problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
In triangle $XYZ$, sides $XY$, $YZ$, and $ZX$ are tangent to a circle with center $O$. Given that $\angle XYZ = 80^\circ$ and $\angle YZX = 78^\circ$, find $\angle YXO$, in degrees. | 11^\circ | math | 60 |
Find all positive integers \(a, b, c\), where \(1 < a < b < c\), such that \((a-1)(b-1)(c-1)\) is a divisor of \(abc - 1\). | (2, 4, 8) | math | 51 |
Point \( M \) lies on the parabola \( y = 2x^2 - 3x + 4 \), and point \( F \) lies on the line \( y = 3x - 4 \).
Find the minimum value of \( MF \). | \frac{7 \sqrt{10}}{20} | math | 58 |
Given that one root of the equation $x^2+mx+3=0$ is 1, find the other root and the value of $m$. | -4 | math | 33 |
The line $y = 2$ intersects the graph of $y = 3x^2 + 2x - 5$ at the points $C$ and $D$. The distance between $C$ and $D$ can be written as $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are positive integers that do not share any factors other than one. Find the value of $p - q$. | 85 | math | 93 |
Consider the ellipse $25x^2 +9 y^2 = 225.$ A hyperbola is drawn, using the foci of the ellipse as its vertices and the endpoints of the major axis of the ellipse as its foci. Let $(s, t)$ be a point where the hyperbola and ellipse intersect. Compute $s^2.$ | \frac{81}{41} | math | 76 |
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. Determine the smallest possible value of the length of the side of the original square. | 4 | math | 49 |
Given that $f(x)$ and $g(x)$ are even and odd functions defined on $\mathbb{R}$, respectively, and $f(x) - g(x) = 2^{1-x}$, find the value of $g(-1)$. | -\frac{3}{2} | math | 54 |
Given that a certain ninth-grade school held a basketball game where each pair of classes played one game, and a total of $45$ games were played, calculate the number of participating classes. | 10 | math | 39 |
Given the functions $f(x)=\ln x$, $g(x)= \frac {a}{x}(a > 0)$, and let $F(x)=f(x)+g(x)$.
$(1)$ Find the intervals of monotonicity for the function $F(x)$.
$(2)$ If the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ on the graph of the function $y=F(x)(x\in(0,3])$ always satisfies $k\leqslant \frac {1}{... | a_{min}= \frac {1}{2} | math | 136 |
Given $a > 1$, $b > 1$, and $$\frac {1}{a-1} + \frac {1}{b-1} = 1$$, find the minimum value of $a + 4b$. | 14 | math | 51 |
In $\triangle ABC$, the sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$, respectively. If $a$, $b$, and $c$ form a geometric sequence and $\cos B = \frac{4}{5}$, find the value of $\frac{1}{\tan A} + \frac{1}{\tan C}$. | \frac{5}{3} | math | 84 |
The equation of the circle passing through the intersection points of the circles $x^{2}+y^{2}+6x-4=0$ and $x^{2}+y^{2}+6y-28=0$, with its center lying on the line $x-y-4=0$, is ____. | x^{2}+y^{2}-x+7y-32=0 | math | 69 |
The parabola $y = ax^2 + bx + c$ has a vertex at $(q,q)$ and a $y$-intercept at $(0, -2q)$, where $q \neq 0$. What is the value of $b$? | \frac{6}{q} | math | 58 |
Find the sum of the squares of the natural divisors of the number 1800. (For example, the sum of the squares of the natural divisors of the number 4 is \(1^{2} + 2^{2} + 4^{2} = 21\)). | 5035485 | math | 63 |
Given that $x$, $y$, and $z$ are positive numbers and they satisfy the equation $x^{2}+y^{2}+z^{2}=1$, find the minimum value of $S= \frac {1+z}{2xyz}$. | 4 | math | 55 |
For let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For instance, $\clubsuit(9)=9$ and $\clubsuit(234)=2+3+4=9$. Determine the number of two-digit values of $x$ for which $\clubsuit(\clubsuit(x))=2$. | 9 | math | 72 |
A point $P$ is outside a circle and is $26$ inches from the center. A secant from $P$ cuts the circle at $Q$ and $R$ so that the external segment of the secant $PQ$ is $15$ inches and $QR$ is $9$ inches. An additional tangent from $P$ touches the circle at point $S$. Find the radius of the circle.
A) $\sqrt{144}$ inche... | \sqrt{316} | math | 132 |
The average value of numbers from 1, 2, 3, …, 60 that can be divided by 3 but not by 4 is. | 30 | math | 34 |
Given sets $A = \{x | x^2 - 4 = 0\}$ and $B = \{x | ax - 2 = 0\}$. If $B \subseteq A$, find the set of real number values for $a$. | \{0, 1, -1\} | math | 56 |
Any six points are taken inside or on a square with side length $2$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$. | \sqrt{2} | math | 64 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively.
(1) If $A$, $B$, and $C$ form an arithmetic sequence, find the range of values for $\cos A + \cos C$;
(2) If $a$, $b$, and $c$ form a geometric sequence, and $\cos B = \frac{4}{5}$, find the value of $\frac{1}{\tan... | \frac{5}{3} | math | 124 |
Given that $A_{11}^{m}=11\times10\times\dots\times6\times5$, find the value of $m$. | 7 | math | 35 |
The negation of the proposition "There exists an $x \in \mathbb{R}$ such that $2^x > 0$" is "____". | \forall x \in \mathbb{R}, 2^x \leq 0 | math | 34 |
What is the largest integer \( k \) such that \( k+1 \) divides
\[ k^{2020} + 2k^{2019} + 3k^{2018} + \cdots + 2020k + 2021? \ | 1010 | math | 67 |
13. Convert the parametric equation $\left\{\begin{matrix} x=3+4\cos \theta \\ y=-2+4\sin \theta \end{matrix}\right.$ (with $\theta$ as the parameter) into its Cartesian form. | (x-3)^2+(y+2)^2=16 | math | 56 |
Four people, A, B, C, and D, went fishing and caught a total of 25 fish. The number of fish each person caught, ranked from most to least, is A, B, C, D. It is known that the number of fish A caught is equal to the sum of the number of fish B and C caught, and the number of fish B caught is equal to the sum of the numb... | 11, \, 7, \, 4, \, 3 | math | 102 |
In triangle $\triangle ABC$, the sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ are three consecutive even numbers, and $C=2A$. Find the value of $a$. | 8 | math | 49 |
Given a triangle with side lengths 17 cm, 144 cm, and 145 cm, determine the area of the triangle in square centimeters. | 1224 | math | 35 |
Triangle $PQR$ has side lengths $PQ=7$, $QR=8$, and $PR=9$. Two ants start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$? | 5 | math | 62 |
Given the positive real numbers \(x\) and \(y\) such that
\[ x^5 + 5x^3y + 5x^2y^2 + 5xy^3 + y^5 = 1, \]
what positive values can \(x + y\) take? | 1 | math | 62 |
Find all triples of natural numbers $(a, b, c)$ for which the number $$ 2^a + 2^b + 2^c + 3 $$ is the square of an integer. | (1, 1, 1) | math | 47 |
Steaks are mainly divided into filet steak, sirloin steak, ribeye steak, and T-bone steak. A beef purchaser randomly selects 100 boxes from a batch of steaks and obtains the following data based on the classification standard of steaks:
| Type of Steak | Filet Steak | Sirloin Steak | Ribeye Steak | T-bone Steak |
|---... | \frac{3}{5} | math | 234 |
Given an ellipse $(C)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ that passes through point $A(2,3)$ with its right focus at $F(2,0)$.
1. Find the equation of ellipse $(C)$.
2. Assume the origin as $O$, and a line $(l)$ parallel to $OA$ that intersects with ellipse $(C)$. The distance between $OA$ and $(... | y= \frac{3}{2}x ± \frac{13}{2} | math | 133 |
Given that the square roots of a positive number are $a$ and $2a-6$.
$(1)$ Find the value of $a$ and determine the positive number.
$(2)$ Find the cube root of $10a+7$. | 3 | math | 53 |
In a triangle, one angle measures $45^\circ$, and the triangle is scalene. An exterior angle at one vertex is $135^\circ$. Find the sum of the two possible values for the smallest angle in the triangle. | 90^\circ | math | 50 |
The tangent line to the curve $y=x^3+x-2$ at point P is parallel to the line $y=4x-1$. Find the equation of this tangent line. | y=4x-4 \text{ or } y=4x | math | 39 |
Consider a square with side length $2a$. Inside the square, a smaller square with side length $b$ is placed in one corner, such that its sides are flush with two sides of the larger square. The remaining area of the large square is divided into four congruent rectangles. Find the perimeter, in units, of one of these fo... | 4a | math | 83 |
Find the equation of the hyperbola that shares the same asymptotes with the hyperbola ${x^{2}}-4{{y}^{2}}=4$ and passes through the point $M\left( 2,{ }\sqrt{5} \right)$. | \frac{{y}^{2}}{4}-\frac{{x}^{2}}{16}=1 | math | 57 |
There are 18 identical cars in a train. In some cars, exactly half of the seats are free, in others, exactly one-third of the seats are free, and in the remaining cars, all seats are occupied. At the same time, exactly one-ninth of all seats in the whole train are free. How many cars have all seats occupied? | 13 | math | 73 |
Find the slope of the tangent line to the curve \( y = x^3 \) at the point \( C(-2, -8) \). | 12 | math | 31 |
Emily surveyed a group of people about their knowledge of bats. To the nearest tenth of a percent, she found that $74.5\%$ of the people surveyed thought bats carried diseases. Of the people who thought bats carried diseases, $52.3\%$ said that bats frequently carried Lyme disease. Since bats do not frequently carry Ly... | 87 | math | 90 |
Calculate the result of $2^{-3}$. | \frac{1}{8} | math | 10 |
Let set $M=\{-1, 0, 1\}$, and set $N=\{a, a^2\}$. Find the real number $a$ such that $M \cap N = N$. | -1 | math | 47 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given $a=2$, $C=\frac{\pi}{4}$, $\cos B=\frac{3}{5}$,
(I) Find the values of $\sin B$ and $\sin A$.
(II) Find the area $S$ of $\triangle ABC$. | \frac{8}{7} | math | 88 |
You have 5 different shirts and 6 different pairs of pants. However, one special shirt cannot be paired with two specific pants. How many unique outfits can you make? | 28 | math | 35 |
Given the function $f(x) = x^4 - 4x^3 + 10x^2$, calculate the number of roots of the equation $f(x) = 0$ in the interval $[1, 2]$. | 0 | math | 52 |
Given an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e= \frac{\sqrt{2}}{2}$ and one of its foci at $(-1,0)$.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) If a line $l$ with slope $1$ intersects the ellipse $C$ at points $A$ and $B$, and $|AB| = \frac{4\sqrt{2}}{3}$, find the eq... | y = x + 1 \text{ or } y = x - 1 | math | 141 |
The sum \( A = \left\lfloor \frac{8}{9} \right\rfloor + \left\lfloor \frac{8^2}{9} \right\rfloor + \cdots + \left\lfloor \frac{8^{2014}}{9} \right\rfloor \) when divided by 63 leaves a remainder of \(\qquad\). (\(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to the real number \( x \)). | 56 | math | 117 |
If the root of a linear equation is the solution of a system of linear inequalities, then the linear equation is called the "associated equation" of the inequality system. In the equations ① $5x-2=0$, ② $\frac{3}{4}x+1=0$, ③ $x-\left(3x+1\right)=-5$, the associated equation of the inequality system $\left\{\begin{array... | ③ | math | 126 |
Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{2023} - 1 = 0$. Find the probability that
\[
\sqrt{3} \le \left| v + w \right|.
\] | \frac{337}{1011} | math | 57 |
Given two circles with radii 5 units and 7 units, determine the maximum number of different values for the number of lines that can be tangent to both circles simultaneously. | 5 | math | 35 |
Given plane vectors $\overrightarrow{a}=(-1,-1)$ and $\overrightarrow{b}=(-3,4)$, calculate the projection vector of $5\overrightarrow{a}$ onto $\overrightarrow{b}$. | (\frac{3}{5},-\frac{4}{5}) | math | 48 |
Given vectors $\overrightarrow{a}=( \frac {1}{3},\tan \alpha)$ and $\overrightarrow{b}=(\cos \alpha,1)$, and knowing that $\overrightarrow{a} \parallel \overrightarrow{b}$, calculate the value of $\cos 2\alpha$. | \frac {7}{9} | math | 66 |
What is the sum of the 2010 fractions of the form $\frac{3}{n(n+3)}$ if the values of $n$ range from 1 to 2010? Express your answer as a decimal to the nearest thousandth. | 1.832 | math | 56 |
Given that $x$ is a perfect square, determine the expression for the next larger perfect square. | x+2\sqrt{x}+1 | math | 20 |
Let $S = \{1, 2, \ldots, 2016\}$ , and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest
positive integer such that $f^{(n)}(1) = 1$ , where $f^{(i)}(x) = f(f^{(i-1)}(x))$ . What is the expected value of $n$ ?
| \frac{2017}{2} | math | 111 |
Consider a right triangle $\triangle ABC$ with sides of lengths $9, 12,$ and $15$. A circle of radius $2$ rolls around the inside of this triangle, always touching at least one side of the triangle. Calculate the total distance traveled by the center $P$ of the circle during a complete return to its original starting p... | 11 + 5 + 8 = 24 | math | 73 |
Given that $a$ and $b$ are opposite numbers, and $a \neq 0$, $c$ and $d$ are reciprocals, $|m|=3$, find the value of $m^{2}-\left(-1\right)+|a+b|-cdm$. | 7 \text{ or } 13 | math | 63 |
The distance from the focus of the parabola $y=\frac{1}{4}x^2$ to the asymptote of the hyperbola $y^2-\frac{x^2}{3}=1$ is $\frac{\sqrt{3}}{2}$. | \frac{\sqrt{3}}{2} | math | 58 |
Find $c$ such that $\lfloor c \rfloor$ satisfies
\[3x^2 - 9x - 30 = 0\]
and $\{ c \} = c - \lfloor c \rfloor$ satisfies
\[4x^2 - 8x + 1 = 0.\] | 6 - \frac{\sqrt{3}}{2} | math | 70 |
Calculate the number of increasing sequences of positive integers $b_1 \le b_2 \le b_3 \le \cdots \le b_7 \le 2010$ such that $b_i - i$ is odd for $1 \le i \le 7$. The result can be expressed as ${p \choose q}$ for some integers $p > q$. Determine the remainder when $p$ is divided by 1000. | 8 | math | 98 |
Given the general term formula of the sequence $\{a_n\}$, $a_n = n \cos \frac{n\pi}{2}$, and its first $n$ terms sum $S_n$, find the value of $S_{2016}$. | 1008 | math | 56 |
Given the line $y=2x+\frac{p}{2}$ and the parabola ${x}^{2}=2py\ (p > 0)$, find the distance between their intersection points $A$ and $B$. | 10p | math | 50 |
The quadratic $x^2 + 2200x + 4200$ can be written in the form $(x+b)^2 + c$, where $b$ and $c$ are constants. What is $\frac{c}{b}$? | -1096 | math | 55 |
Given that $\theta$ is an angle in the third quadrant, and $sin^{4}\theta+cos^{4}\theta= \frac {5}{9}$, then $sin2\theta= \_\_\_\_\_\_$. | \frac {2 \sqrt {2}}{3} | math | 50 |
Let $\omega$ be a nonreal root of $z^4 = 1.$ Let $b_1, b_2, \dots, b_n$ be real numbers such that
\[
\frac{1}{b_1 + \omega} + \frac{1}{b_2 + \omega} + \dots + \frac{1}{b_n + \omega} = 3 - 4i.
\]
Compute
\[
\frac{2b_1 - 1}{b_1^2 - b_1 + 1} + \frac{2b_2 - 1}{b_2^2 - b_2 + 1} + \dots + \frac{2b_n - 1}{b_n^2 - b_n + 1}.
\] | 6 | math | 172 |
Given the following four propositions:
\\(①\\) A symmetry axis of the function \\(y=2\sin(2x- \frac{\pi}{3})\\) is \\(x= \frac{5\pi}{12}\\);
\\(②\\) The graph of the function \\(y=\tan x\\) is symmetric about the point \\((\frac{\pi}{2},0)\\);
\\(③\\) The sine function is increasing in the first quadrant;
\\(④\... | ①② | math | 208 |
A three-dimensional rectangular box has its dimensions labeled as $X$, $Y$, and $Z$. The surface areas of its faces are $18$, $18$, $36$, $36$, $54$, and $54$ square units respectively. Determine $X + Y + Z$. | 11 | math | 64 |
Determine an increasing interval for the function $f(x)=\sin (\omega x+\varphi)$ where $\omega > 0$ and $0 < \varphi < \pi$. | \left[- \frac{\pi}{3}, \frac{\pi}{6}\right] | math | 39 |
The number of extreme points of the function $f(x)=\sin x-x$ $(x\in \left( -\frac{\pi}{2},\frac{\pi}{2} \right))$ is __________. | 0 | math | 47 |
If the graph of the function $y=3\sin(2x+\phi)$ $(0 < \phi < \pi)$ is symmetric about the point $\left(\frac{\pi}{3},0\right)$, then $\phi=$ ______. | \frac{\pi}{3} | math | 52 |
Given that the sequence $\{a\_n\}$ is a geometric sequence, the sum of the first $n$ terms is $S\_n$, and $a\_5 = 2S\_4 + 3$, $a\_6 = 2S\_5 + 3$, find the common ratio $q=$ _____ . | 3 | math | 69 |
Given that \( A \) and \( B \) are polynomials of degree 672, and \( C \) is a polynomial of degree 671, the degrees of \( A+B \) and \( A-C \) are \( m \) and \( n \) respectively. Calculate:
\[
|m - n| + |2m - n - 672| + |-3m - 2|
\] | 2018 | math | 93 |
Let $a$ and $b$ be positive integers for which $ab - 8a + 7b = 637$. What is the minimal possible value of $|a - b|$? | 3 | math | 43 |
Given the function $f(x)=\cos 2x+2\sin x$.
(Ⅰ) Find the value of $f\left(-\frac{\pi}{6}\right)$;
(Ⅱ) When $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, find the maximum value of the function $f(x)$ and the corresponding value of $x$. | \frac{3}{2} | math | 90 |
Calculate the remainder of $10!$ when divided by $13$. | 6 | math | 16 |
The derivative of the function $y=\sin^4\frac{x}{4}+\cos^4\frac{x}{4}$ is | -\frac{\sin x}{4} | math | 28 |
Given that circle M passes through the intersection points of the circles $x^2+y^2+6x-4=0$ and $x^2+y^2+6y-28=0$,
(I) If the center of the circle is on the line $x-2y-3=0$, find the equation of circle M.
(II) If the area of the circle is minimal, find the equation of circle M. | \left(x+ \frac{7}{2}\right)^2+\left(y- \frac{1}{2}\right)^2= \frac{25}{2} | math | 94 |
Let $a$, $b$, and $c$ be complex numbers such that $|a| = |b| = |c| = 1$ and
\[
\frac{a^3}{bc} + \frac{b^3}{ac} + \frac{c^3}{ab} = -3.
\]
Find all possible values of $|a + b + c|$. | 1 | math | 85 |
For what value of $m$ does the quadratic equation in $x$, $(2m+1)x^{2}+4mx+2m-3=0$, have:
$(1)$ two distinct real roots;
$(2)$ two equal real roots;
$(3)$ no real roots. | m \in (-\infty, -\frac{3}{4}) | math | 60 |
The graph of the function $y= \sqrt {3}\sin 2x-\cos 2x$ can be obtained by shifting the graph of the function $y=2\sin (2x+ \frac {\pi}{6})$ to the right by at least \_\_\_\_\_\_ units. | \frac {\pi}{6} | math | 66 |
Given the ellipse $C_1: \frac{x^2}{9} + \frac{y^2}{5} = 1$ and the hyperbola $C_2: x^2 - \frac{y^2}{3} = 1$, let the intersection point of $C_1$ and $C_2$ in the first quadrant be P. Then, the distance from point P to the left focus of the ellipse is _______. | 4 | math | 97 |
The fraction $\frac{{x-1}}{{x+1}}$ is meaningful, determine the range of values of $x$. | x\neq -1 | math | 26 |
Consider the sum $\frac{4+8+12+\cdots + 60}{6+12+18+\cdots+72}=$
A) $\frac{15}{12}$
B) $\frac{15}{14}$
C) $\frac{20}{19}$
D) $\frac{40}{39}$
E) $\frac{44}{43}$ | \frac{40}{39} | math | 95 |
Let \( n \in \mathbf{Z}_{+} \). Does there exist a positive real number \(\varepsilon=\varepsilon(n)\) such that for any \( x_{1}, x_{2}, \cdots, x_{n} > 0 \), the following inequality holds?
$$
\begin{array}{l}
\left(\prod_{i=1}^{n} x_{i}\right)^{\frac{1}{n}} \\
\leqslant(1-\varepsilon) \frac{1}{n} \sum_{i=1}^{n} x_{i... | \varepsilon = \frac{1}{n} | math | 171 |
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $3:1$, and the ratio of the rectangle's length to its width is $3:2$. Find the percentage of the rectangle's area occupied by the square. | \left(\frac{s^2}{13.5s^2}\right) \times 100 = \left(\frac{1}{13.5}\right) \times 100 = 7.41 | math | 58 |
On the segment $A C$, point $B$ is given such that $A B=14$ and $B C=28$. Semicircles are constructed on the segments $A B$, $B C$, and $A C$ as diameters in the same half-plane relative to the line $A B$. Find the radius of the circle that touches all three semicircles. | 6 | math | 81 |
In equilateral triangle ABC, the projection of $\overrightarrow{{AB}}$ in the direction of $\overrightarrow{{BC}}$ is -1, and $\overrightarrow{{AD}}$ = 2$\overrightarrow{{DC}}$. Find the area of △ABD. | \frac{2\sqrt{3}}{3} | math | 58 |
Given the number $11700000$, express it in scientific notation. | 1.17 \times 10^{7} | math | 19 |
The square with vertices $(-a, -a), (a, -a), (-a, a), (a, a)$ is cut by the line $y = x$ into two quadrilaterals. What is the perimeter of one of these quadrilaterals divided by $a$? Express your answer in simplified radical form. | 4 + 2\sqrt{2} | math | 68 |
Given the expression $3^{3+3^{3}}$, determine the number of distinct values that can be obtained by changing the order of operations and parentheses. | 3 | math | 32 |
Suppose $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence ${a_{n}}$, if $\frac{a_{8}}{a_{7}} = \frac{13}{5}$, then calculate $\frac{S_{15}}{S_{13}}$. | 3 | math | 67 |
Find a general formula for the sequence $$1, \frac {2}{3}, \frac {1}{2}, \frac {2}{5}, \ldots$$ | a_{n}= \frac {2}{n+1} | math | 35 |
In trapezoid $JKLM$, sides $\overline{JK}$ and $\overline{LM}$ are parallel, $\angle J = 3\angle M$, and $\angle L = 2\angle K$. Find $\angle K$. | 72^\circ | math | 51 |
How many four-digit numbers \( N \) are there, such that half of the number \( N \) is divisible by 2, a third of \( N \) is divisible by 3, and a fifth of \( N \) is divisible by 5? | 10 | math | 55 |
Kuba made an agreement with a shepherd to take care of his sheep. The shepherd promised Kuba that after a year of service, he would receive twenty gold coins and one sheep. However, Kuba resigned right after the seventh month of service. Even so, the shepherd fairly compensated him by paying him five gold coins and giv... | 16 \text{ gold coins} | math | 82 |
Sides of a triangle form an arithmetic sequence with common difference $2$ , and its area is $6 \text{ cm }^2$ . Find its
sides. | 2\sqrt{6} - 2, 2\sqrt{6}, 2\sqrt{6} + 2 | math | 39 |
Equilateral $\triangle DEF$ has side length $300$. Points $R$ and $S$ lie outside the plane of $\triangle DEF$ and are on opposite sides of the plane. Furthermore, $RA=RB=RC$, and $SA=SB=SC$, and the planes containing $\triangle RDE$ and $\triangle SDE$ form a $150^{\circ}$ dihedral angle. There is a point $M$ whose di... | 300 | math | 123 |
In triangle $ABC$, $AB=15$ and $AC=18$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive inte... | 16 | math | 109 |
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