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 | Given a polar coordinate system with the origin \(O\) as the pole and the non-negative half of the \(x\)-axis as the polar axis, where the length units in both coordinate systems are the same, the polar equation of curve \(C_1\) is \(\rho\sin^2\theta - 4\cos\theta = 0\), and the parametric equation of curve \(C_2\) is ... | \sqrt{3} | 286 | 5 |
math | Given that $a$ and $b$ are integers, and the distinct roots of the equation $ax^2+bx-2008=0$ are both prime numbers, calculate the value of $3a+b$. | 1000 | 47 | 4 |
math | Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? | 400 | 75 | 3 |
math | Given the complex number $z= \frac{(m^{2}-m-2)+(m^{2}+m)i}{1+i}$ (where $m \in \mathbb{R}$ and $i$ is the imaginary unit) is a purely imaginary number.
1. Find the value of $m$.
2. If the complex number $w$ satisfies $|w-z|=1$, find the maximum value of $|w|$. | 3 | 93 | 1 |
math | Evaluate $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{34}\rceil$$ | 127 | 54 | 3 |
math | A obtuse triangle has obtuse angles; each angle of an equilateral triangle is °. | 1, 60 | 26 | 5 |
math | Find the number of integers between 1 and 2013 with the property that the sum of its digits equals 9. | 101 | 27 | 3 |
math | Given the function $f(x)=2\sqrt{3}\sin x \cos x - 2\cos^{2}x$ where $x\in \mathbb{R}$.
$(1)$ Find the range of the function $f(x)$.
$(2)$ In triangle $ABC$, with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, if $f(A)=-2$ and $a=\sqrt{3}$, find the maximum value of the area $S$ of triangle $ABC$... | \frac{\sqrt{3}}{4} | 121 | 10 |
math | Define the sequence \( (y_n) \) by \( y_1 = 200 \) and \( y_k = y_{k - 1}^2 - y_{k - 1} \) for all \( k \ge 2 \). Compute
\[
\frac{1}{y_1 + 1} + \frac{1}{y_2 + 1} + \frac{1}{y_3 + 1} + \dotsb.
\] | \frac{1}{200} | 105 | 9 |
math | Among the natural numbers from 1 to 100, how many different pairs of distinct numbers can be selected such that their sum is greater than 100? | 2500 | 35 | 4 |
math | Given a grasshopper jumps either $364$ or $715$ units on the real number line and it starts from the point $0$, find the smallest distance the grasshopper can be away from the point $2010$. | 5 | 53 | 1 |
math | A journalist visits a company with $\mathrm{N}$ people. He knows that there is a person $\mathrm{Z}$ in this company who knows everyone else, but no one knows him. The journalist can ask each member of the company the question: "Do you know such-and-such person?" Find the smallest number of questions required to defini... | \mathrm{N}-1 | 93 | 6 |
math | What is the sum of the digits of the decimal representation of $3^{2005} \times 7^{2007} \times 2$? | 17 | 36 | 2 |
math | In a workshop, there are 28 workers producing special bolts and nuts. Each bolt is paired with a nut on each end. Each worker can produce either 12 bolts or 18 nuts per day on average. How many workers need to produce bolts and how many workers need to produce nuts so that the bolts and nuts produced in one day can be ... | 16 | 76 | 2 |
math | Given the equation concerning $x$, $(m-1)x^{m^{2}+1}+2x-3=0$, the value of $m$ is ________. | -1 | 37 | 2 |
math | Two schools, Xiaoxin and Xiaoxiao, have the same number of students participating in the Golden Cup Competition. The schools use cars to transport the students to the competition site. Xiaoxin's cars each carry 15 people, while Xiaoxiao's cars each carry 13 people. As a result, Xiaoxiao needs one more car than Xiaoxin.... | 184 | 143 | 3 |
math | Let $n$ be an integer of the form $a^2 + b^2$ , where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$ , then $p$ divides $ab$ . Determine all such $n$ . | 5, 13 | 83 | 5 |
math | If $x$ cows produce $x+2$ cans of milk in $x+4$ days, determine the number of days it will take $x+4$ cows to produce $x+6$ cans of milk. | \frac{x(x+4)(x+6)}{(x+2)(x+4)} | 47 | 20 |
math | In $\triangle ABC$, it is given that $D$ is a point on side $AC$, $BD = DC$, and $\angle BCD = 70^\circ$. Determine the measure of $\angle ADB$. | 140^\circ | 46 | 5 |
math | 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 | 123 | 3 |
math | Let $A$ and $B$ be the sums of the odd-position and even-position terms, respectively, in the expansion of $(x+a)^{n}$. Find $A^{2}-B^{2}$. | (x^2 - a^2)^n | 45 | 9 |
math | In triangle $XYZ$, $XY = 12$, $YZ = 16$, and $XZ = 20$, with $ZD$ as the angle bisector. Find the length of $ZD$. | \frac{16\sqrt{10}}{3} | 47 | 14 |
math | Calculate the value of the sum
\[\binom{101}{0} - \binom{101}{2} + \binom{101}{4} - \dots - \binom{101}{100}.\] | -2^{50} | 58 | 6 |
math | Given the plane vectors $\overrightarrow{AC}=(1,2)$ and $\overrightarrow{BD}=(-2,2)$, the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{CD}$ is ______. | - \frac {9}{4} | 49 | 8 |
math | Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$ | \frac{\sqrt{7}}{5} | 66 | 10 |
math | Let $f(x)$ be the function defined for $x\geq 0$ which satisfies the following conditions.
(a) $f(x)=\begin{cases}x \ \ \ \ \ \ \ \ ( 0\leq x<1) 2-x \ \ \ (1\leq x <2) \end{cases}$
(b) $f(x+2n)=f(x) \ (n=1,2,\cdots)$
Find $\lim_{n\to\infty}\int_{0}^{2n}f(x)e^{-x}\ dx.$ | 1 | 134 | 1 |
math | A $10$-cm-by-$10$-cm square is partitioned as shown. Points $C$ and $D$ are the midpoints of two opposite sides of the square. Two diagonal lines are drawn from each corner of the square to the opposite midpoint, forming a new shaded diamond-shaped region in the center. What is the area of this new shaded region?
[asy... | 50 \text{ square cm} | 233 | 8 |
math | The product of the digits of a four-digit number is 810. If none of the digits is repeated, find the sum of the digits. | 23 | 31 | 2 |
math | Two cars left point $A$ for point $B$ and from point $B$ to point $A$ simultaneously and met after 3 hours. The first car arrived at point $B$ 1.1 hours later than the second one arrived at point $A$. How many times greater is the speed of the second car compared to the speed of the first car? | \frac{6}{5} | 76 | 7 |
math | What are the first three digits to the right of the decimal point in the decimal representation of $(10^{100} + 1)^{5/3}$? | 666 | 36 | 3 |
math | Given a positive integer $a$ satisfying $192 | a^3 + 191$ and $a < 2009$, find the sum of all possible positive integers $a$ that meet the condition. | 10571 | 48 | 5 |
math | Suppose that for some positive integer $n$ , the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$ . | 31 | 67 | 2 |
math | The range of the function $$y=\log_{\frac{1}{3}}(-x^2+2x+8)$$ is \_\_\_\_\_\_. | [-2, +\infty) | 36 | 8 |
math | Let $\mathbf{a} = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix}.$
Determine the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$ | \begin{pmatrix} 5 \\ 3 \\ 1 \end{pmatrix} | 130 | 20 |
math | The function $$f(x)= \begin{cases} \sin{x}, x\leq a, \\ \frac {1}{x}, x>a, \end{cases}$$ has a range of $[-1, 1]$. Determine the range of values for $a$. | [1, +\infty) | 59 | 8 |
math | From the numbers 1 to 200, one or more numbers were selected to form a group with the following property: if the group contains at least two numbers, then the sum of any two numbers in this group is divisible by 5. What is the maximum number of numbers that can be in the group with this property? | 40 | 68 | 2 |
math | Given line $l\_1$: $x+y-3=0$ intersects the $x$-axis at point $A$, line $l\_2$ has equation $y=2x$, and intersects line $l\_1$ at point $B$. Point $C$ is on the negative half of the $y$-axis, and $AC=2\sqrt{3}$.
(1) Find the area of $\triangle ABC$.
(2) Find the equation of the circumcircle of $\triangle ABC$. | {x}^{2}+{y}^{2}-2x-3=0 | 109 | 18 |
math | Given the complex number $z=\frac{4a^{2}-3a-1}{a+3}+(a^{2}+2a-3)i$ $(a\in \mathbb{R})$.
$(1)$ If $z=\bar{z}$, find $a$;
$(2)$ For what value of $a$ is $z$ a purely imaginary number. | a=-\frac{1}{4} | 83 | 9 |
math | Five rays $\overrightarrow{OA}$ , $\overrightarrow{OB}$ , $\overrightarrow{OC}$ , $\overrightarrow{OD}$ , and $\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that $\angle EOD = 2\angle COB$ , $\angle COB = 2\angle BOA$ , while $\angle DOC = 3\angle BOA$ . ... | 90 | 154 | 2 |
math | From the following infinite list of numbers, how many are integers? $$\sqrt{10000}, \sqrt[3]{10000}, \sqrt[4]{10000}, \sqrt[5]{10000}, \sqrt[6]{10000}, \ldots$$ | 2 | 71 | 1 |
math | Given that the moving circle $P$ is tangent to the circle $M: (x+3)^{2}+y^{2}=1$ externally and tangent to the circle $N: (x-3)^{2}+y^{2}=81$ internally.
$(1)$ Find the equation of the trajectory of the center of the moving circle $P$.
$(2)$ Find the range of values of $\frac{1}{PM}+\frac{1}{PN}$. | [\frac{2}{5}, \frac{5}{8}] | 103 | 14 |
math | Find the interval in which the zero point of the function $f(x)=\ln x- \frac{2}{x}$ is located. | (2,3) | 29 | 5 |
math | In triangle $\triangle ABC$, $a=3$, $b-c=2$, $\cos B=-\frac{1}{2}$. Find the values of $b$ and $c$. Also, find the value of $\sin \left(B+C\right)$. | \sin(B+C) = \frac{3\sqrt{3}}{14} | 56 | 19 |
math | A sequence of diagrams consists of equilateral triangles, each subsequently divided into smaller equilateral triangles. In the first diagram, there are no shaded triangles. In the second diagram, 1 out of 4 small triangles are shaded. In the third diagram, 4 out of 9 small triangles are shaded. If this pattern of shadi... | \frac{25}{36} | 85 | 9 |
math | If the eccentricity of the hyperbola $\frac{x^{2}}{m+9} + \frac{y^{2}}{9} = 1$ is 2, find the value of $m$. | -36 | 46 | 3 |
math | Let $\mathrm {P}$ be the product of the roots of the polynomial $z^7+z^6+z^5+z^3+z+1=0$ that have a positive imaginary part, where $\mathrm {P} = r (\cos{\theta^\circ} + i \sin{\theta^\circ})$, with $r > 0$ and $0 \leq \theta < 360$. Find $\theta$. | 180^\circ | 91 | 5 |
math | Arrange $\frac{1}{2}, \frac{1}{3}, $\frac{1}{4}, $\frac{1}{5}, $\frac{1}{6}, $\frac{1}{7}$ and the average of these 6 fractions in increasing order. Determine the position of the average in this ordered list. | 5 | 66 | 1 |
math | Let $a_0 = -3, b_0 = 2$, and for $n \geq 0$, let
\begin{align*}
a_{n+1} &= 2a_n + 2b_n + \sqrt{a_n^2 + b_n^2}, \\
b_{n+1} &= 2a_n + 2b_n - \sqrt{a_n^2 + b_n^2}.
\end{align*}
Find $\frac{1}{a_{2012}} + \frac{1}{b_{2012}}.$ | \frac{1}{6} | 128 | 7 |
math | The square quilt block shown is made from sixteen unit squares, where eight of these squares have been divided in half diagonally to form triangles. Each triangle is shaded. What fraction of the square quilt is shaded? Express your answer as a common fraction. | \frac{1}{4} | 50 | 7 |
math | In the octagon COMPUTER exhibited below, all interior angles are either $90^{\circ}$ or $270^{\circ}$ and we have $C O=O M=M P=P U=U T=T E=1$. Point $D$ (not to scale in the diagram) is selected on segment $R E$ so that polygons COMPUTED and $C D R$ have the same area. Find $D R$. | 2 | 91 | 1 |
math | What is the largest value of $x$ such that the expression \[\dfrac{x+2}{4x^3 - 40x^2 + 36x - 8}\] is undefined? | 4 + \sqrt{15} | 45 | 8 |
math | Let \( S \) be the smallest positive multiple of 15 that comprises exactly \( 3k \) digits with \( k \) '0's, \( k \) '3's, and \( k \) '8's. Find the remainder when \( S \) is divided by 11. | 6 | 65 | 1 |
math | Suppose \(a\), \(b\), \(c\), and \(d\) are integers satisfying:
- \(2(a-b+c) = 10\)
- \(2(b-c+d) = 12\)
- \(2(c-d+a) = 6\)
- \(2(d-a+b) = 4\).
What is the value of \(a+b+c+d\)? | 8 | 84 | 1 |
math | Given a triangle $\triangle ABC$ with its three interior angles $A$, $B$, and $C$, and let $\alpha = A + B$, $\beta = C + A$, $\gamma = C + B$. Determine the maximum number of acute angles among $\alpha$, $\beta$, and $\gamma$. | 1 | 63 | 1 |
math | Given that the function $f(x)$ is an even function defined on $[-2,2]$, and when $0 \leq x \leq 2$, $f(x) = \frac{1}{2}x^2 + x$.
$(Ⅰ)$ Find the analytical expression of the function $f(x)$.
$(Ⅱ)$ If $f(a+1) - f(2a-1) > 0$, find the range of real number $a$. | (0,1] | 103 | 5 |
math | Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), let F be the right focus of the hyperbola. A perpendicular line from point F to the x-axis intersects the two asymptotes at points A and B, and intersects the hyperbola in the first quadrant at point P. Let O be the origin of the coordinate s... | \frac{2\sqrt{3}}{3} | 166 | 12 |
math | A line parallel to side \( AB \) of triangle \( ABC \) intersects side \( BC \) at point \( M \) and side \( AC \) at point \( N \). The area of triangle \( MCN \) is twice the area of trapezoid \( ABMN \). Find the ratio \( CM:MB \). | 2 + \sqrt{6} | 71 | 7 |
math | Find the total area of metal wasted when a circular disc is cut out from a square metallic sheet of side length \( s \), and then a rectangle (with the rectangle's longer side equal to the circle's diameter and the shorter side half of the longer side) is cut from this circular disc. | \frac{s^2}{2} | 60 | 8 |
math | Complex numbers $p, q, r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ | 768 | 49 | 3 |
math | Given that circle $C$ passes through two points $A(-1,1)$ and $B(3,5)$, and the center of the circle $C$ lies on the line $2x-y-5=0$.
$(1)$ Find the equation of circle $C$;
$(2)$ A line $l$ passing through point $P(1,5)$ intersects circle $C$ at points $M$ and $N$, and $|MN|=4\sqrt{3}$. Find the equation of line $l$. | x=1 \text{ or } 3x+4y-23=0 | 113 | 19 |
math | Given $\cos (75^{\circ}+\alpha)= \frac {5}{13}$, where $\alpha$ is an angle in the third quadrant,
(1) Find the value of $\sin (75^{\circ}+\alpha)$.
(2) Find the value of $\cos (\alpha-15^{\circ})$.
(3) Find the value of $\sin (195^{\circ}-\alpha)+\cos (105^{\circ}-\alpha)$. | -\frac{10}{13} | 110 | 9 |
math | For three numbers $a$, $b$, $c$, let $M\left\{ a,b,c \right\}$ represent the median of these three numbers, and let $\max \left\{ a,b,c \right\}$ represent the maximum of these three numbers. For example: $M\left\{ -5,0,3 \right\}=0$, $\max \left\{ -1,2,5 \right\}=5$, $\max \left\{ -2,-1,a\right\}=\left\{ \begin{array}... | -5 \leqslant x \leqslant 4 | 295 | 15 |
math | Solve the equations:<br/>$(1)\left(2x-1\right)^{2}=9$.<br/>$(2)x^{2}-4x-12=0$. | x_1 = 6, x_2 = -2 | 41 | 13 |
math | In $\triangle ABC$, $AB= \sqrt {2}$, $BC=1$, $\cos C= \frac {3}{4}$.
$(1)$ Find the value of $\sin A$;
$(2)$ Find $AC$. | 2 | 52 | 1 |
math | The number $n$ is a three-digit positive integer and is the product of three factors: $x$, $y$, and $10x+y$, where $x$ is a prime digit, $x + y$ is not a multiple of 3, and $10x+y$ forms a prime number. What is the largest possible value of $n$ given these conditions? | 795 | 80 | 3 |
math | A woman weaves 5 feet of cloth on the first day, and in one month (considered to be 30 days), she weaves a total of 390 feet of cloth. What is the length of cloth she weaves on the fifth day? | \frac{209}{29} | 56 | 10 |
math | Given that \(x^{2} + y^{2} = 25\), find the maximum value of the function \(z = \sqrt{8y - 6x + 50} + \sqrt{8y + 6x + 50}\). | 6\sqrt{10} | 58 | 7 |
math | A boss schedules a meeting at a cafe with two of his staff, planning to arrive randomly between 1:00 PM and 4:00 PM. Each staff member also arrives randomly within the same timeframe. If the boss arrives and any staff member isn't there, he leaves immediately. Each staff member will wait for up to 90 minutes for the ot... | \frac{1}{4} | 91 | 7 |
math | Expand the binomial ${(\sqrt{x}-\frac{2}{x})^{n}}$:
(1) If $n=6$, find the second to last term.
(2) If the coefficient ratio of the 5th term to the 3rd term is $56:3$, find the sum of all binomial coefficients. | 1024 | 72 | 4 |
math | (This question is worth 12 points.) There are 6 students standing in a row. Calculate:
(1) How many different arrangements are there if student A does not stand at the head or the tail of the row?
(2) How many different arrangements are there if students A, B, and C are not adjacent to each other? (Formulate an expre... | 144 | 80 | 3 |
math | How many positive integers \( x \) with \( 200 \leq x \leq 600 \) have exactly one digit that is a prime number? | 156 | 37 | 3 |
math | Given $\triangle ABC$ where $AB=2$ and $AC= \sqrt {3} \times BC$, find the maximum area of $\triangle ABC$. | S_{max} = \sqrt {3} | 33 | 10 |
math | There are 3 boys and 4 girls. Calculate the total number of different arrangements under the following conditions:
(1) Arrange in two rows, with 3 people in the front row and 4 people in the back row.
(2) Everyone stands in a row, with person A neither at the head nor at the tail of the line.
(3) Everyone stands in a r... | 1440 | 105 | 4 |
math | If each side of a regular hexagon consists of 6 toothpicks, and there are 6 sides, calculate the total number of toothpicks used to build the hexagonal grid. | 36 | 39 | 2 |
math | Given a geometric sequence $\{a_n\}$ with a common ratio $q$, and the product of its first $n$ terms is $T_n$, where the first term $a_1 > 1$, and $a_{2014}a_{2015} - 1 > 0$, $\frac{a_{2014} - 1}{a_{2015} - 1} < 0$, find the largest natural number $n$ such that $T_n > 1$. | 4028 | 113 | 4 |
math | The values of $f$, $g$, $h$ and $j$ are 4, 5, 6, and 7, but not necessarily in that order. What is the largest possible value of the sum of the four products $fg$, $gh$, $hj$, and $fj$? | 120 | 66 | 3 |
math | Given that line $l$ is the tangent line of the curve $y=x^{2}+x-2$ at point $(1,0)$, and $m$ is another tangent line of the curve, with $l \perp m$.
(1) Find the equation of line $m$.
(2) Find the area of the triangle formed by lines $l$, $m$, and the $x$-axis. | \frac{125}{12} | 91 | 10 |
math | The side $AB$ of triangle $ABC$ is divided into $n$ equal parts (with division points $B_0 = A, B_1, B_2, \ldots, B_n = B$), and the side $AC$ of this triangle is divided into $n+1$ equal parts (with division points $C_0 = A, C_1, C_2, \ldots, C_{n+1} = C$). The triangles $C_i B_i C_{i+1}$ are shaded. What fraction of ... | \frac{1}{2} | 127 | 7 |
math | Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and vectors $\overrightarrow{m}=(a,\cos A)$, and $\overrightarrow{n}=(\sin B,\sqrt{3}b)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
$(1)$ Find angle $A$;
$(2)$ If $a=\sqrt{7}$ and $b+c=3$... | \frac{\sqrt{3}}{2} | 122 | 10 |
math | Solve the problem where the equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$ must be reduced to the equivalent equation $\cos ax \cos bx \cos cx = 0$ for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$ | 14 | 82 | 2 |
math | If the consecutive integers from $60$ to $1$ were written as $$605958...,$$ what would be the $77^{\text{th}}$ digit to be written? | 3 | 45 | 1 |
math | Jessica now has a 3 by 3 grid and three colors to choose from. If non-overlapping regions that share a side cannot be the same color, how many possible ways are there for Jessica to color the grid? | 768 | 45 | 3 |
math | If the degree measures of the $n$ interior angles of a convex $n$-gon are in the ratio 1:2:3: $\ldots: n$, find the number of possible values of $n$. | 2 | 46 | 1 |
math | Let $0 \le a, b, c \le 5$ be integers. For how many ordered triples $(a,b,c)$ is $a^2b+b^2c+c^2a-ab^2-bc^2-ca^2 = 0$? | 96 | 57 | 2 |
math | Given the planar vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ which are not collinear, and $| \overrightarrow {a}|=1$, $\overrightarrow {a} \cdot \overrightarrow {b} = 1$, find the maximum value of $| \overrightarrow {a} - \overrightarrow {b}|$. | \sqrt{3} | 75 | 5 |
math | Lottery. (For 7th grade, 3 points) It so happened that Absent-Minded Scientist has only 20 rubles left, but he needs to buy a bus ticket to get home. The bus ticket costs 45 rubles. Nearby the bus stop, instant lottery tickets are sold for exactly 10 rubles each. With a probability of $p = 0.1$, a ticket contains a win... | 0.19 | 141 | 4 |
math | Let $M(m,4), m > 0$ be a point on the parabola $x^2 = 2py, p > 0$, and $F$ be its focus. Given that $|FM|=5$,
(1) Find the values of $m$ and $p$.
(2) Draw a tangent line to the parabola at point $M$, which intersects the $y$-axis at point $N$. Calculate the area of $\triangle FMN$. | 10 | 105 | 2 |
math | Keisha's basketball team must decide on a new uniform. The seventh-graders will pick the color of the shorts (black, gold, or red) and the eighth-graders will pick the color of the jersey (black, white, gold, or blue), and each group will not confer with the other. Additionally, the ninth-graders will choose whether to... | \frac{5}{6} | 122 | 7 |
math | Determine the values of $t$ for which the expression $\sqrt{49 - (t-3)^2} - 7$ equals zero.
A) $t = 0$
B) $t = 3$
C) $t = \pm 3$
D) $t = \pm 7$ | t = 3 | 68 | 4 |
math | Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence. | 1039 | 54 | 4 |
math | Given the function $f(x) = e^{2x} + (a-2)e^{x} - ax$.
$(1)$ Discuss the monotonicity of $f(x)$.
$(2)$ If $f(x)$ has two zeros, find the range of values for $a$. | (0, 1) | 62 | 6 |
math | Let \( C \) be the curve \( y^2 = x^3 \) (where \( x \) takes all non-negative real values). Let \( O \) be the origin, and \( A \) be the point where the gradient is 1. Find the length of the curve from \( O \) to \( A \). | \frac{8}{27} (2\sqrt{2} - 1) | 71 | 19 |
math | Crude oil needs to be cooled and heated to refine it into various products such as gasoline, diesel, plastic, etc. If the temperature of the crude oil (in degrees Celsius) at the $x^{th}$ hour is given by the function $y = f(x) = x^2 - 7x + 15$ ($0 \leq x \leq 8$), then the instantaneous rate of change of the temperatu... | 1 | 141 | 1 |
math | The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n + 2017} {1 + b_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$. | 2018 | 70 | 4 |
math | 12 real numbers x and y satisfy \( 1 + \cos^2(2x + 3y - 1) = \frac{x^2 + y^2 + 2(x+1)(1-y)}{x-y+1} \). Find the minimum value of xy. | \frac{1}{25} | 62 | 8 |
math | For how many positive integers $n$ does $\frac{1}{n}$ yield a terminating decimal with a non-zero tenths digit? | 4 | 28 | 1 |
math | Let's call a number greater than 25 semi-prime if it is the sum of two distinct prime numbers. What is the maximum number of consecutive natural numbers that can be semi-prime? | 5 | 40 | 1 |
math | The coefficients of the polynomial
\[a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0\] are all integers, and its roots $s_1, s_2, \dots, s_6$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^6 + a_1 x^5 + a_2 x^4 + a_3 x^3 + a_4 x^2 + a_5 x + a_6 = 0\] are also $s_1, s_2, \dots, s_6.... | 7 | 187 | 1 |
math | In the triangle shown below, what is the measure of $\angle ACB$? Assume the following angles are given in the diagram:
1. $\angle BAD = 150^\circ$
2. $\angle BAC = 80^\circ$
[asy]
size(250);
draw((0,0)--(64.3,76.6)--(166,0)--cycle);
draw((0,0)--(32.15,38.3)--(166,0));
label("$A$", (64.3,76.6), N);
label("$150^\circ$"... | 70^\circ | 190 | 4 |
math | Given that $y=f(x)$ is a continuous and differentiable function on $\mathbb{R}$, and $xf'(x)+f(x) > f'(x)$, the number of zeros of the function $g(x)=(x-1)f(x)+ \frac {1}{2}$ on $(1,+\infty)$ is ______. | 0 | 72 | 1 |
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