Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
14,500 |
On the lateral edges \(AA_1\), \(BB_1\), and \(CC_1\) of a triangular prism \(ABC A_1 B_1 C_1\), points \(M\), \(N\), and \(P\) are located respectively such that \(AM: AA_1 = B_1N: BB_1 = C_1P: CC_1 = 3:4\). On the segments \(CM\) and \(A_1N\), points \(E\) and \(F\) are located respectively such that \(EF \parallel ... | 1/3 | 3.90625 |
14,501 | Let \( n \) be a positive integer with at least four different positive divisors. Let the four smallest of these divisors be \( d_{1}, d_{2}, d_{3}, d_{4} \). Find all such numbers \( n \) for which
\[ d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}=n \] | 130 | 85.9375 |
14,502 | There are 8 white balls and 2 red balls in a bag. Each time a ball is randomly drawn and then a white ball is put back. What is the probability that all red balls are drawn exactly at the 4th draw? | 0.0434 | 0 |
14,503 | Find the number of solutions in natural numbers to the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\). | 110 | 43.75 |
14,504 | Let \( A \) be a 4-digit integer. When both the first digit (left-most) and the third digit are increased by \( n \), and the second digit and the fourth digit are decreased by \( n \), the new number is \( n \) times \( A \). Find the value of \( A \). | 1818 | 25.78125 |
14,505 | Given that positive real numbers \( a \) and \( b \) satisfy \( ab(a+b)=4 \), find the minimum value of \( 2a + b \). | 2\sqrt{3} | 33.59375 |
14,506 | Find all positive integers \( n > 1 \) such that any of its positive divisors greater than 1 has the form \( a^r + 1 \), where \( a \) is a positive integer and \( r \) is a positive integer greater than 1. | 10 | 0.78125 |
14,507 | A circle is constructed on the side $BC$ of triangle $ABC$ as its diameter, and it intersects segment $AB$ at point $D$. Find the ratio of the areas of triangles $ABC$ and $BCD$, given that $AC = 15$, $BC = 20$, and $\angle ABC = \angle ACD$. | 25/16 | 3.125 |
14,508 | Let $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0$, let $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle of $\phi$ counter-clockwise, and let $\mathbf{T}$ be a translation matrix that we will ignore in the computation as translations do not affect dilation or rot... | \frac{12}{5} | 2.34375 |
14,509 | In triangle $ABC$, the angle bisectors $AA_{1}$ and $BB_{1}$ intersect at point $O$. Find the ratio $AA_{1} : OA_{1}$ given $AB=6, BC=5$, and $CA=4$. | 3 : 1 | 3.90625 |
14,510 | A sphere passes through two adjacent vertices of a unit cube and touches the planes of the faces that do not contain these vertices. What is the radius of this sphere? | 2 - \frac{\sqrt{7}}{2} | 0 |
14,511 | Given that the complex number \( z \) satisfies \( |z|=1 \), find the maximum value of \( \left| z^3 - 3z - 2 \right| \). | 3\sqrt{3} | 75 |
14,512 | A farmer builds a rectangular chicken coop that leans against a wall, covering an area of 36m<sup>2</sup>. Due to geographical constraints, the length of the side of the chicken coop, $x$, cannot exceed 7m. The wall height is 2m. The cost of constructing the front of the chicken coop is 40 yuan/m<sup>2</sup>, the cost ... | 2760 | 0 |
14,513 | How many integers from 1 to 2001 have a digit sum that is divisible by 5? | 399 | 79.6875 |
14,514 | Does there exist a natural number \( n \), greater than 1, such that the value of the expression \(\sqrt{n \sqrt{n \sqrt{n}}}\) is a natural number? | 256 | 14.0625 |
14,515 | Given that $\operatorname{tg} \theta$ and $\operatorname{ctg} \theta$ are the real roots of the equation $2x^{2} - 2kx = 3 - k^{2}$, and $\alpha < \theta < \frac{5 \pi}{4}$, find the value of $\cos \theta - \sin \theta$. | -\sqrt{\frac{5 - 2\sqrt{5}}{5}} | 4.6875 |
14,516 | Two ants, one starting at $(-1,1)$ and the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. When the left ant first hits the line $y=\frac{1}{2}$, what is its speed? | 3 \sqrt{3} - 3 | 0 |
14,517 | The numbers from 1 to 9 are divided into three groups of three numbers, and then the numbers in each group are multiplied. $A$ is the largest of the three products. What is the smallest possible value of $A$? | 72 | 71.09375 |
14,518 | In a right triangle ABC, with right angle at A, side AB measures 5 units and side BC measures 13 units. Find $\sin C$. | \frac{12}{13} | 4.6875 |
14,519 | In a convex 1950-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides. What is the greatest number of sides it can have? | 1949 | 42.1875 |
14,520 | A volleyball net is in the shape of a rectangle with dimensions of $50 \times 600$ cells.
What is the maximum number of strings that can be cut so that the net does not fall apart into pieces? | 30000 | 1.5625 |
14,521 | Let \( R \) be the rectangle in the Cartesian plane with vertices at \((0,0)\), \((2,0)\), \((2,1)\), and \((0,1)\). \( R \) can be divided into two unit squares. Pro selects a point \( P \) uniformly at random in the interior of \( R \). Find the probability that the line through \( P \) with slope \(\frac{1}{2}\) wil... | 3/4 | 17.96875 |
14,522 | Let \( n \) be a natural number. Decompose \( n \) into sums of powers of \( p \) (where \( p \) is a positive integer greater than 1), in such a way that each power \( p^k \) appears at most \( p^2 - 1 \) times. Denote by \( C(n, p) \) the total number of such decompositions. For example, for \( n = 8 \) and \( p = 2 ... | 118 | 82.03125 |
14,523 | Calculate: \(\frac{1}{2 \cos \frac{2 \pi}{7}}+\frac{1}{2 \cos \frac{4 \pi}{7}}+\frac{1}{2 \cos \frac{6 \pi}{7}}\). | -2 | 26.5625 |
14,524 | Let \( a_{1}, a_{2}, \cdots, a_{2006} \) be 2006 positive integers (they can be the same) such that \( \frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}} \) are all different from each other. What is the minimum number of distinct numbers in \( a_{1}, a_{2}, \cdots, a_{2006} \)? | 46 | 3.125 |
14,525 | On the clock tower at the train station, there is an electronic clock. Along the boundary of the circular clock face, there are small colorful lights at each minute mark. At 9:35:20 PM, there are how many small colorful lights in the acute angle formed by the minute hand and the hour hand? | 12 | 60.9375 |
14,526 | Consider the function $g(x)$ represented by the line segments in the graph below. The graph consists of four line segments as follows: connecting points (-3, -4) to (-1, 0), (-1, 0) to (0, -1), (0, -1) to (2, 3), and (2, 3) to (3, 2). Find the sum of the $x$-coordinates where $g(x) = x + 2$. | -1.5 | 0 |
14,527 | There are four cards, each with one of the numbers $2$, $0$, $1$, $5$ written on them. Four people, A, B, C, and D, each take one card.
A says: None of the numbers you three have differ by 1 from the number I have.
B says: At least one of the numbers you three have differs by 1 from the number I have.
C says: The numb... | 5120 | 0.78125 |
14,528 | Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Calculate the sum $S$ of this series. | \frac{5}{26} | 0 |
14,529 | Given complex numbers $w$ and $z$ such that $|w+z|=3$ and $|w^2+z^2|=18,$ find the smallest possible value of $|w^3+z^3|.$ | \frac{81}{2} | 0 |
14,530 | Let the set \( \mathrm{S} = \{1, 2, 3, \ldots, 10\} \). The subset \( \mathrm{A} \) of \( \mathrm{S} \) satisfies \( \mathrm{A} \cap \{1, 2, 3\} \neq \emptyset \) and \( \mathrm{A} \cup \{4, 5, 6\} \neq \mathrm{S} \). Find the number of such subsets \( \mathrm{A} \). | 888 | 5.46875 |
14,531 | Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, with $a_{1}=1$, $a_{2}=2$, $a_{3}=3$, and the sequence $\{a_{n}+a_{n+1}+a_{n+2}\}$ is an arithmetic sequence with a common difference of $2$, calculate the value of $S_{25}$. | 269 | 3.125 |
14,532 | Given that $\alpha^{2005}+\beta^{2005}$ can be expressed as a two-variable polynomial in terms of $\alpha + \beta$ and $\alpha \beta$, find the sum of the coefficients of this polynomial. | -1 | 32.8125 |
14,533 | Real numbers \(a, b, c\) satisfy the following system of equations:
\[ \left\{ \begin{array}{l} a^{2}+a b+b^{2}=11 \\ b^{2}+b c+c^{2}=11 \end{array} \right. \]
(a) What is the minimum value that the expression \(c^{2}+c a+a^{2}\) can take?
(b) What is the maximum value that the expression \(c^{2}+c a+a^{2}\) can take... | 44 | 3.125 |
14,534 | Let points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5} \) be located on the unit sphere. Find the maximum value of \( \min \left\{A_{i} A_{j} \mid 1 \leq i < j \leq 5 \right\} \) and determine all cases where this maximum value is achieved. | \sqrt{2} | 27.34375 |
14,535 | Calculate the definite integral:
$$
\int_{6}^{9} \sqrt{\frac{9-2x}{2x-21}} \, dx
$$ | \pi | 25.78125 |
14,536 | In the vertices of a regular 300-gon, the numbers from 1 to 300 are arranged in some order, each number appearing exactly once. It turns out that for each number \(a\), there are as many numbers smaller than \(a\) among the 15 closest numbers to it clockwise as there are among the 15 closest numbers to it counterclockw... | 10 | 21.09375 |
14,537 | On a table, there are 10 cards numbered $1, 1, 2, 2, 3, 3, 4, 4, 5, 5$. These 10 cards are shuffled and arranged in a row from left to right. Then, the number of cards between the two 1s, the two 2s, the two 3s, the two 4s, and the two 5s are counted. What is the maximum sum of these 5 numbers? | 20 | 50.78125 |
14,538 | In a convex quadrilateral \(ABCD\), \(\overrightarrow{BC} = 2 \overrightarrow{AD}\). Point \(P\) is a point in the plane of the quadrilateral such that \(\overrightarrow{PA} + 2020 \overrightarrow{PB} + \overrightarrow{PC} + 2020 \overrightarrow{PD} = \mathbf{0}\). Let \(s\) and \(t\) be the areas of quadrilateral \(AB... | 337/2021 | 0 |
14,539 | The area of the lunar crescent shape bounded by the portion of the circle of radius 5 and center (0,0), the portion of the circle with radius 2 and center (0,2), and the line segment from (0,0) to (5,0). | \frac{21\pi}{4} | 3.90625 |
14,540 | Three circles with radii 2, 3, and 10 units are placed inside a larger circle such that all circles are touching one another. Determine the value of the radius of the larger circle. | 15 | 17.1875 |
14,541 | Given a sequence $\{a_n\}$ that satisfies: $a_1=-13$, $a_6+a_8=-2$, and $a_{n-1}=2a_n-a_{n+1}$ for $n\geqslant 2$, find the sum of the first 13 terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$. | -\frac{1}{13} | 36.71875 |
14,542 | How many numbers between $1$ and $3010$ are integers multiples of $4$ or $5$ but not of $20$? | 1204 | 0.78125 |
14,543 | From the sequence of natural numbers $1, 2, 3, 4, \ldots$, erase every multiple of 3 and 4, but keep every multiple of 5 (for example, 15 and 20 are not erased). After removing the specified numbers, write the remaining numbers in a sequence: $A_{1}=1, A_{2}=2, A_{3}=5, A_{4}=7, \ldots$. Find the value of $A_{1988}$. | 3314 | 100 |
14,544 | Two people are playing "Easter egg battle." In front of them is a large basket of eggs. They randomly pick one egg each and hit them against each other. One of the eggs breaks, the defeated player takes a new egg, and the winner keeps their egg for the next round (the outcome of each round depends only on which egg has... | 11/12 | 30.46875 |
14,545 | All natural numbers from 1 to 2017 inclusive were written in a row. How many times was the digit 7 written? | 602 | 32.03125 |
14,546 | On the side of a triangle, a point is taken such that an angle equals another angle. What is the smallest possible distance between the centers of the circles circumscribed around triangles, if \( BC = 1 \)? | 1/2 | 40.625 |
14,547 | Three candles can burn for 30, 40, and 50 minutes, respectively (but are not ignited simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How long are exactly two candles burning simultaneously? | 35 | 19.53125 |
14,548 | Add $452_8$ and $167_8$ in base $8$, then subtract $53_8$ from the result. | 570_8 | 21.09375 |
14,549 | Find the natural number \( N \) such that it is divisible by 5 and 49, and it has exactly 10 divisors, including 1 and \( N \). | 12005 | 32.8125 |
14,550 | Given eight students, including Abby and Bridget, are randomly assigned to the 12 spots arranged in three rows of four as shown, calculate the probability that Abby and Bridget are seated directly adjacent to each other (in the same row or same column). | \frac{17}{66} | 1.5625 |
14,551 | Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$ , number $3$ written on the face opposite number $5$ , and number $4$ written on the face opposite number $6$ . The eight cubes... | 24 | 0.78125 |
14,552 |
Let \( a_{1}, a_{2}, \cdots, a_{k}\left(k \in \mathbf{Z}_{+}\right) \) be integers greater than 1, and they satisfy
\[
\left(a_{1}!\right)\left(a_{2}!\right) \cdots\left(a_{k}!\right) \mid 2017!
\]
Determine the maximum value of \( \sum_{i=1}^{k} a_{i} \) as \( k \) varies. | 5024 | 0 |
14,553 | The number \( N \) has the smallest positive divisor 1, the second largest positive divisor \( k \), and the third largest positive divisor \( m \). Moreover, \( k^k + m^m = N \). What is \( N \)? | 260 | 19.53125 |
14,554 | The side \( AD \) of the rectangle \( ABCD \) is equal to 2. On the extension of the side \( AD \) beyond point \( A \), a point \( E \) is taken such that \( EA = 1 \), and \( \angle BEC = 30^\circ \). Find \( BE \). | 2\sqrt{3} | 21.09375 |
14,555 | Given a machine that transforms a positive integer \( N \) based on the rule: if \( N = 7 \), the machine outputs \( 3 \times 7 + 1 = 22 \).
By inputting the result back into the machine and repeating five times, the output sequence is:
\[ 7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52 \r... | 83 | 7.03125 |
14,556 | Consider a 5-minute interval. In this period, an average of 5 bites occur on the first fishing rod, and 1 bite on the second fishing rod. Therefore, the total average number of bites on both rods during these 5 minutes is 6. Determine the average waiting time for the first bite. | 50 | 4.6875 |
14,557 | In a math competition, there are 5 problems, each with a different natural number score. The smaller the problem number, the lower its score (for example, the score for problem 1 is less than the score for problem 2). Xiao Ming solved all the problems correctly. The total score for the first 2 problems is 10 points, an... | 35 | 36.71875 |
14,558 | In a plane, 100 points are marked. It turns out that 40 marked points lie on each of two different lines \( a \) and \( b \). What is the maximum number of marked points that can lie on a line that does not coincide with \( a \) or \( b \)? | 23 | 0 |
14,559 | Each of the eight vertices of a rectangular prism is truncated, and each face of the prism is divided by a diagonal cut into two triangles. Calculate the total number of edges of the altered shape. | 42 | 21.875 |
14,560 | We will call a ticket with a number from 000000 to 999999 excellent if the difference between some two adjacent digits of its number is 5.
Find the number of excellent tickets. | 409510 | 96.09375 |
14,561 | Along an alley, 75 trees consisting of maples and larches were planted in a single row. It is known that there are no two maples with exactly 5 trees between them. What is the maximum number of maples that could have been planted along the alley? | 39 | 3.125 |
14,562 | The sum \( b_{6} + b_{7} + \ldots + b_{2018} \) of the terms of the geometric progression \( \left\{b_{n}\right\} \) with \( b_{n}>0 \) is equal to 6. The sum of the same terms taken with alternating signs \( b_{6} - b_{7} + b_{8} - \ldots - b_{2017} + b_{2018} \) is equal to 3. Find the sum of the squares of these ter... | 18 | 26.5625 |
14,563 | On a strip of size \(1 \times N\), 25 checkers are placed on the first 25 squares on the left. A checker can move to the adjacent right empty square or jump over the adjacent right checker to the next square (if that square is empty). Movement to the left is not allowed. What is the smallest \(N\) such that all checker... | 50 | 19.53125 |
14,564 | Petya inscribed two squares in a right triangle with sides 3, 4, and 5. One vertex of the first square coincides with the right-angle vertex of the triangle, while one side of the second square lies on the hypotenuse. Petya found the side lengths of each square, expressed their ratio as an irreducible fraction, and fou... | 19 | 10.9375 |
14,565 | How many integer pairs $(x,y)$ are there such that \[0\leq x < 165, \quad 0\leq y < 165 \text{ and } y^2\equiv x^3+x \pmod {165}?\] | 99 | 0.78125 |
14,566 | The distances between the points are given as $A B = 30$, $B C = 80$, $C D = 236$, $D E = 86$, $E A = 40$. What is the distance $E C$? | 150 | 13.28125 |
14,567 | What is the maximum length of a closed self-avoiding polygon that can travel along the grid lines of an $8 \times 8$ square grid? | 80 | 23.4375 |
14,568 | It is known that the only solution to the equation
$$
\pi / 4 = \operatorname{arcctg} 2 + \operatorname{arcctg} 5 + \operatorname{arcctg} 13 + \operatorname{arcctg} 34 + \operatorname{arcctg} 89 + \operatorname{arcctg}(x / 14)
$$
is a natural number. Find it. | 2016 | 27.34375 |
14,569 | Consider a modified sequence rule:
1) If a number is 30 or less, triple the number.
2) If a number is more than 30, subtract 15 from it.
Let $G$ be the first number in a sequence generated by the new rule. $G$ is a "magic number" if 18 is not a term in the sequence that starts with $G$. Determine how many of the whole... | 12 | 0 |
14,570 | How many positive integer solutions does the equation have $$ \left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1? $$ ( $\lfloor x \rfloor$ denotes the integer part of $x$ , for example $\lfloor 2\rfloor = 2$ , $\lfloor \pi\rfloor = 3$ , $\lfloor \sqrt2 \rfloor =1$ ) | 110 | 51.5625 |
14,571 | Suppose a sequence of positive numbers $\left\{a_{n}\right\}$ satisfies: $a_{0} = 1, a_{n} = a_{n+1} + a_{n+2},$ for $n = 0, 1, 2, \ldots$. Find $a_{1}$. | \frac{\sqrt{5} - 1}{2} | 21.09375 |
14,572 | A cashier from Aeroflot has to deliver tickets to five groups of tourists. Three of these groups live in the hotels "Druzhba", "Rossiya", and "Minsk". The fourth group's address will be given by tourists from "Rossiya", and the fifth group's address will be given by tourists from "Minsk". In how many ways can the cashi... | 30 | 19.53125 |
14,573 | Let \(a, b, c \in (0,1]\) and \(\lambda\) be a real number such that \(\frac{\sqrt{3}}{\sqrt{a+b+c}} \geq 1+\lambda(1-a)(1-b)(1-c)\) is always satisfied. Find the maximum value of \(\lambda\). | 64/27 | 0 |
14,574 | Give an example of a number $x$ for which the equation $\sin 2017 x - \operatorname{tg} 2016 x = \cos 2015 x$ holds. Justify your answer. | \frac{\pi}{4} | 14.84375 |
14,575 | Three workers are digging a pit. They work in shifts, with each worker working as long as it takes for the other two to dig half of the pit. Working in this manner, they dug the pit. How many times faster would the three workers dig the same pit if they worked simultaneously? | 2.5 | 0 |
14,576 | The odd function $y=f(x)$ has a domain of $\mathbb{R}$, and when $x \geq 0$, $f(x) = 2x - x^2$. If the range of the function $y=f(x)$, where $x \in [a, b]$, is $[\frac{1}{b}, \frac{1}{a}]$, then the minimum value of $b$ is ______. | -1 | 5.46875 |
14,577 | Given the origin of the rectangular coordinate system xOy as the pole and the positive semi-axis of the x-axis as the polar axis, establish a polar coordinate system with the same unit length. The parametric equation of the line l is $$\begin{cases} \overset{x=2+t}{y=1+t}\end{cases}$$ (t is the parameter), and the pola... | \sqrt{2} | 76.5625 |
14,578 | How many values of $x$, $-30<x<120$, satisfy $\cos^2 x + 3\sin^2 x = 1$? | 48 | 41.40625 |
14,579 | In the company, there are elves, fairies, and dwarves. Each elf is friends with all fairies except for three of them, and each fairy is friends with twice as many elves. Each elf is friends with exactly three dwarves, and each fairy is friends with all the dwarves. Each dwarf is friends with exactly half of the total n... | 12 | 4.6875 |
14,580 | For lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$ to be parallel, $a=$ ______. | -7 | 34.375 |
14,581 | The region consisting of all points in three-dimensional space within 4 units of line segment $\overline{CD}$, plus a cone with the same height as $\overline{CD}$ and a base radius of 4 units, has a total volume of $448\pi$. Find the length of $\textit{CD}$. | 17 | 21.09375 |
14,582 | Let $S = \{1, 2, \ldots, 2005\}$. If any set of $n$ pairwise coprime numbers from $S$ always contains at least one prime number, find the smallest value of $n$. | 16 | 1.5625 |
14,583 | Given $$\sqrt {2 \frac {2}{3}}=2 \sqrt { \frac {2}{3}}$$, $$\sqrt {3 \frac {3}{8}}=3 \sqrt { \frac {3}{8}}$$, $$\sqrt {4 \frac {4}{15}}=4 \sqrt { \frac {4}{15}}$$, ..., if $$\sqrt {6 \frac {a}{t}}=6 \sqrt { \frac {a}{t}}$$ (where $a$, $t$∈$R^*$), then $a=$ \_\_\_\_\_\_ , $t=$ \_\_\_\_\_\_ . | 35 | 16.40625 |
14,584 | On Monday, Knight Milivoj traveled 25 miles and spent the night in Zubín. The next day, Tuesday, he reached Veselín. On the way back, he traveled 6 miles more on Thursday than on Monday and spent the night in Kostín. On Friday, he traveled the remaining 11 miles to Rytířov. Determine the distance between Zubín and Vese... | 17 | 7.8125 |
14,585 | Buses travel along a country road, at equal intervals in both directions and at equal speeds. A cyclist, traveling at $16 \, \mathrm{km/h}$, begins counting buses from the moment when two buses meet beside him. He counts the 31st bus approaching from the front and the 15th bus from behind, both meeting the cyclist agai... | 46 | 5.46875 |
14,586 | We are laying railway tracks that are $15 \text{ meters }$ long at a temperature of $t = -8^{\circ} \text{C}$. What gap should we leave between each rail if the maximum expected temperature is $t = 60^{\circ} \text{C}$? The coefficient of expansion for the rail is $\lambda = 0.000012$. | 12.24 | 10.9375 |
14,587 | Given that \(a, b, c\) are positive integers and the quadratic equation \(a x^{2}+b x+c=0\) has two real roots whose absolute values are both less than \(\frac{1}{3}\), find the minimum value of \(a + b + c\). | 25 | 4.6875 |
14,588 | If \( p = \frac{21^{3}-11^{3}}{21^{2}+21 \times 11+11^{2}} \), find \( p \).
If \( p \) men can do a job in 6 days and 4 men can do the same job in \( q \) days, find \( q \).
If the \( q \)-th day of March in a year is Wednesday and the \( r \)-th day of March in the same year is Friday, where \( 18 < r < 26 \), fin... | 27 | 0 |
14,589 | The class monitor wants to buy soda in batches for all 50 students and teachers in the class for the sports day. According to the store's policy, every 5 empty bottles can be exchanged for one soda bottle, so there is no need to buy 50 bottles of soda. Then, the minimum number of soda bottles that need to be bought to ... | 40 | 2.34375 |
14,590 | The circles \(O_{1}\) and \(O_{2}\) touch the circle \(O_{3}\) with radius 13 at points \(A\) and \(B\) respectively and pass through its center \(O\). These circles intersect again at point \(C\). It is known that \(OC = 12\). Find \(AB\). | 10 | 30.46875 |
14,591 | Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in $S$ ? | 21 | 20.3125 |
14,592 | Two irreducible fractions have their denominators equal to 600 and 700. Find the minimum value for the denominator of the sum of the fractions. | 168 | 17.1875 |
14,593 | Four different natural numbers, of which one is 1, have the following properties: the sum of any two of them is a multiple of 2, the sum of any three of them is a multiple of 3, and the sum of all four numbers is a multiple of 4. What is the minimum possible sum of these four numbers? | 40 | 8.59375 |
14,594 | In triangle $XYZ$, where $\angle X = 90^\circ$, the hypotenuse $YZ = 13$, and $\tan Z = 3\cos Y$. What is the length of side $XY$? | \frac{2\sqrt{338}}{3} | 0 |
14,595 | In triangle \(ABC\), the sides opposite to angles \(A, B,\) and \(C\) are denoted by \(a, b,\) and \(c\) respectively. Given that \(c = 10\) and \(\frac{\cos A}{\cos B} = \frac{b}{a} = \frac{4}{3}\). Point \(P\) is a moving point on the incircle of triangle \(ABC\), and \(d\) is the sum of the squares of the distances ... | 160 | 0 |
14,596 | Suppose \( a \) is an integer. A sequence \( x_1, x_2, x_3, x_4, \ldots \) is constructed with:
- \( x_1 = a \),
- \( x_{2k} = 2x_{2k-1} \) for every integer \( k \geq 1 \),
- \( x_{2k+1} = x_{2k} - 1 \) for every integer \( k \geq 1 \).
For example, if \( a = 2 \), then:
\[ x_1 = 2, \quad x_2 = 2x_1 = 4, \quad x_3 =... | 1409 | 7.03125 |
14,597 | The numbers \(a, b, c, d\) belong to the interval \([-6.5 ; 6.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 182 | 6.25 |
14,598 | Workshop A and Workshop B together have 360 workers. The number of workers in Workshop A is three times that of Workshop B. How many workers are there in each workshop? | 270 | 2.34375 |
14,599 | Four chess players - Ivanov, Petrov, Vasiliev, and Kuznetsov - played a round-robin tournament (each played one game against each of the others). A victory awards 1 point, a draw awards 0.5 points to each player. It was found that the player in first place scored 3 points, and the player in last place scored 0.5 points... | 36 | 7.03125 |
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