Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
15,400 | Find the largest possible number in decimal notation where all the digits are different, and the sum of its digits is 37. | 976543210 | 15.625 |
15,401 |
Given a triangle \(ABC\) where \(AB = AC\) and \(\angle A = 80^\circ\). Inside triangle \(ABC\) is a point \(M\) such that \(\angle MBC = 30^\circ\) and \(\angle MCB = 10^\circ\). Find \(\angle AMC\). | 70 | 89.0625 |
15,402 | Among all the simple fractions where both the numerator and the denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{3}{5}$. Provide the numerator of this fraction in your answer. | 59 | 39.84375 |
15,403 | Let \( A \) be a set containing only positive integers, and for any elements \( x \) and \( y \) in \( A \), \(|x-y| \geq \frac{x y}{30}\). Determine at most how many elements \( A \) may contain. | 10 | 7.8125 |
15,404 | Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.
Anton drove the car for half the time Vasya did, and Sasha drove for as long as Anton and Dima together. Dima was at the wheel for only one-tenth of the distance. What frac... | 0.4 | 77.34375 |
15,405 | Four problems were attempted by 100 contestants in a Mathematics competition. The first problem was solved by 90 contestants, the second by 85 contestants, the third by 80 contestants, and the fourth by 75 contestants. What is the smallest possible number of contestants who solved all four problems? | 30 | 35.15625 |
15,406 | Calculate the sum:
\[
\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2016}\right)+\left(\frac{2}{3}+\frac{2}{4}+\cdots+\frac{2}{2016}\right)+\left(\frac{3}{4}+\frac{3}{5}+\cdots+\frac{3}{2016}\right)+\cdots+\left(\frac{2014}{2015}+\frac{2014}{2016}\right)+\frac{2015}{2016}.
\] | 1015560 | 8.59375 |
15,407 | Given a convex quadrilateral \(ABCD\), \(X\) is the midpoint of the diagonal \(AC\). It is known that \(CD \parallel BX\). Find \(AD\), given that \(BX = 3\), \(BC = 7\), and \(CD = 6\). | 14 | 7.8125 |
15,408 | If the inequality \( ab + b^2 + c^2 \geq \lambda(a + b)c \) holds for all positive real numbers \( a, b, c \) that satisfy \( b + c \geq a \), then the maximum value of the real number \( \lambda \) is \(\quad\) . | \sqrt{2} - \frac{1}{2} | 7.8125 |
15,409 | In 2006, the revenues of an insurance company increased by 25% and the expenses increased by 15% compared to the previous year. The company's profit (revenue - expenses) increased by 40%. What percentage of the revenues were the expenses in 2006? | 55.2 | 32.03125 |
15,410 | Three people, Jia, Yi, and Bing, participated in a competition and they took the top 3 places (with no ties). Jia said: "I am first", Yi said: "I am not first", and Bing said: "I am not third". Only one of them is telling the truth. If the rankings of Jia, Yi, and Bing are respectively $A, B, C$, then the three-digit n... | 312 | 16.40625 |
15,411 | Nine integers from 1 to 5 are written on a board. It is known that seven of them are at least 2, six are greater than 2, three are at least 4, and one is at least 5. Find the sum of all the numbers. | 26 | 6.25 |
15,412 | Find a 4-digit perfect square, knowing that the number formed by the first two digits is one more than the number formed by the last two digits. | 8281 | 81.25 |
15,413 | How many points on the hyperbola \( y = \frac{2013}{x} \) are there such that the tangent line at those points intersects both coordinate axes at points with integer coordinates? | 48 | 0.78125 |
15,414 | Given that \( M \) is the midpoint of the height \( D D_{1} \) of a regular tetrahedron \( ABCD \), find the dihedral angle \( A-M B-C \) in radians. | \frac{\pi}{2} | 10.9375 |
15,415 | On the edge \(AD\) and the diagonal \(A_1C\) of the parallelepiped \(ABCDA_1B_1C_1D_1\), points \(M\) and \(N\) are taken respectively, such that the line \(MN\) is parallel to the plane \(BDC_1\) and \(AM:AD = 1:5\). Find the ratio \(CN:CA_1\). | 3/5 | 6.25 |
15,416 | Let \( M = \{1, 2, \cdots, 10\} \), and \( A_1, A_2, \cdots, A_n \) be distinct non-empty subsets of \( M \). If \(i \neq j\), then \( A_i \cap A_j \) can have at most two elements. Find the maximum value of \( n \). | 175 | 75 |
15,417 | Four distinct natural numbers, one of which is an even prime number, have the following properties:
- The sum of any two numbers is a multiple of 2.
- The sum of any three numbers is a multiple of 3.
- The sum of all four numbers is a multiple of 4.
Find the smallest possible sum of these four numbers. | 44 | 5.46875 |
15,418 | A ball invites 2018 couples, each assigned to areas numbered $1, 2, \cdots, 2018$. The organizer specifies that at the $i$-th minute of the ball, the couple in area $s_i$ (if any) moves to area $r_i$, and the couple originally in area $r_i$ (if any) exits the ball. The relationship is given by:
$$
s_i \equiv i \pmod{20... | 1009 | 39.0625 |
15,419 | The altitude \(AH\) and the angle bisector \(CL\) of triangle \(ABC\) intersect at point \(O\). Find the angle \(BAC\) if it is known that the difference between the angle \(COH\) and half of the angle \(ABC\) is \(46^\circ\). | 92 | 14.84375 |
15,420 | Olga Ivanovna, the homeroom teacher of class 5B, is organizing a "Mathematical Ballet". She wants to arrange the boys and girls so that exactly 2 boys are at a distance of 5 meters from each girl. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating? | 20 | 0 |
15,421 | Let \( S \) be the set of points whose coordinates \( x \), \( y \), and \( z \) are integers that satisfy \( 0 \leq x \leq 2 \), \( 0 \leq y \leq 3 \), and \( 0 \leq z \leq 4 \). Two distinct points are randomly chosen from \( S \). Find the probability that the midpoint of the two chosen points also belongs to \( S \... | 23/177 | 0.78125 |
15,422 | For positive integer \( n \), let \( f(n) \) denote the unit digit of \( 1+2+3+\cdots+n \). Find the value of \( f(1)+f(2)+\cdots+f(2011) \). | 7046 | 22.65625 |
15,423 | The task is given a finite increasing sequence \( a_{1}, a_{2}, \ldots, a_{n} \) (\(n \geq 3\)) of natural numbers, and for all \( k \leq n-2 \), the equality \( a_{k+2}=3 a_{k+1}-2 a_{k}-1 \) holds. The sequence must necessarily contain the term \( a_{k}=2021 \). Determine the maximum number of three-digit numbers div... | 36 | 12.5 |
15,424 | Determine the value of the following product with a short calculation:
$$
\frac{6 \cdot 27^{12}+2 \cdot 81^{9}}{8000000^{2}} \cdot \frac{80 \cdot 32^{3} \cdot 125^{4}}{9^{19}-729^{6}}
$$ | 10 | 53.90625 |
15,425 | Let \( u, v, w \) be positive real numbers, all different from 1. If
\[
\log_{u}(vw) + \log_{v}(w) = 5 \quad \text{and} \quad \log_{v}(u) + \log_{w}(v) = 3,
\]
find the value of \( \log_{w}(u) \). | \frac{4}{5} | 10.15625 |
15,426 | \(\cos \frac{\pi}{11} - \cos \frac{2 \pi}{11} + \cos \frac{3 \pi}{11} - \cos \frac{4 \pi}{11} + \cos \frac{5 \pi}{11} = \) (Answer with a number). | \frac{1}{2} | 47.65625 |
15,427 | Write the product of the digits of each natural number from 1 to 2018 (for example, the product of the digits of the number 5 is 5; the product of the digits of the number 72 is \(7 \times 2=14\); the product of the digits of the number 607 is \(6 \times 0 \times 7=0\), etc.). Then find the sum of these 2018 products. | 184320 | 100 |
15,428 | Teacher Shi distributed cards with the numbers 1, 2, 3, and 4 written on them to four people: Jia, Yi, Bing, and Ding. Then the following conversation occurred:
Jia said to Yi: "The number on your card is 4."
Yi said to Bing: "The number on your card is 3."
Bing said to Ding: "The number on your card is 2."
Ding said ... | 2341 | 78.125 |
15,429 | There are 120 different five-digit numbers composed of the digits 1, 2, 3, 4, and 5. Arrange them in descending order. The 95th number is $\quad$ | 21354 | 84.375 |
15,430 | Given that for any positive integer \( n \), \( 9^{2n} - 8^{2n} - 17 \) is always divisible by \( m \), find the largest positive integer \( m \). | 2448 | 71.09375 |
15,431 | There are 300 children in the "Young Photographer" club. In a session, they divided into 100 groups of 3 people each, and in every group, each member took a photograph of the other two members in their group. No one took any additional photographs. In total, there were 100 photographs of "boy+boy" and 56 photographs of... | 72 | 9.375 |
15,432 | One day, a group of young people came to the Platonic Academy located in the outskirts of Athens. The academy's gate was closed, and above the gate a sign read: "No one ignorant of geometry may enter!" Next to the sign was a diagram with four small rectangles of areas $20, 40, 48, \text{and } 42$ forming a larger recta... | 150 | 3.90625 |
15,433 | Which number has the property that if it is multiplied by $1, 2, 3, 4, 5$, or $6$, the resulting product contains only the digits that appear in the original number? | 142857 | 88.28125 |
15,434 | If the six-digit number $\overline{201 a b 7}$ is divisible by 11 and 13, then the two-digit number $\overline{a b}$ equals: | 48 | 64.84375 |
15,435 |
Two circles of radius \( r \) touch each other. Additionally, each of them is externally tangent to a third circle of radius \( R \) at points \( A \) and \( B \) respectively.
Find the radius \( r \), given that \( AB = 12 \) and \( R = 8 \). | 24 | 0 |
15,436 | The archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island and that among any 7 islands, there are always two that are connected by a bridge. What is the maximum possible value of $N$? | 36 | 1.5625 |
15,437 | There is a ten-digit number. From left to right:
- Its first digit indicates how many zeros are in the number.
- Its second digit indicates how many ones are in the number.
- Its third digit indicates how many twos are in the number.
- $\cdots \cdots$
- Its tenth digit indicates how many nines are in the number.
Find ... | 6210001000 | 83.59375 |
15,438 | Several young men and women are seated around a round table. It is known that to the left of exactly 7 women, there are women, and to the left of 12 women, there are men. It is also known that for 75% of the young men, there are women to their right. How many people are seated at the table? | 35 | 3.125 |
15,439 | The triangle $\triangle ABC$ is equilateral, and the point $P$ is such that $PA = 3 \, \text{cm}$, $PB = 4 \, \text{cm}$, and $PC = 5 \, \text{cm}$. Calculate the length of the sides of the triangle $\triangle ABC$. | \sqrt{25 + 12 \sqrt{3}} | 0.78125 |
15,440 | A \(10 \times 1\) rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct tilings of the rectangle are there, if at le... | 1022 | 36.71875 |
15,441 | Petya and Vasya calculated that if they walk at a speed of 4 km per hour to the neighboring village, which is 4 kilometers away, they will be 10 minutes late for the football match held there for the district championship. How should they proceed to arrive at the match on time and achieve the greatest time gain, having... | 10 | 37.5 |
15,442 | Let \( P \) be a regular polygon with 2006 sides. A diagonal of \( P \) is called good if its endpoints divide the perimeter of \( P \) into two parts, each having an odd number of sides of \( P \). The sides of \( P \) are also called good. Suppose \( P \) has been subdivided into triangles by 2003 diagonals that do n... | 1003 | 89.0625 |
15,443 | A line \( l \) passes through the focus \( F \) of the parabola \( y^2 = 4x \) and intersects the parabola at points \( A \) and \( B \). Point \( M \) is given as \( (4,0) \). Extending \( AM \) and \( BM \) intersects the parabola again at points \( C \) and \( D \), respectively. Find the value of \(\frac{S_{\triang... | 16 | 0 |
15,444 | In triangle \( \triangle ABC \), \( AB = 8 \), \( BC = 11 \), and \( AC = 6 \). The points \( P \) and \( Q \) are on \( BC \) such that \( \triangle PBA \) and \( \triangle QAC \) are each similar to \( \triangle ABC \). What is the length of \( PQ \)? | \frac{21}{11} | 58.59375 |
15,445 | For real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right]=1994\). | 312 | 95.3125 |
15,446 | The sequence \(\{a_n\}\) has consecutive terms \(a_n\) and \(a_{n+1}\) as the roots of the equation \(x^2 - c_n x + \left(\frac{1}{3}\right)^n = 0\), with initial term \(a_1 = 2\). Find the sum of the infinite series \(c_1, c_2, \cdots, c_n, \cdots\). | \frac{9}{2} | 12.5 |
15,447 | For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)? | 256 | 61.71875 |
15,448 |
An isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypotenuse, and the opposite sides of these vert... | 324 | 0.78125 |
15,449 | The numbers $1,2, \ldots, 2016$ are grouped into pairs in such a way that the product of the numbers in each pair does not exceed a certain natural number $N$. What is the smallest possible value of $N$ for which this is possible? | 1017072 | 77.34375 |
15,450 | For each positive integer \( n \), define \( A_{n} = \frac{20^{n} + 11^{n}}{n!} \), where \( n! = 1 \times 2 \times \cdots \times n \). Find the value of \( n \) that maximizes \( A_{n} \). | 19 | 77.34375 |
15,451 | In a notebook, there is a grid rectangle of size $3 \times 7$. Igor's robot was asked to trace all the lines with a marker, and it took him 26 minutes (the robot draws lines at a constant speed). How many minutes will it take him to trace all the lines in a $5 \times 5$ grid square with the marker? | 30 | 23.4375 |
15,452 | Point \( M \) divides the side \( BC \) of parallelogram \( ABCD \) in the ratio \( BM: MC = 1: 3 \). Line \( AM \) intersects diagonal \( BD \) at point \( K \). Find the area of quadrilateral \( CMKD \) if the area of parallelogram \( ABCD \) is 1. | \frac{19}{40} | 4.6875 |
15,453 | Dad says he is exactly 35 years old, not counting weekends. How old is he really? | 49 | 16.40625 |
15,454 | In an isosceles trapezoid, one base is \(40 \text{ cm}\) and the other is \(24 \text{ cm}\). The diagonals of this trapezoid are mutually perpendicular. Find its area. | 1024 | 14.0625 |
15,455 | How many 9-digit numbers that are divisible by 5 can be formed by permuting the digits of the number 377353752? | 1120 | 53.90625 |
15,456 | Let \(X_{0}\) be the interior of a triangle with side lengths 3, 4, and 5. For all positive integers \(n\), define \(X_{n}\) to be the set of points within 1 unit of some point in \(X_{n-1}\). The area of the region outside \(X_{20}\) but inside \(X_{21}\) can be written as \(a\pi + b\), for integers \(a\) and \(b\). C... | 4112 | 0 |
15,457 | A pyramid with a triangular base has edges of unit length, and the angles between its edges are \(60^{\circ}, 90^{\circ},\) and \(120^{\circ}\). What is the volume of the pyramid? | \frac{\sqrt{2}}{12} | 22.65625 |
15,458 | Several island inhabitants gather in a hut, with some belonging to the Ah tribe and the rest to the Uh tribe. Ah tribe members always tell the truth, while Uh tribe members always lie. One inhabitant said, "There are no more than 16 of us in the hut," and then added, "All of us are from the Uh tribe." Another said, "Th... | 15 | 3.125 |
15,459 | Let point \( P \) lie on the face \( ABC \) of a tetrahedron \( ABCD \) with edge length 2. The distances from \( P \) to the planes \( DAB \), \( DBC \), and \( DCA \) form an arithmetic sequence. Find the distance from \( P \) to the plane \( DBC \). | \frac{2\sqrt{6}}{9} | 61.71875 |
15,460 | There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a "cluster" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster. However, one night an ... | 153 | 89.84375 |
15,461 | Person A and Person B start walking towards each other from points $A$ and $B$ respectively, which are 10 kilometers apart. If they start at the same time, they will meet at a point 1 kilometer away from the midpoint of $A$ and $B$. If Person A starts 5 minutes later than Person B, they will meet exactly at the midpoin... | 10 | 40.625 |
15,462 | Given fixed points \( A(3,0) \), \( B(0,4) \), and point \( P \) on the incircle of triangle \( \triangle AOB \) (where \( O \) is the origin), find the maximum value of \( |PA|^2 + |PB|^2 + |PO|^2 \). | 22 | 3.90625 |
15,463 | In tetrahedron $ABCD$, $AD > AB$, $AD \perp AB$, $AD \perp AC$, $\angle BAC = \frac{\pi}{3}$. Let the areas of triangles $ADB$, $ADC$, $ABC$, and $BCD$ be $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$, respectively. It is known that $S_{1} + S_{2} = S_{3} + S_{4}$. Find the value of $\frac{S_{3}}{S_{1}} + \frac{S_{3}}{S_{2}}$... | 3/2 | 1.5625 |
15,464 | Find the maximum value of the expression
$$
\frac{a}{x} + \frac{a+b}{x+y} + \frac{a+b+c}{x+y+z}
$$
where \( a, b, c \in [2,3] \), and the triplet of numbers \( x, y, z \) is some permutation of the triplet \( a, b, c \). | 15/4 | 0 |
15,465 | Given \(2x^2 + 3xy + 2y^2 = 1\), find the minimum value of \(f(x, y) = x + y + xy\). | -\frac{9}{8} | 49.21875 |
15,466 | Let \( T \) be the set of all positive divisors of \( 60^{100} \). \( S \) is a subset of \( T \) such that no number in \( S \) is a multiple of another number in \( S \). Find the maximum value of \( |S| \). | 10201 | 16.40625 |
15,467 | Point \( M \) lies on the side of a regular hexagon with a side length of 10. Find the sum of the distances from point \( M \) to the lines containing the other sides of the hexagon. | 30\sqrt{3} | 25.78125 |
15,468 | Given real numbers \(a\) and \(b\) that satisfy \(0 \leqslant a, b \leqslant 8\) and \(b^2 = 16 + a^2\), find the sum of the maximum and minimum values of \(b - a\). | 12 - 4\sqrt{3} | 13.28125 |
15,469 | Let \( \triangle DEF \) be a triangle and \( H \) the foot of the altitude from \( D \) to \( EF \). If \( DE = 60 \), \( DF = 35 \), and \( DH = 21 \), what is the difference between the minimum and the maximum possible values for the area of \( \triangle DEF \)? | 588 | 10.15625 |
15,470 | We start with 5000 forints in our pocket to buy gifts, visiting three stores. In each store, we find a gift that we like and purchase it if we have enough money. The prices in each store are independently 1000, 1500, or 2000 forints, each with a probability of $\frac{1}{3}$. What is the probability that we are able to ... | 17/27 | 43.75 |
15,471 | Find the largest natural number whose all digits in its decimal representation are different and which decreases 5 times if you cross out the first digit. | 3750 | 0 |
15,472 | Divide each natural number by the sum of the squares of its digits (for single-digit numbers, divide by the square of the number). Is there a smallest quotient among the obtained quotients, and if so, which one is it? | 1/9 | 63.28125 |
15,473 | Suppose \(A, B\) are the foci of a hyperbola and \(C\) is a point on the hyperbola. Given that the three sides of \(\triangle ABC\) form an arithmetic sequence, and \(\angle ACB = 120^\circ\), determine the eccentricity of the hyperbola. | 7/2 | 21.09375 |
15,474 | For the pair of positive integers \((x, y)\) such that \(\frac{x^{2}+y^{2}}{11}\) is an integer and \(\frac{x^{2}+y^{2}}{11} \leqslant 1991\), find the number of such pairs \((x, y)\) (where \((a, b)\) and \((b, a)\) are considered different pairs if \(a \neq b\)). | 131 | 8.59375 |
15,475 | Let \( M_{n} = \left\{ 0 . \overline{a_{1} a_{2} \cdots a_{n}} \mid a_{i} \ \text{is either 0 or 1 for} \ i=1,2, \cdots, n-1, \ a_{n}=1 \right\} \). \( T_{n} \) is the number of elements in \( M_{n} \) and \( S_{n} \) is the sum of all elements in \( M_{n} \). Find \( \lim_{n \rightarrow \infty} \frac{S_{n}}{T_{n}} \). | 1/18 | 27.34375 |
15,476 | When measuring a part, random errors occur that follow a normal distribution with a parameter $\sigma=10$ mm. Find the probability that the measurement is made with an error not exceeding $15$ mm. | 0.8664 | 86.71875 |
15,477 | We write on the board the equation
$$
(x-1)(x-2) \cdots(x-2016)=(x-1)(x-2) \cdots(x-2016),
$$
where there are 2016 linear factors on each side. What is the smallest positive value of $k$ such that we can omit exactly $k$ of these 4032 linear factors in such a way that there is at least one linear factor on each side,... | 2016 | 50.78125 |
15,478 | In a kindergarten's junior group, there are two identical small Christmas trees and five children. The teachers want to divide the children into two circles around each tree, with at least one child in each circle. The teachers distinguish the children but do not distinguish the trees: two such divisions into circles a... | 50 | 27.34375 |
15,479 | Let \( M = \{1, 2, \cdots, 17\} \). If there exist four distinct numbers \( a, b, c, d \in M \) such that \( a + b \equiv c + d \pmod{17} \), then \( \{a, b\} \) and \( \{c, d\} \) are called a balanced pair of the set \( M \). Find the number of balanced pairs in the set \( M \). | 476 | 99.21875 |
15,480 | There is a simple pendulum with a period of $T=1$ second in summer. In winter, the length of the pendulum shortens by 0.01 centimeters. How many seconds faster is this pendulum in winter over a 24-hour period compared to summer? (Round to the nearest second).
Note: The formula for the period of a simple pendulum is $T... | 17 | 78.90625 |
15,481 | On a highway, there are checkpoints D, A, C, and B arranged in sequence. A motorcyclist and a cyclist started simultaneously from A and B heading towards C and D, respectively. After meeting at point E, they exchanged vehicles and each continued to their destinations. As a result, the first person spent 6 hours traveli... | 340 | 1.5625 |
15,482 | The country Omega grows and consumes only vegetables and fruits. It is known that in 2014, 1200 tons of vegetables and 750 tons of fruits were grown in Omega. In 2015, 900 tons of vegetables and 900 tons of fruits were grown. During the year, the price of one ton of vegetables increased from 90,000 to 100,000 rubles, a... | -9.59 | 7.03125 |
15,483 | Two identical cylindrical vessels are connected by a small tube with a valve at the bottom. Initially, the valve is closed, and water is poured into the first vessel while oil is poured into the second vessel, such that the liquid levels are equal and are $h=40$ cm. At what level will the water be in the first vessel i... | 32.94 | 27.34375 |
15,484 | In a triangle with sides 6 cm, 10 cm, and 12 cm, an inscribed circle is tangent to the two longer sides. Find the perimeter of the resulting triangle formed by the tangent line and the two longer sides. | 16 | 12.5 |
15,485 | At 30 palm trees on different parts of an uninhabited island, a sign is attached.
- On 15 of them it says: "Exactly under 15 signs a treasure is buried."
- On 8 of them it says: "Exactly under 8 signs a treasure is buried."
- On 4 of them it says: "Exactly under 4 signs a treasure is buried."
- On 3 of them it says: "... | 15 | 18.75 |
15,486 | Five monkeys share a pile of peanuts. The first monkey divides the peanuts into five piles, leaving one peanut which it eats, and takes away one pile. The second monkey then divides the remaining peanuts into five piles, leaving exactly one peanut, eats it, and takes away one pile. This process continues in the same ma... | 3121 | 50 |
15,487 | Two skiers started from the same point one after another with an interval of 9 minutes. The second skier caught up with the first one 9 km from the starting point. After reaching the “27 km” mark, the second skier turned back and met the first skier at a distance of 2 km from the turning point. Find the speed of the se... | 15 | 16.40625 |
15,488 | In parallelogram \(ABCD\), \(AB = 1\), \(BC = 4\), and \(\angle ABC = 60^\circ\). Suppose that \(AC\) is extended from \(A\) to a point \(E\) beyond \(C\) so that triangle \(ADE\) has the same area as the parallelogram. Find the length of \(DE\). | 2\sqrt{3} | 0.78125 |
15,489 | The teacher asked the students to calculate \(\overline{AB} . C + D . E\). Xiao Hu accidentally missed the decimal point in \(D . E\), getting an incorrect result of 39.6; while Da Hu mistakenly saw the addition sign as a multiplication sign, getting an incorrect result of 36.9. What should the correct calculation resu... | 26.1 | 0 |
15,490 | There is a house at the center of a circular field. From it, 6 straight roads radiate, dividing the field into 6 equal sectors. Two geologists start their journey from the house, each choosing a road at random and traveling at a speed of 4 km/h. Determine the probability that the distance between them after an hour is ... | 0.5 | 28.90625 |
15,491 | Find all positive integers \( n \) such that \( n \) is equal to 100 times the number of positive divisors of \( n \). | 2000 | 87.5 |
15,492 | The scent of blooming lily of the valley bushes spreads within a radius of 20 meters around them. How many blooming lily of the valley bushes need to be planted along a straight 400-meter-long alley so that every point along the alley can smell the lily of the valley? | 10 | 61.71875 |
15,493 | The sequence $\left\{a_{n}\right\}$ is defined such that $a_{n}$ is the last digit of the sum $1 + 2 + \cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Calculate $S_{2016}$. | 7066 | 50.78125 |
15,494 | In the plane Cartesian coordinate system, the area of the region corresponding to the set of points $\{(x, y) \mid(|x|+|3 y|-6)(|3 x|+|y|-6) \leq 0\}$ is ________. | 24 | 30.46875 |
15,495 | A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles? | 1002 | 12.5 |
15,496 | In an isosceles triangle \(ABC\) with base \(AC\), point \(D\) divides side \(BC\) in the ratio \(3:1\) starting from vertex \(B\), and point \(E\) is the midpoint of segment \(AD\). It is known that \(BE = \sqrt{7}\) and \(CE = 3\). Find the radius of the circumcircle of triangle \(ABC\). | \frac{8}{3} | 6.25 |
15,497 | Find the maximum constant $k$ such that $\frac{k a b c}{a+b+c} \leqslant (a+b)^{2} + (a+b+4c)^{2}$ holds for all positive real numbers $a, b, c$. | 100 | 0 |
15,498 | At 1:00 PM, two identical recreational boats set off in opposite directions from a pier on a river. At the same time, a raft also departed from the pier. An hour later, one of the boats turned around and started moving back. The other boat did the same at 3:00 PM. What is the speed of the current if, at the moment the ... | 2.5 | 35.9375 |
15,499 | Ten points are given in the plane, and no three points are collinear. Four distinct segments connecting pairs of these points are chosen at random, all with the same probability. What is the probability that three of the chosen segments will form a triangle? | 16/473 | 18.75 |
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