Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
13,900 | The numbers \( x \) and \( y \) are such that the equalities \( \operatorname{ctg} x - \operatorname{ctg} y = 2 \) and \( 5 \sin (2x - 2y) = \sin 2x \sin 2y \) hold. Find \( \operatorname{tg} x \operatorname{tg} y \). | -\frac{6}{5} | 23.4375 |
13,901 | The diameters of two pulleys with parallel axes are 80 mm and 200 mm, respectively, and they are connected by a belt that is 1500 mm long. What is the distance between the axes of the pulleys if the belt is tight (with millimeter precision)? | 527 | 0 |
13,902 | In the trapezoid \(MPQF\), the bases are \(MF = 24\) and \(PQ = 4\). The height of the trapezoid is 5. Point \(N\) divides the side into segments \(MN\) and \(NP\) such that \(MN = 3NP\).
Find the area of triangle \(NQF\). | 22.5 | 2.34375 |
13,903 | The numbers \( 62, 63, 64, 65, 66, 67, 68, 69, \) and \( 70 \) are divided by, in some order, the numbers \( 1, 2, 3, 4, 5, 6, 7, 8, \) and \( 9 \), resulting in nine integers. The sum of these nine integers is \( S \). What are the possible values of \( S \)? | 187 | 0.78125 |
13,904 | Suppose a regular tetrahedron \( P-ABCD \) has all edges equal in length. Using \(ABCD\) as one face, construct a cube \(ABCD-EFGH\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \( PA \) and \( CF \). | \frac{2 + \sqrt{2}}{4} | 0 |
13,905 | Arrange all positive integers whose digits sum to 8 in ascending order to form a sequence $\{a_n\}$, called the $P$ sequence. Then identify the position of 2015 within this sequence. | 83 | 0 |
13,906 | Given an equilateral triangle with side length \( n \), divided into unit triangles as illustrated, let \( f(n) \) be the number of paths from the top-row triangle to the triangle in the center of the bottom row. The path must move through adjacent triangles sharing a common edge, never revisiting any triangle, and nev... | 2011! | 0 |
13,907 | Given that (1+ex)<sup>2019</sup>=a<sub>0</sub>+a<sub>1</sub>x+a<sub>2</sub>x<sup>2</sup>+……+a<sub>2019</sub>x<sup>2019</sup>, find the value of:
- $$\frac {a_{1}}{e}$$+ $$\frac {a_{2}}{e^{2}}$$\- $$\frac {a_{3}}{e^{3}}$$+ $$\frac {a_{4}}{e^{4}}$$\-……- $$\frac {a_{2019}}{e^{2019}}$$ | -1 | 7.03125 |
13,908 | Petya has seven cards with the digits 2, 2, 3, 4, 5, 6, 8. He wants to use all the cards to form the largest natural number that is divisible by 12. What number should he get? | 8654232 | 42.96875 |
13,909 | From an external point \(A\), a tangent \(AB\) and a secant \(ACD\) are drawn to a circle. Find the area of triangle \(CBD\), given that the ratio \(AC : AB = 2 : 3\) and the area of triangle \(ABC\) is 20. | 25 | 83.59375 |
13,910 | Given a six-digit phone number, how many different seven-digit phone numbers exist such that, by crossing out one digit, you obtain the given six-digit number? | 70 | 45.3125 |
13,911 | The diagonals of a trapezoid are mutually perpendicular, and one of them is 13. Find the area of the trapezoid if its height is 12. | 1014/5 | 0 |
13,912 | For a natural number $N$, if at least five out of the nine natural numbers $1-9$ can divide $N$, then $N$ is called a "five-divisible number". What is the smallest "five-divisible number" greater than 2000? | 2004 | 98.4375 |
13,913 | How many days have passed from March 19, 1990, to March 23, 1996, inclusive? | 2197 | 12.5 |
13,914 | In triangle \(ABC\), angle \(C\) is \(60^\circ\) and the radius of the circumcircle of this triangle is \(2\sqrt{3}\).
A point \(D\) is taken on the side \(AB\) such that \(AD = 2DB\) and \(CD = 2\sqrt{2}\). Find the area of triangle \(ABC\). | 3\sqrt{2} | 0 |
13,915 | The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers. | 10 | 17.1875 |
13,916 | On the continuation of side \( BC \) of parallelogram \( ABCD \), a point \( F \) is taken beyond point \( C \). Segment \( AF \) intersects diagonal \( BD \) at point \( E \) and side \( CD \) at point \( G \), where \( GF=3 \) and \( AE \) is 1 more than \( EG \). What part of the area of parallelogram \( ABCD \) is ... | \frac{1}{6} | 14.0625 |
13,917 | There is a basket of apples. After dividing them into three equal parts, 2 apples remain. Taking out two of these parts, and dividing them into three equal parts again, 2 apples remain. After taking out two of these parts again and dividing them into three equal parts, 2 apples remain. How many apples are in the basket... | 23 | 57.8125 |
13,918 | There is a strip with a length of 100, and each cell of the strip contains a chip. You can swap any two adjacent chips for 1 ruble, or you can swap any two chips that have exactly three chips between them for free. What is the minimum number of rubles needed to rearrange the chips in reverse order? | 50 | 26.5625 |
13,919 | The equation \(x^{2}+5x+1=0\) has roots \(x_{1}\) and \(x_{2}\). Find the value of the expression
\[
\left(\frac{x_{1} \sqrt{6}}{1+x_{2}}\right)^{2}+\left(\frac{x_{2} \sqrt{6}}{1+x_{1}}\right)^{2}
\] | 220 | 52.34375 |
13,920 | Given fifty distinct natural numbers, twenty-five of which do not exceed 50, and the remaining are greater than 50 but do not exceed 100. Additionally, no two of these numbers differ by exactly 50. Find the sum of these numbers. | 2525 | 8.59375 |
13,921 | Find all the roots of the equation
\[ 1 - \frac{x}{1} + \frac{x(x-1)}{2!} - \frac{x(x-1)(x-2)}{3!} + \frac{x(x-1)(x-2)(x-3)}{4!} - \frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0 \]
(Where \( n! = 1 \cdot 2 \cdot 3 \cdots n \))
In the answer, specify the sum of the found roots. | 21 | 21.09375 |
13,922 | The function \( f \) satisfies the equation \((x-1) f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x-1}\) for each value of \( x \) not equal to 0 and 1. Find \( f\left(\frac{2016}{2017}\right) \). | 2017 | 14.84375 |
13,923 | Find all positive real numbers \(c\) such that the graph of \(f: \mathbb{R} \rightarrow \mathbb{R}\) given by \(f(x) = x^3 - cx\) has the property that the circle of curvature at any local extremum is centered at a point on the \(x\)-axis. | \frac{\sqrt{3}}{2} | 39.84375 |
13,924 | Winnie the Pooh decided to give Piglet a birthday cake in the shape of a regular hexagon. On his way, he got hungry and cut off 6 pieces from the cake, each containing one vertex and one-third of a side of the hexagon (see the illustration). As a result, he gave Piglet a cake weighing 900 grams. How many grams of the c... | 112.5 | 2.34375 |
13,925 | A notebook sheet is painted in 23 colors, with each cell in the sheet painted in one of these colors. A pair of colors is called "good" if there exist two adjacent cells painted in these colors. What is the minimum number of good pairs? | 22 | 35.9375 |
13,926 | A large batch of tires contains $1.5\%$ defects. What should be the sample size for the probability of finding at least one defective tire in the sample to be more than $0.92 ?$ | 168 | 14.0625 |
13,927 | If Greg rolls five fair eight-sided dice, what is the probability that he rolls more 1's than 8's? | \frac{10246}{32768} | 0 |
13,928 | For which values of the parameter \(a\) does the equation \(x^{4} - 40 x^{2} + 144 = a(x^{2} + 4x - 12)\) have exactly three distinct solutions? | 48 | 0.78125 |
13,929 | Three children need to cover a distance of 84 kilometers using two bicycles. Walking, they cover 5 kilometers per hour, while bicycling they cover 20 kilometers per hour. How long will it take for all three to reach the destination if only one child can ride a bicycle at a time? | 8.4 | 6.25 |
13,930 |
On an island, there are two tribes: knights and liars. Knights always tell the truth, and liars always lie. One day, 80 people sat at a round table, and each of them declared: "Among the 11 people sitting immediately after me in a clockwise direction, there are at least 9 liars." How many knights are sitting at the ro... | 20 | 2.34375 |
13,931 | Two vertices of a square with an area of \( 256 \, \text{cm}^2 \) lie on a circle, while the other two vertices lie on a tangent to this circle. Find the radius of the circle. | 10 | 29.6875 |
13,932 | $ABCD$ forms a rhombus. $E$ is the intersection of $AC$ and $BD$ . $F$ lie on $AD$ such that $EF$ is perpendicular to $FD$ . Given $EF=2$ and $FD=1$ . Find the area of the rhombus $ABCD$ | 20 | 6.25 |
13,933 | At the end of the school year, teachers of the third grade met with the parents of some of their students; exactly 31 people were present at this meeting. The Latin teacher was asked questions by 16 parents, the French teacher by 17 parents, the English teacher by 18 parents, and so on up to the Math teacher, who was a... | 23 | 1.5625 |
13,934 | Robot Petya displays three three-digit numbers every minute, which sum up to 2019. Robot Vasya swaps the first and last digits of each of these numbers and then sums the resulting numbers. What is the maximum sum that Vasya can obtain? | 2118 | 4.6875 |
13,935 | Given the parabola \( y^{2} = 2 p x \) with focus \( F \) and directrix \( l \), a line passing through \( F \) intersects the parabola at points \( A \) and \( B \) such that \( |AB| = 3p \). Let \( A' \) and \( B' \) be the projections of \( A \) and \( B \) onto \( l \), respectively. If a point \( M \) is randomly ... | 1/3 | 29.6875 |
13,936 | In square \(ABCD\) with a side length of 10, points \(P\) and \(Q\) lie on the segment joining the midpoints of sides \(AD\) and \(BC\). Connecting \(PA\), \(PC\), \(QA\), and \(QC\) divides the square into three regions of equal area. Find the length of segment \(PQ\). | 20/3 | 8.59375 |
13,937 | How many positive rational numbers less than \(\pi\) have denominator at most 7 when written in lowest terms? (Integers have denominator 1.) | 54 | 91.40625 |
13,938 | Given a quadratic function $f(x) = ax^2 + bx + 1$ that satisfies $f(-1) = 0$, and when $x \in \mathbb{R}$, the range of $f(x)$ is $[0, +\infty)$.
(1) Find the expression for $f(x)$.
(2) Let $g(x) = f(x) - 2kx$, where $k \in \mathbb{R}$.
(i) If $g(x)$ is monotonic on $x \in [-2, 2]$, find the range of the real ... | k = 6 | 12.5 |
13,939 | How many points can be placed inside a circle of radius 2 such that one of the points coincides with the center of the circle and the distance between any two points is not less than 1? | 19 | 5.46875 |
13,940 | Determine the largest natural number \( n \) so that
\[ 4^{995} + 4^{1500} + 4^{n} \]
is a square number. | 2004 | 14.0625 |
13,941 | Find all three-digit integers \( abc = n \) such that \( \frac{2n}{3} = a! \cdot b! \cdot c! \). | 432 | 98.4375 |
13,942 | Mac is trying to fill 2012 barrels with apple cider. He starts with 0 energy. Every minute, he may rest, gaining 1 energy, or if he has \( n \) energy, he may expend \( k \) energy \((0 \leq k \leq n)\) to fill up to \( n(k+1) \) barrels with cider. What is the minimal number of minutes he needs to fill all the barrels... | 46 | 0.78125 |
13,943 | \( n \) is a positive integer that is not greater than 100 and not less than 10, and \( n \) is a multiple of the sum of its digits. How many such \( n \) are there? | 24 | 91.40625 |
13,944 | Given that the probability mass function of the random variable $X$ is $P(X=k)= \frac{k}{25}$ for $k=1, 2, 3, 4, 5$, find the value of $P(\frac{1}{2} < X < \frac{5}{2})$. | \frac{1}{5} | 0 |
13,945 | A school is hosting a Mathematics Culture Festival, and it was recorded that on that day, there were more than 980 (at least 980 and less than 990) students visiting. Each student visits the school for a period of time and then leaves, and once they leave, they do not return. Regardless of how these students schedule t... | 32 | 20.3125 |
13,946 | Given that \( 169(157 - 77x)^2 + 100(201 - 100x)^2 = 26(77x - 157)(1000x - 2010) \), find the value of \( x \). | 31 | 88.28125 |
13,947 | There are 17 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). All of them said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle? | 11 | 38.28125 |
13,948 | If \( f(x) = x^{6} - 2 \sqrt{2006} x^{5} - x^{4} + x^{3} - 2 \sqrt{2007} x^{2} + 2 x - \sqrt{2006} \), then find \( f(\sqrt{2006} + \sqrt{2007}) \). | \sqrt{2007} | 0 |
13,949 | Let's call a natural number "remarkable" if all of its digits are different, it does not start with the digit 2, and by removing some of its digits, the number 2018 can be obtained. How many different seven-digit "remarkable" numbers exist? | 1800 | 1.5625 |
13,950 | The strengths of the two players are equal, meaning they have equal chances of winning each game. They agreed that the prize would go to the first player to win 6 games. They had to stop the game after the first player won 5 games and the second won 3. In what proportion should the prize be fairly divided? | 7:1 | 28.90625 |
13,951 | The base of a pyramid is a parallelogram with sides measuring 10 cm and 18 cm, and an area of 90 cm². The height of the pyramid passes through the intersection point of the diagonals of the base and is 6 cm. Determine the lateral surface area of the pyramid. | 192 | 1.5625 |
13,952 | Let $u_0 = \frac{1}{3}$, and for $k \ge 0$, let $u_{k+1} = \frac{3}{2}u_k - \frac{3}{2}u_k^2$. This sequence tends to a limit; call it $M$. Determine the least value of $k$ such that $|u_k - M| \le \frac{1}{2^{1000}}$. | 10 | 26.5625 |
13,953 | From a plywood circle with a diameter of 30 cm, two smaller circles with diameters of 20 cm and 10 cm are cut out. What is the diameter of the largest circle that can be cut from the remaining piece of plywood? | 20 | 37.5 |
13,954 | In the two regular tetrahedra \(A-OBC\) and \(D-OBC\) with coinciding bases, \(M\) and \(N\) are the centroids of \(\triangle ADC\) and \(\triangle BDC\) respectively. Let \(\overrightarrow{OA}=\boldsymbol{a}, \overrightarrow{OB}=\boldsymbol{b}, \overrightarrow{OC}=\boldsymbol{c}\). If point \(P\) satisfies \(\overrigh... | 439 | 0 |
13,955 | The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home for the pen and then goes to school at t... | \frac{1}{4} | 3.125 |
13,956 | Given points $A\left(\begin{matrix} \cos \alpha , & \sin \alpha \end{matrix}\right)$ and $B\left(\begin{matrix} \cos \beta , & \sin \beta \end{matrix}\right)$, where $\alpha$ and $\beta$ are acute angles, and the distance between $A$ and $B$ is $\frac{\sqrt{10}}{5}$.
$(1)$ Find the value of $\cos (\alpha -\beta)$; $(2... | \frac{24}{25} | 64.84375 |
13,957 | A positive integer n is called *primary divisor* if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$ , $2$ , $4$ , and $8$ each differ by $1$ from a prime number ( $2$ , $3$ , $5$ , a... | 48 | 0 |
13,958 | Given vectors $\overrightarrow{a} = (x, -3)$, $\overrightarrow{b} = (-2, 1)$, $\overrightarrow{c} = (1, y)$ on a plane. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b} - \overrightarrow{c}$, and $\overrightarrow{b}$ is parallel to $\overrightarrow{a} + \overrightarrow{c}$, find the projection of $\overr... | -\sqrt{5} | 0.78125 |
13,959 | A fair six-sided die is rolled twice. Let $a$ and $b$ be the numbers obtained from the first and second roll respectively. Determine the probability that three line segments of lengths $a$, $b$, and $5$ can form an isosceles triangle. | \frac{7}{18} | 6.25 |
13,960 | Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and to the three edges that meet at $B$? | 2 - \sqrt{2} | 1.5625 |
13,961 | The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When is it advisable to swap the tires so that they wear out equally? (Assume that the tires are swapped only once, although in practice drivers do this more frequently.) | 9375 | 46.09375 |
13,962 | Find the least prime factor of the number represented by \(1 \underbrace{0000 \cdots 00}_{2010 \text{-many}} 1\). | 11 | 56.25 |
13,963 | How many different positive three-digit integers can be formed using only the digits in the set $\{4, 4, 5, 6, 6, 7, 7\}$, with no digit used more times than it appears in the set? | 42 | 2.34375 |
13,964 | Add together all natural numbers less than 1980 for which the sum of their digits is even! | 979605 | 82.8125 |
13,965 | In a circle, there are two mutually perpendicular chords $AB$ and $CD$. Determine the distance between the midpoint of segment $AD$ and the line $BC$, given that $AC=6$, $BC=5$, and $BD=3$. If necessary, round the answer to two decimal places. | 4.24 | 0 |
13,966 | The simple (i.e., non-intersecting) quadrilateral \(ABCD\) has sides \(AB\), \(BC\), and \(CD\) with lengths 4, 5, and 20, respectively. If the angles \(B\) and \(C\) are obtuse, and \(\sin C = -\cos B = \frac{3}{5}\), then what is the length of the side \(AD\)?
(Note: Taken from the 30th annual American High School M... | 25 | 0.78125 |
13,967 | On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$ . We have such strange rules:
(1) Define $P(n)$ : the number of prime numbers that are smaller than $n$ . Only when $P(i)\equiv j\pmod7$ , wolf $i$ may eat sheep $j$ (he can also choose not to eat the s... | 288 | 14.0625 |
13,968 | The center of a semicircle, inscribed in a right triangle such that its diameter lies on the hypotenuse, divides the hypotenuse into segments of 30 and 40. Find the length of the arc of the semicircle that is enclosed between the points where it touches the legs. | 12\pi | 0 |
13,969 | Is it possible to append two digits to the right of the number 277 so that the resulting number is divisible by any number from 2 to 12? | 27720 | 9.375 |
13,970 | For a 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 | 92.1875 |
13,971 | The overall idea is a common method in mathematical problem-solving. Below is the train of thought for factoring the polynomial $(a^{2}+2a)(a^{2}+2a+2)+1$: Consider "$a^{2}+2a$" as a whole, let $a^{2}+2a=x$, then the expression $=x(x+2)+1=x^{2}+2x+1=(x+1)^{2}$, then restore "$x$" to "$a^{2}+2a$". The solution process i... | 2024 | 46.09375 |
13,972 | Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:
\[ J P Y_{22} = 1388.01 - 41.89 - 2.77 - 309.72 - 554.91 - 0.24 = 478.4... | -12.6 | 80.46875 |
13,973 | $ f\left( x \right) \equal{} \frac {x^5}{5x^4 \minus{} 10x^3 \plus{} 10x^2 \minus{} 5x \plus{} 1}$ . $ \sum_{i \equal{} 1}^{2009} f\left( \frac {i}{2009} \right) \equal{} ?$ | 1005 | 11.71875 |
13,974 | In a race, four cars each independently run timed laps around a circuit. Each car's lap time is discretely measured in seconds and can be any integer value between 150 and 155 seconds, uniformly distributed. The winner is the car with the shortest lap time. In case of a tie, the involved cars re-run the lap until a sin... | \frac{1}{3} | 7.03125 |
13,975 | If the graph of the function $f(x) = (1-x^2)(x^2+ax+b)$ is symmetric about the line $x = -2$, then the maximum value of $f(x)$ is \_\_\_\_\_\_\_\_. | 16 | 56.25 |
13,976 | Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 5. Point $K$ is such that $\overrightarrow{L K}=\overrightarrow{F B}-3 \overrightarrow{A B}$. Determine whether point $K$ lies inside, on the boundary, or outside of $A B C D E F$, and also find the le... | \frac{5 \sqrt{3}}{3} | 0 |
13,977 | Given a sequence of 0s and 1s of length 23 that begins with a 0, ends with a 0, contains no two consecutive 0s, and contains no four consecutive 1s, determine the number of such sequences. | 200 | 0 |
13,978 | How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer? | 12 | 21.09375 |
13,979 | In a plane, there are 10 lines, among which 4 lines are parallel to each other. Then, these 10 lines can divide the plane into at most how many parts? | 50 | 3.125 |
13,980 | Little Pang, Little Dingding, Little Ya, and Little Qiao's four families, totaling 8 parents and 4 children, went to the amusement park together. The ticket prices are as follows: adult tickets are 100 yuan per person; children's tickets are 50 yuan per person; if there are 10 or more people, they can buy group tickets... | 800 | 3.90625 |
13,981 | A mother gives pocket money to her children sequentially: 1 ruble to Anya, 2 rubles to Borya, 3 rubles to Vitya, then 4 rubles to Anya, 5 rubles to Borya, and so on until Anya receives 202 rubles, and Borya receives 203 rubles. How many more rubles will Anya receive compared to Vitya? | 68 | 30.46875 |
13,982 | A bottle of cola costs 2 yuan, and two empty bottles can be exchanged for one more bottle of cola. With 30 yuan, what is the maximum number of bottles of cola that you can drink? | 29 | 47.65625 |
13,983 | It is known that the ellipse $C_1$ and the parabola $C_2$ have a common focus $F(1,0)$. The center of $C_1$ and the vertex of $C_2$ are both at the origin. A line $l$ passes through point $M(4,0)$ and intersects the parabola $C_2$ at points $A$ and $B$ (with point $A$ in the fourth quadrant).
1. If $|MB| = 4|AM|$, find... | \sqrt{34} | 2.34375 |
13,984 | Find all values of \( n \in \mathbf{N} \) for which there exist a number \( m \in \mathbf{N} \), a triangle \( ABC \) with sides \( AB = 33 \), \( AC = 21 \), \( BC = n \), and points \( D \), \( E \) on sides \( AB \), \( AC \) respectively, satisfying the conditions \( AD = DE = EC = m \). | 30 | 0.78125 |
13,985 | The focus of a vertically oriented, rotational paraboloid-shaped tall vessel is at a distance of 0.05 meters above the vertex. If a small amount of water is poured into the vessel, what angular velocity $\omega$ is needed to rotate the vessel around its axis so that the water overflows from the top of the vessel? | 9.9 | 0 |
13,986 | Let's call an integer "extraordinary" if it has exactly one even divisor other than 2. How many extraordinary numbers exist in the interval $[1 ; 75]$? | 11 | 53.125 |
13,987 | A semicircular sponge with a diameter of $20 \text{ cm}$ is used to wipe a corner of a room's floor such that the ends of the diameter continuously touch the two walls forming a right angle. What area does the sponge wipe? | 100\pi | 0 |
13,988 | Find the area in the plane contained by the graph of
\[
|x + 2y| + |2x - y| \le 6.
\] | 5.76 | 0 |
13,989 | The edges meeting at one vertex of a rectangular parallelepiped are in the ratio of $1: 2: 3$. What is the ratio of the lateral surface areas of the cylinders that can be circumscribed around the parallelepiped? | \sqrt{13} : 2\sqrt{10} : 3\sqrt{5} | 15.625 |
13,990 |
Circles \(\omega_{1}\) and \(\omega_{2}\) intersect at points \(A\) and \(B\). Segment \(PQ\) is tangent to \(\omega_{1}\) at \(P\) and to \(\omega_{2}\) at \(Q\), and \(A\) is closer to \(PQ\) than \(B\). Point \(X\) is on \(\omega_{1}\) such that \(PX \parallel QB\), and point \(Y\) is on \(\omega_{2}\) such that \(... | 2 - \sqrt{3} | 2.34375 |
13,991 | In the diagram, \(ABCD\) is a right trapezoid with \(AD = 2\) as the upper base, \(BC = 6\) as the lower base. Point \(E\) is on \(DC\). The area of triangle \(ABE\) is 15.6 and the area of triangle \(AED\) is 4.8. Find the area of trapezoid \(ABCD\). | 24 | 1.5625 |
13,992 | Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$ .) If $M$ is the maximum possible final value remaining on the board, and if m is the mini... | 2014 | 6.25 |
13,993 | Anton thought of a three-digit number, and Alex is trying to guess it. Alex successively guessed the numbers 109, 704, and 124. Anton observed that each of these numbers matches the thought number exactly in one digit place. What number did Anton think of? | 729 | 50 |
13,994 | For the set $\{1,2,\cdots,n\}$ and each of its non-empty subsets, define a unique "alternating sum" as follows: Arrange the numbers in each subset in descending order, then start from the largest number and alternately subtract and add subsequent numbers to obtain the alternating sum (for example, the alternating sum o... | 1024 | 71.875 |
13,995 | Given the function $f(x)=2\sqrt{3}\sin ^{2}x+2\sin x\cos x-\sqrt{3}$, where $x\in\left[ \frac{\pi}{3}, \frac{11\pi}{24}\right]$.
(1) Find the range of the function $f(x)$.
(2) Suppose that the lengths of two sides of an acute-angled triangle $ABC$ are the maximum and minimum values of the function $f(x)$, respectivel... | \sqrt{2} | 15.625 |
13,996 | Fill the numbers $1,2,\cdots,36$ into a $6 \times 6$ grid, placing one number in each cell, such that the numbers in each row are in increasing order from left to right. What is the minimum possible sum of the six numbers in the third column? | 108 | 53.90625 |
13,997 | We have 10 springs, each originally $0.5 \mathrm{~m}$ long with a spring constant of $200 \mathrm{~N}/\mathrm{m}$. A mass of $2 \mathrm{~kg}$ is hung on each spring, and the springs, along with the masses, are hung in a series. What is the length of the resulting chain? (Neglect the mass of the springs.) | 10.39 | 0 |
13,998 | A teacher drew a rectangle $ABCD$ on the board. A student named Petya divided this rectangle into two rectangles with a line parallel to side $AB$. It turned out that the areas of these parts are in the ratio 1:2, and their perimeters are in the ratio 3:5 (in the same order). Another student named Vasya divided this re... | 20/19 | 0 |
13,999 | Given the family of curves
$$
2(2 \sin \theta - \cos \theta + 3) x^{2} - (8 \sin \theta + \cos \theta + 1) y = 0,
$$
where $\theta$ is a parameter. Find the maximum length of the chord that these curves cut on the line $y = 2 x$. | 8\sqrt{5} | 16.40625 |
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