Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
16,000 | Determine the value of the following expression:
$$
\left\lfloor\frac{11}{2010}\right\rfloor+\left\lfloor\frac{11 \times 2}{2010}\right\rfloor+\left\lfloor\frac{11 \times 3}{2010}\right\rfloor+\\left\lfloor\frac{11 \times 4}{2010}\right\rfloor+\cdots+\left\lfloor\frac{11 \times 2009}{2010}\right\rfloor,
$$
where \(\lfl... | 10045 | 86.71875 |
16,001 | Square \(ABCD\) has side length 2, and \(X\) is a point outside the square such that \(AX = XB = \sqrt{2}\). What is the length of the longest diagonal of pentagon \(AXB\)?
| \sqrt{10} | 14.84375 |
16,002 | In the expression \((x+y+z)^{2030}+(x-y-z)^{2030}\), the parentheses were expanded, and like terms were collected. How many monomials of the form \(x^{a} y^{b} z^{c}\) have a nonzero coefficient? | 1032256 | 41.40625 |
16,003 | The brakes of a car allow it to stay stationary on an inclined asphalt surface with a base angle not exceeding $30^{\circ}$. Determine the minimum braking distance of this car when traveling at a speed of $30 \, \text{m/s}$ on a flat horizontal road with the same surface. The acceleration due to gravity is $g=10 \, \te... | 78 | 6.25 |
16,004 | 27 identical dice were glued together to form a $3 \times 3 \times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube? | 189 | 17.96875 |
16,005 | Find the smallest positive integer $n$ that satisfies the following conditions: For $n$, there exists a positive integer $k$ such that $\frac{8}{15} < \frac{n}{n+k} < \frac{7}{13}$. | 15 | 0.78125 |
16,006 | Two circles are externally tangent to each other at point \( A \), and both are tangent to a third circle at points \( B \) and \( C \). The extension of chord \( AB \) of the first circle intersects the second circle at point \( D \), and the extension of chord \( AC \) intersects the first circle at point \( E \). Th... | 13 | 1.5625 |
16,007 | A number of trucks with the same capacity were requested to transport cargo from one place to another. Due to road issues, each truck had to carry 0.5 tons less than planned, which required 4 additional trucks. The mass of the transported cargo was at least 55 tons but did not exceed 64 tons. How many tons of cargo wer... | 2.5 | 35.9375 |
16,008 | Someone forms an integer by writing the integers from 1 to 82 in ascending order, i.e. 1234567891011 ...808182. Find the sum of the digits of this integer. | 667 | 34.375 |
16,009 | Given a polygon drawn on graph paper with a perimeter of 2014 units, and whose sides follow the grid lines, what is the maximum possible area of this polygon? | 253512 | 16.40625 |
16,010 | Into each row of a \( 9 \times 9 \) grid, Nigel writes the digits \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \) in order, starting at one of the digits and returning to 1 after 9: for example, one row might contain \( 7, 8, 9, 1, 2, 3, 4, 5, 6 \). The grid is gorgeous if each nine-digit number read along a row or column or along the... | 9^8 | 0 |
16,011 | On the hypotenuse \( AB \) of a right triangle \( ABC \), square \( ABDE \) is constructed externally with \( AC=2 \) and \( BC=5 \). In what ratio does the angle bisector of angle \( C \) divide side \( DE \)? | 2 : 5 | 33.59375 |
16,012 | Find the number of the form $7x36y5$ that is divisible by 1375. | 713625 | 35.9375 |
16,013 | Given a tetrahedron \(ABCD\). Points \(M\), \(N\), and \(K\) lie on edges \(AD\), \(BC\), and \(DC\) respectively, such that \(AM:MD = 1:3\), \(BN:NC = 1:1\), and \(CK:KD = 1:2\). Construct the section of the tetrahedron with the plane \(MNK\). In what ratio does this plane divide the edge \(AB\)? | 2/3 | 0.78125 |
16,014 | Three friends are driving cars on a road in the same direction. At a certain moment, they are positioned relative to each other as follows: Andrews is at a certain distance behind Brooks, and Carter is at a distance twice the distance from Andrews to Brooks, ahead of Brooks. Each driver is traveling at a constant speed... | 6.666666666666667 | 0 |
16,015 | Given is a regular tetrahedron of volume 1. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection? | \frac{1}{2} | 60.15625 |
16,016 | What fraction of the volume of a parallelepiped is the volume of a tetrahedron whose vertices are the centroids of the tetrahedra cut off by the planes of a tetrahedron inscribed in the parallelepiped? | 1/24 | 0.78125 |
16,017 | Given that points $\mathbf{A}$ and $\mathbf{B}$ lie on the curves $C_{1}: x^{2} - y + 1 = 0$ and $C_{2}: y^{2} - x + 1 = 0$ respectively, determine the minimum value of $|AB|$. | \frac{3 \sqrt{2}}{4} | 76.5625 |
16,018 | Mice built an underground house consisting of chambers and tunnels:
- Each tunnel leads from one chamber to another (i.e., none are dead ends).
- From each chamber, exactly three tunnels lead to three different chambers.
- From each chamber, it is possible to reach any other chamber through tunnels.
- There is exactly... | 10 | 13.28125 |
16,019 | Every second, the computer displays a number equal to the sum of the digits of the previous number multiplied by 31. On the first second, the number 2020 was displayed. What number will be displayed on the screen on the 2020th second? | 310 | 10.9375 |
16,020 | In the center of a circular field stands a geologists' house. From it, 8 straight roads extend, dividing the field into 8 equal sectors. Two geologists embark on a journey from their house at a speed of 5 km/h, each choosing a road at random. Determine the probability that the distance between them will be more than 8 ... | 0.375 | 0 |
16,021 | Given a triangular pyramid \( S-ABC \) with vertex \( S \). The projection of \( S \) onto the base \( \triangle ABC \) is the orthocenter \( H \) of \( \triangle ABC \). Additionally, \( BC = 2 \), \( SB = SC \), and the dihedral angle between the face \( SBC \) and the base is \( 60^\circ \). Determine the volume of ... | \frac{\sqrt{3}}{3} | 21.09375 |
16,022 | In a box, there are 100 balls of different colors: 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls. How many balls must be drawn randomly from the box to ensure that at least 15 of them are of the same color? | 75 | 78.90625 |
16,023 | The regular hexagon \(ABCDEF\) has diagonals \(AC\) and \(CE\). The internal points \(M\) and \(N\) divide these diagonals such that \(AM: AC = CN: CE = r\). Determine \(r\) if it is known that points \(B\), \(M\), and \(N\) are collinear. | \frac{1}{\sqrt{3}} | 0.78125 |
16,024 | Given four points \(O, A, B, C\) on a plane, such that \(OA = 4\), \(OB = 3\), \(OC = 2\), and \(\overrightarrow{OB} \cdot \overrightarrow{OC} = 3\), find the maximum value of the area \(S_{\triangle ABC}\). | 2\sqrt{7} + \frac{3\sqrt{3}}{2} | 0 |
16,025 | Find the number of ways to pave a $1 \times 10$ block with tiles of sizes $1 \times 1, 1 \times 2$ and $1 \times 4$, assuming tiles of the same size are indistinguishable. It is not necessary to use all the three kinds of tiles. | 169 | 86.71875 |
16,026 | Two digits of a number were swapped, and as a result, it increased by more than 3 times. The resulting number is 8453719. Find the original number. | 1453789 | 14.84375 |
16,027 | In quadrilateral \(ABCD\), \(\angle DAC = 98^\circ\), \(\angle DBC = 82^\circ\), \(\angle BCD = 70^\circ\), and \(BC = AD\). Find \(\angle ACD\). | 28 | 21.875 |
16,028 | Inside the cube \(ABCD A_{1}B_{1}C_{1}D_{1}\) there is a sphere with center \(O\) and radius 10. The sphere intersects the face \(AA_{1}D_{1}D\) in a circle of radius 1, the face \(A_{1}B_{1}C_{1}D_{1}\) in a circle of radius 1, and the face \(CD D_{1}C_{1}\) in a circle of radius 3. Find the length of the segment \(OD... | 17 | 30.46875 |
16,029 | There are 100 chairs arranged in a circle. If \( n \) people are sitting on these chairs, such that any new person sitting down will always sit on a chair adjacent to one of the \( n \) people, what is the minimum value of \( n \)? | 34 | 72.65625 |
16,030 | Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\sin x)^{2}}
$$ | \frac{1}{3} | 19.53125 |
16,031 | In triangle \( ABC \), let \( E \) be the point where the side \( AC \) is divided into quarters closest to \( C \), and let \( F \) be the midpoint of side \( BC \). The line passing through points \( E \) and \( F \) intersects line \( AB \) at point \( D \). What percentage of the area of triangle \( ABC \) is the a... | 112.5 | 26.5625 |
16,032 | Given the sets \( M=\{x, xy, \lg(xy)\} \) and \( N=\{0, |x|, y\} \), and that \( M=N \), find 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 | 24.21875 |
16,033 | Let $L$ be the intersection point of the diagonals $CE$ and $DF$ of a regular hexagon $ABCDEF$ with side length 2. Point $K$ is defined such that $\overrightarrow{LK} = \overrightarrow{AC} - 3 \overrightarrow{BC}$. Determine whether point $K$ lies inside, on the boundary, or outside of $ABCDEF$, and find the length of ... | \frac{2\sqrt{3}}{3} | 6.25 |
16,034 | Find the number of 2's in the factorization of the number $2011 \cdot 2012 \cdot 2013 \cdot \ldots \cdot 4020$. Provide the answer in the given field. | 2010 | 44.53125 |
16,035 | The bases \(AB\) and \(CD\) of the trapezoid \(ABCD\) are 41 and 24 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\). | 984 | 50.78125 |
16,036 | Find the minimum value for \(a, b > 0\) of the expression
$$
\frac{|2a - b + 2a(b - a)| + |b + 2a - a(b + 4a)|}{\sqrt{4a^2 + b^2}}
$$ | \frac{\sqrt{5}}{5} | 6.25 |
16,037 | In \( \triangle ABC \), \( AB = 4 \), \( BC = 7 \), \( CA = 5 \). Let \(\angle BAC = \alpha\). Find the value of \( \sin^6 \frac{\alpha}{2} + \cos^6 \frac{\alpha}{2} \). | 7/25 | 83.59375 |
16,038 | If a positive integer has eight positive divisors and the sum of these eight positive divisors is 3240, it is called a "good number." For example, 2006 is a good number because the sum of its positive divisors $1, 2, 17, 34, 59, 118, 1003, 2006$ is 3240. Find the smallest good number. | 1614 | 31.25 |
16,039 | For a natural number \( N \), if at least eight out of the nine natural numbers from 1 to 9 can divide \( N \), then \( N \) is called a "Ba Xian number". What is the smallest "Ba Xian number" greater than 2000? | 2016 | 91.40625 |
16,040 | There are exactly 120 ways to color five cells in a \( 5 \times 5 \) grid such that each row and each column contains exactly one colored cell.
There are exactly 96 ways to color five cells in a \( 5 \times 5 \) grid without the corner cell such that each row and each column contains exactly one colored cell.
How man... | 78 | 66.40625 |
16,041 | In rectangle \(ABCD\), \(AB = 2\) and \(AD = 1\). Let \(P\) be a moving point on side \(DC\) (including points \(D\) and \(C\)), and \(Q\) be a moving point on the extension of \(CB\) (including point \(B\)). The points \(P\) and \(Q\) satisfy \(|\overrightarrow{DP}| = |\overrightarrow{BQ}|\). What is the minimum value... | 3/4 | 68.75 |
16,042 | The difference between the cube and the square of a number has the form $a b c a b c$ (in the decimal system). What is this number? | 78 | 81.25 |
16,043 |
Calculate the volumes of the solids formed by rotating the regions bounded by the graphs of the functions around the y-axis.
$$
y = \arcsin x, \quad y = \arccos x, \quad y = 0
$$ | \frac{\pi}{2} | 3.90625 |
16,044 | An archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are always at least two connected by a bridge. What is the maximum possible value of $N$? | 36 | 6.25 |
16,045 | In trapezoid \(ABCD\) with \(BC \parallel AD\), it is known that \(AD = 3 \cdot BC\). A line intersects the non-parallel sides of the trapezoid at points \(M\) and \(N\) such that \(AM:MB = 3:5\) and \(CN:ND = 2:7\). Find the ratio of the areas of quadrilaterals \(MBCN\) and \(AMND\). | 9/23 | 0 |
16,046 | Given the sequence: $\frac{2}{3}, \frac{2}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}, \frac{2}{27}, \frac{4}{27}, \cdots$, $\frac{26}{27}, \cdots, \frac{2}{3^{n}}, \frac{4}{3^{n}}, \cdots, \frac{3^{n}-1}{3^{n}}, \cdots$. Find the position of $\frac{2018}{2187}$ in the sequence. | 1552 | 23.4375 |
16,047 | Two right triangles \( \triangle AXY \) and \( \triangle BXY \) have a common hypotenuse \( XY \) and side lengths (in units) \( AX=5 \), \( AY=10 \), and \( BY=2 \). Sides \( AY \) and \( BX \) intersect at \( P \). Determine the area (in square units) of \( \triangle PXY \). | 25/3 | 1.5625 |
16,048 | Petya can draw only 4 things: a sun, a ball, a tomato, and a banana. Today he drew several things, including exactly 15 yellow items, 18 round items, and 13 edible items. What is the maximum number of balls he could have drawn?
Petya believes that all tomatoes are round and red, all balls are round and can be of any c... | 18 | 8.59375 |
16,049 | Let \( f(x) = \left\{ \begin{array}{cc} 1 & 1 \leqslant x \leqslant 2 \\ x-1 & 2 < x \leqslant 3 \end{array} \right. \). For any \( a \,(a \in \mathbb{R}) \), define \( v(a) = \max \{ f(x) - a x \mid x \in [1,3] \} - \min \{ f(x) - a x \mid x \in [1,3] \} \). Draw the graph of \( v(a) \) and find the minimum value of \... | \frac{1}{2} | 30.46875 |
16,050 | Find the flux of the vector field \(\mathbf{a} = y^2 \mathbf{j} + z \mathbf{k}\) through the part of the surface \(z = x^2 + y^2\), cut off by the plane \(z=2\). The normal vector is taken to be outward with respect to the region bounded by the paraboloid. | -2\pi | 27.34375 |
16,051 | Find all integers \( n \) such that \( n^{4} + 6n^{3} + 11n^{2} + 3n + 31 \) is a perfect square. | 10 | 88.28125 |
16,052 | Let \( p(x) = x^{4} + a x^{3} + b x^{2} + c x + d \), where \( a, b, c, d \) are constants, and \( p(1) = 1993 \), \( p(2) = 3986 \), \( p(3) = 5979 \). Calculate \( \frac{1}{4}[p(11) + p(-7)] \). | 5233 | 38.28125 |
16,053 | A production team in a factory is manufacturing a batch of parts. Initially, when each worker is on their own original position, the task can be completed in 9 hours. If the positions of workers $A$ and $B$ are swapped, and other workers' efficiency remains the same, the task can be completed one hour earlier. Similarl... | 108 | 7.8125 |
16,054 | Someone wrote down two numbers $5^{2020}$ and $2^{2020}$ consecutively. How many digits will the resulting number contain? | 2021 | 28.90625 |
16,055 | In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 12-digit number is divisible by 12. How many ways can this be done? | 5184 | 51.5625 |
16,056 | The common ratio of the geometric sequence \( a+\log _{2} 3, a+\log _{1} 3, a+\log _{8} 3 \) is ______. | \frac{1}{3} | 23.4375 |
16,057 | Find the smallest positive integer \( m \) such that the equation regarding \( x, y, \) and \( z \):
\[ 2^x + 3^y - 5^z = 2m \]
has no positive integer solutions. | 11 | 40.625 |
16,058 | Let \( S = \{1, 2, 3, 4, \ldots, 50\} \). A 3-element subset \(\{a, b, c\}\) of \(S\) is said to be good if \(a + b + c\) is divisible by 3. Determine the number of 3-element subsets of \(S\) which are good. | 6544 | 85.15625 |
16,059 |
In an isosceles trapezoid \(ABCD\), the larger base \(AD = 12\) and \(AB = 6\). Find the distance from point \(O\), the intersection of the diagonals, to point \(K\), the intersection of the extensions of the lateral sides, given that the extensions of the lateral sides intersect at a right angle. | \frac{12(3 - \sqrt{2})}{7} | 0 |
16,060 | Given a four-digit number $\overline{A B C D}$ that satisfies the following properties: $\overline{A B}$, $\overline{B C}$, and $\overline{C D}$ are all perfect squares (a perfect square is a number that can be expressed as the square of an integer, such as $4 = 2^2$ and $81 = 9^2$). What is the sum of all four-digit n... | 13462 | 70.3125 |
16,061 | What is the minimum number of points that can be chosen on a circle with a circumference of 1956 so that for each of these points there is exactly one chosen point at a distance of 1 and exactly one at a distance of 2 (distances are measured along the circle)? | 1304 | 0.78125 |
16,062 | Positive numbers \(a\), \(b\), and \(c\) satisfy the following equations:
\[ a^{2} + a b + b^{2} = 1 \]
\[ b^{2} + b c + c^{2} = 3 \]
\[ c^{2} + c a + a^{2} = 4 \]
Find \(a + b + c\). | \sqrt{7} | 54.6875 |
16,063 | For each positive integer $n$, an associated non-negative integer $f(n)$ is defined to satisfy the following three rules:
i) $f(a b)=f(a)+f(b)$.
ii) $f(n)=0$ if $n$ is a prime greater than 10.
iii) $f(1)<f(243)<f(2)<11$.
Given that $f(2106)<11$, determine the value of $f(96)$. | 31 | 48.4375 |
16,064 | There are two types of tables in a restaurant: a square table can seat 4 people, and a round table can seat 9 people. The restaurant manager calls a number a "wealth number" if the total number of diners can exactly fill a certain number of tables. How many "wealth numbers" are there among the numbers from 1 to 100? | 88 | 100 |
16,065 | The product of three consecutive even numbers has the form $87XXXXX8$. Provide the 5 missing digits. ($X$ does not necessarily represent the same digits.) | 52660 | 7.8125 |
16,066 | Given ten 0's and ten 1's, how many 0-1 binary sequences can be formed such that no three or more consecutive 0's are together? For example, 01001001010011101011 is such a sequence, but the sequence 01001000101001110111 does not satisfy this condition. | 24068 | 47.65625 |
16,067 | "Sun", "Xing", and "Zhe" each represent a digit. The product of the two three-digit numbers 'SunXingZhe' and 'ZheXingSun' is 78445. Can you find the sum of these two three-digit numbers? | 686 | 10.9375 |
16,068 | The kindergarten teacher evenly distributed 270 apples, 180 pears, and 235 oranges to the larger group of children. The remaining quantities of apples, pears, and oranges are in the ratio $3:2:1$. How many children are there in the larger group? | 29 | 12.5 |
16,069 | The equations of the sides of a quadrilateral are:
$$
y=-x+7, \quad y=\frac{x}{2}+1, \quad y=-\frac{3}{2} x+2 \quad \text {and} \quad y=\frac{7}{4} x+\frac{3}{2}.
$$
Determine the area of the quadrilateral. | \frac{327}{52} | 7.03125 |
16,070 | For the set \( \{x \mid a \leqslant x \leqslant b\} \), we define \( b-a \) as its length. Let the set \( A=\{x \mid a \leqslant x \leqslant a+1981\} \), \( B=\{x \mid b-1014 \leqslant x \leqslant b\} \), and both \( A \) and \( B \) are subsets of the set \( U=\{x \mid 0 \leqslant x \leqslant 2012\} \). The minimum le... | 983 | 15.625 |
16,071 | Investment funds A, B, and C claim that they can earn profits of 200%, 300%, and 500% respectively in one year. Tommy has $90,000 and plans to invest in these funds. However, he knows that only one of these funds can achieve its claim while the other two will close down. He has thought of an investment plan which can g... | 30000 | 6.25 |
16,072 | There are 2006 positive integers \( a_{1}, a_{2}, \cdots, a_{2006} \) (which can be the same) such that the ratios \( \frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}} \) are all distinct. What is the minimum number of distinct numbers among \( a_{1}, a_{2}, \cdots, a_{2006} \)? | 46 | 3.90625 |
16,073 | In a scalene triangle with integer side lengths $a, b, c$, the following relation holds. What is the smallest height of the triangle?
$$
\frac{a^{2}}{c}-(a-c)^{2}=\frac{b^{2}}{c}-(b-c)^{2}
$$ | 2.4 | 1.5625 |
16,074 | Let \(CD\) be a chord of a circle \(\Gamma_{1}\) and \(AB\) a diameter of \(\Gamma_{1}\) perpendicular to \(CD\) at \(N\) with \(AN > NB\). A circle \(\Gamma_{2}\) centered at \(C\) with radius \(CN\) intersects \(\Gamma_{1}\) at points \(P\) and \(Q\), and the segments \(PQ\) and \(CD\) intersect at \(M\). Given that ... | 78 | 0 |
16,075 | How many four-digit numbers can be formed using three 1s, two 2s, and five 3s? | 71 | 14.84375 |
16,076 | Define the lengths of intervals $(m, n)$, $[m, n)$, $(m, n]$, and $[m, n]$ to be $n - m$ ($n, m \in \mathbf{R}$ and $n > m$). Find the sum of the lengths of the intervals for real numbers $x$ that satisfy the inequality
\[
\frac{1}{x-20}+\frac{1}{x-17} \geqslant \frac{1}{512}
\] | 1024 | 48.4375 |
16,077 | It is given that \( a = 103 \times 97 \times 10009 \). Find \( a \). | 99999919 | 97.65625 |
16,078 | In a trapezoid, the lengths of the bases are 5 and 15, and the lengths of the diagonals are 12 and 16. Find the area of the trapezoid. | 96 | 17.96875 |
16,079 | Numbers $1,2,3,4,5,6,7,$ and $8$ are placed at the vertices of a cube such that the sum of any three numbers belonging to any face of the cube is not less than 10. Find the minimum possible sum of four numbers belonging to one face. | 16 | 89.0625 |
16,080 | In a $10 \times 5$ grid, an ant starts from point $A$ and can only move right or up along the grid lines but is not allowed to pass through point $C$. How many different paths are there from point $A$ to point $B$? | 1827 | 20.3125 |
16,081 | Two identical cylindrical vessels are connected at the bottom with a small-diameter tube with a valve. While the valve was closed, water was poured into the first vessel and oil was poured into the second vessel, so that the liquid levels were identical and equal to $h = 40 \text{ cm}$. At what level will the water est... | 32.94 | 19.53125 |
16,082 | In a taxi, a passenger can sit in the front and three passengers can sit in the back. In how many ways can four passengers sit in a taxi if one of these passengers wants to sit by the window? | 18 | 60.15625 |
16,083 | Determine the number $ABCC$ (written in decimal system) given that
$$
ABCC = (DD - E) \cdot 100 + DD \cdot E
$$
where $A, B, C, D,$ and $E$ are distinct digits. | 1966 | 3.125 |
16,084 | Twelve mayoral candidates each made a statement about how many times lies had been told before their turn. The first candidate said, "Before me, one lie was told." The second candidate said, "Now, two lies have been told." The third candidate said, "Now, three lies have been told," and so on, until the twelfth candidat... | 11 | 34.375 |
16,085 | Calculate the definite integral:
$$
\int_{-1}^{0}(x+2)^{3} \cdot \ln ^{2}(x+2) \, dx
$$ | 4 \ln^{2} 2 - 2 \ln 2 + \frac{15}{32} | 0 |
16,086 | Given the function \( f(n) \) defined on the set of natural numbers \(\mathbf{N}\), and satisfies:
\[
\begin{array}{l}
f(1) = f(2) = 1, \\
f(3n) = 3 f(n) - 2, \\
f(3n+1) = 3 f(n) + 1, \\
f(3n+2) = 3 f(n) + 4 \quad (n \in \mathbf{N}).
\end{array}
\]
Determine the largest positive integer \( n \) less than or equal to ... | 1093 | 100 |
16,087 | Points \( M \) and \( N \) are located on side \( BC \) of triangle \( ABC \), and point \( K \) is on side \( AC \), with \( BM : MN : NC = 1 : 1 : 2 \) and \( CK : AK = 1 : 4 \). Given that the area of triangle \( ABC \) is 1, find the area of quadrilateral \( AMNK \). | 13/20 | 1.5625 |
16,088 | The real numbers \( x, y, z \) satisfy the equations \( x + y + z = 2 \) and \( xy + yz + zx = 1 \). Find the maximum possible value of \( x - y \). | \frac{2 \sqrt{3}}{3} | 40.625 |
16,089 | Subset \( S \subseteq \{1, 2, 3, \ldots, 1000\} \) is such that if \( m \) and \( n \) are distinct elements of \( S \), then \( m + n \) does not belong to \( S \). What is the largest possible number of elements in \( S \)? | 501 | 6.25 |
16,090 | In triangle \(ABC\), a point \(D\) is marked on side \(AC\) such that \(BC = CD\). Find \(AD\) if it is known that \(BD = 13\) and angle \(CAB\) is three times smaller than angle \(CBA\). | 13 | 35.15625 |
16,091 | Consider the permutation of $1, 2, \cdots, 20$ as $\left(a_{1} a_{2} \cdots a_{20}\right)$. Perform the following operation on this permutation: swap the positions of any two numbers. The goal is to transform this permutation into $(1, 2, \cdots, 20)$. Let $k_{a}$ denote the minimum number of operations needed to reach... | 19 | 0.78125 |
16,092 | Given that the volume of the tetrahedron \(ABCD\) is 40, \(AB = 7\), \(BC = 8\), \(AC = 9\), and the orthogonal projection of vertex \(D\) onto the plane \(ABC\) is precisely the incenter \(H\) of triangle \(ABC\), what is the surface area of the tetrahedron? | 60 + 12 \sqrt{5} | 0.78125 |
16,093 | From the first 539 positive integers, we select some such that their sum is at least one-third of the sum of the original numbers. What is the minimum number of integers we need to select for this condition to be satisfied? | 99 | 40.625 |
16,094 | An elevator containing 9 passengers can stop at ten different floors. The passengers exit in groups of two, three, and four people. In how many ways can this happen? | 10 * 9 * 8 * 36 * 35 | 0 |
16,095 | A cashier, upon checking the account before leaving work, finds that the cash is 153 yuan less than the account book. She knows the actual amount collected cannot be wrong, so it must be due to a decimal point error during bookkeeping. What is the actual amount of the cash that was recorded incorrectly? | 17 | 31.25 |
16,096 | Each vertex of the parallelogram $ABCD$ lies on the same side of the plane $S$ such that the distances of the vertices $A, B$, and $C$ from the plane $S$ are 4 cm, 6 cm, and 8 cm, respectively. The area of the projection of the parallelogram onto the plane $S$, which forms the quadrilateral $A'B'C'D'$, is $10 \text{ cm... | 60 | 54.6875 |
16,097 | How many five-digit numbers are there that are divisible by 5 and do not contain repeating digits? | 5712 | 28.125 |
16,098 | From the 20 numbers 11, 12, 13, 14, ... 30, how many numbers must be chosen to ensure that there are at least two numbers whose sum is a multiple of 10? | 11 | 39.0625 |
16,099 | Cátia leaves school every day at the same time and rides her bicycle home. When she pedals at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at $4:30$ PM. If she pedals at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at $5:15$ PM. At what speed should she pedal to arrive home at $5:00$ PM? | 12 | 22.65625 |
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