Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
15,100 | Consider \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that
\[ A + A^{\times} = I_{2020} \]
\[ A \cdot A^{\times} = I_{2020} \]
where \(A^{\times}\) is the adjugate matrix of \(A\), i.e., the matrix whose elements are \(a_{ij} = (-1)^{i+j} d_{ji}\), where \(d_{ji}\) is the determinant obtained from \(A\), eliminating t... | 673 | 0 |
15,101 | A quadrilateral pyramid \(SABCD\) is given, with a base that is a trapezoid \(ABCD\). The ratio of the bases \(AD\) and \(BC\) of this trapezoid is 2. Construct the cross-section of the pyramid with a plane passing through point \(D\) and the midpoints of the edges \(SA\) and \(SB\). In what ratio does this plane divid... | 2:1 | 29.6875 |
15,102 | If \( n \) is a positive integer such that \( n^{6} + 206 \) is divisible by \( n^{2} + 2 \), find the sum of all possible values of \( n \). | 32 | 52.34375 |
15,103 | If the positive real numbers \( x \) and \( y \) satisfy \( x - 2 \sqrt{y} = \sqrt{2x - y} \), then the maximum value of \( x \) is ____ . | 10 | 36.71875 |
15,104 | Does there exist a positive integer \( m \) such that the equation \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = \frac{m}{a+b+c}\) has infinitely many solutions in positive integers \( (a, b, c) \)? | 12 | 8.59375 |
15,105 | Find the number of natural numbers \( k \) not exceeding 242400, such that \( k^2 + 2k \) is divisible by 303. | 3200 | 53.90625 |
15,106 | The side edge of a regular tetrahedron \( S-ABC \) is 2, and the base is an equilateral triangle with side length 1. A section passing through \( AB \) divides the volume of the tetrahedron into two equal parts. Find the cosine of the dihedral angle between this section and the base. | \frac{2}{\sqrt{15}} | 0 |
15,107 | Given the function \( f(x) = x^2 + x + \sqrt{3} \), if for all positive numbers \( a, b, c \), the inequality \( f\left(\frac{a+b+c}{3} - \sqrt[3]{abc}\right) \geq f\left(\lambda \left(\frac{a+b}{2} - \sqrt{ab}\right)\right) \) always holds, find the maximum value of the positive number \( \lambda \). | \frac{2}{3} | 16.40625 |
15,108 | Mila and Zhenya each came up with a number and wrote down all the natural divisors of their numbers on the board. Mila wrote down 10 numbers, Zhenya wrote down 9 numbers, and the number 6 appeared twice. How many distinct numbers are on the board in total? | 18 | 62.5 |
15,109 | Let \( f(x) = x^3 + 3x + 1 \), where \( x \) is a real number. Given that the inverse function of \( f \) exists and is given by
\[ f^{-1}(x) = \left( \frac{x - a + \sqrt{x^2 - bx + c}}{2} \right)^{1/3} + \left( \frac{x - a - \sqrt{x^2 - bx + c}}{2} \right)^{1/3} \]
where \( a \), \( b \), and \( c \) are positive cons... | 521 | 0.78125 |
15,110 | Given that \( m \) and \( n \) are two distinct positive integers and the last four digits of \( 2019^{m} \) and \( 2019^{n} \) are the same, find the minimum value of \( m+n \). | 502 | 54.6875 |
15,111 | In a tetrahedron \( ABCD \), \( AB = AC = AD = 5 \), \( BC = 3 \), \( CD = 4 \), \( DB = 5 \). Find the volume of this tetrahedron. | 5\sqrt{3} | 31.25 |
15,112 | There are 4 problems in a mathematics competition. The scores are allocated as follows: 2 marks for a correct answer, -1 mark for a wrong answer, and 0 marks for a blank answer. To ensure that 3 candidates will have the same scores, how many candidates, denoted as $S$, must there be at least in the competition? Find th... | 25 | 3.90625 |
15,113 | Chester is traveling from Hualien to Lugang, Changhua, to participate in the Hua Luogeng Golden Cup Mathematics Competition. Before setting off, his father checked the car's odometer, which read a palindromic number of 69,696 kilometers (a palindromic number remains the same when read forward or backward). After drivin... | 82.2 | 10.9375 |
15,114 | If for a number \( x \) you calculate the sum of its digits and repeat this process two more times with the resulting number, you get a sequence of four numbers. Find the smallest \( x \) for which all four numbers are distinct and the last number is 2. | 2999 | 89.0625 |
15,115 | Find the sum of the digits of the number \( A \), if \( A=2^{63} \cdot 4^{25} \cdot 5^{106}-2^{22} \cdot 4^{44} \cdot 5^{105}-1 \). | 959 | 17.96875 |
15,116 | Let \( x_{1}, y_{1}, x_{2}, y_{2} \) be real numbers satisfying the equations \( x_{1}^{2}+5 x_{2}^{2}=10 \), \( x_{2} y_{1}-x_{1} y_{2}=5 \) and \( x_{1} y_{1}+5 x_{2} y_{2}=\sqrt{105} \). Find the value of \( y_{1}^{2}+5 y_{2}^{2} \). | 23 | 33.59375 |
15,117 | A chocolate bar originally weighed 400 grams and cost 150 rubles. Recently, to save money, the manufacturer reduced the weight of the bar to 300 grams and increased its price to 180 rubles. By what percentage did the manufacturer's revenue increase? | 60 | 65.625 |
15,118 | Given a sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) such that \( a_{1}=a_{2}=1 \), \( a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \). Additionally, it holds that \( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2} + a_{n+3} \). Determine the value of \( a_{1} + a_{2} ... | 200 | 60.9375 |
15,119 | In a certain city, the rules for selecting license plate numbers online are as follows: The last five characters of the plate must include two English letters (with the letters "I" and "O" not allowed), and the last character must be a number. How many possible combinations meet these requirements? | 3456000 | 0.78125 |
15,120 | During a fireworks display, a body is launched upwards with an initial velocity of $c=90 \mathrm{m/s}$. We hear its explosion $t=5$ seconds later. At what height did it explode if the speed of sound is $a=340 \mathrm{m/s}$? (Air resistance is neglected.) | 289 | 0 |
15,121 | A rectangular table of size \( x \) 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 1 cm higher and 1 cm to the right of the previous one. The last sheet is adjacent to the top-right ... | 77 | 29.6875 |
15,122 | A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?
The perimeter of a shape is the sum of its side lengths. | 52 | 12.5 |
15,123 | When Ma Xiaohu was doing a subtraction problem, he mistakenly wrote the units digit of the minuend as 5 instead of 3, and the tens digit as 0 instead of 6. Additionally, he wrote the hundreds digit of the subtrahend as 2 instead of 7. The resulting difference was 1994. What should the correct difference be? | 1552 | 20.3125 |
15,124 | Given that \(a\) and \(b\) are real numbers, and the equation \( x^{4} + a x^{3} + b x^{2} + a x + 1 = 0 \) has at least one real root, find the minimum value of \(a^{2} + b^{2}\). | 4/5 | 31.25 |
15,125 | Dima and Sergey were picking berries from a raspberry bush that had 900 berries. Dima alternated his actions: he put one berry in the basket and ate the next one. Sergey also alternated his actions: he put two berries in the basket and ate the next one. It is known that Dima picks berries twice as fast as Sergey. At so... | 100 | 19.53125 |
15,126 | Jolene and Tia are playing a two-player game at a carnival. In one bin, there are five red balls numbered 5, 10, 15, 20, and 25. In another bin, there are 25 green balls numbered 1 through 25. In the first stage of the game, Jolene chooses one of the red balls at random. Next, the carnival worker removes the green ball... | 13/40 | 11.71875 |
15,127 | A package of milk with a volume of 1 liter cost 60 rubles. Recently, for the purpose of economy, the manufacturer reduced the package volume to 0.9 liters and increased its price to 81 rubles. By what percentage did the manufacturer's revenue increase? | 50 | 75.78125 |
15,128 | In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Glafira discovered that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it in 30 days. How many days will it take for one cow to drink this pon... | 75 | 54.6875 |
15,129 | Inside triangle \(ABC\), a random point \(M\) is chosen. What is the probability that the area of one of the triangles \(ABM\), \(BCM\), and \(CAM\) will be greater than the sum of the areas of the other two? | 0.75 | 0 |
15,130 | The volume of a hemispherical soup bowl is 8 liters. How much soup fills the bowl up to half its height? | 2.5 | 67.96875 |
15,131 | Find the sum of all the roots of the equation \( 4x^{2} - 58x + 190 = (29 - 4x - \log_{2} x) \cdot \log_{2} x \). | 12 | 6.25 |
15,132 | How many natural numbers are there whose square and cube together require 10 digits to describe? | 53 | 40.625 |
15,133 | Given real numbers \( x, y, z, w \) satisfying \( x + y + z + w = 1 \), find the maximum value of \( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \). | \frac{3}{2} | 54.6875 |
15,134 | Vasya wrote a note on a piece of paper, folded it in quarters, and wrote "MAME" on top. He then unfolded the note, added something more, folded it again randomly along the crease lines (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" re... | 1/8 | 10.15625 |
15,135 | Let \( g(n) = (n^2 - 2n + 1)^{1/3} + (n^2 - 1)^{1/3} + (n^2 + 2n + 1)^{1/3} \). Find \( \frac{1}{g(1)} + \frac{1}{g(3)} + \frac{1}{g(5)} + \ldots + \frac{1}{g(999999)} \). | 50 | 0 |
15,136 | Let \(C\) be a cube with side length 4 and center \(O\). Let \(S\) be the sphere centered at \(O\) with radius 2. Let \(A\) be one of the vertices of the cube. Let \(R\) be the set of points in \(C\) but not in \(S\), which are closer to \(A\) than to any other vertex of \(C\). Find the volume of \(R\). | 8 - \frac{4\pi}{3} | 50.78125 |
15,137 | For each vertex of the triangle \(ABC\), the angle between the altitude and the angle bisector drawn from that vertex was determined. It turned out that these angles at vertices \(A\) and \(B\) are equal to each other and are less than the angle at vertex \(C\). What is the measure of angle \(C\) in the triangle? | 60 | 17.1875 |
15,138 | Given the constraints \(x + 2y \leq 5\), \(2x + y \leq 4\), \(x \geq 0\), and \(y \geq 0\), find the coordinates \((x, y)\) where \(3x + 4y\) achieves its maximum value, and determine that maximum value. | 11 | 14.84375 |
15,139 | How many natural numbers greater than one have a product with their smallest prime divisor that is not greater than 100? | 33 | 0 |
15,140 | The denominator of the fraction $15 \cdot 18$ in simplest form is 30. Find the sum of all such positive rational numbers less than 10. | 400 | 25 |
15,141 | An even perfect square in the decimal system is of the form: $\overline{a b 1 a b}$. What is this perfect square? | 76176 | 16.40625 |
15,142 | Find the smallest positive integer \( n \) for which there are exactly 2323 positive integers less than or equal to \( n \) that are divisible by 2 or 23, but not both. | 4644 | 69.53125 |
15,143 | Find the only value of \( x \) in the open interval \((- \pi / 2, 0)\) that satisfies the equation
$$
\frac{\sqrt{3}}{\sin x} + \frac{1}{\cos x} = 4.
$$ | -\frac{4\pi}{9} | 4.6875 |
15,144 | A number is reduced by 5 times and then increased by 20 times to get 40. What is this number? | 10 | 91.40625 |
15,145 | In the isosceles trapezoid \(ABCD\) with bases \(AD\) and \(BC\), perpendiculars \(BH\) and \(DK\) are drawn from vertices \(B\) and \(D\) to the diagonal \(AC\). It is known that the feet of the perpendiculars lie on the segment \(AC\) and that \(AC = 20\), \(AK = 19\), and \(AH = 3\). Find the area of the trapezoid \... | 120 | 0 |
15,146 | The numbers \( a, b, c, d \) belong to the interval \([-5, 5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 110 | 96.875 |
15,147 | In a regular \( n \)-gon, \( A_{1} A_{2} A_{3} \cdots A_{n} \), where \( n > 6 \), sides \( A_{1} A_{2} \) and \( A_{5} A_{4} \) are extended to meet at point \( P \). If \( \angle A_{2} P A_{4}=120^\circ \), determine the value of \( n \). | 18 | 1.5625 |
15,148 | Li Yun is sitting by the window in a train moving at a speed of 60 km/h. He sees a freight train with 30 cars approaching from the opposite direction. When the head of the freight train passes the window, he starts timing, and he stops timing when the last car passes the window. The recorded time is 18 seconds. Given t... | 44 | 3.125 |
15,149 | If any two adjacent digits of a three-digit number have a difference of at most 1, it is called a "steady number". How many steady numbers are there? | 75 | 85.9375 |
15,150 | The product of all the positive integer divisors of an integer is $2^{120} \cdot 3^{60} \cdot 5^{90}$. What could this integer be? | 18000 | 60.15625 |
15,151 | If the height on the base of an isosceles triangle is $18 \mathrm{~cm}$ and the median on the leg is $15 \mathrm{~cm}$, what is the area of this isosceles triangle? | 144 | 14.84375 |
15,152 |
Calculate the definite integral:
$$
\int_{1}^{e^{2}} \frac{\ln ^{2} x}{\sqrt{x}} \, dx
$$ | 24e - 32 | 10.9375 |
15,153 | On the island of Liars and Knights, a circular arrangement is called correct if everyone standing in the circle can say that among his two neighbors there is a representative of his tribe. One day, 2019 natives formed a correct arrangement in a circle. A liar approached them and said: "Now together we can also form a c... | 1346 | 0 |
15,154 | What is the smallest sum that nine consecutive natural numbers can have if this sum ends in 2050306? | 22050306 | 13.28125 |
15,155 | Let \( n \) be a positive integer not exceeding 1996. If there exists a \( \theta \) such that \( (\sin \theta + i \cos \theta)^{n} = \sin \theta + i \cos n \theta \), find the number of possible values for \( n \). | 499 | 39.84375 |
15,156 | A five-digit number is called a "hill" if its first three digits are in ascending order and its last three digits are in descending order. For example, 13760 and 28932 are hills, whereas 78821 and 86521 are not hills. How many hills exist that are greater than the number 77777? | 36 | 67.1875 |
15,157 | In triangle \( ABC \), side \( AC \) is the largest. Points \( M \) and \( N \) on side \( AC \) are such that \( AM = AB \) and \( CN = CB \). It is known that angle \( \angle NBM \) is three times smaller than angle \( \angle ABC \). Find \( \angle ABC \). | 108 | 21.875 |
15,158 | Find the numbers \( x \) between 0 and 30 for which the sine of \( x \) degrees equals the sine of \( x \) radians. How many such numbers exist between 30 and 90? | 10 | 0.78125 |
15,159 | Given that the four vertices of the triangular pyramid $P-ABC$ lie on the surface of the sphere $O$, and $PA = PB = PC$. The triangle $ABC$ is an equilateral triangle with side length 2. Points $E$ and $F$ are the midpoints of $AC$ and $BC$ respectively, and $\angle EPF = 60^\circ$. Find the surface area of the sphere ... | 6 \pi | 3.90625 |
15,160 | Let's call a year interesting if a person turns the number of years equal to the sum of the digits of the year of their birth in that year. A certain year turned out to be interesting for Ivan, who was born in the 20th century, and for Vovochka, who was born in the 21st century. What is the difference in their ages?
N... | 18 | 25.78125 |
15,161 | Call an integer \( n > 1 \) radical if \( 2^n - 1 \) is prime. What is the 20th smallest radical number? | 4423 | 38.28125 |
15,162 | There are 10 different natural numbers, their sum is 604, and these 10 numbers have the same sum of digits. What is the largest number among these 10 numbers? $\qquad | 109 | 2.34375 |
15,163 | There are three sets of cards in red, yellow, and blue, with five cards in each set, labeled with the letters $A, B, C, D,$ and $E$. If 5 cards are drawn from these 15 cards, with the condition that all letters must be different and all three colors must be included, how many different ways are there to draw the cards? | 150 | 11.71875 |
15,164 | Find the maximum value of the expression \( (\sin 2x + \sin 3y + \sin 4z)(\cos 2x + \cos 3y + \cos 4z) \). | 4.5 | 0.78125 |
15,165 | The denominators of two irreducible fractions are 600 and 700. What is the smallest possible value of the denominator of their sum when written as an irreducible fraction?
Note: We say that the fraction \( p / q \) is irreducible if the integers \( p \) and \( q \) do not have common prime factors in their factorizati... | 168 | 31.25 |
15,166 | Let \(ABCD\) be a convex trapezoid such that \(\angle BAD = \angle ADC = 90^{\circ}\), \(AB = 20\), \(AD = 21\), and \(CD = 28\). Point \(P \neq A\) is chosen on segment \(AC\) such that \(\angle BPD = 90^{\circ}\). Compute \(AP\). | 143/5 | 17.1875 |
15,167 | In a round-robin hockey tournament, 2016 teams participated. According to the tournament rules, 3 points are awarded for a win. | 6093360 | 28.125 |
15,168 | Given the natural numbers $1,2,3,\ldots,10,11,12$, divide them into two groups such that the quotient of the product of all numbers in the first group by the product of all numbers in the second group is an integer and takes on the smallest possible value. What is this quotient? | 231 | 51.5625 |
15,169 | Name the smallest four-digit number in which all digits are different and the second digit is 6. | 1602 | 76.5625 |
15,170 |
Two people are flipping a coin: one flipped it 10 times, and the other 11 times.
What is the probability that the second person gets more heads than the first person? | \frac{1}{2} | 78.90625 |
15,171 | A circle with radius 1 is tangent to a circle with radius 3 at point \( C \). A line passing through point \( C \) intersects the smaller circle at point \( A \) and the larger circle at point \( B \). Find \( AC \), given that \( AB = 2\sqrt{5} \). | \frac{\sqrt{5}}{2} | 2.34375 |
15,172 | In the pyramid \(ABCD\), points \(M\), \(F\), and \(K\) are the midpoints of edges \(BC\), \(AD\), and \(CD\) respectively. Points \(P\) and \(Q\) are chosen on lines \(AM\) and \(CF\) respectively such that \(PQ \parallel BK\). Find the ratio \(PQ : BK\). | 2:5 | 0.78125 |
15,173 | Nikola had one three-digit number and one two-digit number. Each of these numbers was positive and made up of different digits. The difference between Nikola's numbers was 976.
What was their sum? | 996 | 9.375 |
15,174 | A confectionery factory received 5 rolls of ribbon, each 50 meters long, for packing cakes. How many cuts are needed to obtain pieces of ribbon that are 2 meters each?
| 120 | 3.90625 |
15,175 | Let \(ABCD\) be a quadrilateral with \(BC = CD = DA = 1\), \(\angle DAB = 135^\circ\), and \(\angle ABC = 75^\circ\). Find \(AB\). | \frac{\sqrt{6}-\sqrt{2}}{2} | 3.90625 |
15,176 | Consider the expression \(1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6\).
Each star in the expression is to be replaced with either ' + ' or ' \times '.
\(N\) is the largest possible value of the expression. What is the largest prime factor of \(N\)? | 103 | 5.46875 |
15,177 | There are 456 natives on an island, each of whom is either a knight who always tells the truth or a liar who always lies. All residents have different heights. Once, each native said, "All other residents are shorter than me!" What is the maximum number of natives who could have then said one minute later, "All other r... | 454 | 5.46875 |
15,178 | There are two fair dice and their sides are positive integers \( a_{1}, \ldots, a_{6} \) and \( b_{1}, \ldots, b_{6} \), respectively. After throwing them, the probability of getting a sum of \( 2, 3, 4, \ldots, 12 \) respectively is the same as that of throwing two normal fair dice. Suppose that \( a_{1}+\cdots+a_{6} ... | 15 | 25 |
15,179 | Inside the cube \( ABCD A_1B_1C_1D_1 \), there is a center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( AA_1D_1D \) along a circle of radius 1, the face \( A_1B_1C_1D_1 \) along a circle of radius 1, and the face \( CDD_1C_1 \) along a circle of radius 3. Find the length of the segment \( ... | 17 | 44.53125 |
15,180 | A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers that are both multiples of 45? | 9090 | 54.6875 |
15,181 | Suppose \( a, b \), and \( c \) are real numbers with \( a < b < 0 < c \). Let \( f(x) \) be the quadratic function \( f(x) = (x-a)(x-c) \) and \( g(x) \) be the cubic function \( g(x) = (x-a)(x-b)(x-c) \). Both \( f(x) \) and \( g(x) \) have the same \( y \)-intercept of -8 and \( g(x) \) passes through the point \( (... | \frac{8}{3} | 22.65625 |
15,182 | In the rhombus \(A B C D\), the angle at vertex \(A\) is \(60^{\circ}\). Point \(N\) divides side \(A B\) in the ratio \(A N: B N = 2: 1\). Find the tangent of angle \(D N C\). | \sqrt{\frac{243}{121}} | 0 |
15,183 | Two differentiable real functions \( f(x) \) and \( g(x) \) satisfy
\[ \frac{f^{\prime}(x)}{g^{\prime}(x)} = e^{f(x) - g(x)} \]
for all \( x \), and \( f(0) = g(2003) = 1 \). Find the largest constant \( c \) such that \( f(2003) > c \) for all such functions \( f, g \). | 1 - \ln 2 | 0.78125 |
15,184 | Suppose you have three children and 40 pieces of candy. How many ways are there to distribute the candy such that each child gets more than one but fewer than 20 pieces? | 171 | 46.09375 |
15,185 | In rectangle \(ABCD\), a point \(E\) is marked on the extension of side \(CD\) beyond point \(D\). The bisector of angle \(ABC\) intersects side \(AD\) at point \(K\), and the bisector of angle \(ADE\) intersects the extension of side \(AB\) at point \(M\). Find \(BC\) if \(MK = 8\) and \(AB = 3\). | \sqrt{55} | 14.0625 |
15,186 | Find a four-digit number that is a perfect square, knowing that the first two digits, as well as the last two digits, are each equal to each other. | 7744 | 96.875 |
15,187 | A point is randomly thrown on the segment [12, 17] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+k-90\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | 2/3 | 53.90625 |
15,188 | Let \( S = \{1, 2, \ldots, 1963\} \). What is the maximum number of elements that can be chosen from \( S \) such that the sum of any two chosen numbers is not divisible by their difference? | 655 | 53.90625 |
15,189 | The chord \( AB \) divides the circle into two arcs, with the smaller arc being \( 130^{\circ} \). The larger arc is divided by chord \( AC \) in the ratio \( 31:15 \) from point \( A \). Find the angle \( BAC \). | 37.5 | 20.3125 |
15,190 | A three-digit number has a remainder of 2 when divided by 4, 5, and 6. If three digits are appended to this three-digit number to make it a six-digit number divisible by 4, 5, and 6, what is the smallest six-digit number that meets this condition? | 122040 | 40.625 |
15,191 | A rectangular table with dimensions $x$ cm $\times 80$ cm is covered with identical sheets of paper measuring 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 sheet. The last sheet is positi... | 77 | 28.125 |
15,192 | On a table, there are 2020 boxes. Some of them contain candies, while others are empty. The first box has a label that reads: "All boxes are empty." The second box reads: "At least 2019 boxes are empty." The third box reads: "At least 2018 boxes are empty," and so on, up to the 2020th box, which reads: "At least one bo... | 1010 | 19.53125 |
15,193 | In a 12-hour interval (from 0 hours to 12 hours), how many minutes are there when the value of the hour is greater than the value of the minutes? | 66 | 4.6875 |
15,194 | There are 90 children in a chess club. During a session, they were divided into 30 groups of 3 people each, and in each group, everyone played one game with everyone else. No other games were played. A total of 30 "boy vs. boy" games and 14 "girl vs. girl" games were played. How many "mixed" groups were there, i.e., gr... | 23 | 17.1875 |
15,195 | Given a regular square pyramid \( P-ABCD \) with a base side length \( AB=2 \) and height \( PO=3 \). \( O' \) is a point on the segment \( PO \). Through \( O' \), a plane parallel to the base of the pyramid is drawn, intersecting the edges \( PA, PB, PC, \) and \( PD \) at points \( A', B', C', \) and \( D' \) respec... | 16/27 | 3.90625 |
15,196 | Given \( f(x) = \max \left| x^3 - a x^2 - b x - c \right| \) for \( 1 \leq x \leq 3 \), find the minimum value of \( f(x) \) as \( a, b, \) and \( c \) range over all real numbers. | 1/4 | 60.15625 |
15,197 | A container holds one liter of wine, and another holds one liter of water. From the first container, we pour one deciliter into the second container and mix thoroughly. Then, we pour one deciliter of the mixture back into the first container. Calculate the limit of the amount of wine in the first container if this proc... | \frac{1}{2} | 14.0625 |
15,198 | Given the function
\[ f(x) = x^2 - (k^2 - 5ak + 3)x + 7 \quad (a, k \in \mathbb{R}) \]
for any \( k \in [0, 2] \), if \( x_1, x_2 \) satisfy
\[ x_1 \in [k, k+a], \quad x_2 \in [k+2a, k+4a], \]
then \( f(x_1) \geq f(x_2) \). Find the maximum value of the positive real number \( a \). | \frac{2 \sqrt{6} - 4}{5} | 0 |
15,199 | Four cars $A$, $B$, $C$, and $D$ start simultaneously from the same point on a circular track. Cars $A$ and $B$ travel clockwise, while cars $C$ and $D$ travel counterclockwise. All cars move at constant but distinct speeds. Exactly 7 minutes after the race starts, $A$ meets $C$ for the first time, and at the same mome... | 53 | 7.03125 |
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