Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
14,700 | Simplify the expression $\frac{\sqrt{10} + \sqrt{15}}{\sqrt{3} + \sqrt{5} - \sqrt{2}}$.
A) $\frac{2\sqrt{30} + 5\sqrt{2} + 11\sqrt{5} + 5\sqrt{3}}{6}$
B) $\sqrt{3} + \sqrt{5} + \sqrt{2}$
C) $\frac{\sqrt{10} + \sqrt{15}}{6}$
D) $\sqrt{3} + \sqrt{5} - \sqrt{2}$ | \frac{2\sqrt{30} + 5\sqrt{2} + 11\sqrt{5} + 5\sqrt{3}}{6} | 56.25 |
14,701 | A polynomial \( P \) is of the form \( \pm x^{6} \pm x^{5} \pm x^{4} \pm x^{3} \pm x^{2} \pm x \pm 1 \). Given that \( P(2)=27 \), what is \( P(3) \)? | 439 | 10.9375 |
14,702 | Find the smallest \( n > 4 \) for which we can find a graph on \( n \) points with no triangles and such that for every two unjoined points we can find just two points joined to both of them. | 16 | 3.125 |
14,703 | In the Cartesian coordinate plane \(xOy\), the circle \(\Omega\) and the parabola \(\Gamma: y^2 = 4x\) have exactly one common point, and the circle \(\Omega\) is tangent to the \(x\)-axis at the focus \(F\) of the parabola \(\Gamma\). Find the radius of the circle \(\Omega\). | \frac{4 \sqrt{3}}{9} | 3.125 |
14,704 | How many multiples of 4 are there between 200 and 500? | 74 | 0 |
14,705 | In the plane Cartesian coordinate system \( xOy \), an ellipse \( C \) : \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) \( (a>b>0) \) has left and right foci \( F_{1} \) and \( F_{2} \) respectively. Chords \( ST \) and \( UV \) are parallel to the \( x \)-axis and \( y \)-axis respectively, intersecting at point \( P... | \sqrt{15} | 1.5625 |
14,706 | Point $F$ is taken on the extension of side $AD$ of rectangle $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 40$ and $GF = 15$, then $BE$ equals:
[Insert diagram similar to above, with F relocated, set different values for EF and GF] | 20 | 1.5625 |
14,707 | A subset \( H \) of the set of numbers \(\{1, 2, \ldots, 100\}\) has the property that if an element is in \( H \), then ten times that element is not in \( H \). What is the maximum number of elements that \( H \) can have? | 91 | 46.875 |
14,708 | Consider the cubes whose vertices lie on the surface of a given cube. Which one is the smallest among them? | \frac{1}{\sqrt{2}} | 0.78125 |
14,709 | Find all values of the parameter \(a\) for which the quadratic trinomial \(\frac{1}{3} x^2 + \left(a+\frac{1}{2}\right) x + \left(a^2 + a\right)\) has two roots, the sum of the cubes of which is exactly 3 times their product. In your answer, specify the largest of such \(a\). | -1/4 | 0 |
14,710 | There are several white rabbits and gray rabbits. When 6 white rabbits and 4 gray rabbits are placed in a cage, there are still 9 more white rabbits remaining, and all the gray rabbits are placed. When 9 white rabbits and 4 gray rabbits are placed in a cage, all the white rabbits are placed, and there are still 16 gray... | 159 | 0 |
14,711 | There are 200 candies. What is the minimum number of schoolchildren that these candies can be distributed to so that, no matter how the candies are distributed, there are always at least two schoolchildren who receive the same number of candies (possibly none)? | 21 | 27.34375 |
14,712 | Given the function $f(x) = \frac{e^x - 1}{e^x + 1}$, let $g(x) = f(x - 1) + 1$. Define the sequence $\{a_n\}$ such that $a_n = g\left(\frac{1}{n}\right) + g\left(\frac{2}{n}\right) + g\left(\frac{3}{n}\right) + \dots + g\left(\frac{2n - 1}{n}\right)$, where $n$ is a positive integer. The sum of the first $n$ terms of s... | k = 18 | 0.78125 |
14,713 | On the radius \( AO \) of a circle with center \( O \), point \( M \) is chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \( BC \), given that the radius of the circle is 15 and \(\sin \alpha = \frac{\sqrt{21}}{5}\)? | 12 | 17.1875 |
14,714 | John drove continuously from 8:15 a.m. until 2:45 p.m. of the same day and covered a distance of 210 miles. What was his average speed in miles per hour? | 32.31 | 82.8125 |
14,715 | Two circles with radii $\sqrt{5}$ and $\sqrt{2}$ intersect at point $A$. The distance between the centers of the circles is 3. A line through point $A$ intersects the circles at points $B$ and $C$ such that $A B = A C$ (point $B$ does not coincide with $C$). Find $A B$. | \frac{6\sqrt{5}}{5} | 0 |
14,716 | A four-digit number \(\overline{abcd} (1 \leqslant a \leqslant 9, 0 \leqslant b, c, d \leqslant 9)\) is called a \(P\) type number if \(a > b, b < c, c > d\). It is called a \(Q\) type number if \(a < b, b > c, c < d\). Let \(N(P)\) and \(N(Q)\) be the number of \(P\) type and \(Q\) type numbers respectively. Find the ... | 285 | 99.21875 |
14,717 | Misha made himself a homemade dartboard at the summer house. The round board is divided into sectors by circles - it can be used to throw darts. Points are awarded according to the number written in the sector, as indicated in the diagram.
Misha threw 8 darts 3 times. The second time, he scored twice as many points as... | 48 | 0 |
14,718 | In triangle \( \triangle ABC \), the sides opposite to the angles \( A \), \( B \), and \( C \) are of lengths \( a \), \( b \), and \( c \) respectively. Point \( G \) satisfies
$$
\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} = \mathbf{0}, \quad \overrightarrow{GA} \cdot \overrightarrow{GB} = 0.
$$
... | \frac{1}{2} | 0 |
14,719 | Calculate the area of the shape bounded by the lines given by the equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=t-\sin t \\
y=1-\cos t
\end{array}\right. \\
& y=1 \quad (0<x<2\pi, \, y \geq 1)
\end{aligned}
$$ | \frac{\pi}{2} + 2 | 0.78125 |
14,720 | On a sheet of paper, points \( A, B, C, D \) are marked. A recognition device can perform two types of operations with absolute precision: a) measuring the distance in centimeters between two given points; b) comparing two given numbers. What is the minimum number of operations needed for this device to definitively de... | 10 | 27.34375 |
14,721 | Given the function \( f(x) = x^2 \cos \frac{\pi x}{2} \), and the sequence \(\left\{a_n\right\}\) in which \( a_n = f(n) + f(n+1) \) where \( n \in \mathbf{Z}_{+} \). Find the sum of the first 100 terms of the sequence \(\left\{a_n\right\}\), denoted as \( S_{100} \). | 10200 | 14.84375 |
14,722 | Zhenya took a $3 \times 3$ board and placed a column of blue and red cubes on each cell. Then he drew a diagram of the resulting arrangement: he wrote down the number of cubes of both colors in each column (the order of the cubes is unknown).
What is the maximum number of blue cubes Zhenya can see if he looks at the c... | 12 | 3.125 |
14,723 | In the village of Matitika, five friends live along a straight road in the following order: Alya, Bella, Valya, Galya, and Dilya. Each of them calculated the sum of distances (in meters) from her house to the houses of the others. Bella reported the number 700, Valya reported 600, Galya reported 650. How many meters ar... | 150 | 22.65625 |
14,724 | Given two moving points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) on the parabola \( y^2 = 6x \), where \( x_1 \neq x_2 \) and \( x_1 + x_2 = 4 \). The perpendicular bisector of segment \( AB \) intersects the \( x \)-axis at point \( C \). Find the maximum area of triangle \( \triangle ABC \). | \frac{14}{3} \sqrt{7} | 0 |
14,725 | Two ants crawled along their own closed routes on a $7 \times 7$ board. Each ant crawled only along the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the minimum possible number of such sides that both the first and the second ant crawled along? | 16 | 3.90625 |
14,726 | Find all five-digit numbers \(\overline{abcde}\) that are divisible by 9, and \(\overline{ace} - \overline{bda} = 760\). | 81828 | 0 |
14,727 | Seven students are standing in a row for a graduation photo. Among them, student A must stand in the middle, and students B and C must stand together. How many different arrangements are there? | 192 | 0 |
14,728 | Adia writes a list in increasing order of the integers between 1 and 100, inclusive, that cannot be written as the product of two consecutive positive integers. What is the 40th integer in her list? | 46 | 82.03125 |
14,729 |
Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin ^{2} x(1-\cos x)}
$$ | 55/96 | 22.65625 |
14,730 | A right triangular prism \( ABC-A_{1}B_{1}C_{1} \) has 9 edges of equal length. Point \( P \) is the midpoint of \( CC_{1} \). The dihedral angle \( B-A_{1}P-B_{1} \) is \( \alpha \). What is \( \sin \alpha \)? | \frac{\sqrt{10}}{4} | 12.5 |
14,731 | Given that $17^{-1} \equiv 26 \pmod{53}$, find $36^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) | 27 | 67.1875 |
14,732 | Inside a cube with edge length 1, an inscribed sphere \( O_1 \) is drawn. Another smaller sphere \( O_2 \) is drawn inside the cube such that it is externally tangent to the larger sphere and simultaneously tangent to three faces of the cube. What is the surface area of the smaller sphere \( O_2 \)? | (7-4\sqrt{3})\pi | 0.78125 |
14,733 | From the set \( M = \{1, 2, \cdots, 2008\} \) of the first 2008 positive integers, a \( k \)-element subset \( A \) is chosen such that the sum of any two numbers in \( A \) cannot be divisible by the difference of those two numbers. What is the maximum value of \( k \)? | 670 | 34.375 |
14,734 | How many irreducible fractions with a numerator of 2015 are there that are less than \( \frac{1}{2015} \) and greater than \( \frac{1}{2016} \)? | 1440 | 73.4375 |
14,735 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $(\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{a}$, and $(2\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{4} | 89.84375 |
14,736 | A certain product was bought in the fall, and 825 rubles were paid for it. In the fall, the price per kilogram of this product was 1 ruble cheaper than in the spring. Therefore, for the same amount in the spring, 220 kg less was bought. How much does 1 kg of the product cost in the spring, and how much was bought in th... | 550 | 19.53125 |
14,737 | In the triangle \( \triangle ABC \), the maximum value of \( \sin A + \sin B + 2 \sqrt{7} \sin C \) is ______. | \frac{27}{4} | 0 |
14,738 | The students' written work has a binary grading system, i.e., a work will either be accepted if it is done well or not accepted if done poorly. Initially, the works are checked by a neural network which makes an error in 10% of the cases. All works identified as not accepted by the neural network are then rechecked man... | 66 | 0 |
14,739 | The Evil League of Evil plans to set out from their headquarters at (5,1) to poison two pipes: one along the line \( y = x \) and the other along the line \( x = 7 \). They wish to determine the shortest distance they can travel to visit both pipes and then return to their headquarters. | 4\sqrt{5} | 14.84375 |
14,740 | Convert the following radians to degrees: convert degrees to radians:
(1) $\frac{\pi}{12} =$ \_\_\_\_\_\_ ; (2) $\frac{13\pi}{6} =$ \_\_\_\_\_\_ ; (3) $- \frac{5}{12}\pi =$ \_\_\_\_\_\_ .
(4) $36^{\circ} =$ \_\_\_\_\_\_ $rad$ ; (5) $-105^{\circ} =$ \_\_\_\_\_\_ $rad$. | -\frac{7\pi}{12} | 83.59375 |
14,741 | There are a certain number of identical plastic bags that can be nested within each other. If all the other bags are inside one of the bags, we call this situation a "bag of bags." Calculate the number of ways to make a "bag of bags" from 10 bags.
Explanation: Use parentheses to denote a bag.
If we had one bag, the w... | 16796 | 88.28125 |
14,742 | Find the largest natural number in which each digit, starting from the third, is equal to the sum of all previous digits of the number. | 101248 | 0.78125 |
14,743 | Given that circle $A$ has radius $150$, and circle $B$, with an integer radius $r$, is externally tangent to circle $A$ and rolls once around the circumference of circle $A$, determine the number of possible integer values of $r$. | 11 | 3.125 |
14,744 | How many kilometers will a traveler cover in 17 days, spending 10 hours a day on this, if he has already covered 112 kilometers in 29 days, traveling 7 hours each day? | 93.79 | 0 |
14,745 | Usain runs one lap around the school stadium at a constant speed, and photographers Arina and Marina are positioned near the track. After the start, for 4 seconds, Usain was closer to Arina, then for 21 seconds he was closer to Marina, and then until the finish, he was again closer to Arina. How long does it take for U... | 42 | 2.34375 |
14,746 | Given a triangle $\triangle ABC$ with area $S$ and sides $a$, $b$, $c$ that satisfy the equations: $S=a^{2}-(b-c)^{2}$, $b+c=8$, find the maximum value of the area $S$ of $\triangle ABC$. | \frac {64}{17} | 16.40625 |
14,747 | Let the set \( T = \{0, 1, \dots, 6\} \),
$$
M = \left\{\left.\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4} \right\rvert\, a_i \in T, i=1,2,3,4\right\}.
$$
If the elements of the set \( M \) are arranged in decreasing order, what is the 2015th number? | \frac{386}{2401} | 0.78125 |
14,748 | Calculate the definite integral:
$$
\int_{0}^{\frac{2\pi}{3}} \frac{\cos^2 x \, dx}{(1 + \cos x + \sin x)^2}
$$ | \frac{\sqrt{3}}{2} - \ln 2 | 5.46875 |
14,749 | A natural number greater than 1 is called "good" if it is equal to the product of its distinct proper divisors (excluding 1 and the number itself). Find the sum of the first ten "good" natural numbers. | 182 | 61.71875 |
14,750 | Find the coefficient of \(x^5\) in the expansion of \(\left(1+2x+3x^2+4x^3\right)^5\). | 1772 | 23.4375 |
14,751 | Simplify first, then evaluate: $\left(\frac{2}{m-3}+1\right) \div \frac{2m-2}{m^2-6m+9}$, and then choose a suitable number from $1$, $2$, $3$, $4$ to substitute and evaluate. | -\frac{1}{2} | 23.4375 |
14,752 | Given that an isosceles trapezoid is circumscribed around a circle, find the ratio of the area of the trapezoid to the area of the circle if the distance between the points where the circle touches the non-parallel sides of the trapezoid is related to the radius of the circle as $\sqrt{3}: 1$. | \frac{8\sqrt{3}}{3\pi} | 0 |
14,753 | Given that \( P \) is a point on the hyperbola \( C: \frac{x^{2}}{4} - \frac{y^{2}}{12} = 1 \), and \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C \), and \( M \) and \( I \) are the centroid and incenter of the triangle \( \triangle P F_{1} F_{2} \) respectively. If \( M I \perp x \)-axi... | \sqrt{6} | 1.5625 |
14,754 | In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of \(AB\) and \(AC\), respectively, and points \(P\) and \(Q\) trisect \(BC\). Given that \(A\), \(M\), \(N\), \(P\), and \(Q\) lie on a circle and \(BC = 1\), compute the area of triangle \(ABC\). | \frac{\sqrt{7}}{12} | 0.78125 |
14,755 | On the base \(AC\) of an isosceles triangle \(ABC\), a point \(E\) is taken, and on the sides \(AB\) and \(BC\), points \(K\) and \(M\) are taken such that \(KE \parallel BC\) and \(EM \parallel AB\). What fraction of the area of triangle \(\mathrm{ABC}\) is occupied by the area of triangle \(KEM\) if \(BM:EM = 2:3\)? | 6/25 | 8.59375 |
14,756 | At an international table tennis championship, 200 participants arrived. The tournament follows a single-elimination format, meaning that in each match two players compete, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who have won at least three... | 66 | 28.125 |
14,757 | A geometric sequence of positive integers starts with a first term of 4 and the fourth term is 324. What is the fifth term of the sequence? | 324 | 0 |
14,758 | For what smallest natural $n$ is each of the fractions
$$
\frac{7}{n+9}, \frac{8}{n+10}, \ldots, \frac{31}{n+33}
$$
in its simplest form? | 35 | 64.0625 |
14,759 | Santa Claus arrived at the house of Arnaldo and Bernaldo carrying ten distinct toys numbered from 1 to 10 and said to them: "Toy number 1 is for you, Arnaldo, and toy number 2 is for you, Bernaldo. But this year, you may choose to keep more toys as long as you leave at least one for me." Determine in how many ways Arna... | 6305 | 0 |
14,760 | A magician and their assistant are planning to perform the following trick. A spectator writes a sequence of $N$ digits on a board. The magician's assistant covers two adjacent digits with a black circle. Then the magician enters. Their task is to guess both of the covered digits (and the order in which they are arrang... | 101 | 11.71875 |
14,761 | Find the arithmetic square root of $4$, the square root of $5$, and the cube root of $-27$. | -3 | 27.34375 |
14,762 | Given triangle \( ABC \). On the side \( AC \), which is the largest in the triangle, points \( M \) and \( N \) are marked such that \( AM = AB \) and \( CN = CB \). It turns out that angle \( NBM \) is three times smaller than angle \( ABC \). Find \( \angle ABC \). | 108 | 27.34375 |
14,763 | A workshop produces products of types $A$ and $B$. Producing one unit of product $A$ requires 10 kg of steel and 23 kg of non-ferrous metals, while producing one unit of product $B$ requires 70 kg of steel and 40 kg of non-ferrous metals. The profit from selling one unit of product $A$ is 80 thousand rubles, and for pr... | 2180 | 0.78125 |
14,764 | On the $xy$-plane, find the number of triangles whose vertices have integer coordinates $(x, y)$ satisfying $1 \leq x \leq 4$ and $1 \leq y \leq 4$. | 516 | 32.03125 |
14,765 | We regularly transport goods from city $A$ to city $B$, which is $183 \mathrm{~km}$ away. City $A$ is $33 \mathrm{~km}$ from the river, while city $B$ is built on the riverbank. The cost of transportation per kilometer is half as much on the river as on land. Where should we build the road to minimize transportation co... | 11\sqrt{3} | 14.0625 |
14,766 | The café "Buratino" operates 6 days a week with Mondays off. Kolya said that from April 1 to April 20, the café was open for 17 days, and from April 10 to April 30, it was open for 18 days. It is known that he made a mistake once. What was the date of the last Tuesday in April? | 29 | 8.59375 |
14,767 | Given the complex number $z$ satisfies $|z+3-\sqrt{3} i|=\sqrt{3}$, what is the minimum value of $\arg z$? | $\frac{5}{6} \pi$ | 0 |
14,768 | There are \( n \) pieces of paper, each containing 3 different positive integers no greater than \( n \). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper. | 84 | 3.125 |
14,769 | Whole numbers that read the same from left to right and right to left are called symmetrical. For example, the number 513315 is symmetrical, whereas 513325 is not. How many six-digit symmetrical numbers exist such that adding 110 to them leaves them symmetrical? | 81 | 92.1875 |
14,770 | Given the line \( y = x - 1 \) intersects the ellipse \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2} - 1} = 1 \) (where \( a > 1 \)) at points \( A \) and \( B \). If the circle with diameter \( AB \) passes through the left focus of the ellipse, find the value of \( a \). | \frac{\sqrt{6} + \sqrt{2}}{2} | 0 |
14,771 | A metro network has at least 4 stations on each line, with no more than three transfer stations per line. No transfer station has more than two lines crossing. What is the maximum number of lines such a network can have if it is possible to travel from any station to any other station with no more than two transfers? | 10 | 2.34375 |
14,772 | How many different ways are there to split the number 2004 into natural summands that are approximately equal? There can be one or several summands. Numbers are considered approximately equal if their difference is no more than 1. Ways that differ only by the order of summands are considered the same. | 2004 | 15.625 |
14,773 | The volume of a cylinder circumscribed around a sphere with radius $r$ is $V_{1}$, and the volume of a cone circumscribed around the same sphere is $V_{2}$. What is the minimum value of the ratio $V_{2} / V_{1}$? | 4/3 | 0 |
14,774 | Find the number of 5-digit numbers where the ten-thousands place is not 5, the units place is not 2, and all digits are distinct. | 21840 | 30.46875 |
14,775 | A solid triangular prism is made up of 27 identical smaller solid triangular prisms. The length of every edge of each of the smaller prisms is 1. If the entire outer surface of the larger prism is painted, what fraction of the total surface area of all the smaller prisms is painted? | 1/3 | 22.65625 |
14,776 | In a regular 2019-gon, numbers are placed at the vertices such that the sum of the numbers in any nine consecutive vertices is 300. It is known that the number at the 19th vertex is 19, and the number at the 20th vertex is 20. What number is at the 2019th vertex? | 61 | 3.125 |
14,777 | Points \( M \) and \( N \) divide side \( AC \) of triangle \( ABC \) into three equal parts, each of which is 5, with \( AB \perp BM \) and \( BC \perp BN \). Find the area of triangle \( ABC \). | \frac{75 \sqrt{3}}{4} | 37.5 |
14,778 | Given a regular quadrilateral pyramid $S-ABCD$ with side edges of length $4$ and $\angle ASB = 30^\circ$, a plane passing through point $A$ intersects the side edges $SB$, $SC$, and $SD$ at points $E$, $F$, and $G$ respectively. Find the minimum perimeter of the cross-section $AEFG$. | 4\sqrt{3} | 1.5625 |
14,779 | Four boys, four girls, and a coach are positioned on a circular track. Each girl is diametrically opposite to one of the boys. The length of the track is 50 meters. On the coach's signal, they all run towards the coach by the shortest path along the track. What is the total distance run by all the children together? | 100 | 4.6875 |
14,780 | Nadia bought a compass and after opening its package realized that the length of the needle leg is $10$ centimeters whereas the length of the pencil leg is $16$ centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least $30$ degrees but the need... | 12 | 0.78125 |
14,781 | The sequence \( a_{0}, a_{1}, \cdots, a_{n} \) satisfies:
\[
a_{0}=\sqrt{3}, \quad a_{n+1}=[a_{n}]+\frac{1}{\{a_{n}\}}
\]
where \( [x] \) denotes the greatest integer less than or equal to the real number \( x \), and \( \{x\}=x-[x] \). Find the value of \( a_{2016} \). | 3024 + \sqrt{3} | 6.25 |
14,782 | János, a secretary of a rural cooperative, travels to Budapest weekly. His wife leaves home at 4 o'clock to meet him at the station, arriving at exactly the same time as the train. They are home by 5 o'clock. One day, the train arrived earlier, unbeknownst to his wife, so she encountered him on the way home. They arriv... | 3.5 | 12.5 |
14,783 | In the trapezoid \(ABCD \) with \( AD \parallel BC \), the angle \( \angle ADB \) is twice the angle \( \angle ACB \). It is known that \( BC = AC = 5 \) and \( AD = 6 \). Find the area of the trapezoid. | 22 | 8.59375 |
14,784 | What is the largest factor of $130000$ that does not contain the digit $0$ or $5$ ? | 26 | 87.5 |
14,785 | Evaluate the series $$\frac{2^1}{8^1 - 1} + \frac{2^2}{8^2 - 1} + \frac{2^3}{8^3 - 1} + \frac{2^4}{8^4 - 1} + \cdots.$$ | \frac{1}{3} | 35.15625 |
14,786 | The area of the floor in a rectangular room is 360 square feet. The length of the room is twice its width. The homeowners plan to cover the floor with 8-inch by 8-inch tiles. How many tiles will be in each row along the length of the room? | 18\sqrt{5} | 0 |
14,787 | The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$ .
*Proposed by Lewis Chen* | 1500 | 1.5625 |
14,788 | Given the radii of the inner and outer circles are $4$ and $8$, respectively, with the inner circle divided into regions with point values 3, 1, 1, and the outer circle divided into regions with point values 2, 3, 3, calculate the probability that the score sum of two darts hitting this board is odd. | \frac{4}{9} | 21.09375 |
14,789 | Three chiefs of Indian tribes are sitting by a fire with three identical pipes. They are holding a war council and smoking. The first chief can finish a whole pipe in ten minutes, the second in thirty minutes, and the third in an hour. How should the chiefs exchange the pipes among themselves in order to prolong their ... | 20 | 27.34375 |
14,790 | Seven people are standing in two rows, with 3 people in the front row and 4 people in the back row. Three people, A, B, and C, are added to the queue, with one person added to the front row and two people added to the back row, while the positions of the other people remain unchanged. Calculate the number of different ... | 360 | 5.46875 |
14,791 | A parallelogram is generated by the vectors $\begin{pmatrix} 3 \\ 1 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix}$. Find the cosine of the angle $\theta$ between the diagonals of the parallelogram. | -\frac{\sqrt{3}}{3} | 0 |
14,792 | A certain electronic device contains three components, with probabilities of failure for each component being $0.1, 0.2, 0.3$, respectively. If the probabilities of the device failing when one, two, or three components fail are $0.25, 0.6, 0.9$, respectively, find the probability that the device fails. | 0.1601 | 31.25 |
14,793 | Evaluate or simplify:
1. $\frac{\sqrt{1 - 2\sin 15^{\circ}\cos 15^{\circ}}}{\cos 15^{\circ} - \sqrt{1 - \cos^{2} 165^{\circ}}}$
2. Given $|\vec{a}| = 4$, $|\vec{b}| = 2$, and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{2\pi}{3}$, find the value of $|\vec{a} + \vec{b}|$. | 2\sqrt{3} | 93.75 |
14,794 | Two water droplets fall freely one after another from a $300 \mathrm{~m}$ high cliff. The first droplet has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts falling.
How many millimeters apart will the two droplets be at the moment the first one reaches the base of the cliff? (The result should ... | 34.6 | 12.5 |
14,795 | Given the sets \( A=\{x \mid 5x - a \leqslant 0\} \) and \( B=\{x \mid 6x - b > 0\} \), where \( a, b \in \mathbf{N} \), and \( A \cap B \cap \mathbf{N} = \{2, 3, 4\} \), determine the number of integer pairs \( (a, b) \). | 30 | 56.25 |
14,796 | Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).
Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \... | 31 | 84.375 |
14,797 |
In a quadrilateral pyramid \(S A B C D\):
- The areas of the lateral faces \(S A B, S B C, S C D, S D A\) are 9, 9, 27, 27 respectively.
- The dihedral angles at the edges \(A B, B C, C D, D A\) are equal.
- The quadrilateral \(A B C D\) is inscribed in a circle, with an area of 36.
Find the volume of the pyramid \(... | 54 | 7.03125 |
14,798 | Liu Yulei bought 4 packs of yogurt and 4 packs of fresh milk at the supermarket, paying a total of 14 yuan. Later, she returned 2 packs of yogurt and bought 4 more packs of fresh milk, and the cashier gave her 1 yuan back. The price of each pack of yogurt is ____ yuan. | 2.5 | 15.625 |
14,799 | A square with sides of 10 inches is shown. If $P$ is a point such that the segments $\overline{PA}$, $\overline{PB}$, and $\overline{PC}$ are equal in length, and segment $\overline{PC}$ is perpendicular to segment $\overline{GD}$, what is the area, in square inches, of triangle $APB$? Here, $G$ is the midpoint of side... | \frac{75}{4} | 10.15625 |
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