Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
10,500 | Calculate the definite integral:
$$
\int_{\pi / 4}^{\operatorname{arctg} 3} \frac{d x}{(3 \operatorname{tg} x+5) \sin 2 x}
$$ | \frac{1}{10} \ln \frac{12}{7} | 2.34375 |
10,501 | A Yule log is shaped like a right cylinder with height $10$ and diameter $5$ . Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$ . Compute $a$ . | 100 | 63.28125 |
10,502 | An alloy consists of zinc and copper in the ratio of $1:2$, and another alloy contains the same metals in the ratio of $2:3$. How many parts of the two alloys can be combined to obtain a third alloy containing the same metals in the ratio of $17:27$? | 9/35 | 0 |
10,503 | A rectangular table of dimensions \( x \) cm \(\times 80\) cm is covered with identical sheets of paper of size \( 5 \) cm \(\times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet ... | 77 | 26.5625 |
10,504 | For any real numbers \( a \) and \( b \), the inequality \( \max \{|a+b|,|a-b|,|2006-b|\} \geq C \) always holds. Find the maximum value of the constant \( C \). (Note: \( \max \{x, y, z\} \) denotes the largest among \( x, y, \) and \( z \).) | 1003 | 97.65625 |
10,505 | For how many integers $n$ between 1 and 150 is the greatest common divisor of 18 and $n$ equal to 6? | 17 | 52.34375 |
10,506 | Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\triangle ABC$ respectively, and $2\sin \frac{7\pi }{6}\sin (\frac{\pi }{6}+C)+ \cos C=-\frac{1}{2}$.
(1) Find $C$;
(2) If $c=2\sqrt{3}$, find the maximum area of $\triangle ABC$. | 3\sqrt{3} | 63.28125 |
10,507 | Given the function $f(x)=\frac{1}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))$ where $n \in \mathbb{N}^{*}$, vector $\overrightarrow{a}=(0,1)$, and $\theta_{n}$ is the angle between vector $\overrightarrow{OA}_{n}$ and $\overrightarrow{a}$. Compute the value of $\frac{\cos \theta_{1}}{\sin \theta_{1... | \frac{2016}{2017} | 75.78125 |
10,508 | Let \( P \) be any point inside a regular tetrahedron \( ABCD \) with side length \( \sqrt{2} \). The distances from point \( P \) to the four faces are \( d_1, d_2, d_3, d_4 \) respectively. What is the minimum value of \( d_1^2 + d_2^2 + d_3^2 + d_4^2 \)? | \frac{1}{3} | 74.21875 |
10,509 | Given one hundred numbers: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{100}\). We compute 98 differences: \(a_{1} = 1 - \frac{1}{3}, a_{2} = \frac{1}{2} - \frac{1}{4}, \ldots, a_{98} = \frac{1}{98} - \frac{1}{100}\). What is the sum of all these differences? | \frac{14651}{9900} | 62.5 |
10,510 | Given that $a$ and $b$ are positive real numbers satisfying $a + 2b = 1$, find the minimum value of $a^2 + 4b^2 + \frac{1}{ab}$. | \frac{17}{2} | 44.53125 |
10,511 | Calculate the definite integral:
$$
\int_{1}^{e} \sqrt{x} \cdot \ln^{2} x \, dx
$$ | \frac{10e\sqrt{e} - 16}{27} | 0 |
10,512 | Simplify $\sin ^{2}\left( \alpha-\frac{\pi}{6} \right)+\sin ^{2}\left( \alpha+\frac{\pi}{6} \right)-\sin ^{2}\alpha$. | \frac{1}{2} | 74.21875 |
10,513 | A and B play a guessing game where A first thinks of a number denoted as $a$, and then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. If $|a-b| \leqslant 1$, then A and B are considered to have a "telepathic connection". If two people are randomly chose... | \frac{7}{25} | 92.96875 |
10,514 | Given a sequence $\{a_n\}$ with the sum of its first n terms $S_n$, if $S_2 = 4$, and $a_{n+1} = 2S_n + 1$ for $n \in N^*$, find the values of $a_1$ and $S_5$. | 121 | 53.90625 |
10,515 | Compute the square of 1085 without using a calculator. | 1177225 | 100 |
10,516 | Let the function
$$
f(x) = x^3 + ax^2 + bx + c \quad \text{for } x \in \mathbf{R},
$$
where \( a \), \( b \), and \( c \) are distinct non-zero integers, and
$$
f(a) = a^3 \quad \text{and} \quad f(b) = b^3.
$$
Find \( a + b + c = \quad \). | 18 | 50 |
10,517 | In a number matrix as shown, the three numbers in each row are in arithmetic progression, and the three numbers in each column are also in arithmetic progression. Given that \( a_{22} = 2 \), find the sum of all 9 numbers in the matrix. | 18 | 71.875 |
10,518 | The recruits stood in a row one behind the other, all facing the same direction. Among them were three brothers: Peter, Nicholas, and Denis. Ahead of Peter, there were 50 people; ahead of Nicholas, there were 100; ahead of Denis, there were 170. When the command "About-face!" was given, everyone turned to face the oppo... | 211 | 58.59375 |
10,519 | In the geometric sequence $\{a_n\}$, if $a_2a_5 = -\frac{3}{4}$ and $a_2 + a_3 + a_4 + a_5 = \frac{5}{4}$, calculate the value of $\frac{1}{a_2} + \frac{1}{a_3} + \frac{1}{a_4} + \frac{1}{a_5}$. | -\frac{5}{3} | 12.5 |
10,520 | Patrícia wrote, in ascending order, the positive integers formed only by odd digits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, ... What was the 157th number she wrote?
A) 997
B) 999
C) 1111
D) 1113
E) 1115 | 1113 | 43.75 |
10,521 | How many positive integers, not exceeding 200, are multiples of 2 or 5 but not 6? | 87 | 3.90625 |
10,522 | Lei Lei bought some goats and sheep. If she had bought 2 more goats, the average price of each sheep would increase by 60 yuan. If she had bought 2 fewer goats, the average price of each sheep would decrease by 90 yuan. Lei Lei bought $\qquad$ sheep in total. | 10 | 7.03125 |
10,523 | Find the smallest constant $ C$ such that for all real $ x,y$
\[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\]
holds. | 4/3 | 3.90625 |
10,524 | In how many ways can 9 identical items be distributed into four boxes? | 220 | 90.625 |
10,525 | Given that the sum of the coefficients of the expansion of $(\frac{3}{x}-\sqrt{x})^n$ is $512$. Find:<br/>
$(1)$ The coefficient of the term containing $x^{3}$ in the expansion;<br/>
$(2)$ The constant term in the expansion of $(1+\frac{1}{x})(2x-1)^n$. | 17 | 95.3125 |
10,526 | What is the sum of all the integers from 100 to 2000? | 1996050 | 31.25 |
10,527 | For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"? | 2016 | 78.125 |
10,528 | Certain integers, when divided by $\frac{3}{5}, \frac{5}{7}, \frac{7}{9}, \frac{9}{11}$, result in a mixed number where the fractional part is $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}$, respectively. Find the smallest integer greater than 1 that satisfies these conditions. | 316 | 53.125 |
10,529 | In a certain country, the airline system is arranged in such a way that any city is connected by airlines with no more than three other cities, and from any city to any other, one can travel with no more than one stopover. What is the maximum number of cities that can be in this country? | 10 | 73.4375 |
10,530 | $A, B, C, D, E, F, G$ are seven people sitting around a circular table. If $d$ is the total number of ways that $B$ and $G$ must sit next to $C$, find the value of $d$. | 48 | 35.9375 |
10,531 | In \(\triangle ABC\), \(AB = 13\), \(BC = 14\), and \(CA = 15\). \(P\) is a point inside \(\triangle ABC\) such that \(\angle PAB = \angle PBC = \angle PCA\). Find \(\tan \angle PAB\). | \frac{168}{295} | 85.15625 |
10,532 | Comparing two rectangular parallelepiped bars, it was found that the length, width, and height of the second bar are each 1 cm greater than those of the first bar, and the volume and total surface area of the second bar are 18 cm³ and 30 cm² greater, respectively, than those of the first one. What is the total surface ... | 22 | 46.875 |
10,533 | $ (a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $ a_n \not \equal{} 0$ , $ a_na_{n \plus{} 3} \equal{} a_{n \plus{} 2}a_{n \plus{} 5}$ and $ a_1a_2 \plus{} a_3a_4 \plus{} a_5a_6 \equal{} 6$ . So $ a_1a_2 \plus{} a_3a_4 \plus{} \cdots \plus{}a_{41}a_{42} \equal{} ?$ | 42 | 61.71875 |
10,534 | Let $f(x)$ have a derivative, and satisfy $\lim_{\Delta x \to 0} \frac{f(1)-f(1-2\Delta x)}{2\Delta x}=-1$. Find the slope of the tangent line at point $(1,f(1))$ on the curve $y=f(x)$. | -1 | 46.875 |
10,535 | The famous skater Tony Hawk is riding a skateboard (segment $A B$) in a ramp, which is a semicircle with a diameter $P Q$. Point $M$ is the midpoint of the skateboard, and $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $P Q$. What values can the angle $\angle A C M$ take, if it is known tha... | 12 | 68.75 |
10,536 | Solve the fractional equation application problem.
On the eve of Children's Day, a certain shopping mall purchased a certain electric toy for $7200. Due to good sales, after a period of time, the mall purchased the same toy again for $14800. The quantity purchased the second time was twice the quantity purchased the f... | 100 | 53.125 |
10,537 | Mr. Wang drives from his home to location $A$. On the way there, he drives the first $\frac{1}{2}$ of the distance at a speed of 50 km/h and increases his speed by $20\%$ for the remaining distance. On the way back, he drives the first $\frac{1}{3}$ of the distance at a speed of 50 km/h and increases his speed by $32\%... | 330 | 64.0625 |
10,538 | I have 7 books, two of which are identical copies of a science book and another two identical copies of a math book, while the rest of the books are all different. In how many ways can I arrange them on a shelf, and additionally, how many of these arrangements can be made if I decide to highlight exactly two books (not... | 26460 | 11.71875 |
10,539 | To meet the shopping needs of customers during the "May Day" period, a fruit supermarket purchased cherries and cantaloupes from the fruit production base for $9160$ yuan, totaling $560$ kilograms. The purchase price of cherries is $35$ yuan per kilogram, and the purchase price of cantaloupes is $6 yuan per kilogram. ... | 35 | 5.46875 |
10,540 | Calculate $7.45 + 2.56$ as a decimal. | 10.01 | 89.84375 |
10,541 | Find the largest integer \( k \) such that for at least one natural number \( n > 1000 \), the number \( n! = 1 \cdot 2 \cdot \ldots \cdot n \) is divisible by \( 2^{n+k+2} \). | -3 | 11.71875 |
10,542 | A natural number undergoes the following operation: the rightmost digit of its decimal representation is discarded, and then the number obtained after discarding is added to twice the discarded digit. For example, $157 \mapsto 15 + 2 \times 7 = 29$, $5 \mapsto 0 + 2 \times 5 = 10$. A natural number is called ‘good’ if ... | 19 | 6.25 |
10,543 | Any six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible value such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$. | \frac{\sqrt{5}}{2} | 77.34375 |
10,544 | **p1.** The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus?**p2.** In ... | 23 | 20.3125 |
10,545 | In a bucket, there are $34$ red balls, $25$ green balls, $23$ yellow balls, $18$ blue balls, $14$ white balls, and $10$ black balls. Find the minimum number of balls that must be drawn from the bucket without replacement to guarantee that at least $20$ balls of a single color are drawn. | 100 | 48.4375 |
10,546 | Given two lines $l_{1}$: $(a+2)x+(a+3)y-5=0$ and $l_{2}$: $6x+(2a-1)y-5=0$ are parallel, then $a=$ . | -\dfrac{5}{2} | 25.78125 |
10,547 | Given Jane lists the whole numbers $1$ through $50$ once and Tom copies Jane's numbers, replacing each occurrence of the digit $3$ by the digit $2$, calculate how much larger Jane's sum is than Tom's sum. | 105 | 3.125 |
10,548 | In Lhota, there was an election for the mayor. Two candidates ran: Mr. Schopný and his wife, Dr. Schopná. The village had three polling stations. In the first and second stations, Dr. Schopná received more votes. The vote ratios were $7:5$ in the first station and $5:3$ in the second station. In the third polling stati... | 24 : 24 : 25 | 34.375 |
10,549 | Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet. | 375 | 82.8125 |
10,550 | The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 55 and 31 respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\). | 1705 | 42.96875 |
10,551 | A number is composed of 10 ones, 9 tenths (0.1), and 6 hundredths (0.01). This number is written as ____, and when rounded to one decimal place, it is approximately ____. | 11.0 | 28.90625 |
10,552 | Simplify first, then evaluate: $(1-\frac{m}{{m+3}})÷\frac{{{m^2}-9}}{{{m^2}+6m+9}}$, where $m=\sqrt{3}+3$. | \sqrt{3} | 86.71875 |
10,553 | Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 17x\]for all integers $x.$ Find the smallest possible value of $a.$ | 17 | 85.15625 |
10,554 | Given the function $f(x)=\cos^2x+\cos^2\left(x-\frac{\pi}{3}\right)-1$, where $x\in \mathbb{R}$,
$(1)$ Find the smallest positive period and the intervals of monotonic decrease for $f(x)$;
$(2)$ The function $f(x)$ is translated to the right by $\frac{\pi}{3}$ units to obtain the function $g(x)$. Find the expression ... | - \frac{\sqrt{3}}{4} | 63.28125 |
10,555 | At a school cafeteria, Sam wants to buy a lunch consisting of one main dish, one beverage, and one snack. The table below lists Sam's choices in the cafeteria. How many distinct possible lunches can he buy if he avoids pairing Fish and Chips with Soda due to dietary restrictions?
\begin{tabular}{ |c | c | c | }
\hline... | 14 | 71.09375 |
10,556 | Let the function $f(x) = (\sin x + \cos x)^2 - \sqrt{3}\cos 2x$.
(Ⅰ) Find the smallest positive period of $f(x)$;
(Ⅱ) Find the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ and the corresponding value of $x$ when the maximum value is attained. | \frac{5\pi}{12} | 69.53125 |
10,557 | Calculate the sum of the square of the binomial coefficients: $C_2^2+C_3^2+C_4^2+…+C_{11}^2$. | 220 | 25 |
10,558 | Given that 3 people are to be selected from 5 girls and 2 boys, if girl A is selected, determine the probability that at least one boy is selected. | \frac{3}{5} | 65.625 |
10,559 | Find the distance \( B_{1} H \) from point \( B_{1} \) to the line \( D_{1} B \), given \( B_{1}(5, 8, -3) \), \( D_{1}(-3, 10, -5) \), and \( B(3, 4, 1) \). | 2\sqrt{6} | 68.75 |
10,560 | Given \( f(x)=\frac{2x+3}{x-1} \), the graph of the function \( y=g(x) \) is symmetric with the graph of the function \( y=f^{-1}(x+1) \) with respect to the line \( y=x \). Find \( g(3) \). | \frac{7}{2} | 78.90625 |
10,561 | $JKLM$ is a square and $PQRS$ is a rectangle. If $JK$ is parallel to $PQ$, $JK = 8$ and $PS = 2$, then the total area of the shaded regions is: | 48 | 50 |
10,562 | Let the function $f(x)$ be defined on $\mathbb{R}$ and satisfy $f(2-x) = f(2+x)$ and $f(7-x) = f(7+x)$. Also, in the closed interval $[0, 7]$, only $f(1) = f(3) = 0$. Determine the number of roots of the equation $f(x) = 0$ in the closed interval $[-2005, 2005]$. | 802 | 50 |
10,563 |
Ali Baba and the 40 thieves decided to divide a treasure of 1987 gold coins in the following manner: the first thief divides the entire treasure into two parts, then the second thief divides one of the parts into two parts, and so on. After the 40th division, the first thief picks the largest part, the second thief pi... | 49 | 18.75 |
10,564 | Find all three-digit numbers that are equal to the sum of all their digits plus twice the square of the sum of their digits. List all possible numbers in ascending order without spaces and enter the resulting concatenated multi-digit number. | 171465666 | 42.1875 |
10,565 | Given the sets $A=\{x|x^{2}-px-2=0\}$ and $B=\{x|x^{2}+qx+r=0\}$, if $A\cup B=\{-2,1,5\}$ and $A\cap B=\{-2\}$, find the value of $p+q+r$. | -14 | 78.125 |
10,566 | There are 85 beads in total on a string, arranged in the pattern "three green, four red, one yellow, three green, four red, one yellow, ...". How many red beads are there? | 42 | 78.125 |
10,567 | A snowball with a temperature of $0^{\circ} \mathrm{C}$ is launched at a speed $v$ towards a wall. Upon impact, $k=0.02\%$ of the entire snowball melts. Determine what percentage of the snowball will melt if it is launched towards the wall at a speed of $\frac{v}{2}$? The specific heat of fusion of snow is $\lambda = 3... | 0.005 | 92.1875 |
10,568 | The polynomial
\[px^4 + qx^3 + rx^2 + sx + t = 0\]
has coefficients that are all integers, and roots $-3$, $4$, $6$, and $\frac{1}{2}$. If $t$ is a positive integer, find its smallest possible value. | 72 | 69.53125 |
10,569 | A certain type of beverage with a prize promotion has bottle caps printed with either "reward one bottle" or "thank you for your purchase". If a bottle is purchased and the cap is printed with "reward one bottle", it is considered a winning bottle, with a winning probability of $\frac{1}{6}$. Three students, A, B, and ... | \frac{25}{27} | 79.6875 |
10,570 | Given that the length of the major axis of the ellipse is 4, the left vertex is on the parabola \( y^2 = x - 1 \), and the left directrix is the y-axis, find the maximum value of the eccentricity of such an ellipse. | \frac{2}{3} | 21.09375 |
10,571 | Find the smallest prime \( p > 100 \) for which there exists an integer \( a > 1 \) such that \( p \) divides \( \frac{a^{89} - 1}{a - 1} \). | 179 | 78.90625 |
10,572 | Using only the digits $2,3$ and $9$ , how many six-digit numbers can be formed which are divisible by $6$ ? | 81 | 28.90625 |
10,573 | Given that $a_1, a_2, b_1, b_2, b_3$ are real numbers, and $-1, a_1, a_2, -4$ form an arithmetic sequence, $-4, b_1, b_2, b_3, -1$ form a geometric sequence, calculate the value of $\left(\frac{a_2 - a_1}{b_2}\right)$. | \frac{1}{2} | 79.6875 |
10,574 | When $x=1$, the value of the expression $px^3+qx-10$ is 2006; when $x=-1$, find the value of the expression $px^3+qx-10$. | -2026 | 97.65625 |
10,575 | In the rectangular coordinate system xOy, the parametric equation of curve C is $$\begin{cases} x=3\cos\theta \\ y=3\sin\theta \end{cases}$$ (θ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate e... | \frac { \sqrt {2}}{8} | 0 |
10,576 | Given $A=a^{2}-2ab+b^{2}$, $B=a^{2}+2ab+b^{2}$, where $a\neq b$. <br/>$(1)$ Determine the sign of $A+B$ and explain the reason; <br/>$(2)$ If $ab$ are reciprocals of each other, find the value of $A-B$. | -4 | 99.21875 |
10,577 | Calculate: $$\frac {\cos 2^\circ}{\sin 47^\circ} + \frac {\cos 88^\circ}{\sin 133^\circ}$$. | \sqrt{2} | 78.90625 |
10,578 | What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ? | 18 | 97.65625 |
10,579 | Given 8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder, and the recorder is seated between the leader and vice leader, determine the number of different seating arrangements possible, considering that arrangements that can be obtained by rotation are id... | 240 | 71.09375 |
10,580 | Given that the function f(x) defined on the set of real numbers ℝ satisfies f(x+1) = 1/2 + √(f(x) - f^2(x)), find the maximum value of f(0) + f(2017). | 1+\frac{\sqrt{2}}{2} | 32.8125 |
10,581 | Given the sets
$$
\begin{array}{c}
M=\{x, xy, \lg (xy)\} \\
N=\{0, |x|, y\},
\end{array}
$$
and that \( M = N \), determine the value of
$$
\left(x+\frac{1}{y}\right)+\left(x^2+\frac{1}{y^2}\right)+\left(x^3+\frac{1}{y^3}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right).
$$ | -2 | 17.1875 |
10,582 | Let the function \( f(x) \) satisfy the following conditions:
(i) If \( x > y \), and \( f(x) + x \geq w \geq f(y) + y \), then there exists a real number \( z \in [y, x] \), such that \( f(z) = w - z \);
(ii) The equation \( f(x) = 0 \) has at least one solution, and among the solutions of this equation, there exists ... | 2004 | 94.53125 |
10,583 | A cyclist is riding on a track at a constant speed. It is known that at 11:22, he covered a distance that is 1.4 times greater than the distance he covered at 11:08. When did he start? | 10:33 | 23.4375 |
10,584 | In an opaque bag, there are four small balls labeled with the Chinese characters "阳", "过", "阳", and "康" respectively. Apart from the characters, the balls are indistinguishable. Before each draw, the balls are thoroughly mixed.<br/>$(1)$ If one ball is randomly drawn from the bag, the probability that the character on ... | \frac{1}{3} | 78.125 |
10,585 | Given the function $f(x)=( \frac {1}{3})^{x}$, the sum of the first $n$ terms of the geometric sequence $\{a\_n\}$ is $f(n)-c$, and the first term of the sequence $\{b\_n\}_{b\_n > 0}$ is $c$. The sum of the first $n$ terms, $S\_n$, satisfies $S\_n-S_{n-1}= \sqrt {S\_n}+ \sqrt {S_{n-1}}(n\geqslant 2)$.
(I) Find the gen... | 252 | 35.9375 |
10,586 | Find the maximum number of Permutation of set { $1,2,3,...,2014$ } such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation $b$ comes exactly after $a$ | 1007 | 1.5625 |
10,587 | Given that $\tan \alpha = -\frac{1}{3}$ and $\cos \beta = \frac{\sqrt{5}}{5}$, with $\alpha, \beta \in (0, \pi)$, find:
1. The value of $\tan(\alpha + \beta)$;
2. The maximum value of the function $f(x) = \sqrt{2} \sin(x - \alpha) + \cos(x + \beta)$. | \sqrt{5} | 19.53125 |
10,588 | In the expansion of \((-xy + 2x + 3y - 6)^6\), what is the coefficient of \(x^4 y^3\)? (Answer with a specific number) | -21600 | 3.125 |
10,589 | Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\{a_n^2\}$ is $S_n$. A geometric sequence $\{b_n\}$ has a common ratio $q$ less than 1 and consists of rational sine values, with the first term $b_1 = d^2$, and the s... | \frac{1}{2} | 24.21875 |
10,590 | The height \(CH\), dropped from the vertex of the right angle of the triangle \(ABC\), bisects the angle bisector \(BL\) of this triangle. Find the angle \(BAC\). | 30 | 14.84375 |
10,591 | A bus with programmers departed from Novosibirsk to Pavlodar. After traveling 70 km, another car with Pavel Viktorovich left Novosibirsk on the same route and caught up with the bus in Karasuk. After that, Pavel traveled another 40 km, while the bus traveled only 20 km in the same time. Find the distance from Novosibir... | 140 | 80.46875 |
10,592 | On the sides \( AB, BC \), and \( AC \) of triangle \( ABC \), points \( M, N, \) and \( K \) are taken respectively so that \( AM:MB = 2:3 \), \( AK:KC = 2:1 \), and \( BN:NC = 1:2 \). In what ratio does the line \( MK \) divide the segment \( AN \)? | 6:7 | 42.96875 |
10,593 | The reform pilot of basic discipline enrollment, also known as the Strong Foundation Plan, is an enrollment reform project initiated by the Ministry of Education, mainly to select and cultivate students who are willing to serve the country's major strategic needs and have excellent comprehensive quality or outstanding ... | \frac{5}{3} | 11.71875 |
10,594 | There are two types of camels: dromedary camels with one hump on their back and Bactrian camels with two humps. Dromedary camels are taller, with longer limbs, and can walk and run in the desert; Bactrian camels have shorter and thicker limbs, suitable for walking in deserts and snowy areas. In a group of camels that h... | 15 | 87.5 |
10,595 | For a natural number $n \geq 1$, it satisfies: $2002 \times n$ is a perfect cube, and $n \div 2002$ is a perfect square. The smallest such $n$ is | 2002^5 | 0 |
10,596 | In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $b^{2}=ac$ and $a^{2}-c^{2}=ac-bc$.
1. Find the measure of $\angle A$.
2. Let $f(x)=\cos (wx-\frac{A}{2})+\sin (wx) (w > 0)$ and the smallest positive period of $f(x)$ is $\pi$. Find the maximum value of $f(x)$ on ... | \sqrt{3} | 82.8125 |
10,597 | What is the smallest whole number larger than the perimeter of any triangle with a side of length $7$ and a side of length $24$? | 62 | 20.3125 |
10,598 | Olga Ivanovna, the homeroom teacher of class 5B, is staging a "Mathematical Ballet". She wants to arrange the boys and girls so that every girl has exactly 2 boys at a distance of 5 meters from her. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating? | 20 | 0 |
10,599 | \(a_n\) is the last digit of \(1 + 2 + \ldots + n\). Find \(a_1 + a_2 + \ldots + a_{1992}\). | 6984 | 39.84375 |
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