problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Let the function $f(x)= \frac {3}{2}x^{2}-2ax$ ($a > 0$) and $g(x)=a^{2}\ln x+b$ have a common point, and the equations of tangent lines at the common point are the same. Find the maximum value of the real number $b$. | b= \frac {1}{2e^{2}} | math | 71 |
Given the parabola $y=ax^2$ passes through point A(1, 2), then $a= \_\_\_\_\_\_$, and the equation of the directrix is $\_\_\_\_\_\_$. | a=2; y=-\frac{1}{8} | math | 49 |
Consider the quadratic equation $5x^2 - 6x - 12 = 0$. The positive difference between the two roots of this equation can be written as $\frac{\sqrt{p}}{q}$, where $q$ is an integer and $p$ is an integer not divisible by the square of any prime number. Find $p + q$. | 74 | math | 76 |
Let $S = \{1, 22, 333, \dots , 999999999\}$ . For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$ ? | 14 | math | 82 |
Given vectors $\overrightarrow{a}=(-2,-1,3)$, $\overrightarrow{b}=(-1,1,2)$, $\overrightarrow{c}=(x,2,2)$.
$(1)$ When $|\overrightarrow{c}|=2\sqrt{2}$, if the vector $k\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$, find the values of the real numbers $x$ and $k$.
$(2)$ If the vecto... | x=-\frac{4}{5} | math | 149 |
Given that the shortest distance from a point on the circle $(x-2)^2+y^2=1$ to the line $y= \sqrt {3}x+b$ is $\sqrt {3}$, find the value of $b$. | 2 | math | 51 |
For which integers \( n \geq 3 \) does there exist a regular \( n \)-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system? | 4 | math | 39 |
Let the complex number \(z\) satisfy \(z=\dfrac{-1+i}{1+i}\), calculate the modulus of \(z\). | 1 | math | 29 |
For the elective course "Coordinate System and Parametric Equations," determine the length of the chord cut by the line
$$
\begin{cases}
x=1+4t,
\\
y=-1-3t
\end{cases}
\quad (t \text{ is a parameter})
$$
from the curve
$$
\rho = \sqrt{2}\cos\left(\theta + \frac{\pi}{4}\right).
$$ | \frac{7}{5} | math | 95 |
$|5x^2-\tfrac25|\le|x-8|$ if and only if $x$ is in the interval $[a, b]$ . There are relatively prime positive integers $m$ and $n$ so that $b -a =\tfrac{m}{n}$ . Find $m + n$ . | 18 | math | 82 |
**Text**: The function $f(x)$ is an even function defined on $\mathbb{R}$, and its graph is symmetric about the line $x = 2$. When $x \in (-2,2)$, $f(x) = -x^2 + 1$. Then, when $x \in (-4, -2)$, the expression for $f(x)$ is __________. | -(x + 4)^2 + 1 | math | 85 |
A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with a radius of 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$ | \sqrt{59} | math | 59 |
Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0, b > 0$), with one of its foci at F$(c, 0)$, and one endpoint of the imaginary axis at B$(0, b)$. If the line FB is perpendicular to the asymptote $y = \frac{b}{a}x$ of the hyperbola, find the eccentricity of this hyperbola. | e = \frac{1 + \sqrt{5}}{2} | math | 113 |
Given the function $f(x) = 4x^2 - 4ax + a^2 - 2a + 2$ has a maximum value of 3 on the interval $[0, 2]$. Find the value of the real number $a$. | 1 + \sqrt{2} | math | 57 |
Consider the sequence whose nth term is $(-1)^n\cdot (n+1)$. Calculate the average of the first 150 terms of this sequence. | -0.5 | math | 35 |
In the arithmetic sequence {a<sub>n</sub>}, a<sub>3</sub>+a<sub>4</sub>+a<sub>5</sub>=84, a<sub>9</sub>=73.
(I) Find the general term formula for the sequence {a<sub>n</sub>};
(II) For any m∈N*, let b<sub>m</sub> denote the number of terms in the sequence {a<sub>n</sub>} that fall within the interval (9<sup>m</sup>,9<s... | S_{m}=\frac {9^{2m+1}-10×9^{m}+1}{80} | math | 152 |
How long are the midlines of a triangle if the sides of the triangle are given? | s_{a} = \frac{1}{2} b, \quad s_{b} = \frac{1}{2} c, \quad s_{c} = \frac{1}{2} a | math | 18 |
(1) Find the coefficient of $x^3$ in the expansion of $(\frac{1}{2} - x)^5$ and the sum of all coefficients in the expansion.
(2) From the numbers 0, 2, 3, 4, 5, 6, select any 4 to form a 4-digit number without repeating digits. Find the number of 4-digit numbers that meet the condition. | 300 | math | 91 |
Given the sequence $\{a_n\}$ where the sum of the first $n$ terms $S_n=n^2+1$, and the sequence $\{b_n\}$ where $b_n=\frac{2}{a_n+1}$, find the value of $b_5$. | \frac{1}{5} | math | 61 |
Given the expression \( \left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\ldots\left(1-\frac{1}{12^{2}}\right) \), compute its value. | \frac{13}{24} | math | 60 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a=2$, $3b\sin C-5c\sin B\cos A=0$, find the maximum area of $\triangle ABC$. | 2 | math | 66 |
In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $a\cos B = 4$ and $b\sin A = 3$.
(I) Find $\tan B$ and the value of side $a$;
(II) If the area of triangle $ABC$ is $S = 9$, find the perimeter of triangle $ABC$. | 11 + \sqrt{13} | math | 101 |
Let \( n \in \mathbf{Z}_{+} \). When \( n > 100 \), the first two digits of the decimal part of \( \sqrt{n^{2}+3n+1} \) are ______. | 50 | math | 52 |
If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - 3px + 4q = 0$, and $\tan{\alpha} \cdot \tan{\beta}$ is the value of $s$ in the equation $x^2 - 7rx + s = 0$ where $\cot{\alpha} + \cot{\beta} = r$, determine the value of $rs$. | 3p | math | 91 |
What is the increasing interval of the function $y=\sin (-2x+ \frac{\pi}{4})$? | \left[k\pi+ \frac{3}{8}\pi,k\pi+ \frac{7}{8}\pi\right] | math | 25 |
Given proposition p: The function $y=x^2-3ax+4$ is increasing on the interval $[1, +\infty)$, and proposition q: The function $y=(2a-1)^x$ is a decreasing function. If "p and q" is a false proposition, determine the range of values for the real number $a$. | (-\infty, \frac{1}{2}] \cup (\frac{2}{3}, +\infty) | math | 76 |
At a conference, the 2016 participants were registered from P1 to P2016. Each participant from P1 to P2015 shook hands with exactly the same number of participants as the number on their registration form. How many hands did the 2016th participant shake? | 1008 | math | 65 |
In triangle \(ABC\) with the ratio of the sides \(AB: AC = 5: 2\), the angle bisector of \(\angle BAC\) intersects side \(BC\) at point \(L\). Find the length of segment \(AL\) if the length of the vector \(2 \cdot \overrightarrow{AB} + 5 \cdot \overrightarrow{AC}\) is 2016. | 288 | math | 88 |
Evaluate the value of $\frac{(2210-2137)^2 + (2137-2028)^2}{64}$. | 268.90625 | math | 36 |
For the cubic function $f(x) = ax^3 + bx^2 + cx + d$ ($a \neq 0$), a definition is given: Let $f'(x)$ be the derivative of the function $y = f(x)$, and $f''(x)$ the derivative of $f'(x)$. If the equation $f''(x) = 0$ has real solutions $x_0$, then the point $(x_0, f(x_0))$ is called the "inflection point" of the functi... | 2012 | math | 288 |
The points $P(3,-2), Q(3,1), R(7,1)$, and $S$ form a rectangle. What are the coordinates of $S$? | (7,-2) | math | 39 |
There are exactly 120 ways to color five cells in a $5 \times 5$ grid such that exactly one cell in each row and each column is colored.
There are exactly 96 ways to color five cells in a $5 \times 5$ grid without the corner cell, such that exactly one cell in each row and each column is colored.
How many ways are th... | 78 | math | 114 |
Given a positive integer $n$ not exceeding $100$ such that if $n \leq 60$, the probability of choosing $n$ is $p$, and if $n > 60$, the probability of choosing $n$ is $2p$. Find the probability that a randomly chosen integer is a perfect square. | \frac{3}{35} | math | 71 |
The graphs of $y = -|x-a|^2 + b$ and $y = |x-c|^2 + d$ intersect at points $(1,8)$ and $(9,4)$. Find $a+c$. | 10 | math | 47 |
Convert 89 to binary. | 1011001_{(2)} | math | 7 |
Let x and y be two-digit positive integers with a mean of 45. Find the maximum value of the ratio x/y. | 8 | math | 27 |
Determine the number of whole number values for the cost $x$ of each ticket, such that the total cost of tickets for the 9th graders is $60, the total cost of tickets for the 10th graders is $90, and the total cost of tickets for the 11th graders is $49$. | 1 | math | 74 |
Given the function f(x) = -x^3 - 6x^2 - 9x + 3.
(I) Find the interval(s) where the function is strictly decreasing.
(II) Find the maximum and minimum values of the function on the interval [-4, 2]. | 7 | math | 60 |
Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. This room is the first in a sequence of 2 indistinguishable rooms. In each room, 1 door leads to the next room in the sequence (or, for the second room, into Bowser's level), while the other 3 do... | 20 | math | 132 |
Given that the sequence $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, $a_1+1$, $a_2+1$, $a_4+1$ form a geometric sequence, and $a_2+a_3=-12$, then $a_n=$ ______. | -2n-1 | math | 66 |
Given the ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a＞b＞0)$$, with the right focus $$F( \sqrt {3}，0)$$, and eccentricity $$e= \frac { \sqrt {3}}{2}$$.
(1) Find the equation of ellipse C.
(2) A line l passing through F and having an inclination angle of 45° intersects the ellipse at two distinct points M... | \frac {2 \sqrt {6}}{5} | math | 131 |
Given a tetrahedron P-ABC, where PA = 4, AB = AC = $2\sqrt{3}$, BC = 6, and PA is perpendicular to plane ABC, the radius of the circumscribed sphere of this tetrahedron is __________. | 4 | math | 59 |
During a flood control emergency, a large gasoline canister drifting downstream is to be detonated using a shooting method. It is known that there are only 5 bullets available, and the first successful shot can only cause gasoline to leak, while a second successful hit is required for detonation. The probability of hit... | \frac{232}{243} | math | 96 |
Calculate the value of the polynomial $f(x) = x^{6} - 12x^{5} + 60x^{4} - 160x^{3} + 240x^{2} - 192x + 64$ using Horner's Rule when $x = 2$. Find the value of $v_{4}$. | 80 | math | 82 |
What is the base ten equivalent of $45327_8$? | 19159 | math | 17 |
For some positive integer $j$, when 75 is divided by $j^2$, the remainder is 3. What is the remainder when 130 is divided by $j$? | 1 | math | 41 |
The product of two consecutive page numbers is $20{,}412$. What is the sum of these two page numbers? | 285 | math | 28 |
How many sets of integers \( (a, b, c, d) \) satisfy:
\[ 0 < a < b < c < d < 500, \]
\[ a + d = b + c, \]
and
\[ bc - ad = 93? \]
(The 11th USA Mathematical Talent Search, 1993) | 405 + 465 = 870 | math | 78 |
Find four positive integers such that, when adding them three at a time, the sums are 6, 7, 8, and 9. | 1, 2, 3, 4 | math | 31 |
In a geometric sequence $\{a_n\}$, it is known that $S_6=48$ and $S_{12}=60$. Find the value of $S_{24}$. | \frac{255}{4} | math | 44 |
Given that $\lg 2 = 0.3010$, determine the number of digits in the integer $2^{2015}$. | 607 | math | 32 |
Given that the password is a four-digit number composed of one 2, one 9, and two 6s, calculate the maximum number of attempts needed to find the document password. | 12 | math | 38 |
How many times should two dice be rolled so that the probability of getting two sixes at least once is greater than $1/2$? | 25 | math | 30 |
Solve the following equation in the set of real numbers:
$$
1+2 \sqrt{x}-\sqrt[3]{x}-2 \sqrt[6]{x}=0
$$ | x = 1 \quad \text{and} \quad x = \frac{1}{64} | math | 39 |
Rationalize the denominator of \(\frac{35}{\sqrt[3]{35}}\). | \sqrt[3]{1225} | math | 23 |
In trapezoid \(ABCD\), a point \(X\) is taken on the base \(AD\) such that segments \(XB\) and \(XC\) divide the trapezoid into three similar, yet pairwise unequal, non-isosceles triangles. The side \(AB\) has a length of 6. Find \(AX \cdot DX\). | 36 | math | 74 |
The function $f(x) = \cos x$ is transformed such that the abscissa of each point on the graph becomes $\frac{1}{\omega}$ times the original ($\omega > 0$), while the ordinate remains unchanged. The resulting graph is then shifted to the right by $\frac{\pi}{12}$ units. The new graph is symmetric about the line $x = \fr... | \omega = 6 | math | 99 |
Given the universal set $U=\mathbb{R}$, set $A=\{x\mid y=\frac{1}{\sqrt{a-x}}\}$, and set $B=\{x\mid x^2-x-6=0\}$.
1. If $a=-1$, find $A\cap B$.
2. If $(\neg_{U}A)\cap B=\varnothing$, find the range of values for the real number $a$. | a\in(3,+\infty) | math | 101 |
Given the sequence: $$( \frac{1}{1}), ( \frac{1}{2}, \frac{2}{1}), ( \frac{1}{3}, \frac{2}{2}, \frac{3}{1}), ( \frac{1}{4}, \frac{2}{3}, \frac{3}{2}, \frac{4}{1}), \ldots, ( \frac{1}{n}, \frac{2}{n-1}, \frac{3}{n-2}, \ldots, \frac{n-1}{2}, \frac{n}{1})$$, denote the elements of the sequence as: $a_1, a_2, a_3, a_4, a_5... | 7 | math | 178 |
A boy buys oranges at the rate of 4 oranges for 15 cents and sells them at the rate of 6 oranges for 30 cents. Calculate the number of oranges he needs to sell to make a profit of 200 cents. | 160 | math | 52 |
Find the smallest natural number \( n \) for which \( 1999! \) is not divisible by \( 34^n \). | 124 | math | 31 |
(1) Use the Euclidean algorithm to find the greatest common divisor (GCD) of 117 and 182, and verify it using the method of successive subtraction.
(2) Use the Horner's method to calculate the value of the polynomial \\(f(x)=1-9x+8x^{2}-4x^{4}+5x^{5}+3x^{6}\\) at \\(x=-1\\). | 12 | math | 97 |
In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His erroneous product was $161$. What is the correct value of the product of $a$ and $b$? | 224 | math | 50 |
In a certain school, there are 5000 students. Each student is assigned an ID number from 0001 to 5000. No two students can have the same ID number. If a student is selected uniformly at random, what is the probability that the ID number of the student does not contain any 2s among its digits? | \frac{729}{1250} | math | 75 |
Given the constant $a > 1$ and real numbers $x$ and $y$ that satisfy the equation $\log_{a}x + 2\log_{x}a + \log_{x}y = -3$, if the maximum value of $y$ is $\sqrt{2}$, find the value of $x$. | x = \frac{1}{8} | math | 71 |
What is the coefficient of $x^2$ when $-5x^3 - 5x^2 - 7x + 1$ is multiplied by $-x^2 - 6x + 1$ and the like terms are combined? | 36 | math | 54 |
Let $a$ and $b$ be real numbers. Consider the following five statements:
1. $a < b$
2. $b < 0$
3. $a < 0$
4. $\frac{1}{a} < \frac{1}{b}$
5. $a^2 < b^2$
What is the maximum number of these statements that can be true for any values of $a$ and $b$? | 3 | math | 94 |
For $ n \equal{} 1,\ 2,\ 3,\ \cdots$ , let $ (p_n,\ q_n)\ (p_n > 0,\ q_n > 0)$ be the point of intersection of $ y \equal{} \ln (nx)$ and $ \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1$ .
(1) Show that $ 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2}$ to find $ \lim_{n\to\infty} q_n$ .
... | 1 | math | 190 |
Yura was walking along the road and met a tractor pulling a long pipe. Yura decided to measure the length of the pipe. He walked along it "against the movement of the tractor" and counted 20 steps. Then he walked along the pipe "with the movement of the tractor" and counted 140 steps. Knowing that his step is 1 meter, ... | 35 | math | 105 |
Calculate the sum $$\frac{2^1}{8^1 - 1} + \frac{2^2}{8^2 - 1} + \frac{2^3}{8^3 - 1} + \frac{2^4}{8^4 - 1} + \cdots.$$ | \frac{1}{3} | math | 68 |
Solve the system of equations:
\[
\begin{cases}
x - y + z = 1 \\
y - z + u = 2 \\
z - u + v = 3 \\
u - v + x = 4 \\
v - x + y = 5
\end{cases}
\] | (0,6,7,3,-1) | math | 66 |
In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(15,20)$ that does not go inside the circle $(x-9)^{2}+(y-12)^{2}= 36$?
A) $6\sqrt{21} + \pi$
B) $3\sqrt{21} + 2\pi$
C) $6\sqrt{21} + 2\pi$
D) $12 + 4\pi$
E) $9\sqrt{21} + \pi$ | 6\sqrt{21} + 2\pi | math | 127 |
Given points $(-3,{y}_{1}),(1,{y}_{2}),(-\frac{1}{2},{y}_{3})$ are all on the graph of the function $y=x^{2}-2x+3$, compare the values of $y_1, y_2$, and $y_3$. | y_{2} < y_{3} < y_{1} | math | 68 |
What is the value of $0.\overline{234} + 0.\overline{567} - 0.\overline{891}$? Express your answer as a fraction in lowest terms. | \frac{-10}{111} | math | 48 |
When Tanya excluded all numbers from 1 to 333 that are divisible by 3 but not by 7, and all numbers that are divisible by 7 but not by 3, she ended up with 215 numbers. Did she solve the problem correctly? | 205 | math | 58 |
If the sequence $\{a_{n}\}$ is a sequence of positive terms, and $\sqrt{{a_1}}+\sqrt{{a_2}}+⋯+\sqrt{{a_n}}={n^2}+3n(n\in N^{*})$, then $a_{2}=$______,$\frac{{{a_1}}}{2}+\frac{{{a_2}}}{3}+⋯+\frac{{{a_n}}}{{n+1}}=\_\_\_\_\_\_.$ | 2n^{2}+6n | math | 107 |
Find the point where the line passing through $(2, -1, 3)$ and $(6, -5, 7)$ intersects the $xz$-plane. | (1, 0, 2) | math | 35 |
Given that the slant height of a cone is 2, and its net is a semicircle, what is the area of the cross section of the axis of the cone? | \sqrt{3} | math | 37 |
In the rectangular coordinate system $xoy$, the parametric equations of circle $C$ are given by $\begin{cases}x=1+\cos φ \\\\ y=\sin φ\end{cases}$ ($φ$ is the parameter). Establish a polar coordinate system with $O$ as the pole and the non-negative half of the $x$-axis as the polar axis.
(1) Find the Cartesian equatio... | 2 \sqrt{3} | math | 192 |
Let $z$ be a complex number such that
\[z^3 + \frac{1}{z^3} = 52.\]Find all possible values of
\[z + \frac{1}{z}.\]Enter all possible values, separated by commas. | 4, -2 + 3i, -2 - 3i | math | 58 |
Given that ${F_1}$ and ${F_2}$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, respectively. $A$ and $B$ are the right and upper vertices of the ellipse, respectively. $P$ is a point on the ellipse, $O$ is the coordinate origin, $OP$ is parallel to $AB$, $PF_1$ is perpend... | \frac{x^2}{10} + \frac{y^2}{5} = 1 | math | 147 |
Given $a_{n}=2n-1$, $b_{n}=\left\{\begin{array}{l}{2{a}_{n},n\text{ is even}}\\{-{a}_{n},n\text{ is odd}}\end{array}\right.$, determine the formula for $S_{2n}$. | 2n^2+3n | math | 72 |
Given that the function $f(x-1)$ is an odd function, and the function $f(x+3)$ is an even function, with $f(0)=1$, then $f(8)=$ ? | -1 | math | 45 |
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | 118 | math | 54 |
The equation of the tangent line to the function \(y = xe^x\) at the point \((1, e)\) is \(y = xe^x \) is tangent to \(y = x - 1 + e\) at the point \((x, y)\), determine the general equation of the tangent line. | y = 2ex - e | math | 67 |
Given that \\(f(x)\\) is an odd function, and when \\(x < 0\\), \\(f(x) = x^{4} - x\\), then the equation of the tangent line to the curve \\(y = f(x)\\) at \\(x = 1\\) is \_\_\_\_\_\_. | 5x + y - 3 = 0 | math | 74 |
Given the quadratic function $f\left(x\right)=ax^{2}+bx+c$ passes through the point $\left(0,3\right)$, and the solution set of the inequality $ax^{2}+bx+c\leqslant 0$ is $\{x\left|\right.1\leqslant x\leqslant 3\}$.
$(1)$ Find the analytical expression of $f\left(x\right)$;
$(2)$ If $g\left(x\right)=f\left(x\right)... | \lambda \in \left(\frac{2}{3}, +\infty \right) | math | 223 |
A rectangular parallelepiped with dimensions \( m \times n \times k \) is divided into unit cubes. How many parallelepipeds in total (including the original one) are formed? | \frac{m n k (m+1)(n+1)(k+1)}{8} | math | 40 |
Given \(| \overrightarrow{a}|=3 \sqrt {2}\) and \(| \overrightarrow{b}|=6\), and \(\overrightarrow{a}+ \overrightarrow{b}\) is perpendicular to \(\overrightarrow{a}\), find the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\). | 135^{\circ} | math | 78 |
The length, width, and height of a certain rectangular prism are $4$, $2$, and $1$ respectively, calculate the surface area of the circumscribed sphere of the rectangular prism. | 21\pi | math | 40 |
Given that the sequence $\{a\_n\}$ is an arithmetic sequence, $a\_1+a\_5=-20$, $a\_3+a\_8=-10$.
1. Find the general term of the sequence $\{a\_n\}$.
2. For what value of $n$ does the sum of the first $n$ terms of the sequence $\{a\_n\}$, denoted as $S\_n$, reach its minimum? Also, find this minimum value. | -56 | math | 105 |
Given the function $f(x)=\sin(2ωx-\frac{π}{6})$, the graph is shifted to the left by $\frac{π}{4}$ units to obtain the graph of function $g(x)$. The graph of function $g(x)$ is symmetric about the $y$-axis. If $ω$ is the smallest positive number for which the transformation is valid, then $ω=$ \_\_\_\_\_\_. | ω=\frac{4}{3} | math | 93 |
Let $f(x)=x^{2}+6 x+7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x$. | 23 | math | 36 |
Find the point \( M' \) symmetrical to the point \( M \) with respect to the line.
\( M(3, 3, 3) \)
\[ \frac{x-1}{-1} = \frac{y-1.5}{0} = \frac{z-3}{1} \] | (1, 0, 1) | math | 70 |
\(\frac{\sin 6 \alpha}{\sin 2 \alpha} + \frac{\cos (6 \alpha - \pi)}{\cos 2 \alpha}\). | 2 | math | 38 |
There are \_\_\_\_\_\_ six-digit numbers composed of three distinct odd numbers, two 2's, and one 0. (Answer with a number) | 3000 | math | 35 |
Given the equation of the parabola $y=-2x^{2}+1$, determine the equation of the parabola after it is shifted $3$ units to the right and then $1$ unit down. | -2(x-3)^{2} | math | 46 |
$\triangle ABC$ has a right angle at $C$ and $\angle A = 30^\circ$. If $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, then find the measure of $\angle BDC$. | 60^\circ | math | 57 |
Find the number of positive integers $n \le 1500$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor = n\]
for some real number $x.$ | 856 | math | 63 |
If the graph of the function $f(x) = |x+m| + |nx+1|$ is symmetric about $x=2$, then the set $\{x | x = m+n\} = \quad$. | \{-4\} | math | 46 |
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