problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
If the quadratic inequality $x^2 + mx + 2 > 0$ has a solution in the interval $[1, 2]$, determine the range of the real number $m$. | (-3, +\infty) | math | 41 |
Given that each of the numbers $n$ , $n+1$ , $n+2$ , $n+3$ is divisible by its sum of digits in its decimal representation and the number in ones column of $n$ is $8$ , determine the number of different values the tens column of $n$ can have. | 1 | math | 79 |
What is the angle opposite side \( c \) in a triangle with sides \( a, b, c \), perimeter \( 2s \), and area \( T \), if
$$
T+\frac{a b}{2}=s(s-c) ?
$$ | 45^\circ | math | 55 |
Serge and Tanya want to show Masha a magic trick. Serge leaves the room. Masha writes down a sequence $(a_1, a_2, \ldots , a_n)$ , where all $a_k$ equal $0$ or $1$ . After that Tanya writes down a sequence $(b_1, b_2, \ldots , b_n)$ , where all $b_k$ also equal $0$ or $1$ . Then Masha either does nothing or... | n | math | 258 |
Let $O$ be the origin. A variable plane has a distance of 2 from the origin, and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the centroid of triangle $ABC.$ Find
\[\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}.\] | 2.25 | math | 109 |
How many rows of Pascal's Triangle contain the number $67$? | 1 | math | 15 |
In the polar coordinate system, find the polar equation of the circle with center at point $P(2, \frac{\pi}{3})$ and tangent to the line $l$: $\rho\sin (\theta- \frac{\pi}{3})=2$. | \rho=4\sin (\theta+ \frac{\pi}{6}) | math | 55 |
Given the equation in terms of $x$, $4+3ax=2a-7$, has a unique solution, and the equation in terms of $y$, $2+y=(b+1)y$, has no solution, determine the situation of the solution for the equation $az=b$ in terms of $z$. | z=0 | math | 66 |
Find all pairs of positive integers $(m,n)$ such that $\frac{n^2+1}{2m}$ and $\sqrt{2^{n-1}+m+4}$ are both integers. | (m, n) = (1, 3) | math | 48 |
Let the function $f(x) = |x-1| + |2x+4|$.
(1) Find the minimum value of $y=f(x)$;
(2) Find the solution set of the inequality $|f(x)-6| \leq 1$. | \left[-\frac{10}{3}, -\frac{8}{3}\right] \cup [0, \frac{4}{3}] | math | 58 |
Determine the range of values for the real number $a$ in the inequality $ax^2 - |x + 1| + 3a \geq 0$, where the solution set for $x$ is $R$. | [\frac{1}{2}, +\infty) | math | 49 |
Find the maximum value of the parameter \(a\) for which the equation \((|x-2|+2a)^{2}-3(|x-2|+2a)+4a(3-4a)=0\) has three solutions. Specify the largest value in your answer. | 0.5 | math | 60 |
Given an acute triangle $ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. Choose any one of the three conditions below to solve the following problems (if multiple conditions are used, follow the first solution):
$(1)$ Find angle $B$;
$(2)$ If $b=2$ and the area of triangle $ABC$ is $\sqrt{3... | 2 | math | 98 |
Suppose $r \ge 2$ is an integer, and let $m_1, n_1, m_2, n_2, \dots, m_r, n_r$ be $2r$ integers such that $$ \left|m_in_j-m_jn_i\right|=1 $$ for any two integers $i$ and $j$ satisfying $1 \le i<j \le r$ . Determine the maximum possible value of $r$ .
*Proposed by B Sury* | 3 | math | 120 |
Line $m$ is parallel to line $n$ and the measure of $\angle 1$ is $\frac{1}{4}$ the measure of $\angle 2$. What is the degree measure of $\angle 5$?
[asy] size(100); defaultpen(linewidth(0.7)+fontsize(10));
path m = (-1.35,0.72)--(0.45,0.72), n = (-1,0)--(1,0), k = (-0.67,1.09)--(0.27,-0.48);
pair A = intersectionpoi... | 36^\circ | math | 290 |
Given $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles, calculate the number of distinguishable arrangements in a row from left to right. | 420 | math | 41 |
Solve the integral equation:
a) \(y(\tau)=\int_{0}^{t} y(\tau) \, d\tau + 1\);
b) \(\int_{0}^{t} y(\tau) \sin(t-\tau) \, d\tau = 1 - \cos t\). | y(t) = 1 | math | 70 |
Given an unfair die that has a probability 4 times as likely to result in an odd number as an even number, three such dice are rolled. Find the probability that the sum of the numbers rolled is odd. | \frac{76}{125} | math | 43 |
Janine is creating a design using a large square and inside it are placed six small circles. Each circle is tangent to one side of the square and the three circles adjacent to it. The side length of the square in Janine's design is 30 inches. How many square inches will be shaded in her design?
[asy]
size(120);
draw((... | 900 - 150\pi \text{ square inches} | math | 251 |
For a number $x$, we use $\left(x\right]$ to represent the largest integer less than $x$, for example: $\left(1.6\right]=1$, $\left(-4\right]=-5$.
① Fill in the blanks: $(0 ]=\_\_\_\_\_\_,(-2023 ]=\_\_\_\_\_\_$;
② If $|\left(x\right]-3|=6$, then the range of values for $x$ is ______. | 9 < x \leqslant 10 \text{ or } -3 < x \leqslant -2 | math | 106 |
If the function $f(x)=\sin \omega x + \sqrt{3}\cos \omega x (\omega > 0)$ has a minimum positive period of $\pi$, then one value of $\varphi$ that satisfies the condition "$f(x+\varphi)$ is an even function" is ______ (provide one value of $\varphi$ that satisfies the condition). | \frac{\pi}{12} | math | 77 |
Positive real numbers \( x, y, z \) satisfy
\[
\left\{
\begin{array}{l}
\frac{2}{5} \leqslant z \leqslant \min \{x, y\}, \\
xz \geqslant \frac{4}{15}, \\
yz \geqslant \frac{1}{5}.
\end{array}
\right.
\]
Find the maximum value of \( \frac{1}{x} + \frac{2}{y} + \frac{3}{z} \). | 13 | math | 123 |
Given two non-zero vectors $\overrightarrow{a}, \overrightarrow{b}$ with an angle of $60^{\circ}$ between them, and $|\overrightarrow{a} - \overrightarrow{b}| = 1$, determine the maximum value of $|\overrightarrow{a} + \overrightarrow{b}|$. | \sqrt{3} | math | 70 |
Given the parabola $C: y=ax^{2}(a > 0)$, the distance from the focus to the directrix is $\dfrac{1}{4}$, and two points $A(x_{1},y_{1})$, $B(x_{2},y_{2})$ on $C$ are symmetric about the line $y=x+m$, and $x_{1}x_{2}=-\dfrac{1}{2}$. Find the value of $m$. | \dfrac{3}{2} | math | 104 |
Let $d(n)$ denote the number of positive divisors of the positive integer $n$. What is the smallest positive real value of $c$ such that $d(n) \leq c \cdot \sqrt{n}$ holds for all positive integers $n$? | \sqrt{3} | math | 55 |
Given a polar coordinate system with the origin as the pole and the non-negative semi-axis of the $x$-axis as the polar axis, the polar coordinate equation of the line $l$ is $ρ\cos(θ-\frac{π}{4})=5+\sqrt{2}$. The parametric equation of the curve $C$ is $ \begin{cases} x=2+2\cos α \\ y=2\sin α \end{cases}$ ($α$ is the ... | 3 | math | 201 |
Real numbers \( x, y, \) and \( z \) satisfy the equation:
\[ 3(x + y + z) = x^2 + y^2 + z^2. \]
Let \( N \) be the maximum value of \( xy + xz + yz \), and let \( n \) be the minimum value of \( xy + xz + yz \). Find \( N + 5n \). | 27 | math | 92 |
In right triangle $DEF$ where the hypotenuse $\overline{DE}=13$ and leg $\overline{DF}=12$, find the point $D_1$ where the bisector of angle $D$ meets the opposite side $\overline{EF}$. Construct a new right triangle $XYZ$ with hypotenuse $\overline{XY} = D_1E$ and leg $\overline{XZ} = D_1F$. Calculate the length of $X... | \frac{12}{25} | math | 192 |
Distinct lines \(\ell\) and \(m\) lie in the xy-plane. They intersect at the origin. Point \(Q(3, -2)\) is first reflected about line \(m\), then the resulting point \(Q'\) is reflected about line \(\ell\) to point \(Q''\). The equation of line \(\ell\) is \(2x - 5y = 0\), and the coordinates of \(Q''\) are \((-2, 3)\)... | 5x + 2y = 0 | math | 111 |
Find the polynomial \( p(x) \), with real coefficients, such that \( p(3) = 10 \) and
\[ p(x) p(y) = p(x) + p(y) + p(xy) - 3 \]
for all real numbers \( x \) and \( y \). | x^2 + 1 | math | 65 |
Given that point $P$ lies on the line $y=2x$, if there exist two points $A$ and $B$ on the circle $C(x-3)^2+y^2=4$ such that $PA \perp PB$, then the range of the abscissa $x_{0}$ of point $P$ is _______. | \left[ \frac{1}{5},1\right] | math | 74 |
$(1)(-5)+9$;
$(2)12-(-16)+(-2)-1$;
$(3)6\div (-2)\times (-\frac{1}{3})$;
$(4)(-15)\times (\frac{1}{3}+\frac{1}{5})$;
$(5)(-2)^{3}-(-8)\div |-\frac{4}{3}|$;
$(6)-1^{2022}-(\frac{1}{2}-\frac{1}{3})\times 3$. | -\frac{3}{2} | math | 127 |
Given the acute angles $α$ and $β$ that satisfy $\sin α=\frac{\sqrt{10}}{10}$ and $\cos β=\frac{2\sqrt{5}}{5}$, find the value of $α+β$. | \frac{\pi}{4} | math | 54 |
Given an ellipse $\Gamma: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passing through point $E(\sqrt{3}, \frac{1}{2})$, with an eccentricity of $\frac{\sqrt{3}}{2}$.
(1) Find the equation of ellipse $\Gamma$;
(2) A line $l$ is tangent to circle $O: x^2 + y^2 = b^2$ at point $M$ and intersects ellipse $\Gamma$ ... | S_{\triangle OAB} = \frac{1}{2} \times 2 \times 1 = 1 | math | 144 |
Tom’s graduating class has 300 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 12 rows and at least 18 students in each row, what is the sum of all possible values of $x$ where $x$ is the number of students in each row? | 45 | math | 78 |
The lengths of the sides of a trapezoid are 2, 10, 10, and 20. Find the length of the segment connecting the midpoints of the diagonals.
(It is known that these points are distinct). If there are multiple possible answers, list them in ascending order separated by commas or semicolons. | 9 | math | 72 |
Given that $\text{1 mile} = \text{6 furlongs}$ and $\text{1 furlong} = \text{60 rods}$, calculate the number of rods in one mile. | 360 | math | 44 |
Given the function $g(x)=x^{2}-2 (x \in \mathbb{R})$ and $f(x) = \begin{cases} g(x)+x+4, & x < g(x) \\ g(x)-x, & x \geq g(x) \end{cases}$, determine the range of $f(x)$. | [-2.25, 0] \cup (2, +\infty) | math | 76 |
The highest degree term of the polynomial $-\frac{4}{5}x^{2}y+\frac{2}{3}x^{4}y^{2}-x+1$ is ______, and the coefficient of the linear term is ______. | -1 | math | 52 |
Given that $|$$\overrightarrow {a}$$| = 2|$ $\overrightarrow {b}$$|, $|$ $\overrightarrow {b}$$| \neq 0$, and the quadratic equation $x^2 + |$ $\overrightarrow {a}$$|x - $$ $\overrightarrow {a}$ \cdot$ $\overrightarrow {b}$ = 0 has two equal real roots, find the angle between the vectors $\overrightarrow {a}$ and $\ove... | \frac{2\pi}{3} | math | 108 |
For how many different values of \( k \) is the 4-digit number \( 7k52 \) divisible by 12? | 3 | math | 30 |
A student passes through three intersections with traffic lights, labeled \\(A\\), \\(B\\), and \\(C\\), on the way to school. It is known that the probabilities of encountering a red light at intersections \\(A\\), \\(B\\), and \\(C\\) are \\( \dfrac {1}{3} \\), \\( \dfrac {1}{4} \\), and \\( \dfrac {3}{4} \\) respect... | \dfrac {235}{3} | math | 217 |
Find all real numbers $x$ such that \[4 \le \frac{2x}{3x-7} < 9.\] (Give your answer in interval notation.) | (\frac{63}{25}, 2.8] | math | 38 |
Given a sequence $\{a_n\}$ that satisfies $a_1=2$, $a_{n+1}=\frac{2(n+2)}{n+1}a_n$ for $n\in \mathbb{N^*}$, find the value of $\frac{a_{2017}}{a_{1}+a_{2}+\cdots+a_{2016}}$. | \frac{1009}{1008} | math | 90 |
Given two circles $C_1: (x-a)^2 + (y+2)^2 = 4$ and $C_2: (x+b)^2 + (y+2)^2 = 1$ intersect, find the equation of the line where their common chord lies. | (2a+2b)x + 3 + b^2 - a^2 = 0 | math | 60 |
Two numbers between $0$ and $1$ on a number line are to be chosen at random. What is the probability that the second number chosen will exceed the first number chosen by a distance greater than $\frac 14$ unit on the number line? Express your answer as a common fraction. | \frac{9}{32} | math | 61 |
$O$ is the origin, $P$ is on the circle $C\left(x-2\right)^{2}+\left(y-1\right)^{2}=1$, $OP$ is tangent to the circle $C$. Determine the length of segment $|OP|$. | 2 | math | 61 |
Given the universal set U = {-2, -1, 0, 1, 2}, and set A = {y | y = |x|, x ∈ U}, determine the complement of A in U, denoted as ∁<sub>U</sub>A. | \{-2, -1\} | math | 59 |
A certain taxi operates in the east-west direction from Gulou in the afternoon. The positive direction is east, and the negative direction is west. The recorded distances (in kilometers) in order are: $+9$, $-3$, $-5$, $+4$, $8$, $+6$, $3$, $-6$, $-4$, $+10$.
$(1)$ How far is the taxi from the starting point at Gulou... | 139.2 \text{ yuan} | math | 142 |
Let \( n \) be a two-digit number written in the decimal system, and \( s \) the sum of the squares of its digits. What are the smallest and largest values of \( n - s \)? | 25 \text{ and } -63 | math | 44 |
The product of three different positive integers is equal to \(5^4\). What is the sum of these integers? | 131 | math | 24 |
A circle with center $A$ and radius four feet is tangent at $C$ to a circle with center $B$, as shown. If point $B$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\pi$. | 48\pi | math | 57 |
The Bank of Springfield has introduced a new plan called the "Ultra Savings Account" which compounds annually at a rate of 2%. If Lisa decides to invest $1500 in this new account, how much interest will she earn after 10 years? | 328.49 | math | 53 |
A positive number and its square sum to 245. Additionally, the sum is three times the number itself. What is the number? | 14 | math | 29 |
There are 5 students and 2 teachers to be seated in a row for a group photo. The teachers cannot sit at either end and must sit together. How many different seating arrangements are possible? | 960 | math | 40 |
Let $f(x)$ be a function such that $f(0) = 1$ and
\[f(xy) = f \left( \frac{x^2 + y^2}{2} \right) + (x - y)^2\]for all real numbers $x$ and $y.$ Find $f(x).$ | 1 - 2x | math | 72 |
Given $f(x)=4x^{5}+3x^{4}+2x^{3}-x^{2}-x-\frac{1}{2}$, use Horner's method to find $f(-2)$. | -\frac{197}{2} | math | 48 |
Let $\boxed{N}$ mean the number of whole number divisors of $N$. For example, $\boxed{3}=2$, because $3$ has two divisors, $1$ and $3.$ Find the value of \[\boxed{\boxed{11}\times\boxed{20}}\] | 12 | math | 66 |
A sequence $\{a\_n\}$ satisfies $a\_1=1$, $\sqrt{\frac{1}{a\_n^2}+2}=\frac{1}{a\_{n+1}} (n\in\mathbb{N}^*)$, and $b\_n=\frac{1}{a\_n^2\cdot 2^n}$. Determine the sum of the first $n$ terms of the sequence $\{b\_n\}$, denoted as $S\_n=$ \_\_\_\_\_\_. | 3-\frac{2n+3}{2^n} | math | 116 |
Given the arithmetic sequence {a_n}, where the sum of the first n terms, S_n, is given by a specific formula, and 4 + a_5 = a_6 + a_4, determine the value of S_9. | 36 | math | 51 |
Suppose $P$ is the point $(5,3)$ and $Q$ is the point $(-3,6)$. Find point $T$ such that $Q$ is the midpoint of segment $\overline{PT}$. | (-11,9) | math | 49 |
Given a cuboid with dimensions 77 × 81 × 100, cut into small cubes with side length 1, find the number of small cubes that a diagonal inside the cuboid passes through. | 256 | math | 45 |
Given a complex number $z=\frac{{3+i}}{{2-i}}$, calculate the modulus of $z$. | \sqrt{2} | math | 23 |
Given $a-b=4$ and $ab=3$, find:<br/>$(1) a^{2}+b^{2}$;<br/>$(2) \left(a-2\right)\left(b+2\right)$. | 7 | math | 51 |
Let
\[ f(x) = x^3 + 4x^2 + 13x + 20. \]
The graphs of $y = f(x)$ and $y = f^{-1}(x)$ intersect at exactly one point $(a,b)$. Enter the ordered pair $(a,b)$. | (-2, -2) | math | 66 |
The 19th Asian Games were successfully held in Hangzhou from September 23 to October 8, 2023. The volunteers of the Hangzhou Asian Games are called "Little Green Lotus." There are a total of 36 Little Green Lotus in a sports venue, including 12 boys and 24 girls. The number of Little Green Lotus who can speak Japanese ... | (6, 12), (7, 14), (8, 16) | math | 260 |
Using five twos, arithmetic operations, and exponentiation, make the numbers from 1 to 5. | 1, 2, 3, 4, 5 | math | 22 |
The number $192000000$ can be expressed in scientific notation. Express it as a product of a number between $1$ and $10$ and an integer power of $10$. | 1.92 \times 10^{8} | math | 46 |
The following are some two-digit subtraction operations: $21-12=9$, $31-13=18$, $32-23=9$, $42-24=18$, $14-41=-27$, $51-15=36$, $26-62=-36$, $\ldots$ Observing the above equations and their results, investigate a special case in two-digit subtraction operations:<br/>$(1)$ Write down another equation that fits the above... | (10a+b)-(10b+a)=9(a-b) | math | 134 |
a) \(1 < \cos \alpha + \cos \beta + \cos \gamma \leq \frac{3}{2}\)
b) \(1 < \sin \left( \frac{\alpha}{2} \right) + \sin \left( \frac{\beta}{2} \right) + \sin \left( \frac{\gamma}{2} \right) \leq \frac{3}{2}\) | 1 < \sin \left( \frac{\alpha}{2} \right) + \sin \left( \frac{\beta}{2} \right) + \sin \left( \frac{\gamma}{2} \right) \leq \frac{3}{2} | math | 93 |
Find all functions $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$ ( $\displaystyle N^\ast = \{ 1,2,3,\ldots \}$ ) with the property that, for all $\displaystyle n \geq 1$ , \[ f(1) + f(2) + \ldots + f(n) \] is a perfect cube $\leq n^3$ .
*Dinu Teodorescu* | f(n) = 3n^2 - 3n + 1 | math | 110 |
There are 6 lamps in a row in the corridor. To save electricity without affecting the lighting, it is required to turn off 2 of them, but 2 adjacent lamps cannot be turned off. How many ways are there to turn off the lamps? | 10 | math | 52 |
Convert the decimal number 23 to binary. | 10111 | math | 10 |
The current population of a city is 1 million people. If the annual natural growth rate is 1.2%,
(1) Write the function relationship between the total population of the city (in ten thousand) and the number of years;
(2) Calculate approximately how many years later the population of the city will reach 1.2 million (acc... | 0.9\% | math | 168 |
A sequence of integers \( a_1, a_2, a_3, \ldots \) is defined by
\[
\begin{array}{c}
a_1 = k, \\
a_{n+1} = a_n + 8n \text{ for all integers } n \geq 1.
\end{array}
\]
Find all values of \( k \) such that every term in the sequence is a square. | 1 | math | 95 |
Compute
\[
\frac{\lfloor \sqrt{1} \rfloor \cdot \lfloor \sqrt{3} \rfloor \cdot \lfloor \sqrt{5} \rfloor \dotsm \lfloor \sqrt{35} \rfloor}{\lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{4} \rfloor \cdot \lfloor \sqrt{6} \rfloor \dotsm \lfloor \sqrt{36} \rfloor}.
\] | \frac{1}{6} | math | 114 |
Given a monotonically increasing geometric sequence $\{a_n\}$ satisfies $a_2 + a_3 + a_4 = 28$, and $a_3 + 2$ is the arithmetic mean of $a_2$ and $a_4$.
(1) Find the general term formula for the sequence $\{a_n\}$.
(2) If $b_n = a_n \log_{\frac{1}{2}} a_n$ and $S_n = b_1 + b_2 + b_3 + \cdots + b_n$, for any positive ... | (-\infty, -1] | math | 163 |
Given a positive integer $n$ that has $72$ divisors and $5n$ has $96$ divisors, find the greatest integer $k$ such that $5^k$ divides $n$. | 2 | math | 46 |
In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks needs to be replaced with any digit from $0,1,2,3,4,5,6,7,8$ (digits can repeat) so that the resulting 10-digit number is divisible by 45. How many ways can this be done? | 1458 | math | 82 |
Given that a real number $a$ is chosen arbitrarily within the interval $(-1,1)$ and a real number $b$ is chosen arbitrarily within the interval $(0,1)$, the probability that the line $ax-by=0$ intersects with the circle $(x-1)^{2}+(y-2)^{2}=1$ is $\_\_\_\_\_\_$. | \frac{5}{16} | math | 81 |
For the real numbers \(a\) and \(b\), it holds that \(a^{2} + 4b^{2} = 4\). How large can \(3a^{5}b - 40a^{3}b^{3} + 48ab^{5}\) be? | 16 | math | 65 |
When \( 4^{m} + 4^{n} \) is a multiple of 100, what is the smallest value of \( m + n \)? (where \( m \) and \( n \) are natural numbers and \( m > n \)) | 7 | math | 57 |
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day $1$ , he gives no
name at all. Every day after that, he alternately adds $2$ words and $4$ words to his name. For example, on day $4$ he
submits an $8\text{-word}$ name. On day $n$ he submits the $44\text{-wor... | 16 | math | 171 |
Determine the smallest positive integer $x$ for which $2520x = M^3$, where $M$ is an integer. | 3675 | math | 30 |
The function $f(x) = (m^2 - 1)x^m$ is a power function, and it is increasing on the interval $(0, +\infty)$. The value of the real number $m$ is ______. | m = \sqrt{2} | math | 51 |
Given $a > 0$, $b > 0$, if $\sqrt{2}$ is the geometric mean of $2^{a}$ and $2^{b}$, calculate the minimum value of $\frac{1}{a} + \frac{1}{b}$. | 4 | math | 57 |
When $\sqrt[4]{2^5 \cdot 5^3}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$? | 252 | math | 52 |
Given the line $x+y+m=0$ and the circle $x^{2}+y^{2}=4$ intersect at two distinct points $A$ and $B$. $O$ is the origin, and $|\overrightarrow{OA}+\overrightarrow{OB}|\geq |\overrightarrow{AB}|$. Determine the range of values for the real number $m$. | (-2\sqrt{2},-2]\cup[2,2\sqrt{2}) | math | 80 |
The line $y = b-x$ with $0 < b < 4$ intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 9:25, what is the value of $b$? Express the answer as a decimal to the nearest tenth.
[asy]
draw((0,-3)--(0,5.5),Arrows);
draw((4,-3.5)--(4,5),Arrows... | 2.5 | math | 305 |
A $5 \times 5$ block of calendar dates is shown. First, the order of the numbers in the third row is reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?
$$
\begin{array}{|c|c|c|c|c|}
\hline
1 & 2 & 3 & 4 & 5 \\
\hline
6 & 7 & 8 & 9 & 10 \\
\hline
1... | 0 | math | 217 |
Given that "$x > k$" is a sufficient but not necessary condition for "$\frac {3}{x+1} < 1$", determine the range of values for $k$. | [2, +\infty) | math | 38 |
Given that \(a_{1}, a_{2}, a_{3}, a_{4}\) are positive real numbers, satisfying:
$$
\sum_{i=1}^{4} i a_{i} \leqslant 10, \quad a_{i} \geqslant \frac{1}{2^{4-i}} \quad (i=1, 2, 3, 4).
$$
Define \(f=\sum_{i=1}^{4} \frac{1}{1+a_{i}^{i}}\). Find the minimum value of \(f\). | 2 | math | 129 |
Using the digits 1, 2, 3, and 4, calculate the number of four-digit numbers that can be formed without repeating any digit. | 24 | math | 32 |
Given the function $f\left(x\right)=x^{3}+x+1$, if $f\left(1-x\right)+f\left(2x\right) > 2$, determine the range of $x$. | (-1,+\infty) | math | 51 |
Let \( A \), \( M \), and \( C \) be nonnegative integers such that \( A+M+C=15 \). Determine the maximum value of the expression \[ A\cdot M\cdot C + A\cdot M + M\cdot C + C\cdot A. \] | 200 | math | 64 |
I own a collection of 18 books, including 4 novels that are each part of a series. To keep the variety on my vacation, I decide not to pick two books from this series. With this restriction, how many different pairs of books can I choose to take? | 147 | math | 57 |
The solution set of the inequality $\dfrac{x}{{x}^{2}-8x+15}\geqslant 2$ is . | \left[ \dfrac{5}{2},3\right) \cup (5,6] | math | 32 |
Given that $\sin{2\alpha} = \frac{2}{3}$, find the value of $\cos^2\left(\alpha + \frac{\pi}{4}\right)$. | \frac{1}{6} | math | 41 |
If we want to write down all the integers from 1 to 10,000, how many times do we have to write a digit, for example, the digit 5? | 4000 | math | 40 |
Given the ellipse with a short axis AB and one of its foci F<sub>1</sub>, find the eccentricity of the ellipse for which triangle ABF<sub>1</sub> is an equilateral triangle. | \frac{\sqrt{3}}{2} | math | 47 |
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