Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
14,400 |
Let the function \( y = f(x) \) satisfy: for all \( x \in \mathbb{R} \), \( y = f(x) \geqslant 0 \), and \( f(x+1) = \sqrt{9 - f(x)^2} \). When \( x \in [0,1) \),
$$
f(x) = \begin{cases}
2^x, & 0 \leqslant x < \frac{1}{2}, \\
\log_{10} (x + 31), & \frac{1}{2} \leqslant x < 1
\end{cases}
$$
Find \( f(\sqrt{1000}) \). | \frac{3\sqrt{3}}{2} | 0.78125 |
14,401 | Let \( n \) be a positive integer. When \( n > 100 \), what are the first two decimal places of the fractional part of \( \sqrt{n^2 + 3n + 1} \)? | 49 | 28.125 |
14,402 | In a Cartesian coordinate system, the "rectangular distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q) = \left|x_{1}-x_{2}\right| + \left|y_{1}-y_{2}\right|$. If the "rectangular distance" from point $C(x, y)$ to points $A(1,3)$ and $B(6,9)$ is equal, where real... | 5(\sqrt{2} + 1) | 0 |
14,403 | A point moving in the positive direction of the $O x$ axis has the abscissa $x(t)=5(t+1)^{2}+\frac{a}{(t+1)^{5}}$, where $a$ is a positive constant. Find the minimum value of $a$ such that $x(t) \geqslant 24$ for all $t \geqslant 0$. | 2 \sqrt{\left( \frac{24}{7} \right)^7} | 0 |
14,404 | What is the smallest number with three different prime factors, none of which can be less than 10? | 2431 | 89.84375 |
14,405 | The digits $0,1,2,3,4,5,6$ are randomly arranged in a sequence. What is the probability of obtaining a seven-digit number that is divisible by four? (The number cannot start with zero.) | 0.25 | 7.03125 |
14,406 | A five-digit number is called a "pit" if its first three digits are in descending order and the last three digits are in ascending order. For example, 73016 and 98346 are pits, while 88012 and 56821 are not pits. How many pits are there that are less than the number 22222? | 36 | 82.8125 |
14,407 | Given $cos(\frac{π}{4}-α)=\frac{3}{5}$ and $sin(\frac{5π}{4}+β)=-\frac{12}{13}$, where $α∈(\frac{π}{4},\frac{3π}{4})$ and $β∈(0,\frac{π}{4})$, find $\frac{tanα}{tanβ}$. | -17 | 28.90625 |
14,408 | Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to school at a certain time. But on Friday, Nikita left home at 7:10 AM and ran in the opposite direction. Uncle Vanya waited for him and at 8:10 AM drove after him, caught up with Nikita, turned around, and took him to school,... | 13 | 0 |
14,409 | Let \( O \) be an interior point of \( \triangle ABC \). Extend \( AO \) to meet \( BC \) at \( D \). Similarly, extend \( BO \) and \( CO \) to meet \( CA \) and \( AB \) respectively at \( E \) and \( F \). Given \( AO = 30 \), \( FO = 20 \), \( BO = 60 \), \( DO = 10 \) and \( CO = 20 \), find \( EO \). | 20 | 1.5625 |
14,410 | One mole of an ideal monatomic gas is first heated isobarically, during which it performs 40 J of work. Then it is heated isothermally, receiving the same amount of heat as in the first case. What work does the gas perform (in Joules) in the second case? | 100 | 60.9375 |
14,411 | Petya wants to create an unusual die, which should have the shape of a cube, with dots drawn on the faces (different numbers of dots on different faces). Additionally, on each pair of adjacent faces, the number of dots must differ by at least two (it is allowed to have more than six dots on some faces). How many dots i... | 27 | 1.5625 |
14,412 | Vasya is inventing a 4-digit password for a combination lock. He does not like the digit 2, so he does not use it. Moreover, he doesn't like when two identical digits stand next to each other. Additionally, he wants the first digit to match the last one. How many possible combinations need to be checked to guarantee gu... | 504 | 10.15625 |
14,413 | In the cells of a $100 \times 100$ square, the numbers $1, 2, \ldots, 10000$ were placed, each exactly once, such that numbers differing by 1 are recorded in adjacent cells along the side. After that, the distances between the centers of each two cells, where the numbers in those cells differ exactly by 5000, were calc... | 50\sqrt{2} | 30.46875 |
14,414 | Someone, when asked for the number of their ticket, replied: "If you add all the six two-digit numbers that can be made from the digits of the ticket number, half of the resulting sum will be exactly my ticket number." Determine the ticket number. | 198 | 3.90625 |
14,415 | A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at 1 meter per second. ... | \frac{\pi}{6} | 0 |
14,416 | A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
| 438 | 0 |
14,417 | Each face of a fair six-sided die is marked with one of the numbers $1, 2, \cdots, 6$. When two such identical dice are rolled, the sum of the numbers on the top faces of these dice is the score for that roll. What is the probability that the product of the scores from three such rolls is divisible by 14? Express your ... | 1/3 | 0 |
14,418 | Attempt to obtain one billion (1,000,000,000) by multiplying two integers, each of which contains no zeros. | 512 * 1953125 | 0 |
14,419 | In the rectangular prism \(A B C D - A_{1} B_{1} C_{1} D_{1}\), \(A B = A A_{1} = 2\), \(A D = 2 \sqrt{3}\). Point \(M\) lies within plane \(B A_{1} C_{1}\). Find the minimum value of \(\overrightarrow{M A} \cdot \overrightarrow{M C}\). | -\frac{16}{7} | 3.90625 |
14,420 | Find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers. | 11 | 73.4375 |
14,421 | Let natural numbers \( k \) and \( n \) be coprime, where \( n \geq 5 \) and \( k < \frac{n}{2} \). A regular \((n ; k)\)-star is defined as a closed broken line formed by replacing every \( k \) consecutive sides of a regular \( n \)-gon with a diagonal connecting the same endpoints. For example, a \((5 ; 2)\)-star ha... | 48432 | 0 |
14,422 |
Let $x, y, z$ be positive numbers satisfying the following system of equations:
$$
\left\{\begin{array}{l}
x^{2} + xy + y^{2} = 12 \\
y^{2} + yz + z^{2} = 9 \\
z^{2} + xz + x^{2} = 21
\end{array}\right.
$$
Find the value of the expression $xy + yz + xz$. | 12 | 95.3125 |
14,423 | What is the largest number of integers that we can choose from the set $\{1, 2, 3, \ldots, 2017\}$ such that the difference between any two of them is not a prime number? | 505 | 39.84375 |
14,424 | On a \(6 \times 6\) chessboard, we randomly place counters on three different squares. What is the probability that no two counters are in the same row or column? | 40/119 | 18.75 |
14,425 | Given a trapezoid \(ABCD\). A point \(M\) is chosen on its lateral side \(CD\) such that \( \frac{CM}{MD} = \frac{4}{3} \). It turns out that segment \( BM \) divides the diagonal \( AC \) into two segments, the ratio of the lengths of which is also \( \frac{4}{3} \). What possible values can the ratio \( \frac{AD}{BC}... | 7/12 | 0.78125 |
14,426 | A die is rolled twice continuously, resulting in numbers $a$ and $b$. What is the probability $p$, in numerical form, that the cubic equation in $x$, given by $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$, has three distinct real roots? | 3/4 | 5.46875 |
14,427 | Thirty-nine students from seven classes came up with 60 problems, with students of the same class coming up with the same number of problems (not equal to zero), and students from different classes coming up with a different number of problems. How many students came up with one problem each? | 33 | 1.5625 |
14,428 | Petya's watch runs 5 minutes fast per hour, and Masha's watch runs 8 minutes slow per hour. At 12:00, they set their watches to the accurate school clock and agreed to meet at the skating rink at 6:30 PM according to their respective watches. How long will Petya wait for Masha if each arrives at the skating rink exactl... | 1.5 | 14.0625 |
14,429 | Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1. | 11111111100000000 | 10.15625 |
14,430 | Given the real number \( x \), \([x] \) denotes the integer part that does not exceed \( x \). Find the positive integer \( n \) that satisfies:
\[
\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994
\] | 312 | 92.96875 |
14,431 | Find
\[
\sum_{n = 1}^\infty \frac{3^n}{1 + 3^n + 3^{n + 1} + 3^{2n + 1}}.
\] | \frac{1}{4} | 7.8125 |
14,432 | Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2023} \right| = \frac{1}{2023}.\] | \frac{4042}{2023} | 15.625 |
14,433 | Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the rural store salesperson, going on break. At two minutes past noon, Anna Kuzminichna looked out the window again, and there was still no one in front of the closed store. Klava was gone for exactly 10 minutes, and when she returned, she found Iva... | 0.75 | 0 |
14,434 | A certain unit has 160 young employees. The number of middle-aged employees is twice the number of elderly employees. The total number of elderly, middle-aged, and young employees is 430. In order to understand the physical condition of the employees, a stratified sampling method is used for the survey. In a sample of ... | 18 | 11.71875 |
14,435 | A $1.4 \mathrm{~m}$ long rod has $3 \mathrm{~kg}$ masses at both ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot with a speed of $1.6 \mathrm{~m} /\mathrm{s}$? | 0.8 | 0 |
14,436 | Calculate the value of the expression
$$
\frac{\left(3^{4}+4\right) \cdot\left(7^{4}+4\right) \cdot\left(11^{4}+4\right) \cdot \ldots \cdot\left(2015^{4}+4\right) \cdot\left(2019^{4}+4\right)}{\left(1^{4}+4\right) \cdot\left(5^{4}+4\right) \cdot\left(9^{4}+4\right) \cdot \ldots \cdot\left(2013^{4}+4\right) \cdot\left(... | 4080401 | 34.375 |
14,437 | How many strictly positive integers less than or equal to 120 are not divisible by 3, 5, or 7? | 54 | 100 |
14,438 | While one lion cub, who is 6 minutes away from the water hole, heads there, another, having already quenched its thirst, heads back along the same road 1.5 times faster than the first. At the same time, a turtle starts towards the water hole along the same road, being 32 minutes away from it. At some point, the first l... | 2.4 | 0.78125 |
14,439 | A certain number of books are distributed among children. If each child gets $m$ books, there are 14 books left. If each child gets 9 books, the last child only gets 6 books. How many children are there in total? And how many books are there? | 150 | 57.03125 |
14,440 | An isosceles right triangle is removed from each corner of a square piece of paper to form a rectangle. If $AB = 15$ units in the new configuration, what is the combined area of the four removed triangles? | 112.5 | 13.28125 |
14,441 | If the direction vector of line $l$ is $\overrightarrow{d}=(1,\sqrt{3})$, then the inclination angle of line $l$ is ______. | \frac{\pi}{3} | 61.71875 |
14,442 | In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\). \(\angle A = \angle D = 45^\circ\), while \(\angle B = \angle C = 135^\circ\). If \(AB = 6\) and the area of \(ABCD\) is 30, find \(BC\). | 2\sqrt{2} | 0 |
14,443 | The kite \( ABCD \) is symmetric with respect to diagonal \( AC \). The length of \( AC \) is 12 cm, the length of \( BC \) is 6 cm, and the internal angle at vertex \( B \) is a right angle. Points \( E \) and \( F \) are given on sides \( AB \) and \( AD \) respectively, such that triangle \( ECF \) is equilateral.
... | 4\sqrt{3} | 7.8125 |
14,444 | Given that the graph of the power function $y=(m^{2}-2m-2)x^{m^{2}+4m}$ is symmetric with respect to the origin and does not intersect the $x$-axis or $y$-axis, determine the value of the integer $m$. | -1 | 24.21875 |
14,445 | Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( \frac{A M}{A A_{1}} = \frac{5}{6}, \frac{B N}{B B_{1}} = \frac{6}{7}, \) and \( \frac{C K}{C C_{1}} = \frac{2}{3} \). Point \( P \) belongs to the ... | 10 | 3.90625 |
14,446 | Hooligan Vasily tore out an entire chapter from a book, with the first page numbered 231, and the number of the last page consisted of the same digits. How many sheets did Vasily tear out of the book? | 41 | 35.15625 |
14,447 | Given that odd prime numbers \( x, y, z \) satisfy
\[ x \mid (y^5 + 1), \quad y \mid (z^5 + 1), \quad z \mid (x^5 + 1). \]
Find the minimum value of the product \( xyz \). | 2013 | 10.15625 |
14,448 | How many solutions in natural numbers does the equation $\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1$ have? | 110 | 48.4375 |
14,449 | Find the area of a trapezoid with bases 11 and 4 and diagonals 9 and 12. | 54 | 5.46875 |
14,450 | The integer $y$ has 24 positive factors. The numbers 20 and 35 are factors of $y$. What is the smallest possible value of $y$? | 1120 | 7.03125 |
14,451 | The equation \( x^{2} + mx + 1 + 2i = 0 \) has real roots. Find the minimum value of the modulus of the complex number \( m \). | \sqrt{2 + 2\sqrt{5}} | 0.78125 |
14,452 | The Euler family consists of four girls, each aged 8 years, and three boys, one aged 10 and twins both aged 12. Calculate the mean age of the children after one year passes. | 10.43 | 38.28125 |
14,453 | If a positive integer \( N \) can be expressed as \( \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor \) for some real number \( x \), then we say that \( N \) is "visible"; otherwise, we say that \( N \) is "invisible". For example, 8 is visible since \( 8 = \lfloor 1.5 \rfloor + \lfloor 2(1.5) \rfloor + \l... | 6034 | 21.09375 |
14,454 | Given that $\dfrac {π}{2} < β < α < \dfrac {3}{4}π$, $\cos (α+β)=- \dfrac {3}{5}$, and $\sin (α-β)= \dfrac {5}{13}$, find $\cos 2β$. | - \dfrac {56}{65} | 16.40625 |
14,455 | A seven-digit natural number \( N \) is called interesting if:
- It consists of non-zero digits;
- It is divisible by 4;
- Any number obtained from \( N \) by permuting its digits is also divisible by 4.
How many interesting numbers exist? | 128 | 1.5625 |
14,456 | Let \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^{3} + a x^{2} + b x \). If there exist three real numbers \( x_{1}, x_{2}, x_{3} \) such that \( x_{1} + 1 \leqslant x_{2} \leqslant x_{3} - 1 \), and \( f(x_{1}) = f(x_{2}) = f(x_{3}) \), find the minimum value of \( |a| + 2|b| \). | \sqrt{3} | 8.59375 |
14,457 | A Tim number is a five-digit positive integer with the following properties:
1. It is a multiple of 15.
2. Its hundreds digit is 3.
3. Its tens digit is equal to the sum of its first three (leftmost) digits.
How many Tim numbers are there? | 16 | 24.21875 |
14,458 | The real number \(x\) satisfies the equation \(x + \frac{1}{x} = \sqrt{3}\). Evaluate the expression \(x^{7} - 5x^{5} + x^{2}\). | -1 | 10.9375 |
14,459 | On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$ . Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$ . $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to... | \pi | 22.65625 |
14,460 | Compute the definite integral:
$$
\int_{0}^{\sqrt{2}} \frac{x^{4} \cdot d x}{\left(4-x^{2}\right)^{3 / 2}}
$$ | 5 - \frac{3\pi}{2} | 41.40625 |
14,461 | In triangle \(ABC\), side \(AB = 6\), \(\angle BAC = 30^\circ\), and the radius of the circumscribed circle is 5. Find side \(AC\). | 3\sqrt{3} + 4 | 0 |
14,462 | Let $M=\{1,2, \cdots, 2005\}$. Subset $A$ of $M$ satisfies the condition: if $x \in A$, then $15x \notin A$. What is the maximum number of elements in $A$? | 1880 | 9.375 |
14,463 | A prize fund is divided into first, second, and third prizes. The prize for each first prize is 3 times that of each second prize, and the prize for each second prize is 3 times that of each third prize. The total prize fund is 10,800 yuan. If the total prize money for the third prize is more than that for the second p... | 2700 | 41.40625 |
14,464 | Find at the point \( M_{0}(1,1,1) \) the direction of the greatest change of the scalar field \( u = xy + yz + xz \) and the magnitude of this greatest change at this point. | 2\sqrt{3} | 41.40625 |
14,465 | A massive vertical plate is fixed to a car moving at a speed of $5 \, \text{m/s}$. A ball is flying towards it at a speed of $6 \, \text{m/s}$ with respect to the ground. Determine the speed of the ball with respect to the ground after a perfectly elastic normal collision. | 16 | 0 |
14,466 | $ABC$ is an isosceles triangle with base $AC$. $CD$ is the bisector of angle $C$, and $\angle ADC = 150^\circ$. Find $\angle B$. | 140 | 33.59375 |
14,467 | In the coordinate plane, a parallelogram $O A B C$ is drawn such that its center is at the point $\left(\frac{19}{2}, \frac{15}{2}\right)$, and the points $A, B,$ and $C$ have natural number coordinates. Find the number of such parallelograms. (Here, $O$ denotes the origin - the point $(0,0)$; two parallelograms with t... | 126 | 15.625 |
14,468 | In a checkered square with a side length of 2018, some cells are painted white and the rest are black. It is known that from this square, one can cut out a 10x10 square where all the cells are white, and a 10x10 square where all the cells are black. What is the smallest value for which it is guaranteed that one can cut... | 10 | 29.6875 |
14,469 | Consider the set \( S = \{1, 2, 3, \cdots, 2010, 2011\} \). A subset \( T \) of \( S \) is said to be a \( k \)-element RP-subset if \( T \) has exactly \( k \) elements and every pair of elements of \( T \) are relatively prime. Find the smallest positive integer \( k \) such that every \( k \)-element RP-subset of \(... | 16 | 3.90625 |
14,470 | Find the maximum value of the expression \( (\sin x + \sin 2y + \sin 3z)(\cos x + \cos 2y + \cos 3z) \). | 4.5 | 6.25 |
14,471 | A class has a group of 7 students, and now select 3 of them to swap seats with each other, while the remaining 4 students keep their seats unchanged. Calculate the number of different ways to adjust their seats. | 70 | 16.40625 |
14,472 | Determine the number of zeros in the quotient $Q = R_{30}/R_6$, where $R_k$ is a number consisting of $k$ repeated digits of 1 in base-ten. | 25 | 0.78125 |
14,473 | In the cells of a 9 × 9 square, there are non-negative numbers. The sum of the numbers in any two adjacent rows is at least 20, and the sum of the numbers in any two adjacent columns does not exceed 16. What can be the sum of the numbers in the entire table? | 80 | 12.5 |
14,474 | Given that the radius of circle \( \odot O \) is 1, the quadrilateral \( ABCD \) is an inscribed square, \( EF \) is a diameter of \( \odot O \), and \( M \) is a point moving along the boundary of the square \( ABCD \). Find the minimum value of \(\overrightarrow{ME} \cdot \overrightarrow{MF} \). | -1/2 | 1.5625 |
14,475 | Riquinho distributed 1000,00 reais among his friends Antônio, Bernardo, and Carlos in the following manner: he gave, successively, 1 real to Antônio, 2 reais to Bernardo, 3 reais to Carlos, 4 reais to Antônio, 5 reais to Bernardo, etc. How much money did Bernardo receive? | 345 | 0.78125 |
14,476 | There are 4 children and 2 coaches in the Chess Interest Group of Hongxing Primary School. The ages of the 4 children differ by 2 years sequentially, and the ages of the 2 coaches differ by 2 years. The sum of the squares of the ages of the 6 people is 2796 years. What is the sum of their ages in years? | 106 | 32.03125 |
14,477 | Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\). | 2020032 | 4.6875 |
14,478 | Let $s$ be the set of all rational numbers $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be represented as a repeating decimal of the form $0.abcabcabc\cdots = 0.\dot{a}b\dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. How many different numerators are there when the elemen... | 660 | 91.40625 |
14,479 | Given four points \( A, B, C, \) and \( D \) in space that are not coplanar, each pair of points is connected by an edge with a probability of \( \frac{1}{2} \). The events of whether any two pairs of points are connected are mutually independent. What is the probability that \( A \) and \( B \) can be connected by a s... | \frac{3}{4} | 7.8125 |
14,480 | The volume of a regular triangular pyramid, whose lateral face is inclined at an angle of $45^{\circ}$ to the base, is $9 \mathrm{~cm}^{3}$. Find the total surface area of the pyramid. | 9 \sqrt{3} (1 + \sqrt{2}) | 0 |
14,481 | John drove continuously from 8:30 a.m. until 2:15 p.m. of the same day and covered a distance of 246 miles. What was his average speed in miles per hour? | 42.78 | 0.78125 |
14,482 | An educational center received an object with a volume of around 150 monoliths (a container designed for 150 monoliths that was almost full). Each monolith is identified either as sandy loam or loam and classified by its genesis as either marine or lake-glacial deposits. The relative frequency (statistical probability)... | 80 | 0 |
14,483 | Four cars, \( A, B, C, \) and \( D \) start simultaneously from the same point on a circular track. \( A \) and \( B \) drive clockwise, while \( C \) and \( D \) drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, \( A \) meets \( C \) for the fir... | 53 | 4.6875 |
14,484 | Given the function $f(x)=4\cos x\sin \left(x+ \dfrac{\pi}{6} \right)$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[- \dfrac{\pi}{6}, \dfrac{\pi}{4} \right]$. | -1 | 14.0625 |
14,485 | Does there exist a point \( M \) on the parabola \( y^{2} = 2px \) such that the ratio of the distance from point \( M \) to the vertex and the distance from point \( M \) to the focus is maximized? If such a point \( M \) exists, find its coordinates and the maximum ratio. If the point \( M \) does not exist, provide ... | \frac{2}{\sqrt{3}} | 0 |
14,486 | In triangle \( \triangle ABC \), it is known that \( \overrightarrow{AB} \cdot \overrightarrow{AC} + 2 \overrightarrow{BA} \cdot \overrightarrow{BC} = 3 \overrightarrow{CA} \cdot \overrightarrow{CB} \). Find the maximum value of \( \sin C \). | \frac{\sqrt{7}}{3} | 0 |
14,487 | If $M(\frac{p}{2}$,$p)$ is a point on the parabola $y^{2}=2px\left(p \gt 0\right)$, and the distance from $M$ to the point $\left(1,0\right)$ is 1 greater than the distance to the $y$-axis, find:<br/>
$(1)$ The equation of the parabola.<br/>
$(2)$ Let line $l$ intersect the parabola at points $A$ and $B$. The circle wi... | 2\sqrt{5} | 0.78125 |
14,488 | Given that quadrilateral \(ABCD\) is a right trapezoid with the upper base \(AD = 8\) cm, the lower base \(BC = 10\) cm, and the right leg \(CD = 6\) cm. Point \(E\) is the midpoint of \(AD\), and point \(F\) is on \(BC\) such that \(BF = \frac{2}{3} BC\). Point \(G\) is on \(DC\) such that the area of triangle \(DEG\)... | 24 | 15.625 |
14,489 | Ivan the Tsarevich is learning to shoot a bow. He placed 14 arrows in his quiver and went to the forest to shoot at cones. He hits a cone with a probability of 0.1, and for each hit cone, the Frog Princess gives him 3 additional arrows. Ivan shoots until he runs out of arrows. Find the expected number of shots that Iva... | 20 | 2.34375 |
14,490 | Let \( A B C D E F \) be a regular hexagon, and let \( P \) be a point inside quadrilateral \( A B C D \). If the area of triangle \( P B C \) is 20, and the area of triangle \( P A D \) is 23, compute the area of hexagon \( A B C D E F \). | 189 | 0.78125 |
14,491 | In $\triangle ABC$, where $\angle C=90^{\circ}$, $\angle B=30^{\circ}$, and $AC=1$, let $M$ be the midpoint of $AB$. $\triangle ACM$ is folded along $CM$ such that the distance between points $A$ and $B$ is $\sqrt{2}$. What is the distance from point $A$ to plane $BCM$? | \frac{\sqrt{6}}{3} | 3.125 |
14,492 | Mrs. Široká was expecting guests in the evening. First, she prepared 25 open-faced sandwiches. She then calculated that if each guest took two sandwiches, three of them would not have enough. She then thought that if she made 10 more sandwiches, each guest could take three, but four of them would not have enough. This ... | 11 | 42.1875 |
14,493 | Calculate the definite integral:
$$
\int_{0}^{\arcsin (3 / \sqrt{10})} \frac{2 \operatorname{tg} x - 5}{(4 \cos x - \sin x)^{2}} \, dx
$$ | \frac{9}{4} - \ln 16 | 0 |
14,494 | A regular 2015-gon \( A_{1} A_{2} \cdots A_{2015} \) is inscribed in a unit circle \( O \). What is the probability that for any two distinct vertices \( A_{i}, A_{j} \), the magnitude \( \left|\overrightarrow{O A_{i}}+\overrightarrow{O A_{j}}\right| \geqslant 1 \) is true? | 671/1007 | 13.28125 |
14,495 | Write 22 as the sum of several distinct natural numbers so that the product of these numbers is maximized. The maximum product is \_\_\_\_\_\_. | 1008 | 2.34375 |
14,496 | Let $A = \left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be a set of numbers, and let the arithmetic mean of all elements in $A$ be denoted by $P(A)\left(P(A)=\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)$. If $B$ is a non-empty subset of $A$ such that $P(B) = P(A)$, then $B$ is called a "balance subset" of $A$. Find the numbe... | 51 | 99.21875 |
14,497 | In how many ways can a tetromino in the shape of the letter $Z$ be placed on a chessboard (size $8 \times 8$ squares) so that it is located exactly on the cells of the board and within the board's boundaries? The tetromino can be rotated and flipped. Justify your answer. | 168 | 6.25 |
14,498 | Two of the altitudes of an acute triangle divide the sides into segments of lengths $7, 4, 3$, and $y$ units, as shown. Calculate the value of $y$. | \frac{12}{7} | 13.28125 |
14,499 | Some positive integers are initially written on a board, where each $2$ of them are different.
Each time we can do the following moves:
(1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+... | -3 | 0 |
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