problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Let these numbers be A and B. Let B be the smallest of these numbers. Then the greater number is $A=10B+n$, where $n$ is the crossed-out digit, and $0 \leq n \leq 9$. According to the condition, $A+B = 11B + n = 2022$. Hence, $n$ is the remainder when 2022 is divided by 11, giving $n = 9$. We get $B = 183$ and $A = 183... | \mathrm{B}=183, \mathrm{~A}=1839 | math | 123 |
Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=20$, find the smallest possible value of $|w^3+z^3|$. | 56 | math | 49 |
Solve the following equations using appropriate methods:<br/>$(1)x^{2}-2x-4=0$;<br/>$(2)x\left(x-2\right)+x-2=0$. | x_{1} = -1, \, x_{2} = 2 | math | 44 |
A rectangular prism has six faces, twelve edges, and eight vertices. A segment, such as $x$, which joins two vertices not joined by an edge, is called a diagonal. Considering the rectanglular prism has dimensions such that the length, height, and width are all different, calculate how many total diagonals it has and ho... | 16 | math | 79 |
What is the modular inverse of $7$, modulo $800$?
Express your answer as an integer from $0$ to $799$, inclusive. | 343 | math | 34 |
Let $m=\underbrace{55555555}_{\text{8 digits}}$ and $n=\underbrace{111111111}_{\text{9 digits}}$.
What is $\gcd(m,n)$? | 1 | math | 56 |
Given the parabola $y=2x^2$, determine the coordinates of its focus. | \left(0, \frac{1}{8}\right) | math | 20 |
A number $x$ is randomly chosen from the interval $[0,π]$. What is the probability that the event $\sin x + \sqrt{3} \cos x \leqslant 1$ occurs? | p = \frac{1}{2} | math | 47 |
Find the distance from the center of the circle $(x+1)^{2}+y^{2}=2$ to the line $y=2x+3$. | \frac{\sqrt{5}}{5} | math | 35 |
Which of the following statements are correct? Select the correct numbers.
$(1)$ A right triangle has only one altitude.
$(2)$ An $n$-sided polygon has a total of $n(n-3)$ diagonals.
$(3)$ Two circles with equal radii are congruent.
$(4)$ If a polygon has all sides equal, then it is a regular polygon.
$(5)$ A... | (3), (5) | math | 107 |
$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$ . Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$ . (See diagram). Given $EC=1$ , find the radius of the circle.
[asy]
size(6cm);
pair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersec... | 16 | math | 233 |
Tossing a pair of dice, let the number obtained on the first throw be $m$ and on the second throw be $n$.
1. Find the probability that $m+n$ is not greater than 4.
2. Find the probability that $m < n+2$. | \frac{13}{18} | math | 60 |
Zibo barbecue is famous throughout China. A specialty barbecue restaurant hopes to make good profits during the National Day holiday. After calculation, the cost of lamb skewers is $3$ yuan per skewer. Drawing on past experience, if each skewer is sold for $10$ yuan, on average, they can sell 300 skewers per day. If th... | 7 \text{ yuan} | math | 194 |
If the three angles \(\alpha\), \(\beta\), and \(\gamma\) form an arithmetic sequence with a common difference of \(\frac{\pi}{3}\), then \(\tan \alpha \cdot \tan \beta + \tan \beta \cdot \tan \gamma + \tan \gamma \cdot \tan \alpha =\) . | -3 | math | 74 |
For how many values of the digit $A$ is it true that $54$ is divisible by $A$ and $273{,}1A4$ is divisible by $4$? | 2 | math | 43 |
Let $a, b \in \mathbb{R}$, and $a \neq 1$. Suppose the odd function $f(x) = \lg \frac{1+ax}{1+x}$ is defined on the interval $(-b, b)$.
$(1)$ Find the value of $a$;
$(2)$ Determine the range of values for $b$;
$(3)$ Solve the inequality $f(x) > 0$. | (-1,0) | math | 97 |
In the rectangular coordinate system, if the equation \( m\left(x^{2}+y^{2}+2y+1\right)=(x-2y+3)^{2} \) represents an ellipse, determine the range of values for \( m \). | (5,+\infty) | math | 56 |
If $f(x+1) = \frac{3}{3+x}$ for all $x>0$, then $2f(x) =$
A) $\frac{6}{2 + x}$
B) $\frac{6}{3 + x}$
C) $\frac{9}{2 + x}$
D) $\frac{9}{3 + x}$
E) $\frac{3}{2 + x}$ | \frac{6}{2 + x} | math | 90 |
Given a $9 \times 9$ chess board, we consider all the rectangles whose edges lie along grid lines (the board consists of 81 unit squares, and the grid lines lie on the borders of the unit squares). For each such rectangle, we put a mark in every one of the unit squares inside it. When this process is completed, how man... | 56 | math | 84 |
Calculate the coefficient of $x^{\frac{3}{2}}$ in the expansion of $\left(2x^2 - \dfrac{3}{x}\right)^9$. | 0 | math | 39 |
If $e^{i \phi} = \frac{3 + i \sqrt{8}}{5},$ then find $\sin 4 \phi.$ | \frac{12\sqrt{8}}{625} | math | 33 |
The solution of the equation $7^{x+7} = 8^x$ can be expressed in the form $x = \log_b 7^7$. What is $b$? | \frac{8}{7} | math | 41 |
Find the sum of the thousands digit and the units digit of the product of the two 101-digit numbers 404040404...040404 and 707070707...070707. | 10 | math | 59 |
Simply the expression
\[\frac{(\sqrt{2} - 1)^{1 - \sqrt{3}}}{(\sqrt{2} + 1)^{1 + \sqrt{3}}},\]writing your answer as $a - b \sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of a prime. | 3 - 2 \sqrt{2} | math | 89 |
Given that the line $l$ passing through point $A(3,1)$ is tangent to circle $C$: $x^{2}+y^{2}-4y-1=0$ at point $B$, find the value of $\overrightarrow{CA} \cdot \overrightarrow{CB}$. | \overrightarrow{CA} \cdot \overrightarrow{CB} = 5 | math | 65 |
Let O be the origin of coordinates, C be the center of the circle $(x-2)^2+y^2=3$, and there is a point M$(x,y)$ on the circle satisfying $\overrightarrow{OM} \cdot \overrightarrow{CM} = 0$. Solve for the value of $\frac{y}{x}$. | \pm \sqrt{3} | math | 72 |
Given that $F_1$ and $F_2$ are the common foci of an ellipse and a hyperbola, $P$ is their common point, and $\angle F_1 P F_2 = \frac{\pi}{3}$, find the minimum value of the product of the eccentricities of the ellipse and the hyperbola. | \frac{\sqrt{3}}{2} | math | 74 |
An odd function $y=f\left(x\right)$ defined on $R$ satisfies $f\left(x+2\right)=-f\left(x\right)$. When $x\in \left(0,1\right]$, $f\left(x\right)=3^{x}-1$. Find $f\left(9.5\right)$. | \sqrt{3}-1 | math | 79 |
Find the roots of the polynomial \(6x^4 + 25x^3 - 59x^2 + 28x\). List your answers with rational numbers as fractions and irrational numbers in simplest radical form, separated by commas. | 0, 1, \frac{-31 + \sqrt{1633}}{12}, \frac{-31 - \sqrt{1633}}{12} | math | 52 |
Find the equation of the Apollonius Circle with foci at A(-2, 0) and B(2, 0), given that the moving point maintains a constant ratio λ = 1/2 to the distances from these foci. | x^2+y^2+\frac{20}{3}x+4=0 | math | 52 |
When a certain biased coin is flipped six times, the probability of getting heads exactly twice is equal to that of getting heads exactly three times. Let $\frac{i}{j}$, in lowest terms, be the probability that the coin comes up heads in exactly $4$ out of $6$ flips. Find $i+j$. | 137089 | math | 66 |
Let $a,$ $b,$ $c,$ $d$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$ If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{\omega^2},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b ... | 4 | math | 165 |
In how many ways, can we draw $n-3$ diagonals of a $n$ -gon with equal sides and equal angles such that: $i)$ none of them intersect each other in the polygonal. $ii)$ each of the produced triangles has at least one common side with the polygonal. | \frac{1}{n-1} \binom{2n-4}{n-2} | math | 69 |
A circle having radius \( r_1 \) centered at point \( N \) is tangent to a circle of radius \( r_2 \) centered at \( M \). Let \( l \) and \( j \) be the two common external tangent lines to the two circles. A circle centered at \( P \) with radius \( r_2 \) is externally tangent to circle \( N \) at the point at which... | 3 | math | 169 |
Given the complex number $z= \frac{2}{1-i}$ (where $i$ is the imaginary unit), find the conjugate of the complex number $z$. | 1 - i | math | 36 |
How many four-character license plates consist of a consonant, followed by a vowel, followed by a consonant, and then a digit? (For this problem, consider Y a vowel.) | 24{,}000 | math | 38 |
Given that the terminal side of angle $α$ passes through point $P$ with coordinates $\left(\sin \frac{2π}{3},\cos \frac{2π}{3}\right)$, determine the smallest positive value of angle $α$. | \dfrac{11π}{6} | math | 53 |
Calculate the limit of the function:
$$\lim _{x \rightarrow \frac{\pi}{2}}(\sin x)^{\frac{18 \sin x}{\operatorname{ctg} x}}$$ | 1 | math | 45 |
A sequence of positive integers is formed by removing all perfect squares from the sequence $1, 2, 3, ...$. What is the $2018^{\text{th}}$ term in this new sequence? | 2063 | math | 47 |
Given vectors $\overrightarrow {a}$ = (cos($\frac {π}{3}$ - x), -sin(x)) and $\overrightarrow {b}$ = (sin(x + $\frac {π}{6}$), sin(x)), and a function f(x) = $\overrightarrow {a}$ • $\overrightarrow {b}$:
(I) Find the smallest positive period and the monotonically decreasing interval of function f(x);
(II) In triangle ... | \sqrt {3} | math | 136 |
The sequence $\{a_{n}\}$ satisfies: $a_{1}=\frac{1}{4}$, $a_{2}=\frac{1}{5}$, and $a_{1}a_{2} + a_{2}a_{3} + \cdots + a_{n}a_{n+1} = n a_{1}a_{n+1}$ for any positive integer $n$. Then, find the value of $\frac{1}{a_{1}} + \frac{1}{a_{2}} + \cdots + \frac{1}{a_{97}}$. | 5044 | math | 131 |
Find the derivative.
\[ y = \cos (\operatorname{ctg} 2) - \frac{1}{16} \cdot \frac{\cos^2 (8x)}{\sin (16x)} \] | \frac{1}{4 \sin^2(8x)} | math | 49 |
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with eccentricity $e = \sqrt{3}$, and $b = \sqrt{2}$.
(I) Find the equation of the hyperbola $C$;
(II) If $P$ is a point on hyperbola $C$ with left and right foci $E$ and $F$, respectively, and $\overrightarrow{PE} \cdot \overrighta... | 2 | math | 142 |
Given the letters in the word BANANA, find the number of distinguishable rearrangements in which all the vowels are at the end and in alphabetical order. | 3 | math | 33 |
A circle with a radius of 3 is centered at point $A$. An equilateral triangle with a side length of 6 units also has one vertex at point $A$. Determine the difference between the area of the region that lies inside the circle but outside of the triangle and the area of the region that lies inside the triangle but outsi... | 9\sqrt{3} - 9\pi | math | 118 |
Given that the derivative of the function $f(x)$ at $x=1$ is $1$, calculate $\lim_{x\to 0} \frac{f(1-x)-f(1+x)}{5x}$. | -\frac{2}{5} | math | 49 |
Given a moving circle passing through a fixed point F(2, 0) and tangent to the line x = -2, determine the trajectory of the center C of the moving circle. | y^2 = 8x | math | 38 |
What is the probability that when we roll five fair 6-sided dice, they won't all show the same number, and the numbers do not form a sequence? | \frac{7530}{7776} | math | 33 |
Given a regular quadrilateral prism \(A B C D - A_{1} B_{1} C_{1} D_{1}\) with a base edge length of 2 and a lateral edge length of 4, where \(E\) is the midpoint of edge \(CD\) and \(F\) is the midpoint of edge \(AA_1\), find the distance from point \(D\) to plane \(EFB_1\). | \frac{2\sqrt{17}}{17} | math | 90 |
The function $y=f(x-1)$ is an odd function, and $y=f(x+1)$ is an even function (both defined on $\mathbb{R}$). If $0 \leq x < 1$, then $f(x) = 2^x$. Find the value of $f(10)$. | 1 | math | 70 |
(2009) Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(2+x) = f(2-x)$. If the equation $f(x) = 0$ has exactly three distinct real roots, and one of them is 0, then the other two roots of the equation $f(x) = 0$ are ____. | 2, 4 | math | 80 |
Let \(a\), \(b\), and \(c\) be the lengths of the three sides of a triangle. Suppose \(a\) and \(b\) are the roots of the equation
\[
x^2 + 4(c + 2) = (c + 4)x,
\]
and the largest angle of the triangle is \(x^\circ\). Find the value of \(x\). | 90 | math | 84 |
Determine the values of $c$ for which $3$ is not in the range of the function $g(x) = x^2 + cx + 4$. | (-2, 2) | math | 35 |
Given an odd function $y=f\left(x\right)$ defined on $R$, which is strictly decreasing on the interval $\left[0,+\infty \right)$. If for any $x\in R$, we always have $f(kx^{2}+2)+f\left(kx+k\right)\leqslant 0$ holds, then the range of real number $k$ is ______. | [0,+\infty) | math | 89 |
Let $a = \pi/2008$. Find the smallest positive integer $n$ such that\[2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]\]is an integer.
| 251 | math | 77 |
Given the function $f(x)=a\ln x$ ($a\in \mathbb{R}$).
- (I) If the function $g(x)=2x+f(x)$ has a minimum value of $0$, find the value of $a$;
- (II) Let $h(x)=f(x)+ax^{2}+(a^{2}+2)x$, find the monotonic intervals of the function $h(x)$;
- (III) Suppose the function $y=f(x)$ and the function $u(x)= \frac {x-1}{2x}$ have... | \frac {1}{2} | math | 161 |
Given a hexagon with certain angles given as $135^\circ$, $105^\circ$, $87^\circ$, $120^\circ$, and $78^\circ$, what is the measure of the unknown angle $Q$? | 195^\circ | math | 55 |
Given a pasture has 10 cows, which were infected due to accidentally consuming feed containing a virus, and the incidence rate of the disease is 0.02, let the number of cows that fall ill be denoted by ξ, then calculate Dξ. | 0.196 | math | 55 |
Compute the definite integral:
$$
\int_{1 / 24}^{1 / 3} \frac{5 \sqrt{x+1}}{(x+1)^{2} \sqrt{x}} \, dx
$$ | 3 | math | 50 |
Given $x = \dfrac{1+i\sqrt{3}}{2}$, where $i = \sqrt{-1}$, calculate $\dfrac{1}{x^2 + x}$. | \frac{-i\sqrt{3}}{3} | math | 43 |
A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn. | 76 | math | 71 |
A book has 525 pages. Aunt Wang plans to read 25 pages every day, and it will take her days to finish. If she reads 21 pages every day, it will take her days to finish. | 21, 25 | math | 54 |
$(1)$ If $x-y=3$ and $xy=2$, find the value of $x^{2}+y^{2}$;<br/>$(2)$ If $a$ satisfies $\left(4-a\right)^{2}+\left(a+3\right)^{2}=7$, find the value of $\left(4-a\right)\left(a+3\right)$. | 21 | math | 85 |
Given $a$ is a real number, and $a^3 + 3a^2 + 3a + 2 = 0$, find the value of $(a+1)^{2008} + (a+1)^{2009} + (a+1)^{2010}$. | 1 | math | 71 |
Given a geometric sequence $\{a_n\}$ satisfies $a_1 + a_2 = 3$ and $a_2 + a_3 = 6$, calculate the value of $a_7$. | 64 | math | 45 |
Given the complex number $\frac{2-bi}{1+2i}$, if its real part and imaginary part are additive inverses of each other, then the real number $b$ is equal to ( ). | -\frac{2}{3} | math | 45 |
A person starts from a certain point, moves forward 20 meters, then turns 30 degrees to the right, moves forward another 20 meters, turns 30 degrees to the right again, and continues this pattern. How many meters has the person walked in total by the time they return to the starting point? | 240 \text{ meters} | math | 67 |
Given points $P(-3, 4)$ and $Q(5, y)$ in a coordinate plane, for what value of $y$ is the slope of the line through $P$ and $Q$ equal to $\frac{1}{2}$? Also, find the coordinates of the midpoint of line segment $PQ$. | (1, 6) | math | 69 |
Given the product of the digits of a 3-digit positive integer equals 30, find the number of such integers. | 12 | math | 25 |
In acute triangle $\triangle ABC$, $\sin A = \sin^2 B + \sin\left( \frac{\pi}{4} + B \right)\sin\left( \frac{\pi}{4} - B \right)$.
(1) Find the value of angle $A$;
(2) If $\overrightarrow{AB} \cdot \overrightarrow{AC} = 12$, find the area of $\triangle ABC$. | 2\sqrt{3} | math | 96 |
Given the function $f(x)=\sin (2x- \frac {\pi}{6})+2\cos ^{2}x-1$.
(I) Find the intervals of increase for the function $f(x)$.
(II) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ respectively, and $a=1$, $b+c=2$, $f(A)= \frac {1}{2}$, find the area of $\triangle ABC$. | \frac { \sqrt {3}}{4} | math | 116 |
Find the one-millionth digit after the decimal point in the decimal representation of the fraction \(3 / 41\). | 7 | math | 25 |
How many $n$-digit numbers can be written using the digits 1 and 2, where no two adjacent digits are both 1? Let the count of such $n$-digit numbers be denoted by $f(n)$. Find $f(10)$. | 144 | math | 58 |
There are 20 cards numbered $1, 2, \cdots, 19, 20$. These cards are placed in a box, and 4 people each draw one card from the box. The two people who draw the two smaller numbers form one group, and the two people who draw the two larger numbers form another group. If two of the people draw the numbers 5 and 14, what i... | \frac{7}{51} | math | 105 |
Given the function $f(x) = x^3 - 6x + 5, x \in \mathbb{R}$.
(1) Find the equation of the tangent line to the function $f(x)$ at $x = 1$;
(2) Find the extreme values of $f(x)$ in the interval $[-2, 2]$. | 5 - 4\sqrt{2} | math | 77 |
The solution set of the inequality $(x+3)(6-x) \geq 0$ is to be determined. | [-3,6] | math | 25 |
Find $546_{8} - 321_{8} - 105_{8}$. Express your answer in base $8$. | 120_8 | math | 33 |
Determine all pairs of non-negative integers \((n, k)\) such that
\[
2023 + 2^n = k^2
\] | (1, 45) | math | 35 |
The function $g$ is given by the table
\[\begin{tabular}{|c||c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline g(x) & 2 & 5 & 3 & 0 & 1 & 4 \\ \hline \end{tabular}\]
If $v_0=0$ and $v_{n+1} = g(v_n)$ for $n \ge 0$, find $v_{2010}$.
A) 0
B) 1
C) 2
D) 3
E) 4 | \text{(A) } 0 | math | 160 |
In the trapezoid \(ABCD\), the side \(CD\) is equal to the diagonal \(AC\). On the smaller arc \(BC\) of the circumcircle of triangle \(BCD\), a point \(E\) is chosen such that \(CD = CE\). Find the angle \(\angle AEB\). | 90^\circ | math | 68 |
Given $g(x)=mx+2$ and $f(x)=x^{2}-2x$, if for all $x\_1\in[-1,2]$ there exists an $x\_0\in[-1,2]$ such that $g(x\_1)=f(x\_0)$ holds true, then the range of values for $m$ is _____. | [-1, \frac{1}{2}] | math | 77 |
Two circles with radii 2 and 6, centered at points \( O_{1} \) and \( O_{2} \) respectively, are externally tangent at point \( C \). A common external tangent and a common internal tangent to the circles are drawn. These tangents intersect at point \( D \). Find the radius of the inscribed circle in triangle \( O_{1} ... | 2(\sqrt{3} - 1) | math | 87 |
The function $f(x)=2\sin (\omega x+ \frac {\pi}{6})$ ($\omega > 0$) is monotonically increasing on the interval $( \frac {\pi}{2},\pi)$. Determine the range of $\omega$. | (0, \frac {1}{3}] | math | 56 |
The difference between the maximum and minimum values of the function $y=2\sin \left( \frac{\pi x}{6}-\frac{\pi }{3} \right)(0\leqslant x\leqslant 9)$ is ______. | 2+ \sqrt{3} | math | 57 |
Let a cube have a side length of $a$, with two parallel square faces labeled $A B C D$ and $E F G H$. Let $M$ be the center of the face $E F G H$. How far is the line $M A$ from the line $B C$? | \frac{2\sqrt{5}}{5} a | math | 63 |
When selecting the first trial point using the 0.618 method during the process, if the experimental interval is $[2000, 3000]$, the first trial point $x_1$ should be chosen at ______. | 2618 | math | 53 |
Given that $\overrightarrow {a}$, $\overrightarrow {b}$, $\overrightarrow {c}$ are three vectors in the same plane, where $\overrightarrow {a} = \left(1, \sqrt {3}\right)$.
(I) If $|\overrightarrow {c}| = 4$ and $\overrightarrow {c}$ is parallel to $\overrightarrow {a}$, find the coordinates of $\overrightarrow {c}$.
... | \dfrac{\pi}{3} | math | 172 |
In the sequence $\{a_n\}$, $a_1=1$, $a_{n+1}=a_n+2^n$ ($n\in\mathbb{N}^+$), then its general formula is. | a_n=2^n-1 | math | 49 |
Use the Horner's method (also known as Qin Jiushao algorithm) to compute the value of the polynomial: $f(x) = 2x^6 + 3x^5 + 5x^3 + 6x^2 + 7x + 1$ when $x = 0.5$. Determine the number of multiplication and addition operations required. | 6 | math | 79 |
A family's telephone has the following probabilities of being answered: 0.1 for the first ring, 0.2 for the second ring, and 0.25 for both the third and fourth rings. Calculate the probability that the phone is answered before the fifth ring. | 0.8 | math | 57 |
Suppose $b$ is an integer such that $0 \le b \le 20$, and $74639281_{85} - b$ is a multiple of $17$. What is $b$? | 1 | math | 52 |
Given the following propositions:
① "a > b" is a sufficient but not necessary condition for "a<sup>2</sup> > b<sup>2</sup>";
② "lga = lgb" is a necessary but not sufficient condition for "a = b";
③ If x, y ∈ R, then "|x| = |y|" is a necessary and sufficient condition for "x<sup>2</sup> = y<sup>2</sup>";
④ In △A... | ③④ | math | 152 |
Given that the probability of a severe flood occurring within 30 years is 0.8, the probability of occurring within 40 years is 0.85, and 30 years have passed without a severe flood in this region, calculate the probability of a severe flood occurring in this region in the next 10 years. | 0.25 | math | 70 |
Suppose $A$, $B$, and $C$ are sets such that $|A| = 7$, $|B| = 7$, and $|C| = 6$. If $n(A) + n(B) + n(C) = n(A \cup B \cup C)$, find the minimum possible value of $|A \cap B \cap C|$. | 1 | math | 82 |
Find all pairs of prime numbers $(p, q)$ such that $pq \mid \left(p^{p} + q^{q} + 1\right)$. | (2, 5) \text{ or } (5, 2) | math | 35 |
The sequence $\left\{a_{n}\right\}$ is defined by the conditions $a_{1}=1$ and $a_{n}=a_{1}+a_{2}+\ldots+a_{n-1}+n$ for $n \geqslant 2$. Find the explicit formula for this sequence. | a_n = 2^n - 1 | math | 71 |
Five plastic bottles are required to make a new bottle. How many new bottles can eventually be made from 625 plastic bottles, assuming that at least three bottles are needed to start a recycling process? Do not include the original 625 bottles in your count. | 156 | math | 55 |
Convert $1024_{10}$ to base 8. | 2000_8 | math | 15 |
Solve the equation \( x'' - 5x' + 4x = 4 \) with initial conditions \( x(0) = 0 \) and \( x'(0) = 2 \). | x(t) = 1 - 2e^t + e^{4t} | math | 45 |
We are fitting new tires on both wheels of a motorcycle. A tire is considered completely worn out if it has run $15000 \mathrm{~km}$ on the rear wheel or $25000 \mathrm{~km}$ on the front wheel. What is the maximum distance the motorcycle can travel until the tires are completely worn out, if we exchange the front tire... | 18750 \text{ km} | math | 89 |
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