task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | Two runners are running in the same direction on a circular track at constant speeds. At a certain moment, runner $A$ is 10 meters ahead of runner $B$, but after $A$ runs 22 meters, runner $B$ catches up.
How many points on the track are there where $B$ can later lap $A$? | 5 | 73 | 1 |
math | Given that $||\vec{a}+\vec{b}||=||\vec{a}-\vec{b}||=2||\vec{a}||$, find the angle between the vectors $(\vec{a}-\vec{b})$ and $\vec{b}$. | \frac{5\pi}{6} | 63 | 9 |
math | Given the functions $f(x)=e^{x}-ax-1$ and $g(x)=\ln (e^{x}-1)-\ln x$, if there exists an $x_0 \in (0,+\infty)$ such that $f(g(x_0)) > f(x_0)$, find the range of values for $a$. | (1,+\infty) | 75 | 7 |
math | Find all rational roots of the polynomial equation:
\[3x^3 - 7x^2 - 8x + 4 = 0.\] | \frac{1}{3} | 32 | 7 |
math | Given $|a|=5$, $|b|=7$, and $|a+b|=a+b$, find the value of $a-b$. | -2 | 30 | 2 |
math | What is the greatest common factor of all three-digit palindromes that are multiples of 3? | 3 | 21 | 1 |
math | In a math class, the teacher gave the students 2 questions. Scoring is as follows: getting one question correct earns 10 points, partially correct earns 5 points, and completely wrong or unanswered earns 0 points. After grading, the teacher found that there were students with each possible score, and for each score, th... | 45 | 91 | 2 |
math | At a certain grocery store, cookies may be bought in boxes of $10$ or $21.$ What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people?
*Author: Ray Li* | 52 | 61 | 2 |
math | Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$ . Determine all possible values of $m$ . | m = 3, 9, 11, 33, 99 | 95 | 19 |
math | Two circles $C_{1}: (x+2)^{2}+(y-m)^{2}=9$ and $C_{2}: (x-m)^{2}+(y+1)^{2}=4$ are externally tangent. Find the value of $m$. | m = 2 | 58 | 4 |
math | Let $g_{1}(x) = \sqrt{x^2 + 3}$, and for integers $n \geq 2$, define \[g_{n}(x) = g_{n-1}(\sqrt{n^2 + x}).\] Let $M$ be the largest value of $n$ for which the domain of $g_n$ is nonempty. Determine the point $c$ for which the domain of $g_M$ consists solely of $\{c\}$. | -25 | 105 | 3 |
math | Determine the range of values for m such that the equation $x^{2}-(m+1)x+4=0$ has two distinct real roots in the interval $(0,3]$. | (3, \frac{10}{3}] | 41 | 11 |
math | How many ways are there to put 4 balls in 3 boxes if the balls are distinguishable but the boxes are not? | 14 | 26 | 2 |
math | Simplify the expression $\frac{\sqrt{3}}{\cos 10^\circ} - \frac{1}{\sin 170^\circ}$. | -4 | 35 | 2 |
math | Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. If $a_5=5S_5=15$, find the sum of the first $100$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$. | \frac{100}{101} | 78 | 11 |
math | A toy company undertakes the production task of the mascot dolls for the 19th Hangzhou Asian Games. The results of a random inspection of a batch of dolls are as follows. Please answer the questions based on the data in the table:<br/>
| Number of dolls sampled $n$ | $10$ | $100$ | $1000$ | $2000$ | $3000$ | $5000$ |
... | 9500 \, \text{dolls} | 294 | 12 |
math | Determine, with proof, the smallest positive integer $n$ with the following property: For every choice of $n$ integers, there exist at least two whose sum or difference is divisible by $2009$ . | n = 1006 | 51 | 8 |
math | Find the general term formula for the sequence $1$, $3$, $6$, $10$, $...$.
A) $n^{2}-n+1$
B) $\frac {1}{2}n(n-1)$
C) $\frac {1}{2}n(n+1)$
D) $2^{n+1}-3$ | \frac {1}{2}n(n+1) | 77 | 12 |
math | In a right triangle JKL, where angle J is the right angle, the length of side JL is 20 units, and side KL is 21 units. Compute $\tan K$. | \frac{20}{\sqrt{41}} | 40 | 12 |
math | Given that $19^{-1} \equiv 31 \pmod{53}$, find $24^{-1} \pmod{53}$, where the result should be a residue modulo 53. | 22 | 48 | 2 |
math | The function $y=f(x)$ is an odd function defined on $\mathbb{R}$ that satisfies $f(3+x)=f(3-x)$. Given that $f(x)=2^{x}$ for $x \in (0,3)$, find $f(x)$ for $x \in (-6,-3)$. | -2^{x+6} | 70 | 7 |
math | The equation
\[
(x-1)(x-2) \cdots(x-2016)=(x-1)(x-2) \cdots(x-2016)
\]
is written on the board. One tries to erase some linear factors from both sides so that each side still has at least one factor, and the resulting equation has no real roots. Find the least number of linear factors one needs to erase to achieve this... | 2016 | 94 | 4 |
math | Given real numbers $\alpha$ and $\beta$ such that $\alpha \beta > 0$, find the set of real numbers $r$ for which there does not exist an infinite sequence $\left\{x_{n}\right\}$ satisfying
$$
x_{0} = r, \quad x_{n+1} = \frac{x_{n}+\alpha}{\beta x_{n}+1}, \quad n = 0, 1, 2, \ldots
$$ | -\frac{1}{\beta} | 105 | 8 |
math | Calculate the result of $\sin 47^{\circ}\cos 17^{\circ}+\cos 47^{\circ}\cos 107^{\circ}$. | \dfrac{1}{2} | 40 | 7 |
math | Given the odd function $f(x)$ that satisfies $f(x+2)=f(x-2)$, and $f(x)=3^{x}$ when $x \in (0,1)$, find the value of $f(\frac{7}{2})$: | - \sqrt {3} | 56 | 6 |
math | The function $\lfloor x\rfloor$ is defined as the largest integer less than or equal to $x$. Consider a function defined by $$f(x) = \lfloor x \rfloor - 2x$$.
What is the range of $f(x)$? Express your answer in interval notation. | (-\infty, 0) | 64 | 8 |
math | Let $n \geq 3$ be a positive integer. Find the smallest positive real $k$ , satisfying the following condition: if $G$ is a connected graph with $n$ vertices and $m$ edges, then it is always possible to delete at most $k(m-\lfloor \frac{n} {2} \rfloor)$ edges, so that the resulting graph has a proper vertex ... | k = \frac{1}{2} | 100 | 9 |
math | There are 2003 circles, each of which is tangent to the sides of $\angle M O N$, and each adjacent pair of circles is externally tangent. Given that $\angle M O N = 60^{\circ}$ and the radius of the smallest circle is 1, find the radius of the 2003rd circle in ascending order of radii. | 3^{2002} | 79 | 7 |
math | Given that the ellipse $C$ passes through point $A(2,3)$, and point $F(2,0)$ is its right focus, find the standard equation of the ellipse $C$. | \frac{x^{2}}{16}+\frac{y^{2}}{12}=1 | 42 | 22 |
math | 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 | 55 | 5 |
math | $M$ and $N$ are real unequal $n\times n$ matrices satisfying $M^3=N^3$ and $M^2N=N^2M$ . Can we choose $M$ and $N$ so that $M^2+N^2$ is invertible? | M^2 + N^2 | 76 | 7 |
math | Determine the standard equations of the conic sections that satisfy the following conditions:
(1) A hyperbola with an eccentricity of $\sqrt{3}$ and foci at $(-5\sqrt{3}, 0)$ and $(5\sqrt{3}, 0)$.
(2) An ellipse with an eccentricity $e = \frac{1}{2}$ and directrices $y = \pm4\sqrt{3}$.
(3) A parabola with its focus on ... | x^2 = 8y | 120 | 7 |
math | Given a line that does not pass through the origin (O) intersects the parabola y^2=2px at points A and B, if the slopes of line OA and AB are 2 and 6 respectively, determine the slope of line OB. | -3 | 53 | 2 |
math | Jason is trying to remember the five digit combination to his safe. He knows that he only used digits 1 through 5 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible combinations does Jason need to try? | 180 | 63 | 3 |
math | Fill in the corresponding braces with the following numbers: $\frac{20}{7}$, $0$, $-10$, $\pi$, $-\frac{2}{3}$, $0.3$, $+50\%$<br/>Set of positive numbers: ______; set of integers: ______; set of fractions: ______; set of non-negative rational numbers: ______. | \frac{20}{7}, \pi, 0.3, +50\%; 0, -10; \frac{20}{7}, -\frac{2}{3}, 0.3, +50\%; \frac{20}{7}, 0, 0.3, +50\% | 82 | 75 |
math | Given the function $f\left( x \right)=a{{x}^{2}}+x-\ln x$, determine the range of values for the real number $a$ such that the function is monotonically increasing on $\left( 2,+\infty \right)$. | \left[ 0, +\infty \right) | 60 | 13 |
math | In the fractions $\frac{b}{8a}$, $\frac{a+b}{a-b}$, $\frac{x-y}{{x}^{2}-{y}^{2}}$, $\frac{x-y}{{x}^{2}+2xy+{y}^{2}}$, how many of them are in simplest form? | 3 | 70 | 1 |
math | How many odd three-digit integers have the property that their digits, read left to right, are in strictly increasing order? | 50 | 24 | 2 |
math | Given \\(f(x)=x^{3}+ax^{2}+bx+a^{2}\\) has an extremum at \\(x=1\\) with the value of \\(10\\), find \\(a+b=\\)_______. | -7 | 55 | 2 |
math | Given the circle with center at $C(4,-6)$ and a radius of $4$, find its equation. | (x-4)^2+(y+6)^2=16 | 24 | 14 |
math | Linh received \$1,500 as a graduation gift. She invests this amount in a savings account with an annual interest rate of 12%, compounded annually. Calculate the total interest Linh will have earned after 4 years. | 862.2 | 50 | 5 |
math | If \( n \) is a positive integer, \( a_{1}=0.8 \) and \( a_{n+1}=a_{n}^{2} \), determine the least value of \( L \) that satisfies \( a_{1} \times a_{2} \times \cdots \times a_{L} < 0.3 \). | 3 | 77 | 1 |
math | Consider the quadratic polynomial \( P(x) = a x^2 + b x + c \) which has distinct positive roots. Vasya wrote four numbers on the board: the roots of \( P(x) \) and the roots of the polynomial \( Q(x) = c x^2 + b x + a \). What is the smallest integer value that the sum of these four numbers can have? | 5 | 83 | 1 |
math | Determine the coefficients $a, b, c$ given that the following system of equations has no solution, but if $|x-1|$ is written instead of $x-1$ in the second equation, the pair $x = \frac{3}{4}, y = \frac{5}{8}$ satisfies the new system of equations.
\[
\begin{aligned}
& a(x-1) + 2y = 1 \\
& b(x-1) + cy = 3
\end{aligned... | a=1, b=2, c=4 | 112 | 11 |
math | Given the complex numbers $z_1 = \sin(2x) + \lambda i$ and $z_2 = m + (m - \sqrt{3} \cos(2x))i$, where $\lambda, m, x \in \mathbb{R}$, and $z_1 = z_2$.
1. If $\lambda = 0$ and $0 < x < \pi$, find the value of $x$.
2. Let $\lambda = f(x)$, and it is known that when $x = \alpha$, $\lambda = \frac{1}{2}$. Find the value o... | -\frac{7}{8} | 149 | 7 |
math | The function \( y = f(x) \) is defined on the set \( (0, +\infty) \) and takes positive values on it. It is known that for any points \( A \) and \( B \) on the graph of the function, the areas of the triangle \( AOB \) and the trapezoid \( ABH_BH_A \) are equal (where \( H_A \) and \( H_B \) are the bases of the perpe... | 1 | 154 | 1 |
math | Find the minimum value of
\[
\sqrt{x^2 + (2 - x)^2} + \sqrt{(x - 1)^2 + (x + 2)^2}
\]
over all real numbers \(x\). | \sqrt{17} | 50 | 6 |
math | Given an arithmetic sequence $\{a_n\}$ with the first term $a_1$ and common difference $d$, and the sum of its first $n$ terms is $S_n$. If the line $y = a_1x$ and the circle $(x - 2)^2 + y^2 = 4$ have two intersection points that are symmetric with respect to the line $x + y + d = 0$, then $S_n = \_\_\_\_\_\_\_\_$. | -n^2 + 2n | 106 | 7 |
math | A line \( l \) intersects the curve \( C : x^2 - y^2 = 1 \) \((x > 0)\) at points \( A \) and \( B \). What is the minimum value of \( f = \overrightarrow{OA} \cdot \overrightarrow{OB} \)? | 1 | 68 | 1 |
math | Consider two lines: line $l$ parametrized as
\begin{align*}
x &= 1 + 4t,\\
y &= 4 + 3t
\end{align*}and the line $m$ parametrized as
\begin{align*}
x &=-5 + 4s\\
y &= 6 + 3s.
\end{align*}Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$ to line $m$.
T... | \begin{pmatrix}-6 \\ 8 \end{pmatrix} | 209 | 16 |
math | In the Cartesian coordinate plane $xOy$, a line is drawn through point P(1, 4) that is tangent to the circle C: $(x - m)^2 + y^2 = m^2 + 5$ ($1 < m < 6$). The tangent points are A and B. The line AB passes through a fixed point; find the coordinates of this point. | (-1, \frac{3}{2}) | 82 | 10 |
math | The central angle of a sector is $60^{\circ}$, and the radius is $R$. Find the radius of the circle inscribed in this sector. | \frac{R}{3} | 34 | 7 |
math | In a certain sequence, the first term is $a_1 = 2010$ and the second term is $a_2 = 2011$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \geq 1$. Determine $a_{1000}$. | 2676 | 90 | 4 |
math | A circle is concentric with and outside a regular hexagon of side length $3$. What should the radius of the circle be so that the probability a randomly chosen point on the circle will completely see exactly four sides of the hexagon is $\frac{1}{3}$?
A) $\frac{3}{4}$
B) $\frac{9}{4}$
C) $\frac{3}{2}$
D) $3$ | \frac{3}{2} | 91 | 7 |
math | Let ellipse $E$ have its center at the origin, with foci on the coordinate axes. Ellipse $E$ passes through points $M(2,\sqrt{2})$ and $N(\sqrt{6},1)$. Let $O$ be the origin.
$(1)$ Find the equation of ellipse $E$;
$(2)$ Is there a circle centered at the origin such that any tangent line to this circle intersects ell... | x^2 + y^2 = \frac{8}{3} | 138 | 15 |
math | The value of $(-0.125)^{2021}\times 8^{2022}$ can be simplified and calculated. | -8 | 32 | 2 |
math | John now draws a regular six-pointed star in the sand, and at each of the 6 outward-pointing points and 6 inward-pointing points he places one of twelve different sea shells. How many ways can he place the shells if reflections and rotations of an arrangement are considered equivalent? | 39916800 | 59 | 8 |
math | 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 | 64 | 2 |
math | Given the function $f(x)=\cos^2\left(x+ \frac{\pi}{12}\right)$, $g(x)=1+ \frac{1}{2}\sin 2x$.
$(I)$ Find the equation of the axis of symmetry for the graph of the function $y=f(x)$.
$(II)$ Find the smallest positive period and the range of the function $h(x)=f(x)+g(x)$. | [1,2] | 94 | 5 |
math | Given the function $f(x)=\ln(ax)-\frac{1}{3}x^3(a≠0)$.<br/>$(Ⅰ)$ When $a=2$, find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $(\frac{1}{2},f(\frac{1}{2}))$;<br/>$(Ⅱ)$ Discuss the monotonicity of the function $f\left(x\right)$;<br/>$(Ⅲ)$ When $a=1$, let $g\left(x\right)=f\left(x\right... | (\frac{1}{3}, +\infty) | 155 | 12 |
math | A Hare and a Wolf are running in the same direction along a circular track with a length of 200 meters, at speeds of \( 5 \, \mathrm{m/s} \) and \( 3 \, \mathrm{m/s} \), respectively. It turns out that the distance between them is exactly the same as it was 40 seconds ago (distance is measured along the shortest of the... | 60 | 132 | 2 |
math | Given a line $y = \frac{\sqrt{3}}{3}x$ and a circle $C$ with its center on the positive x-axis and a radius of 2 intersects the line at points $A$ and $B$ such that $|AB|=2\sqrt{3}$.
(1) Given a point $P(-1, \sqrt{7})$, and $Q$ is any point on circle $C$, find the maximum value of $|PQ|$.
(2) If a ray is drawn from t... | \frac{1}{3} | 147 | 7 |
math | Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20... | 16 | 131 | 2 |
math | Given $\overrightarrow {a}=(\sin x,\cos x)$, $\overrightarrow {b}=(\sin x,\sin x)$, and the function $f(x)=\overrightarrow {a}\cdot \overrightarrow {b}$.
(I) Find the equation of the axis of symmetry of $f(x)$.
(II) Determine the set of values of $x$ for which $f(x)\geqslant 1$.
(III) If the inequality $f(x)-m \lt 2... | (\dfrac{\sqrt {3}-5}{4}, +\infty) | 148 | 17 |
math | Find the coefficient of the $x^3$ term in the expansion of the product $$(3x^3 + 2x^2 + 4x + 5)(4x^3 + 3x^2 + 5x + 6).$$ | 40 | 56 | 2 |
math | In triangle $ABC$, $AB = 18$ and $BC = 12$. Find the largest possible value of $\tan A$. | \frac{2\sqrt{5}}{5} | 30 | 12 |
math | Find a particular solution of the equation \(y^{\prime} + y \tan{x} = 0\), given that \(y = 2\) when \(x = 0\). | y = 2 \cos x | 40 | 7 |
math | α ≠ 1 is a positive real. Find \(\lim_{x \to \infty} \left( \frac{\alpha^x - 1}{\alpha x - x} \right)^{\frac{1}{x}}\). | 1 | 52 | 1 |
math | Given real numbers $a$ and $b$, with $1 < a < b$, find the absolute difference between the average and the median of the four numbers $1$, $a+1$, $2a+b$, and $a+b+1$. | \left| \frac{1}{4} \right| | 52 | 13 |
math | Alice took a mathematics test consisting of 90 questions: 15 arithmetic, 35 algebra, and 40 geometry problems. She answered $80\%$ of the arithmetic, $50\%$ of the algebra, and $70\%$ of the geometry problems correctly. However, she failed to pass the test as she got less than $65\%$ of the total problems right. How ma... | 1 | 112 | 1 |
math | In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square so that W is on JZ, X is on KW, Y is on LX, and Z is on MY. Find the area of quadrilateral WXYZ. | 200 | 100 | 3 |
math | Given that triangle \(ABC\) is isosceles, with \(AB = AC\), and \(\angle ABC = 50^\circ\) and \(\angle DAC = 60^\circ\), calculate the value of \(x\). | 70 | 52 | 2 |
math | Given a regular tetrahedron with each edge length of $4$, there is a small ball $O$ (ignore its weight) inside. Water is poured into the tetrahedron from the top, causing the ball to float upwards. When the volume of the poured water is $\frac{7}{8}$ of the tetrahedron's volume, the ball is tangent to all the tetrahedr... | \frac{2\pi}{3} | 101 | 9 |
math | Given the sequence \( \left\{a_{1}, a_{2}, \cdots \right\} = \left\{ \frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4}, \cdots \right\} \), find the 1988th term \( a_{1988} \). | \frac{29}{35} | 126 | 9 |
math | In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$ , $\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$ . Then find the angle $\angle{DBA}$ | 45 | 69 | 2 |
math | Given the inequality $x_1+x_2+\dots + x_{13}\leq 2006$, determine the number of positive integer tuples $(x_1,x_2,\dots, x_{13})$ satisfying the inequality. | \frac{2006!}{13!1993!} | 53 | 18 |
math | Let the ellipse \\(C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, (a > b > 0)\\) have an eccentricity of \\(\dfrac{2\sqrt{2}}{3}\\), and it is inscribed in the circle \\(x^2 + y^2 = 9\\).
\\((1)\\) Find the equation of ellipse \\(C\\).
\\((2)\\) A line \\(l\\) (not perpendicular to the x-axis) passing through point \\(Q... | -\dfrac{9}{4} | 220 | 8 |
math | Consider a prime number \( p \). We need to design a Parchís tournament subject to the following rules:
- The tournament involves \( p^{2} \) players.
- In each game, \( p \) players participate.
- The tournament is divided into rounds. Each round is divided into games. Each player plays one, or no, game in each round... | p+1 | 121 | 3 |
math | How many distinct sequences of five letters can be made from the letters in COMPUTER if each letter can be used only once, each sequence must begin with M, end with R, and the third letter must be a vowel (A, E, I, O, U)? | 36 | 54 | 2 |
math | The diagram shows a regular pentagon and an equilateral triangle placed inside a square.
What is the value of \( x \)?
A) 24
B) 26
C) 28
D) 30
E) 32 | 24 | 55 | 2 |
math | In the following scenario, $ABCD$ is a square piece of paper with each side measuring 8 cm. Corner $C$ is folded over such that it coincides with $E$, located one-third of the way along $\overline{AD}$ from $A$ to $D$. If $\overline{GF}$ represents the crease created by the fold such that $F$ is on $\overline{CD}$, det... | \frac{32}{9} | 100 | 8 |
math | A sequence $\{a_n\}$ has a sum of its first $n$ terms denoted as $S_n$, where $a_1=1$ and $a_{n+1}=2S_n+1$ for all $n\in\mathbb{N}^*$.
(1) Find the general formula for the $n$-th term of the sequence.
(2) Let $b_n = 2n \cdot a_n$ and $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Find $T_n$. | T_n = \left(n-\frac{1}{2}\right) \cdot 3^n + \frac{1}{2} | 127 | 28 |
math | The point $(2,2,2)$ is rotated $90^\circ$ about the $z$-axis, then reflected through the $xy$-plane, reflected through the $yz$-plane, rotated $90^\circ$ about the $x$-axis, and reflected through the $xy$-plane. Find the coordinates of the point now. | (2, -2, 2) | 77 | 9 |
math | 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 | 83 | 2 |
math | $P = a_{1} a_{2} \cdots a_{n}$ is the product of $n$ numbers. Based on the associative property of multiplication, without altering its order, using only parentheses to indicate the order of multiplication, how many distinct multiplication schemes are there? | C_{n-1} | 58 | 6 |
math | Given that point $P$ is on the curve $y=\frac{4}{e^{x}+1}$, and $\alpha$ is the slant angle of the tangent line at point $P$, determine the range of values for $\alpha$. | \left[\frac{3\pi}{4}, \pi\right) | 52 | 16 |
math | If \\(\alpha\\) and \\(\beta\\) are acute angles, and \\(\tan(\alpha+\beta)=3\\), \\(\tan\beta= \dfrac{1}{2}\\), determine the value of \\(\alpha\\). | \dfrac{\pi}{4} | 52 | 7 |
math | Each of six, standard, six-sided dice is rolled once. What is the probability that there is exactly one pair and one triplet (three dice showing the same value), and the remaining dice show different values? | \frac{25}{162} | 42 | 10 |
math | Find the distance from the center of the circle $x^{2}+y^{2}-2x+2y=0$ to the line $y=x+1$. | \frac{3\sqrt{2}}{2} | 36 | 12 |
math | Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \dots , 4025 $ . Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $ ? | 1006/4025 | 84 | 9 |
math | What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, …, 30\}$? | \frac{2}{3} | 44 | 7 |
math | If a hyperbola passes through the point $(3,\sqrt{2})$, and its asymptote equations are $y=\pm\frac{1}{3}x$, determine the equation of this hyperbola. | y^2-\frac{x^2}{9}=1 | 45 | 12 |
math | A manufacturer plans to hold a promotional event in 2010. After investigation and calculation, the annual sales volume of the product (i.e., the factory's annual output $x$ in ten thousand units) and the annual promotional expenses $m$ (in ten thousand yuan) ($m\geqslant 0$) satisfy $x=3- \dfrac {k}{m+1}$ ($k$ is a con... | 3 | 260 | 1 |
math | On the parabola $y^{2}=2px\left(p \gt 0\right)$ with focus $F$, there is a point $P\left(2,2p\right)$. $O$ is the origin. The coordinates of a point $M$ that satisfies $|MP|=|MO|=|MF|$ are what? | (\frac{1}{4}, \frac{7}{4}) | 74 | 14 |
math | There exists a class of numbers such that both the number and its reversed form are three-digit numbers and perfect squares. How many such numbers exist?
Note: The normal order of a number is read from left to right, and the reverse order is read from right to left. For example, the three-digit number 123 has an order... | 3 | 127 | 1 |
math | A large sphere is placed in a sunny field. At noon, the shadow cast by the sphere extends 12 meters from the point of contact with the ground. At the same time, a pole that is 3 meters tall casts a shadow of 4 meters. Assuming the sun's rays are parallel, find the radius of the sphere. | 9 | 69 | 1 |
math | Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$ .
(Folklore) | 16 | 45 | 2 |
math | Given ten 5-element sets \( A_{1}, A_{2}, \cdots, A_{10} \), where the intersection of any two sets contains at least two elements. Let \( A=\bigcup_{i=1}^{10} A_{i}=\left\{x_{1}, x_{2}, \cdots, x_{n}\right\} \). For any \( x_{i} \in A \), let \( k_{i}(i=1,2, \cdots, n) \) be the number of sets among \( A_{1}, A_{2}, \... | 5 | 192 | 1 |
math | 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 | 98 | 1 |
math | If ten students from Delta school worked for 4 days, six students from Echo school worked for 6 days, eight students from Foxtrot school worked for 3 days, and three students from Golf school worked for 10 days, and the total payment made for the students' work was $1233, with each student receiving the same amount for... | 341.45\text{ dollars} | 92 | 11 |
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