\frac{187}{2}}$
C) $\sqrt{\frac{191}{2}}$
D) $\sqrt{100}$ | \sqrt{\frac{191}{2}} | 17.96875 |
30,580 | Given two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}|=|\overrightarrow{b}|$, and $\overrightarrow{a}\perp(\overrightarrow{a}-2\overrightarrow{b})$, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | \frac{\pi}{3} | 100 |
30,581 | Given the function $f(x)=\ln x+ \frac {1}{2}ax^{2}-x-m$ ($m\in\mathbb{Z}$).
(I) If $f(x)$ is an increasing function, find the range of values for $a$;
(II) If $a < 0$, and $f(x) < 0$ always holds, find the minimum value of $m$. | -1 | 21.875 |
30,582 | In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ Determine the area of $\triangle ABC.$
[asy]
draw((0,0)--(2,5)--(11,5)--(18,0)--cycle);
draw((11,5)--(0,0));
label("$D$",(0,0),W);
label("$A$",(2,5),NW);
label("$B$",(11,5),NE);
label("$C$",(18,0),E);
[/asy] | 4.5 | 75.78125 |
30,583 | In triangle ABC, angle C is a right angle, and CD is the altitude. Find the radius of the circle inscribed in triangle ABC if the radii of the circles inscribed in triangles ACD and BCD are 6 and 8, respectively. | 14 | 14.0625 |
30,584 | The average of five distinct natural numbers is 15, and the median is 18. What is the maximum possible value of the largest number among these five numbers? | 37 | 8.59375 |
30,585 | In a 200-meter race, Sonic is 16 meters ahead of Dash when Sonic finishes the race. The next time they race, Sonic starts 2.5 times this lead distance behind Dash, who is at the starting line. Both runners run at the same constant speed as they did in the first race. Determine the distance Sonic is ahead of Dash when Sonic finishes the second race. | 19.2 | 5.46875 |
30,586 | Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and a die with 8 faces numbered 1 to 8 is rolled. Determine the probability that the product of the numbers on the tile and the die will be a square. | \frac{7}{48} | 41.40625 |
30,587 | Given a cubic polynomial $ 60 x^3 - 80x^2 + 24x - 2 $, which has three roots $\alpha, \beta, \gamma$, all distinct and lying in the interval $(0, 1)$. Find the sum
\[
\frac{1}{1-\alpha} + \frac{1}{1-\beta} + \frac{1}{1-\gamma}.
\] | 22 | 59.375 |
30,588 | Consider a month with 31 days, where the number of the month is a product of two distinct primes (e.g., July, represented as 7). Determine how many days in July are relatively prime to the month number. | 27 | 60.9375 |
30,589 | Point $A$ lies on the line $y=\frac{8}{15} x-6$, and point $B$ on the parabola $y=x^{2}$. What is the minimum length of the segment $AB$? | 1334/255 | 0 |
30,590 | Kanga labelled the vertices of a square-based pyramid using \(1, 2, 3, 4,\) and \(5\) once each. For each face, Kanga calculated the sum of the numbers on its vertices. Four of these sums equaled \(7, 8, 9,\) and \(10\). What is the sum for the fifth face? | 13 | 8.59375 |
30,591 | Find $\frac{12}{15} + \frac{7}{9} + 1\frac{1}{6}$ and simplify the result to its lowest terms. | \frac{247}{90} | 84.375 |
30,592 | Given $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}⊥(\overrightarrow{a}+\overrightarrow{b})$, determine the projection of $\overrightarrow{a}$ onto $\overrightarrow{b}$. | -\frac{1}{2} | 3.125 |
30,593 | Let \( p \) be an integer such that both roots of the equation
\[ 5x^2 - 5px + (66p - 1) = 0 \]
are positive integers. Find the value of \( p \). | 76 | 32.03125 |
30,594 | Given positive real numbers $a$, $b$, $c$, $d$ satisfying $a^{2}-ab+1=0$ and $c^{2}+d^{2}=1$, find the value of $ab$ when $\left(a-c\right)^{2}+\left(b-d\right)^{2}$ reaches its minimum. | \frac{\sqrt{2}}{2} + 1 | 0 |
30,595 | The area of triangle \(ABC\) is 1. Let \(A_1\), \(B_1\), and \(C_1\) be the midpoints of the sides \(BC\), \(CA\), and \(AB\) respectively. Points \(K\), \(L\), and \(M\) are taken on segments \(AB_1\), \(CA_1\), and \(BC_1\) respectively. What is the minimum area of the common part of triangles \(KLM\) and \(A_1B_1C_1\)? | 1/8 | 17.96875 |
30,596 | In $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^{2}+c^{2}-b^{2}=ac$, $c=2$, and point $G$ satisfies $| \overrightarrow{BG}|= \frac { \sqrt {19}}{3}$ and $\overrightarrow{BG}= \frac {1}{3}( \overrightarrow{BA}+ \overrightarrow{BC})$, find the value of $\sin A$. | \frac {3 \sqrt {21}}{14} | 0 |
30,597 | Evaluate the following expression as a simplified fraction. $$1+\cfrac{3}{4+\cfrac{5}{6}}$$ | \frac{47}{29} | 71.09375 |
30,598 | Determine how many ordered pairs $(a, b)$, where $a$ is a positive real number and $b$ is an integer between $1$ and $210$, inclusive, satisfy the equation $(\log_b a)^{2023} = \log_b(a^{2023})$. | 630 | 39.84375 |
30,599 | In parallelogram $ABCD$ , $AB = 10$ , and $AB = 2BC$ . Let $M$ be the midpoint of $CD$ , and suppose that $BM = 2AM$ . Compute $AM$ . | 2\sqrt{5} | 21.875 |
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