Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
15,800 | Calculate the integral \(\int_{0}^{\pi / 2} \frac{\sin^{3} x}{2 + \cos x} \, dx\). | 3 \ln \left(\frac{2}{3}\right) + \frac{3}{2} | 0.78125 |
15,801 | Given a positive number \(r\) such that the set \(T=\left\{(x, y) \mid x, y \in \mathbf{R}\right.\) and \(\left.x^{2}+(y-7)^{2} \leqslant r^{2}\right\}\) is a subset of the set \(S=\{(x, y) \mid x, y \in \mathbf{R}\right.\) and for any \(\theta \in \mathbf{R}\), \(\cos 2\theta + x \cos \theta + y \geqslant 0\},\) deter... | 4 \sqrt{2} | 26.5625 |
15,802 | In the acute triangle \( \triangle ABC \), the sides \( a, b, c \) are opposite to the angles \( \angle A, \angle B, \angle C \) respectively, and \( a, b, c \) form an arithmetic sequence. Also, \( \sin (A - C) = \frac{\sqrt{3}}{2} \). Find \( \sin (A + C) \). | \frac{\sqrt{39}}{8} | 2.34375 |
15,803 | Let \( x_{1}, x_{2}, x_{3}, x_{4} \) be non-negative real numbers satisfying the equation:
\[
x_{1} + x_{2} + x_{3} + x_{4} = 1
\]
Find the maximum value of \( \sum_{1 \leq i < j \leq 4}(x_{i} + x_{j}) \sqrt{x_{i} x_{j}} \), and determine the values of \( x_{1}, x_{2}, x_{3}, x_{4} \) that achieve this maximum value. | 3/4 | 77.34375 |
15,804 | Let the three sides of a triangle be integers \( l \), \( m \), and \( n \) with \( l > m > n \). It is known that \( \left\{\frac{3^l}{10^4}\right\} = \left\{\frac{3^m}{10^4}\right\} = \left\{\frac{3^n}{10^4}\right\} \), where \( \{x\} \) denotes the fractional part of \( x \). Determine the minimum value of the perim... | 3003 | 1.5625 |
15,805 | For each pair of real numbers \((x, y)\) with \(0 \leq x \leq y \leq 1\), consider the set
\[ A = \{ x y, x y - x - y + 1, x + y - 2 x y \}. \]
Let the maximum value of the elements in set \(A\) be \(M(x, y)\). Find the minimum value of \(M(x, y)\). | 4/9 | 53.90625 |
15,806 | 2001 coins, each valued at 1, 2, or 3, are arranged in a row. The coins are placed such that:
- Between any two coins of value 1, there is at least one other coin.
- Between any two coins of value 2, there are at least two other coins.
- Between any two coins of value 3, there are at least three other coins.
What is t... | 501 | 25 |
15,807 | The steamboat "Rarity" travels for three hours at a constant speed after leaving the city, then drifts with the current for an hour, then travels for three hours at the same speed, and so on. If the steamboat starts its journey in city A and goes to city B, it takes it 10 hours. If it starts in city B and goes to city ... | 60 | 13.28125 |
15,808 | Given that \( p \) is a prime number and \( r \) is the remainder when \( p \) is divided by 210, if \( r \) is a composite number that can be expressed as the sum of two perfect squares, find \( r \). | 169 | 4.6875 |
15,809 | How many positive integers less than 2019 are divisible by either 18 or 21, but not both? | 176 | 91.40625 |
15,810 | A collector has \( N \) precious stones. If he takes away the three heaviest stones, then the total weight of the stones decreases by \( 35\% \). From the remaining stones, if he takes away the three lightest stones, the total weight further decreases by \( \frac{5}{13} \). Find \( N \). | 10 | 21.09375 |
15,811 | Two people, A and B, play a "guess the number" game using a fair six-sided die (the faces of the die are numbered $1, 2, \cdots, 6$). Each person independently thinks of a number on the die, denoted as $a$ and $b$. If $|a - b| \leqslant 1$, they are said to be "in sync." What is the probability that A and B are in sync... | 4/9 | 100 |
15,812 | There are exactly 120 ways to color five cells in a $5 \times 5$ grid such that exactly one cell in each row and each column is colored.
There are exactly 96 ways to color five cells in a $5 \times 5$ grid without the corner cell, such that exactly one cell in each row and each column is colored.
How many ways are th... | 78 | 60.15625 |
15,813 | Find the number of integers from 1 to 1000 inclusive that give the same remainder when divided by 11 and by 12. | 87 | 1.5625 |
15,814 | Given a triangle \( ACE \) with a point \( B \) on segment \( AC \) and a point \( D \) on segment \( CE \) such that \( BD \) is parallel to \( AE \). A point \( Y \) is chosen on segment \( AE \), and segment \( CY \) is drawn, intersecting \( BD \) at point \( X \). If \( CX = 5 \) and \( XY = 3 \), what is the rati... | 39/25 | 83.59375 |
15,815 | The function \( f(x) = \max \left\{\sin x, \cos x, \frac{\sin x + \cos x}{\sqrt{2}}\right\} \) (for \( x \in \mathbb{R} \)) has a maximum value and a minimum value. Find the sum of these maximum and minimum values. | 1 - \frac{\sqrt{2}}{2} | 7.8125 |
15,816 | How many ways are there to list the numbers 1 to 10 in some order such that every number is either greater or smaller than all the numbers before it? | 512 | 6.25 |
15,817 | During breaks, schoolchildren played table tennis. Any two schoolchildren played no more than one game against each other. At the end of the week, it turned out that Petya played half, Kolya - a third, and Vasya - one fifth of the total number of games played during the week. What could be the total number of games pla... | 30 | 39.84375 |
15,818 | Given that the interior angles \(A, B, C\) of triangle \(\triangle ABC\) are opposite to the sides \(a, b, c\) respectively, and that \(A - C = \frac{\pi}{2}\), and \(a, b, c\) form an arithmetic sequence, find the value of \(\cos B\). | \frac{3}{4} | 10.9375 |
15,819 | Let \( A = \{ x \mid 5x - a \leqslant 0 \} \) and \( B = \{ x \mid 6x - b > 0 \} \), where \( a, b \in \mathbb{N}_+ \). If \( A \cap B \cap \mathbb{N} = \{ 2, 3, 4 \} \), find the number of integer pairs \((a, b)\). | 30 | 46.09375 |
15,820 | Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{8} x \, dx
$$ | \frac{35\pi}{8} | 85.9375 |
15,821 | A rectangular piece of paper with a length of 20 cm and a width of 12 cm is folded along its diagonal (refer to the diagram). What is the perimeter of the shaded region formed? | 64 | 41.40625 |
15,822 | One day, Xiao Ming took 100 yuan to go shopping. In the first store, he bought several items of product A. In the second store, he bought several items of product B. In the third store, he bought several items of product C. In the fourth store, he bought several items of product D. In the fifth store, he bought several... | 28 | 0 |
15,823 | How many natural numbers with up to six digits contain the digit 1? | 468559 | 47.65625 |
15,824 | A triangle has sides of lengths 20 and 19. If the triangle is not acute, how many possible integer lengths can the third side have? | 16 | 27.34375 |
15,825 | $S$ is a subset of the set $\{1, 2, \cdots, 2023\}$, such that the sum of the squares of any two elements is not a multiple of 9. What is the maximum value of $|S|$? (Here, $|S|$ represents the number of elements in $S$.) | 1350 | 3.90625 |
15,826 | Inside the cube \( ABCD A_1 B_1 C_1 D_1 \) is located the center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( A A_1 D_1 D \) in a circle with a radius of 1, the face \( A_1 B_1 C_1 D_1 \) in a circle with a radius of 1, and the face \( C D D_1 C_1 \) in a circle with a radius of 3. Find th... | 17 | 41.40625 |
15,827 | How can you weigh 1 kg of grain on a balance scale using two weights, one weighing 300 g and the other 650 g? | 1000 | 29.6875 |
15,828 | All vertices of a regular tetrahedron \( A B C D \) are located on one side of the plane \( \alpha \). It turns out that the projections of the vertices of the tetrahedron onto the plane \( \alpha \) are the vertices of a certain square. Find the value of \(A B^{2}\), given that the distances from points \( A \) and \(... | 32 | 6.25 |
15,829 | Let \( p(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants. Given \( p(1) = 1993 \), \( p(2) = 3986 \), \( p(3) = 5979 \), find \( \frac{1}{4} [p(11) + p(-7)] \). | 5233 | 47.65625 |
15,830 | A necklace consists of 80 beads of red, blue, and green colors. It is known that in any segment of the necklace between two blue beads, there is at least one red bead, and in any segment of the necklace between two red beads, there is at least one green bead. What is the minimum number of green beads in this necklace? ... | 27 | 37.5 |
15,831 | Petya cut an 8x8 square along the borders of the cells into parts of equal perimeter. It turned out that not all parts are equal. What is the maximum possible number of parts he could get? | 21 | 0 |
15,832 | Given that $\angle BAC = 90^{\circ}$ and the quadrilateral $ADEF$ is a square with side length 1, find the maximum value of $\frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$. | 2 + \frac{\sqrt{2}}{2} | 7.03125 |
15,833 | Let \( x \) be a positive integer, and write \( a = \left\lfloor \log_{10} x \right\rfloor \) and \( b = \left\lfloor \log_{10} \frac{100}{x} \right\rfloor \). Here \( \lfloor c \rfloor \) denotes the greatest integer less than or equal to \( c \). Find the largest possible value of \( 2a^2 - 3b^2 \). | 24 | 0.78125 |
15,834 |
In a box, there are 3 red, 4 gold, and 5 silver stars. Stars are randomly drawn one by one from the box and placed on a Christmas tree. What is the probability that a red star is placed on the top of the tree, no more red stars are on the tree, and there are exactly 3 gold stars on the tree, if a total of 6 stars are ... | 5/231 | 4.6875 |
15,835 | In the year 2009, there is a property that rearranging the digits of the number 2009 cannot yield a smaller four-digit number (numbers do not start with zero). In what subsequent year does this property first repeat again? | 2022 | 35.15625 |
15,836 | Given \( x, y, z \in (0, +\infty) \) and \(\frac{x^2}{1+x^2} + \frac{y^2}{1+y^2} + \frac{z^2}{1+z^2} = 2 \), find the maximum value of \(\frac{x}{1+x^2} + \frac{y}{1+y^2} + \frac{z}{1+z^2}\). | \sqrt{2} | 54.6875 |
15,837 | On the sides of triangle \(ABC\), points were marked: 10 on side \(AB\), 11 on side \(BC\), and 12 on side \(AC\). None of the vertices of the triangle were marked. How many triangles with vertices at the marked points exist? | 4951 | 2.34375 |
15,838 | The fare in Moscow with the "Troika" card in 2016 is 32 rubles for one trip on the metro and 31 rubles for one trip on ground transportation. What is the minimum total number of trips that can be made at these rates, spending exactly 5000 rubles? | 157 | 27.34375 |
15,839 | The value of the expression \(10 - 10.5 \div [5.2 \times 14.6 - (9.2 \times 5.2 + 5.4 \times 3.7 - 4.6 \times 1.5)]\) is | 9.3 | 51.5625 |
15,840 | In the Cartesian coordinate system \( xOy \), the area of the region corresponding to the set of points \( K = \{(x, y) \mid (|x| + |3y| - 6)(|3x| + |y| - 6) \leq 0 \} \) is ________. | 24 | 13.28125 |
15,841 | Inside triangle \(ABC\), a point \(O\) is chosen such that \(\angle ABO = \angle CAO\), \(\angle BAO = \angle BCO\), and \(\angle BOC = 90^{\circ}\). Find the ratio \(AC : OC\). | \sqrt{2} | 14.84375 |
15,842 | On the board, the number 27 is written. Every minute, the number is erased from the board and replaced with the product of its digits increased by 12. For example, after one minute, the number on the board will be $2 \cdot 7 + 12 = 26$. What number will be on the board after an hour? | 14 | 90.625 |
15,843 | Tanya wrote a certain two-digit number on a piece of paper; to Sveta, who was sitting opposite her, the written number appeared different and was 75 less. What number did Tanya write? | 91 | 29.6875 |
15,844 | What is the maximum number of checkers that can be placed on a $6 \times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) are on the same line (regardless of the angle of inclination)? | 12 | 40.625 |
15,845 | Let \( z \) be a complex number with a modulus of 1. Then the maximum value of \(\left|\frac{z+\mathrm{i}}{z+2}\right|\) is \(\ \ \ \ \ \ \). | \frac{2\sqrt{5}}{3} | 0 |
15,846 | A car left the city for the village, and simultaneously, a cyclist left the village for the city. When the car and the cyclist met, the car immediately turned around and went back to the city. As a result, the cyclist arrived in the city 35 minutes later than the car. How many minutes did the cyclist spend on the entir... | 55 | 15.625 |
15,847 | A polygon is said to be friendly if it is regular and it also has angles that, when measured in degrees, are either integers or half-integers (i.e., have a decimal part of exactly 0.5). How many different friendly polygons are there? | 28 | 34.375 |
15,848 | In how many ways can the number 1024 be factored into three natural factors such that the first factor is a multiple of the second, and the second is a multiple of the third? | 14 | 57.8125 |
15,849 | Two people are flipping a coin: one flipped it 10 times, and the other flipped it 11 times. Find the probability that the second person got heads more times than the first person. | \frac{1}{2} | 61.71875 |
15,850 | Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}$ and $F_{2}$ respectively, draw a line $l$ through the right focus that intersects the ellipse at points $P$ and $Q$. Find the maximum area of the inscribed circle of triangle $F_{1} PQ$. | \frac{9 \pi}{16} | 14.0625 |
15,851 | There are two positive integers \( A \) and \( B \). The sum of the digits of \( A \) is 19, and the sum of the digits of \( B \) is 20. When the two numbers are added together, there are two carries. What is the sum of the digits of \( (A+B) \)? | 21 | 89.0625 |
15,852 | The common ratio of the geometric sequence \( a + \log_{2} 3, a + \log_{4} 3, a + \log_{8} 3 \) is? | \frac{1}{3} | 35.9375 |
15,853 | The school organized a picnic with several attendees. The school prepared many empty plates. Each attendee who arrives will count the empty plates and then take one plate for food (no one can take more than one plate). The first attendee will count all the empty plates, the second will count one fewer, and so on. The l... | 1006 | 65.625 |
15,854 | Regular decagon (10-sided polygon) \(A B C D E F G H I J\) has an area of 2017 square units. Determine the area (in square units) of the rectangle \(C D H I\). | 806.8 | 50 |
15,855 | Out of sixteen Easter eggs, three are red. Ten eggs were placed in a larger box and six in a smaller box at random. What is the probability that both boxes contain at least one red egg? | 3/4 | 21.875 |
15,856 | Given the point \( P \) lies in the plane of the right triangle \( \triangle ABC \) with \( \angle BAC = 90^\circ \), and \( \angle CAP \) is an acute angle. Also given are the conditions:
\[ |\overrightarrow{AP}| = 2, \quad \overrightarrow{AP} \cdot \overrightarrow{AC} = 2, \quad \overrightarrow{AP} \cdot \overrightar... | \frac{\sqrt{2}}{2} | 3.125 |
15,857 | In the plane Cartesian coordinate system \(xOy\), the set of points
$$
\begin{aligned}
K= & \{(x, y) \mid(|x|+|3 y|-6) \cdot \\
& (|3 x|+|y|-6) \leqslant 0\}
\end{aligned}
$$
corresponds to an area in the plane with the measurement of ______. | 24 | 23.4375 |
15,858 | Each of two teams, Team A and Team B, sends 7 players in a predetermined order to participate in a Go contest. The players from both teams compete sequentially starting with Player 1 from each team. The loser of each match is eliminated, and the winner continues to compete with the next player from the opposing team. T... | 3432 | 46.875 |
15,859 | At around 8 o'clock in the morning, two cars left the fertilizer plant one after another, heading toward Happy Village. Both cars travel at a speed of 60 kilometers per hour. At 8:32, the distance the first car had traveled from the fertilizer plant was three times the distance traveled by the second car. At 8:39, the ... | 8:11 | 10.9375 |
15,860 | How many natural five-digit numbers have the product of their digits equal to 2000? | 30 | 63.28125 |
15,861 | Given that \(\frac{x+y}{x-y}+\frac{x-y}{x+y}=3\). Find the value of the expression \(\frac{x^{2}+y^{2}}{x^{2}-y^{2}}+\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\). | 13/6 | 31.25 |
15,862 | Cubes. As is known, the whole space can be filled with equal cubes. At each vertex, eight cubes will converge. Therefore, by appropriately truncating the vertices of the cubes and joining the adjacent truncated parts into a single body, it is possible to fill the space with regular octahedra and the remaining bodies fr... | \frac{1}{6} | 16.40625 |
15,863 | Pi Pi Lu wrote a 2020-digit number: \( 5368 \cdots \cdots \). If any four-digit number taken randomly from this multi-digit number is divisible by 11, what is the sum of the digits of this multi-digit number? | 11110 | 70.3125 |
15,864 | A typesetter scattered part of a set - a set of a five-digit number that is a perfect square, written with the digits $1, 2, 5, 5,$ and $6$. Find all such five-digit numbers. | 15625 | 50 |
15,865 | A child gave Carlson 111 candies. They ate some of them right away, 45% of the remaining candies went to Carlson for lunch, and a third of the candies left after lunch were found by Freken Bok during cleaning. How many candies did she find? | 11 | 17.96875 |
15,866 |
Let \( N \) be the total number of students in the school before the New Year, among which \( M \) are boys, making up \( k \) percent of the total. This means \( M = \frac{k}{100} N \), or \( 100M = kN \).
After the New Year, the number of boys became \( M+1 \), and the total number of students became \( N+3 \). If ... | 197 | 30.46875 |
15,867 | Calculate \( \frac{2}{1} \times \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} \times \frac{6}{5} \times \frac{6}{7} \times \frac{8}{7} \). Express the answer in decimal form, accurate to two decimal places. | 1.67 | 49.21875 |
15,868 |
Given $\boldsymbol{a} = (\cos \alpha, \sin \alpha)$ and $\boldsymbol{b} = (\cos \beta, \sin \beta)$, the relationship between $\boldsymbol{a}$ and $\boldsymbol{b}$ is given by $|k \boldsymbol{a} + \boldsymbol{b}| - \sqrt{3}|\boldsymbol{a} - k \boldsymbol{b}|$, where $k > 0$. Find the minimum value of $\boldsymbol{a} \... | \frac{1}{2} | 46.09375 |
15,869 | On a sheet of graph paper, two rectangles are outlined. The first rectangle has a vertical side shorter than the horizontal side, and for the second rectangle, the opposite is true. Find the maximum possible area of their intersection if the first rectangle contains 2015 cells and the second one contains 2016 cells. | 1302 | 6.25 |
15,870 | Inside a right angle with vertex \(O\), there is a triangle \(OAB\) with a right angle at \(A\). The height of the triangle \(OAB\), dropped to the hypotenuse, is extended past point \(A\) to intersect with the side of the angle at point \(M\). The distances from points \(M\) and \(B\) to the other side of the angle ar... | \sqrt{2} | 3.90625 |
15,871 | Calculate: \( 4\left(\sin ^{3} \frac{49 \pi}{48} \cos \frac{49 \pi}{16} + \cos ^{3} \frac{49 \pi}{48} \sin \frac{49 \pi}{16}\right) \cos \frac{49 \pi}{12} \). | 0.75 | 0 |
15,872 | A box contains one hundred multicolored balls: 28 red, 20 green, 13 yellow, 19 blue, 11 white, and 9 black. What is the minimum number of balls that must be drawn from the box, without looking, to ensure that at least 15 balls of one color are among them? | 76 | 79.6875 |
15,873 | Find the smallest natural number that is greater than the sum of its digits by 1755. | 1770 | 83.59375 |
15,874 | Find the measure of angle \( B \widehat{A} D \), given that \( D \widehat{A C}=39^{\circ} \), \( A B = A C \), and \( A D = B D \). | 47 | 8.59375 |
15,875 |
Let squares of one kind have a side of \(a\) units, another kind have a side of \(b\) units, and the original square have a side of \(c\) units. Then the area of the original square is given by \(c^{2}=n a^{2}+n b^{2}\).
Numbers satisfying this equation can be obtained by multiplying the equality \(5^{2}=4^{2}+3^{2}\... | 15 | 34.375 |
15,876 | Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 units in length. These lines became the wall... | 21 | 53.90625 |
15,877 | In a taxi, one passenger can sit in the front and three in the back. In how many ways can the four passengers be seated if one of them wants to sit by the window? | 12 | 16.40625 |
15,878 | What is the largest number of white and black chips that can be placed on a chessboard so that on each horizontal and each vertical, the number of white chips is exactly twice the number of black chips? | 48 | 48.4375 |
15,879 | Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length $1$, let $P$ be a moving point on the space diagonal $B C_{1}$ and $Q$ be a moving point on the base $A B C D$. Find the minimum value of $D_{1} P + P Q$. | 1 + \frac{\sqrt{2}}{2} | 0 |
15,880 | Find the smallest two-digit number \( N \) such that the sum of digits of \( 10^N - N \) is divisible by 170. | 20 | 99.21875 |
15,881 | An odd six-digit number is called "just cool" if it consists of digits that are prime numbers, and no two identical digits are adjacent. How many "just cool" numbers exist? | 729 | 51.5625 |
15,882 | Given that \( I \) is the incenter of \( \triangle ABC \) and \( 5 \overrightarrow{IA} = 4(\overrightarrow{BI} + \overrightarrow{CI}) \). Let \( R \) and \( r \) be the radii of the circumcircle and the incircle of \( \triangle ABC \) respectively. If \( r = 15 \), then find \( R \). | 32 | 3.90625 |
15,883 | We call a number antitriangular if it can be expressed in the form \(\frac{2}{n(n+1)}\) for some natural number \(n\). For how many numbers \(k\) (where \(1000 \leq k \leq 2000\)) can the number 1 be expressed as the sum of \(k\) antitriangular numbers (not necessarily distinct)? | 1001 | 21.875 |
15,884 | Construct spheres that are tangent to 4 given spheres. If we accept the point (a sphere with zero radius) and the plane (a sphere with infinite radius) as special cases, how many such generalized spatial Apollonian problems exist? | 15 | 28.90625 |
15,885 | One mole of an ideal monatomic gas is first heated isobarically, performing 10 Joules of work. Then it is heated isothermally, receiving the same amount of heat as in the first case. How much work does the gas perform (in Joules) in the second case? | 25 | 56.25 |
15,886 | For how many integers \( n \) is \(\frac{2n^3 - 12n^2 - 2n + 12}{n^2 + 5n - 6}\) equal to an integer? | 32 | 4.6875 |
15,887 | For the numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\), the last two digits are discarded. How many of the first terms in the resulting sequence form an arithmetic progression? | 10 | 28.90625 |
15,888 | Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 W, and for each graphics card, it is 125 W... | 165 | 19.53125 |
15,889 | Points \( A, B, C \), and \( D \) are located on a line such that \( AB = BC = CD \). Segments \( AB \), \( BC \), and \( CD \) serve as diameters of circles. From point \( A \), a tangent line \( l \) is drawn to the circle with diameter \( CD \). Find the ratio of the chords cut on line \( l \) by the circles with di... | \sqrt{6}: 2 | 0 |
15,890 | In rectangle \(ABCD\), \(AB = 20 \, \text{cm}\) and \(BC = 10 \, \text{cm}\). Points \(M\) and \(N\) are taken on \(AC\) and \(AB\), respectively, such that the value of \(BM + MN\) is minimized. Find this minimum value. | 16 | 2.34375 |
15,891 | If \(\sqrt{9-8 \sin 50^{\circ}}=a+b \csc 50^{\circ}\) where \(a, b\) are integers, find \(ab\). | -3 | 13.28125 |
15,892 | Given a real number \(a\), and for any \(k \in [-1, 1]\), when \(x \in (0, 6]\), the inequality \(6 \ln x + x^2 - 8x + a \leq kx\) always holds. Determine the maximum value of \(a\). | 6 - 6 \ln 6 | 11.71875 |
15,893 | In a singing contest, a Rooster, a Crow, and a Cuckoo were contestants. Each jury member voted for one of the three contestants. The Woodpecker tallied that there were 59 judges, and that the sum of votes for the Rooster and the Crow was 15, the sum of votes for the Crow and the Cuckoo was 18, and the sum of votes for ... | 13 | 1.5625 |
15,894 | Given that four integers \( a, b, c, d \) are all even numbers, and \( 0 < a < b < c < d \), with \( d - a = 90 \). If \( a, b, c \) form an arithmetic sequence and \( b, c, d \) form a geometric sequence, then find the value of \( a + b + c + d \). | 194 | 26.5625 |
15,895 | Let \( S = \{1, 2, \cdots, 2009\} \). \( A \) is a 3-element subset of \( S \) such that all elements in \( A \) form an arithmetic sequence. How many such 3-element subsets \( A \) are there? | 1008016 | 85.15625 |
15,896 | Two players, \(A\) and \(B\), play rock-paper-scissors continuously until player \(A\) wins 2 consecutive games. Suppose each player is equally likely to use each hand sign in every game. What is the expected number of games they will play? | 12 | 58.59375 |
15,897 | Given \(0<\theta<\pi\), a complex number \(z_{1}=1-\cos \theta+i \sin \theta\) and \(z_{2}=a^{2}+a i\), where \(a \in \mathbb{R}\), it is known that \(z_{1} z_{2}\) is a pure imaginary number, and \(\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}\). Determine the value of \(\theta\) when \(\bar{a}\) is a negative real number... | \frac{\pi}{2} | 32.03125 |
15,898 | Team A and Team B each have 7 players who compete in a predetermined order in a Go competition. Initially, Player 1 from each team competes. The loser is eliminated, and the winner competes next against the loser's team Player 2, and so on, until all players from one team are eliminated. The remaining team wins. How ma... | 3432 | 63.28125 |
15,899 | By solving the inequality \(\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}\), find the sum of its integer solutions within the interval \([-25, 25]\). | 310 | 0 |
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