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 | Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n= \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 2$ if $b_{n-1}$ is not divisible by 3. Determine how many positive integers $b_1 \le 3000$ are such that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$. | 2000 | 129 | 4 |
math | Several people played a single-round robin tournament in table tennis. At the end of the tournament, it turned out that for any four participants, there are two who scored the same number of points in the games among these four participants. What is the maximum number of tennis players who could have participated in th... | 7 | 86 | 1 |
math | Given that a total of 56 small gifts were exchanged among a group of people, express the number of people attending the gathering, denoted as x, in terms of an equation. | x\left(x-1\right) = 56 | 38 | 13 |
math | Let $[x]$ denote the greatest integer less than or equal to the real number $x$. The equation $\lg ^{2} x-[\lg x]-2=0$ has how many real roots? | 3 | 44 | 1 |
math | For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1 + 2 + 3 + \cdots + n$. What is the greatest possible value of the greatest common divisor of $6T_n$ and $n-1$? | 3 | 64 | 1 |
math | Given the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, which intersects the positive half of the $y$-axis at point $P(0,b)$ and has its right focus at $F(c,0)$, with $O$ being the origin of coordinates, and $\tan \angle PFO = \frac{\sqrt{2}}{2}$.
$(1)$ Find the eccentricity $e$ of the ellipse $C$;... | \frac{x^2}{3} + y^2 = 1 | 215 | 15 |
math | March 12th is Tree Planting Day. A school organizes 65 high school students and their parents to participate in the "Plant a tree, green a land" tree planting activity as a family unit. The activity divides the 65 families into two groups, A and B. Group A is responsible for planting 150 silver poplar seedlings, while ... | \frac{12}{5} | 196 | 8 |
math | A two-digit positive integer is defined as $\emph{snuggly}$ if it equals twice its nonzero tens digit plus the square of its units digit. Calculate the number of two-digit positive integers that are snuggly. | 1 | 46 | 1 |
math | In acute triangle $\triangle ABC$, the sides opposite to the angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\sin ^{2}A=\sin ^{2}B+\sin B\sin C$, then the range of $\frac{c}{b}$ is ______. | \left(1,2\right) | 72 | 9 |
math | Given that non-zero complex numbers \( x \) and \( y \) satisfy \( x^{2} + xy + y^{2} = 0 \), find the value of the algebraic expression \(\left(\frac{x}{x + y}\right)^{2005} + \left(\frac{y}{x + y}\right)^{2005}\). | -1 | 81 | 2 |
math | Line segments connecting an internal point of a convex, unequal-sided $n$-gon with its vertices divide the $n$-gon into $n$ equal triangles.
For what smallest value of $n$ is this possible? | 5 | 46 | 1 |
math | Let \( m = \underbrace{555555555}_{\text{9 digits}} \) and \( n = \underbrace{1111111111}_{\text{10 digits}} \).
What is \( \gcd(m, n) \)? | 1 | 65 | 1 |
math | Let $f(x)$ be an increasing function defined on $(0, +\infty)$ satisfying $f(xy) = f(x)f(y)$ for all positive real numbers $x$ and $y$, and $f(2) = 4$.
(1) Find the values of $f(1)$ and $f(8)$.
(2) Solve the inequality for $x$: $16f\left(\frac{1}{x-3}\right) \geq f(2x+1)$. | (3, \frac{7}{2}] | 110 | 10 |
math | The expression
\[\frac{P+Q}{P-Q}-\frac{P-Q}{P+Q}\]
where $P=x+y$ and $Q=x-y$, is equivalent to: | \frac{x^2-y^2}{xy} | 41 | 11 |
math | A store plans to mix two types of candies priced at 18 yuan per kilogram and 10 yuan per kilogram respectively to create a mixed candy for sale at 15 yuan per kilogram. If the store wants to prepare 100 kilograms of this mixed candy, how many kilograms of each type of candy are needed? | 37.5 | 70 | 4 |
math | In $\triangle ABC$, $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\cos^2{A} - \cos^2{B} + \sin^2{C} = \sin{B}\sin{C} = \frac{1}{4}$, and the area of $\triangle ABC$ is $\sqrt{3}$, find the value of $a$. | a = 2\sqrt{3} | 99 | 9 |
math | Let $\triangle PQR$ be a right triangle with $Q$ as a right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, find the length of $RS$. | 27 | 57 | 2 |
math | The first term of a geometric sequence is 512, and the 8th term is 2. What is the positive, real value for the 6th term? | 16 | 37 | 2 |
math | The number of solutions to the equation $\sin 3x = \cos 2x$ on the closed interval $[0, 2\pi]$ is? | 4 | 34 | 1 |
math | A regular $\triangle ABC$ has each of its sides divided into four equal parts. Through each division point, lines parallel to the other two sides are drawn. These lines intersect with the sides and the other parallel lines, creating 15 lattice points. If $n$ points are chosen from these 15 lattice points, determine the... | 6 | 108 | 1 |
math | A collection of four positive integers has a mean of 6.5, a unique mode of 6, and a median of 7. If a 10 is added to the collection, what is the new median? | 7.0 | 46 | 3 |
math | The graph obtained by shifting the graph of the function $y=\sin \left( \frac {\pi}{6}-2x\right)$ to the right by $\frac {\pi}{12}$ units, determine the equation of the symmetry axis. | \frac{5\pi}{12} | 51 | 10 |
math | Given that $F$ is the focus of the parabola $y^{2}=4x$, and points $A$ and $B$ are on the parabola and located on both sides of the $x$-axis, with $OA \perp OB$ (where $O$ is the origin), find the minimum value of the sum of the areas of $\triangle AOB$ and $\triangle AOF$. | 8 \sqrt {5} | 88 | 6 |
math | The polynomial equation \[x^4 + ax^2 + bx + c = 0,\] where \(a\), \(b\), and \(c\) are rational numbers, has \(3-\sqrt{5}\) as a root. It also has a sum of its roots equal to zero. What is the integer root of this polynomial? | -3 | 72 | 2 |
math | Consider a rectangle \(ABCD\) with \(AB = 2\) and \(BC = 1\). Points \(E\) and \(F\) are on \(\overline{BC}\) and \(\overline{CD}\), respectively, such that \(\triangle AEF\) is equilateral. A rectangle \(A'B'C'D'\) with longer side parallel to \(ABCD\) and one vertex at \(B\) is inscribed such that \(AB' = x\), the sh... | \sqrt{3} | 132 | 5 |
math | Reporters and teachers from the Youth League committee are to be arranged in a row, with the 2 teachers standing next to each other but not at the ends, calculate the total number of different possible arrangements of 5 volunteers and 2 groups of adjacent teachers. | 960 | 53 | 3 |
math | Given the universal set $U={0,1,2,3,4}$, $M={0,1,2}$, $N={2,3}$, find $M\cap (\lnot U\cap N)$. | \{0,1\} | 50 | 7 |
math | $f(x) = \sin x$, where $x \in (0, \pi)$, find the possible number of real roots of the equation $f^2(x) + 2f(x) + a = 0$, where $a \in \mathbb{R}$. | 0, 1, 2 | 60 | 7 |
math | David's quiz scores so far are 85, 88, 90, 82, and 94. What score does he need to get on the sixth quiz to make the arithmetic mean of the six scores equal 90? | 101 | 53 | 3 |
math | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? (Two rectangles are different if they do not share all four vertices.) | 100 | 47 | 3 |
math | If $x\log_{3}4=1$, then $x=$ ______; $4^{x}+4^{-x}=$ ______. | \dfrac{10}{3} | 31 | 8 |
math | A point is randomly thrown on the segment [12, 17] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+k-90\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | 2/3 | 83 | 3 |
math | Construct a new shape by adding an eighth unit cube to the previously described configuration of seven cubes. Place this new cube adjacent to one of the six outlying cubes from the central cube. What is the ratio of the volume in cubic units to the surface area in square units for this new configuration?
A) $\frac{8}{3... | \frac{8}{33} | 102 | 8 |
math | The given arithmetic sequence $\{ a_{n} \}$ satisfies $a_{3} + a_{4} = 4, a_{5} + a_{7} = 6$.
(1) Find the general term formula for $\{ a_{n} \}$.
(2) Let $b_{n} = [a_{n}]$, where $[x]$ denotes the largest integer not greater than $x$. Find the sum of the first 10 terms of the sequence $\{ b_{n} \}$. | 24 | 112 | 2 |
math | Given points A$(x_1, y_1)$ and B$(x_2, y_2)$ are any two points (can be coincident) on the graph of the function $$f(x)= \begin{cases} \frac {2x}{1-2x}, & x\neq \frac {1}{2} \\ -1, & x= \frac {1}{2}\end{cases}$$, and point M is on the line $$x= \frac {1}{2}$$, with $$\overrightarrow {AM} = \overrightarrow {MB}$$.
(Ⅰ) F... | m=1 | 321 | 3 |
math | Let \(ABCD\) be a convex quadrilateral, and let \(M_A,\) \(M_B,\) \(M_C,\) \(M_D\) denote the midpoints of sides \(BC,\) \(CA,\) \(AD,\) and \(DB,\) respectively. Find the ratio \(\frac{[M_A M_B M_C M_D]}{[ABCD]}.\) | \frac{1}{4} | 81 | 7 |
math | Four spheres of radius 1 touch each other pairwise. Find the radius of a sphere that touches all four spheres. | \sqrt{\frac{3}{2}} - 1 | 23 | 12 |
math | A motorcyclist departed from point \( A \) towards point \( B \) at the same time a cyclist departed from point \( B \) towards point \( A \). The motorcyclist arrived at point \( B \) 2 hours after meeting the cyclist, while the cyclist arrived at point \( A \) 4.5 hours after meeting the motorcyclist. How many hours ... | 5 \, \text{hours}; 7.5 \, \text{hours} | 89 | 19 |
math | Given the power function $f(x) = (m^2 - 5m + 7)x^{-m-1}$ ($m \in \mathbb{R}$) is an even function.
(1) Find the value of $f\left( \frac{1}{2} \right)$.
(2) If $f(2a+1) = f(a)$, find the value of the real number $a$. | -\frac{1}{3} | 94 | 7 |
math | 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} | 76 | 9 |
math | **Compute $\begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{pmatrix}^4$.** | \begin{pmatrix} -8 & -8\sqrt{3} \\ 8\sqrt{3} & -8 \end{pmatrix} | 36 | 33 |
math | Calculate the total number of digits used when the first 1500 positive even integers are written. | 5448 | 21 | 4 |
math | Let $f(x) = \log_2(1 + a \cdot 2^x + 4^x)$, where $a$ is a constant.
(1) Find the value of $a$ when $f(2) = f(1) + 2$;
(2) Determine the range of values for $a$ such that for $x \in [1, +\infty)$, the inequality $f(x) \geq x - 1$ always holds. | [-2, +\infty) | 108 | 8 |
math | The interval in which the function $y=2\sin ( \frac {π}{6}-2x)$, where $x\in[0,π]$, is increasing is $\_\_\_\_\_\_$. | [ \frac {π}{3}, \frac {5π}{6}] | 46 | 16 |
math | Let $P(z)$, $Q(z)$, and $R(z)$ be polynomials with real coefficients, having degrees $4$, $5$, and $10$, respectively, and constant terms $3$, $5$, and $8$, respectively. Find the minimum number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) = R(z)$. | 1 | 82 | 1 |
math | A transportation company has purchased a batch of luxury buses for passenger transport. According to market analysis, the total profit per bus, represented as $y$ (in ten thousand dollars), is related to the number of years of operation $x (x \in N^*)$ by the formula $y=-x^2+18x-36$.
(1) How many years should each bus... | 60,000 | 121 | 6 |
math | Ten distinct natural numbers are such that the product of any 5 of them is even, and the sum of all 10 numbers is odd. What is their smallest possible sum? | 65 | 37 | 2 |
math | In triangle \( \triangle ABC \), \( a, b, c \) are the sides opposite to angles \( A, B, C \) respectively, with \( b = 1 \), and \(\cos C + (2a + c) \cos B = 0\).
(1) Find \( B \);
(2) Find the maximum area of \( \triangle ABC \). | \frac{\sqrt{3}}{12} | 82 | 11 |
math | Given a line \( l \) passing through the left focus \( F \) of the ellipse \( C: \frac{x^2}{2} + y^2 = 1 \) and intersecting the ellipse \( C \) at points \( A \) and \( B \), where \( O \) is the origin. If \( OA \perp OB \), find the distance from \( O \) to the line \( AB \). | \frac{\sqrt{6}}{3} | 92 | 10 |
math | How many solutions in nonnegative integers \((a, b, c)\) are there to the equation
\[ 2^{a} + 2^{b} = c! \] | 6 | 39 | 1 |
math | Let $ABCD$ be a rectangle. Determine the set of all points $P$ from the region between the parallel lines $AB$ and $CD$ such that $\angle APB=\angle CPD$ . | y = \frac{b}{2} | 53 | 9 |
math | A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon $5$ centimeters of rain falls in the open field partially... | 60 | 102 | 2 |
math | The accrued salary of a citizen from January to June inclusive was 23,000 rubles per month, and from July to December, it was 25,000 rubles. In August, the citizen, participating in a poetry contest, won a prize and was awarded an e-book worth 10,000 rubles. What amount of personal income tax needs to be paid to the bu... | 39540 | 113 | 5 |
math | Let \( a \) be the fractional part of \( \sqrt{3+\sqrt{5}} - \sqrt{3-\sqrt{5}} \), and let \( b \) be the fractional part of \( \sqrt{6+3 \sqrt{3}} - \sqrt{6-3 \sqrt{3}} \). Find the value of \( \frac{2}{b} - \frac{1}{a} \). | \sqrt{6} - \sqrt{2} + 1 | 91 | 14 |
math | Find all solutions in positive real numbers \( x_i \) (i = 1, 2, 3, 4, 5) of the following system of inequalities:
$$
\begin{array}{l}
\left(x_{1}^{2}-x_{3} x_{5}\right)\left(x_{2}^{2}-x_{3} x_{5}\right) \leq 0, \\
\left(x_{2}^{2}-x_{4} x_{1}\right)\left(x_{3}^{2}-x_{4} x_{1}\right) \leq 0, \\
\left(x_{3}^{2}-x_{5} x_{... | x_{1} = x_{2} = x_{3} = x_{4} = x_{5} | 271 | 24 |
math |
A circle passing through the vertex \( P \) of triangle \( PQR \) touches side \( QR \) at point \( F \) and intersects sides \( PQ \) and \( PR \) at points \( M \) and \( N \), respectively, different from vertex \( P \). Find the ratio \( QF : FR \) if it is known that the length of side \( PQ \) is 1.5 times the l... | 1/2 | 115 | 3 |
math | Given the parametric equation of line $l$ as $\begin{cases} x=-1+ \frac{\sqrt{2}}{2}t \\ y= \frac{\sqrt{2}}{2}t \end{cases}$, where $t$ is the parameter, and the curve $C_{1}$: $\rho^{2}\cos^{2}\theta+3\rho^{2}\sin^{2}\theta-3=0$, with the origin as the pole and the positive $x$-axis as the polar axis, establishing a p... | \left( \frac{3}{2}, -\frac{1}{2}\right) | 217 | 20 |
math | If an integer $n > 8$ is a solution of the equation $x^2 - ax+b=0$ and the representation of $a$ in the base-$n$ number system is $18$, then the base-n representation of $b$ is | 80 | 55 | 2 |
math | Given the quadratic equation \(x^{2} + px + q = 0\). Find the equation whose roots: 1) would differ from the roots of the given equation only by their signs, 2) would be the reciprocals of the roots of the given equation. | q x^2 + p x + 1 = 0 | 58 | 13 |
math | Find the smallest value of $t$ such that \[\frac{16t^3 - 49t^2 + 35t - 6}{4t-3} + 7t = 8t - 2.\] | \frac{3}{4} | 53 | 7 |
math | The imaginary part of $\overline{z}$, where $z=2-i$. | 1 | 18 | 1 |
math | Real numbers \( x, y, z \) satisfy \( x \geq y \geq z \geq 0 \) and \( 6x + 5y + 4z = 120 \). Find the sum of the maximum and minimum values of \( x + y + z \). | 44 | 66 | 2 |
math | Solve the inequality
\[\left| \frac{3x + 2}{x + 1} \right| > 3.\] | (-\infty, -1) \cup \left(-\frac{5}{6}, -1\right) | 31 | 25 |
math | Compute \( A(3, 1) \) with the function \( A(m, n) \) defined as:
\[ A(m,n) = \begin{cases}
n+1 & \text{if } m = 0 \\
A(m-1, 1) & \text{if } m > 0 \text{ and } n = 0 \\
A(m-1, A(m, n-1)) & \text{if } m > 0 \text{ and } n > 0
\end{cases} \] | 13 | 118 | 2 |
math | Two mutually perpendicular axes with a common origin form a Cartesian coordinate system, dividing the plane into how many quadrants? | 4 | 23 | 1 |
math | Given the sequence \(\{a_n\}\):
\[1,1,2,1,2,3,\cdots, 1,2,\cdots, n,\cdots\]
Let \(S_n\) denote the sum of the first \(n\) terms of the sequence \(\{a_n\}\). Find all positive real number pairs \((\alpha, \beta)\) such that
\[
\lim_{n \rightarrow +\infty} \frac{S_n}{n^\alpha} = \beta.
\] | \left( \frac{3}{2}, \frac{\sqrt{2}}{3} \right) | 114 | 23 |
math | Given the geometric sequence $\{a_{n}\}$, if $a_{1} \cdot a_{5} = 16$, then find the value of $a_{3}$. | \pm 4 | 40 | 4 |
math | Solve the system of equations:
\[
\begin{aligned}
x + y - z & = 4 \\
x^{2} + y^{2} - z^{2} & = 12 \\
x^{3} + y^{3} - z^{3} & = 34.
\end{aligned}
\] | x=2, y=3, z=1 \quad \text{and} \quad x=3, y=2, z=1 | 71 | 31 |
math | Given the function $y=\sin \left(3x-\frac{\pi }{4}\right)$ and the function $y=\cos 3x$, determine the horizontal shift required to obtain the graph of the function $y=\sin \left(3x-\frac{\pi }{4}\right)$. | \frac{\pi }{4} | 65 | 8 |
math | Calculate the ratio of the areas of a square, an equilateral triangle, and a regular hexagon inscribed in the same circle. | 8 : 3\sqrt{3} : 6\sqrt{3} | 27 | 17 |
math | Consider the following five propositions:
① If the equation $x^2+(a-3)x+a=0$ has one positive real root and one negative real root, then $a<0$;
② The function $y= \sqrt {x^{2}-1}+ \sqrt {1-x^{2}}$ is an even function, but not an odd function;
③ If the range of the function $f(x)$ is $[-2, 2]$, then the range of t... | ①⑤ | 319 | 4 |
math | Given a square \(ABCD\). Point \(L\) is on side \(CD\) and point \(K\) is on the extension of side \(DA\) past point \(A\) such that \(\angle KBL = 90^{\circ}\). Find the length of segment \(LD\) if \(KD = 19\) and \(CL = 6\). | 7 | 77 | 1 |
math | A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. Calculate the volume of water in the tank, in cubic feet. | 48\pi - 36\sqrt{3} | 49 | 13 |
math | The equation of circle A is $x^2+y^2-2x-2y-7=0$, and the equation of circle B is $x^2+y^2+2x+2y-2=0$. Determine whether circle A and circle B intersect. If they intersect, find the equation of the line passing through the two intersection points and the distance between these two points; if they do not intersect, expla... | \frac{\sqrt{238}}{4} | 91 | 12 |
math | Given that $a$ is an odd multiple of $7771$, find the greatest common divisor of $8a^2 + 57a + 132$ and $2a + 9$. | 9 | 46 | 1 |
math | Determine the interval(s) where the function $f(x)=\ln (x^{2}-5x+6)$ is strictly increasing. | (3,\infty) | 29 | 6 |
math | If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 5,$ then find
\[\begin{vmatrix} 2a - c & b + 2d \\ 2c & d \end{vmatrix}.\] | 10 - 5cd | 59 | 6 |
math | Given a function $f(x)$ that satisfies $f(x+6) + f(x) = 0$ for $x \in \mathbb{R}$, and the graph of $y = f(x-1)$ is symmetric about the point (1, 0). If $f(1) = -2$, then calculate the value of $f(2021)$. | 2 | 82 | 1 |
math | The relationship between the sizes of the numbers $\log _{2} \frac {1}{5}\;,\;2^{0.1}\;,\;2^{-1}$ can be determined by comparing their values. | \log _{2} \frac {1}{5} < 2^{-1} < 2^{0.1} | 45 | 27 |
math | On a rectangular sheet of paper, a picture in the shape of a "cross" was drawn using two rectangles $ABCD$ and $EFGH$, with their sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, $FG=10$. Find the area of the quadrilateral $AFCH$. | 52.5 | 79 | 4 |
math | Let $a$ be a positive number, and the function $f(x)=ax^{2}+bx+c$ satisfies $f(0)=1$ and $f(x)=f(\frac{2}{a}-x)$.<br/>$(1)$ If $f(1)=1$, find $f(x)$;<br/>$(2)$ Let $g(x)=\log _{2}(x-2\sqrt{x}+2)$, if for any real number $t$, there always exist $x_{1}$, $x_{2}\in [t-1,t+1]$, such that $f(x_{1})-f(x_{2})\geqslant g(x_{3}... | [1, +\infty) | 192 | 8 |
math | Jeff received scores in six assignments: 89, 92, 88, 95, 91, and 93. What is the arithmetic mean of these six scores? | 91.3 | 42 | 4 |
math | Apply a $270^\circ$ rotation around the origin in the counter-clockwise direction to the complex number $4 - 2i$. What is the resulting complex number? | 2 - 4i | 37 | 5 |
math | Given $f(x)=x^3f\left( \dfrac {2}{3}\right)^2-x$, the slope of the tangent to the graph of $f(x)$ at the point $\left( \dfrac {2}{3},f\left( \dfrac {2}{3}\right)\right)$ is \_\_\_\_\_\_. | -1 | 76 | 2 |
math | Allison, Charlie, and Emma each have a 6-sided cube. All of the faces on Allison's cube have a 6. The faces on Charlie's cube are numbered 1, 1, 2, 2, 3, and 3. Four of the faces on Emma's cube have a 3 and two of the faces have a 5. All three cubes are rolled. What is the probability that Allison's roll is greater tha... | \frac{2}{3} | 110 | 7 |
math | Given the equation $x^{2}-2(k+1)x+k^{2}+3=0$ has two distinct real roots $x_{1}$ and $x_{2}$.
$(1)$ If $\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}=\frac{6}{7}$, find the value of $k$.
$(2)$ Find the range of values of ${x}_{1}^{2}+{x}_{2}^{2}$. | (8, +\infty) | 107 | 8 |
math | Given a curve $C$ where the distance from a point on the curve to the point $F(0,1)$ is less than its distance to the line $y=-3$ by $2$.
(Ⅰ) Find the equation of curve $C$;
(Ⅱ) A line $l$ passing through point $F$ with a slope of $k$ intersects curve $C$ at points $A$ and $B$, and intersects the circle $F: x^2 + (y-... | 1 | 253 | 1 |
math | Find the number of odd digits in the base-4 representation of $523_{10}$. | 2 | 22 | 1 |
math | Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes "$AB$" and "$CD$", where the digits $A$, $B$, $C$, and $D$ are not necessarily distinct. If each of the two-digit primes is less than 50, how many such house numbers are possible? | 110 | 75 | 3 |
math | Factorize the polynomial $x^{10} - 1$ as a product of non-constant polynomials with real coefficients, and find the largest possible number of such factors. | 4 | 37 | 1 |
math | The graph represented by the equation $(x^2-9)^2(x^2-y^2)^2=0$ has how many solutions in the Cartesian plane. | 4 | 34 | 1 |
math | Given the sequence ${a_n}$, $a_1=1$ and $a_n a_{n+1} + \sqrt{3}(a_n - a_{n+1}) + 1 = 0$. Determine the value of $a_{2016}$. | 2 - \sqrt{3} | 60 | 7 |
math | Find the sum of the roots of $\tan^2 x - 8\tan x + 3 = 0$ that are between $x=0$ and $x=2\pi$ radians. | 3\pi | 43 | 3 |
math | 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 | 47 | 3 |
math | What is the maximum number of sides a polygon can have if each angle is either \( 172^\circ \) or \( 173^\circ \)?
Let the number of angles measuring \( 172^\circ \) be \( a \), and the number of angles measuring \( 173^\circ \) be \( b \). Then the sum of all angles in the polygon is \( 172a + 173b \). On the other h... | 51 | 184 | 2 |
math | Given the function $f(x)=\frac{f''(1)}{e}\cdot e^{x}+\frac{f(0)}{2}\cdot x^{2}-x$, if there exists a real number $m$ such that the inequality $f(m)\leqslant 2n^{2}-n$ holds, determine the range of values for the real number $n$. | \left(-\infty,-\frac{1}{2}\right]\cup\left[1,+\infty\right) | 83 | 28 |
math | There are several bank card payment technologies: chip, magnetic stripe, paypass, cvc. Arrange the actions taken with the bank card in the order corresponding to the payment technologies.
1 - tap
2 - pay online
3 - swipe
4 - insert into terminal | 4312 | 54 | 4 |
math | Mary told John her score on the American High School Mathematics Examination (AHSME), which was over $80$. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over $80$, John could not have determined this. What was Mary's score? (Recall that... | 119 | 142 | 3 |
math | How many necklaces can be made from five white beads and two black beads? | 3 | 16 | 1 |
math | Find all quadratic trinomials \( p(x) \) that attain a minimum value of \(-\frac{49}{4}\) at \(x=\frac{1}{2}\), and the sum of the fourth powers of its roots equals 337. | p(x) = x^2 - x - 12 | 57 | 13 |
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