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76,993 | Task 10.
Find all natural numbers $n$ such that the sum $S(n)$ of the digits in the decimal representation of the number $2^{n}$ is 5. | 5 |
77,006 | ## Task 4 - 320734
Determine the number of all six-digit natural numbers that are divisible by 5 and whose cross sum is divisible by 9! | 20000 |
77,012 | Given a sequence $\{a_{n}\}$ that satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$, then the sum of the first $40$ terms of $\{a_{n}\}$ is ______. | 820 |
77,015 | Xiao Yan spent 185 yuan on a set of clothes. It is known that the price of the shirt is 5 yuan more than twice the price of the trousers. How much does each cost? | 60 |
77,042 | If the function $f(x) = \log_{\text{10}} \frac{ax + 1}{1 - 2x}$ is an odd function, find the value of the real number $a$. | 2 |
77,044 | Example 27 (20th Nordic Mathematical Contest) Given a sequence of positive integers $\left\{a_{n}\right\}$ satisfying $a_{0}=m, a_{n+1}=$ $a_{n}^{5}+487(n \geqslant 0)$. Find the value of $m$ such that the number of perfect squares in $\left\{a_{n}\right\}$ is maximized.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 9 |
77,062 | (Full score: 8 points)
During the 2010 Shanghai World Expo, there were as many as 11 types of admission tickets. Among them, the price for a "specified day regular ticket" was 200 yuan per ticket, and the price for a "specified day concession ticket" was 120 yuan per ticket. A ticket sales point sold a total of 1200 tickets of these two types on the opening day, May 1st, generating a revenue of 216,000 yuan. How many tickets of each type were sold by this sales point on that day? | 300 |
77,069 | Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 12$ and $X$ is an integer, what is the smallest possible value of $X$? | 925 |
77,108 | Given the power function $f(x) = (a-1)x^k$ whose graph passes through the point $(\sqrt{2}, 2)$, find the value of the real number $a+k$. | 4 |
77,246 | G6.1 If $\log _{2} a-2 \log _{a} 2=1$, find $a$. | 4 |
77,248 | Example. Divide 20 people into 4 groups, with each group having $4,5,5,6$ people, how many ways are there to do this? | 4888643760 |
77,310 | 2. Two math teachers are conducting a geometry test, checking each 10th-grade student's ability to solve problems and their knowledge of theory. The first teacher spends 5 and 7 minutes per student, respectively, while the second teacher spends 3 and 4 minutes per student. What is the minimum time required for them to interview 25 students?
# | 110 |
77,317 | For how many positive numbers less than 1000 is it true that among the numbers $2,3,4,5,6,7,8$ and 9 there is exactly one that is not its divisor?
(E. Semerádová) | 4 |
77,324 | A machine can operate at different speeds, and some of the products it produces may have defects. The number of defective products produced per hour varies with the speed of the machine. Let $x$ denote the speed of the machine (in revolutions per second), and let $y$ denote the number of defective products produced per hour. Four observed values of $(x, y)$ are $(8,5)$, $(12,8)$, $(14,9)$, and $(16,11)$. It is known that $y$ has a strong linear correlation with $x$. If the actual production process allows for no more than 10 defective products per hour, what should be the maximum speed of the machine in revolutions per second? (Round to the nearest integer)
Reference formula:
If $(x\_1, y\_1), ..., (x\_n, y\_n)$ are sample points, then $\hat{y} = \hat{b}x + \hat{a}$,
$\overline{x} = \frac{1}{n} \sum\_{i=1}^{n} x\_i$, $\overline{y} = \frac{1}{n} \sum\_{i=1}^{n} y\_i$, $\hat{b} = \frac{\sum\_{i=1}^{n} (x\_i - \overline{x}) (y\_i - \overline{y})}{\sum\_{i=1}^{n} (x\_i - \overline{x})^2} = \frac{\sum\_{i=1}^{n} x\_i y\_i - n \overline{x} \overline{y}}{\sum\_{i=1}^{n} x\_i^2 - n \overline{x}^2}$, $\hat{a} = \overline{y} - \hat{b} \overline{x}$. | 15 |
77,356 | A spider has 8 identical socks and 8 identical shoes. In how many different orders can the spider dress, knowing that, obviously, on each leg, she must put the shoe on after the sock? | 81729648000 |
77,365 | The cafe "Burattino" operates 6 days a week with a day off on Mondays. Kolya made two statements: "from April 1 to April 20, the cafe worked 18 days" and "from April 10 to April 30, the cafe also worked 18 days." It is known that he made a mistake once. How many days did the cafe work from April 1 to April 27? | 23 |
77,400 | A circular coin with a circumference of about 5 cm starts rolling from a point on the boundary of a quadrilateral with a circumference of 20 cm. After making one complete rotation and returning to the starting point, the center of the coin will trace a closed curve. The length of this curve is ____ cm. | 25 |
77,449 | Given that $F_1$ and $F_2$ are the two foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and a line passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the sum of the distances from $F_2$ to $A$ and from $F_2$ to $B$ is $12$, find the length of the segment $AB$. | 8 |
77,503 | Amongst the seven numbers \( 3624, 36024, 360924, 3609924, 36099924, 360999924, \) and \( 3609999924 \), there are \( n \) of them that are divisible by 38. Find the value of \( n \). | 6 |
77,524 | # Task 5. 20 points
A beginner economist-cryptographer received a cryptogram from the ruler, which contained another secret decree on the introduction of a commodity tax on a certain market. The cryptogram specified the amount of tax revenue to be collected. It was also emphasized that a larger amount of tax revenue could not be collected on this market. Unfortunately, the economist-cryptographer decrypted the cryptogram with an error - the digits in the tax revenue amount were determined in the wrong order. Based on the incorrect data, a decision was made to introduce a per-unit tax on the producer in the amount of 90 monetary units per unit of the product. It is known that the market demand is given by $Q_d = 688 - 4P$, and the market supply is linear. Additionally, it is known that a change in price by one unit results in a change in the quantity demanded that is 1.5 times smaller than the change in the quantity supplied. After the tax was introduced, the producer's price decreased to 64 monetary units.
1) Restore the market supply function.
2) Determine the amount of tax revenue collected at the chosen rate.
3) Determine the rate of the quantity tax that would allow the ruler's decree to be fulfilled.
4) What are the tax revenues that the ruler indicated to collect? | 8640 |
77,536 | ## 219. Math Puzzle $8 / 83$
After evaluating satellite images from space, Soviet geologists have discovered a new deposit of 300 million tons of copper ore. It has a metal content of 5 percent. Every year, 10 percent of the reserves are mined.
How many tons of pure copper can be produced in total over 3 years? | 4.065 |
77,588 | Given $a\in \mathbb{R}$, $b\in \mathbb{R}$, if the set $\{a, \frac{b}{a}, 1\} = \{a^2, a+b, 0\}$, then the value of $a^{2023}+b^{2023}$ is ______. | -1 |
77,615 | If \( a \) is the remainder when \( 2614303940317 \) is divided by 13, find \( a \). | 4 |
77,641 | Given the digits 1, 2, 3, 4, 5, how many distinct four-digit even numbers can be formed such that the digits 4 and 5 are adjacent to each other? | 14 |
77,642 | Given that there are two distinct points on the circle $C$: $x^{2}+y^{2}+mx-4=0$ that are symmetric with respect to the line $x-y+4=0$, find the value of the real number $m$. | 8 |
77,650 | Given that point $P$ moves on $y=\sqrt{{x}^{2}+1},x∈[-1,\sqrt{3}]$, and point $Q$ moves on the circle $C:{x}^{2}+(y-a)^{2}=\frac{3}{4}(a＞0)$, and $|PQ|$ has a minimum value of $\frac{3}{2}\sqrt{3}$, then the value of the real number $a$ is ______. | 5 |
77,668 | A middle school is preparing to form an 18-person soccer team. These 18 people will be selected from students in the 10 classes of the first grade. Each class must have at least one student on the team. How many different ways can the team be composed? | 24310 |
77,754 | In a Cartesian coordinate system, the points where both the x-coordinate and y-coordinate are integers are called lattice points. How many lattice points (x, y) satisfy the inequality \((|x|-1)^{2}+(|y|-1)^{2}<2\)? | 16 |
77,763 | A polygon has $n$ sides, and the sum of all its angles except one is $2190^\circ$. What is the value of $n$? | 15 |
77,800 | Calculate the sum of the square of the binomial coefficients: $C_2^2+C_3^2+C_4^2+…+C_{11}^2$. | 220 |
77,818 | Example 4 Find the last three digits of $7^{10000}$ and $7^{9999}$ | 143 |
77,819 | Determine the number of angles \(\theta\) between 0 and \(2\pi\), other than integer multiples of \(\pi / 2\), such that the quantities \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) form a geometric sequence in some order. | 4 |
77,868 | There are 211 students and four different types of chocolates, with each type having more than 633 pieces. Each student can take up to three chocolates or none at all. If the students are grouped according to the type and number of chocolates they take, what is the minimum number of students in the largest group? | 7 |
77,915 | 5. Given that $f(x)$ is a function defined on $\mathbf{R}$, $f(1)=1$ and for any $x \in \mathbf{R}$, we have $f(x+5) \geqslant f(x)+5$, $f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)=$ $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 1 |
77,960 | Given that $F(\sqrt{6},0)$ is the right focus of the ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$, the line $l$ passing through point $F$ intersects the ellipse $C$ at points $A$ and $B$, with $P$ being the midpoint of $AB$ and $O$ being the origin. If $\triangle OFP$ is an isosceles triangle with $OF$ as the base and the circumcircle of $\triangle OFP$ has an area of $2\pi$, then the length of the major axis of the ellipse $C$ is ______. | 6 |
77,961 | Five friends - Kristina, Nadya, Marina, Liza, and Galya - gather in the park every day after buying ice cream from the shop around the corner. One day, they had a conversation.
Kristina: There were five people in front of me.
Marina: I was the first in line!
Liza: There was no one after me.
Nadya: I was standing next to Marina.
Galya: There was only one person after me.
The girls are friends, so they do not lie to each other. How many people were between Kristina and Nadya? | 3 |
78,033 | One movie is 1 hour and 48 minutes long. A second movie is 25 minutes longer than the first. How long is the second movie?
(A) 2 hours and 13 minutes
(B) 1 hour and 48 minutes
(C) 2 hours and 25 minutes
(D) 2 hours and 3 minutes
(E) 2 hours and 48 minutes | 2 |
78,087 | Let \( a \) be an integer. If the inequality \( |x+1| < a - 1.5 \) has no integral solution, find the greatest value of \( a \). | 1 |
78,102 | 70 kg 50 g = kg
3.7 hours = hours minutes. | 42 |
78,107 | Let $a+a_{1}(x+2)+a_{2}(x+2)^{2}+\ldots+a_{12}(x+12)^{12}=(x^{2}-2x-2)^{6}$, where $a_{i}$ are constants. Find the value of $2a_{2}+6a_{3}+12a_{4}+20a_{5}+\ldots+132a_{12}$. | 492 |
78,108 | Let \(a, b, c, d\) be positive integers such that the least common multiple (L.C.M.) of any three of them is \(3^{3} \times 7^{5}\). How many different sets of \((a, b, c, d)\) are possible if the order of the numbers is taken into consideration? | 11457 |
78,188 | Huahua is writing letters to Yuanyuan with a pen. When she finishes the 3rd pen refill, she is working on the 4th letter; when she finishes the 5th letter, the 4th pen refill is not yet used up. If Huahua uses the same amount of ink for each letter, how many pen refills does she need to write 16 letters? | 13 |
78,232 | A unit arranges for 4 people to take night shifts from Monday to Saturday, with each person working at least one day and at most two days. Those who work for two days must work on two consecutive days. The number of different night shift arrangements is (answer in digits). | 144 |
78,301 | Let $P_{1}: y=x^{2}+\frac{101}{100}$ and $P_{2}: x=y^{2}+\frac{45}{4}$ be two parabolas in the Cartesian plane. Let $\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$, find $a+b+c$. | 11 |
78,326 | In the same Cartesian coordinate system, the number of intersection points between the graph of the function $y=\sin \left( x+\frac{\pi}{3} \right)$ $(x\in[0,2\pi))$ and the graph of the line $y=\frac{1}{2}$ is \_\_\_\_. | 2 |
78,363 | Given functions $f(x)=ae^{x}-ax$ and $g(x)=(x-1)e^{x}+x$, let the derivatives of $f(x)$ and $g(x)$ be denoted as $f'(x)$ and $g'(x)$, respectively. If there exists $t \gt 0$ such that $f'(t)=g'(t)$, then the smallest integer value of the real number $a$ is ____. | 3 |
78,427 | Every day from Monday to Friday, an old man went to the blue sea and cast his net. Each day, the number of fish caught in the net was not greater than the number caught the previous day. Over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three days - Monday, Wednesday, and Friday? | 50 |
78,428 | A certain flower shop plans to purchase a batch of carnations and lilies. It is known that the price of each carnation and lily is $5$ yuan and $10$ yuan, respectively. If the flower shop is preparing to purchase a total of $300$ carnations and lilies, and the number of carnations is no more than twice the number of lilies, please design the most cost-effective purchasing plan and explain the reason. | 100 |
78,468 | Observe the number array below, the 20th number in the 20th row is ___.
1
2 3 4
5 6 7 8 9
11 12 13 14 15 16
18 19 20 21 22 23 24 25
… … … … … … … … … | 381 |
78,476 | A digit was removed from a five-digit number, and the resulting four-digit number was added to the original number. The sum turned out to be 54321. Find the original number. | 49383 |
78,495 | Given that \( a + b + c = 5 \),
\[ a^2 + b^2 + c^2 = 15, \quad a^3 + b^3 + c^3 = 47. \]
Find the value of \(\left(a^2 + ab + b^2\right)\left(b^2 + bc + c^2\right)\left(c^2 + ca + a^2\right)\). | 625 |
78,614 | Given the following four propositions:
① The function $f(x)$ is increasing when $x>0$ and also increasing when $x<0$, so $f(x)$ is an increasing function;
② The graph of a direct proportion function must pass through the origin of the Cartesian coordinate system;
③ If the domain of the function $f(x)$ is $[0,2]$, then the domain of the function $f(2x)$ is $[1,2]$;
④ The increasing interval of $y=x^2-2|x|-3$ is $[1,+\infty)$.
Among these, the correct proposition numbers are. (Fill in all the correct proposition numbers) | 2 |
78,628 | Given two positive integers \( m \) and \( n \) satisfying \( 1 \leq m \leq n \leq 40 \), find the number of pairs \( (m, n) \) such that their product \( mn \) is divisible by 33. | 64 |
78,709 | Given $a_1 + a_2 = 1$, $a_2 + a_3 = 2$, $a_3 + a_4 = 3$, ..., $a_{99} + a_{100} = 99$, $a_{100} + a_1 = 100$, find the value of $a_1 + a_2 + a_3 + \ldots + a_{100}$. | 2525 |
78,713 | Given a geometric sequence $\{a_n\}$ with a common ratio $q$, and $\frac {S_{3}}{a_{3}}=3$, then the common ratio $q=$ \_\_\_\_\_\_. | 1 |
78,746 | (Inequality Optional Question)
Given that $a$, $b$, $m$, $n$ are positive numbers, and $a+b=1$, $mn=2$, find the minimum value of $(am+bn)(bm+an)$. | 2 |
78,757 | Given any real numbers \( x \) and \( y \), the function \( f(x) \) satisfies \( f(x) + f(y) = f(x + y) + xy \).
If \( f(1) = m \), how many positive integer pairs \( (m, n) \) satisfy \( f(n) = 2019 \)? | 8 |
78,780 | Given triangle $\triangle ABC$, where the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $a=1$, $c=\sqrt{3}$, and $\angle A=30^\circ$. Find the value of $b$. | 2 |
78,820 | Find the value of $\frac{{\tan 20° + 4\sin 20°}}{{\tan 30°}}$. | 3 |
78,875 | 40. Player A and Player B are playing a game, with 28 pieces of chess, the two players take turns to take the pieces, each time only allowed to take 2, 4, or 8 pieces. The one who takes the last piece wins. If A goes first, then A should take $\qquad$ pieces on the first turn to ensure a win. | 4 |
78,893 | Given $t$ is a constant, the function $y=|x^2-2x-t|$ has a maximum value of 2 on the interval $[0, 3]$. Find the value of $t$. | 1 |
78,906 | Find the smallest natural number \( n \) such that the equation \(\left[\frac{10^{n}}{x}\right]=1989\) has an integer solution \( x \).
(The 23rd All-Soviet Union Math Olympiad, 1989) | 7 |
78,927 | Given that $a$ and $b$ are opposite numbers, $m$ and $n$ are reciprocal numbers ($m-n\neq 0$), and the absolute value of $x$ is $2$, find the value of $-2mn+\frac{b+a}{m-n}-x$. | 0 |
78,937 | Compute the sum of all integers \(1 \leq a \leq 10\) with the following property: there exist integers \(p\) and \(q\) such that \(p, q, p^{2}+a\) and \(q^{2}+a\) are all distinct prime numbers. | 20 |
78,953 | In the Cartesian coordinate system $xOy$, it is known that point $A(x_{1},y_{1})$ is on the curve $C_{1}: y=x^{2}-\ln x$, and point $B(x_{2},y_{2})$ is on the line $x-y-2=0$. The minimum value of $(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}$ is \_\_\_\_\_\_. | 2 |
78,983 | Example 2 In space, there is a convex polyhedron, all of whose vertices are lattice points (each vertex has three integer coordinates). In addition, there are no other integer points inside the polyhedron, on its faces, or on its edges. What is the maximum number of vertices this convex polyhedron can have? | 8 |
78,994 | In a middle school math competition, there were three problems labeled A, B, and C. Among the 25 students who participated, each student solved at least one problem. Among the students who did not solve problem A, the number who solved problem B was twice the number of those who solved problem C. The number of students who solved only problem A was one more than the number of students who solved A along with any other problem. Among the students who solved only one problem, half did not solve problem A. How many students solved only problem B? | 6 |
79,063 | Find the number of all integer solutions of the inequality \( \sqrt{1-\sin \frac{\pi x}{4}-3 \cos \frac{\pi x}{2}}-\sqrt{6} \cdot \sin \frac{\pi x}{4} \geq 0 \) that belong to the interval [1991; 2013]. | 8 |
79,087 | 11. Xiao Hong showed Da Bai the following equations:
$$
\begin{array}{c}
1 \times 2+2 \times 3=2 \times 2 \times 2 \\
2 \times 3+3 \times 4=2 \times 3 \times 3 \\
3 \times 4+4 \times 5=2 \times 4 \times 4 \\
4 \times 5+5 \times 6=2 \times 5 \times 5
\end{array}
$$
Then Xiao Hong asked Da Bai to calculate the following equation:
$$
57 \times 168+58 \times 171-25 \times 14 \times 15-25 \times 15 \times 16=
$$
$\qquad$
Do you know what the answer is? | 8244 |
79,095 | Given $f\left(x\right)=(e^{2x}-e^{ax})\cos x$ is an odd function, find $a=\_\_\_\_\_\_.$ | -2 |
79,099 | A'Niu is riding a horse to cross a river. There are four horses named A, B, C, and D. It takes 2 minutes for horse A to cross the river, 3 minutes for horse B, 7 minutes for horse C, and 6 minutes for horse D. Only two horses can be driven across the river at a time. The question is: what is the minimum number of minutes required to get all four horses across the river? | 18 |
79,147 | Given that $F$ is the left focus of the hyperbola $\frac{x^2}{4} - \frac{y^2}{12} = 1$, and point $A(1,4)$ is given. Let $P$ be a point on the right branch of the hyperbola. Find the minimum value of $|PF|+|PA|$. | 9 |
79,171 | Find the number of real roots of the equation \(\log_{10}^2 x - \left\lfloor \log_{10} x \right\rfloor - 2 = 0\). (Note: \(\lfloor x \rfloor\) denotes the greatest integer not exceeding \(x\).) | 3 |
79,173 | Suppose that $A$ and $B$ are digits in base $d > 6$ such that $\overline{AB}_d + \overline{AA}_d = 162_d$. Find $A_d - B_d$ in base $d$. | 3 |
79,205 | Eliane wants to choose her schedule for swimming. She wants to attend two classes per week, one in the morning and one in the afternoon, not on the same day nor on consecutive days. In the morning, there are swimming classes from Monday to Saturday at 9 AM, 10 AM, and 11 AM, and in the afternoon, from Monday to Friday at 5 PM and 6 PM. How many different ways can Eliane choose her schedule? | 96 |
79,235 | Petya and Masha made apple juice. They produced a total of 10 liters. They poured it into two bottles. However, Masha found it difficult to carry her bottle, so she transferred some juice to Petya's bottle. As a result, Petya ended up with three times more juice, and Masha had three times less. How many liters did Petya have to carry? | 7.5 |
79,249 | 2. [2] A parallelogram has 3 of its vertices at $(1,2),(3,8)$, and $(4,1)$. Compute the sum of all possible $x$ coordinates of the 4 th vertex. | 8 |
79,298 | Given a polygon drawn on graph paper with a perimeter of 2014 units, and whose sides follow the grid lines, what is the maximum possible area of this polygon? | 253512 |
79,344 | Given the complex number $z=(\cos \theta- \frac{4}{5})+(\sin \theta- \frac{3}{5})i$ (where $i$ is the imaginary unit) is purely imaginary, then $\tan (\theta- \frac{\pi}{4})=$ __________. | -7 |
79,355 | The scent of blooming lily of the valley bushes spreads within a radius of 20 meters around them. How many blooming lily of the valley bushes need to be planted along a straight 400-meter-long alley so that every point along the alley can smell the lily of the valley? | 10 |
79,466 | Simplify first, then evaluate: $\frac{4}{5}ab-[{2a{b^2}-4({-\frac{1}{5}ab+3{a^2}b})}]+2a{b^2}$, where $a=-1$ and $b=1$. | 12 |
79,488 | On December $17,$ $1903,$ at Kitty Hawk, N.C., the $1903$ Wright Flyer became the first powered, heavier-than-air machine to achieve controlled, sustained flight with a pilot aboard.
\begin{tabular}[t]{|l|c|c|c|}
\multicolumn{4}{c}{\textbf{December 17, 1903 Flights}}\\\hline
&\textbf{Pilot}&\textbf{Time in Flight}&\textbf{Distance}\\\hline
\textbf{First Flight}&Orville&$12$~seconds&$37$~meters\\\hline
\textbf{Longest Flight}&Wilbur&$59$~seconds&$260$~meters\\\hline
\end{tabular}
The average speed for the first flight was $x$ meters per second. The average speed for the longest flight was $y$ meters per second. What is the average of $x$ and $y?$ Express your answer as a decimal to the nearest tenth. | 3.7 |
79,504 | Given the sequence $a\_n = \frac{n-7}{n-5\sqrt{2}}\ (n\in\mathbb{N}^*)$, if $a\_m$ is the maximum term of the sequence, then $m =$ $\_\_\_\_\_\_$. | 8 |
79,538 | How many triples \((a, b, c)\) of positive integers satisfy the conditions \(6ab = c^2\) and \(a < b < c \leq 35\)? | 8 |
79,632 | Let $f(x)$ be a function defined on $\mathbb{R}$, which is an odd function and symmetric about $x=1$. When $x \in (-1, 0)$, $f(x) = 2^x + \frac{1}{5}$. Find the value of $f(\log_2 20)$. | -1 |
79,634 | Béla has three circular templates: these are suitable for drawing circles with areas of 6, 15, and $83 \mathrm{~cm}^{2}$ respectively. Béla wants to draw some circles, the total area of which should be $220 \mathrm{~cm}^{2}$. How many of each circle should he draw? | 2 |
79,718 | The integers represented by points on the number line that are less than $\sqrt{5}$ units away from the origin are _______. (Write one integer) | 0 |
79,735 | The coefficient of the $x^3$ term in the expansion of $(x^2-x+1)^{10}$ is ______. | -210 |
79,773 | In the arithmetic sequence \(\left(a_{n}\right)\) where \(a_{1}=1\) and \(d=4\),
Calculate
\[
A=\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots+\frac{1}{\sqrt{a_{1579}}+\sqrt{a_{1580}}}
\]
Report the smallest integer greater than \(A\). | 20 |
79,783 | Given \(a\) and \(b\) are real numbers, satisfying:
\[
\sqrt[3]{a} - \sqrt[3]{b} = 12, \quad ab = \left( \frac{a + b + 8}{6} \right)^3.
\]
Find \(a - b\). | 468 |
79,795 | There is a parking lot with $10$ empty spaces. Three different cars, A, B, and C, are going to park in such a way that each car has empty spaces on both sides, and car A must be parked between cars B and C. How many different parking arrangements are there? | 40 |
79,824 | The graph of the function $y=a^{x-4}+1$ always passes through a fixed point $P$, which is also on the graph of the power function $y=f(x)$. Determine the value of $f(16)$. | 4 |
79,909 | The number of sets $A$ that satisfy $\{0,1,2\} \subset A \subseteq \{0,1,2,3,4,5\}$ is ____. | 7 |
79,933 | Given that
$$
S=\left|\sqrt{x^{2}+4 x+5}-\sqrt{x^{2}+2 x+5}\right|,
$$
for real values of \(x\), find the maximum value of \(S^{4}\). | 4 |
79,947 | If \( a \) is the maximum value of \( \frac{1}{2} \sin ^{2} 3 \theta- \frac{1}{2} \cos 2 \theta \), find the value of \( a \). | 1 |
80,012 | Find the number of distinct arrangements in a row of all natural numbers from 1 to 10 such that the sum of any three consecutive numbers is divisible by 3. | 1728 |
80,024 | To test the quality of a certain product, it was decided to use the random number table method to draw 5 samples from 300 products for inspection. The products are numbered from 000, 001, 002, ..., to 299. The following are the 7th and 8th rows of the random number table. If we start from the 5th number in the 7th row and read to the right, the second sample number among the 5 obtained is:
7th row: 84 42 17 53 31 57 24 55 06 88 77 04 74 47 67 21 76 33 50 25 83 92 12 06 76
8th row: 63 01 63 78 59 16 95 55 67 19 98 10 50 71 75 12 86 73 58 07 44 39 52 38 79 | 057 |
80,040 | There are 111 balls in a box, each being red, green, blue, or white. It is known that if 100 balls are drawn, it ensures getting balls of all four colors. Find the smallest integer $N$ such that if $N$ balls are drawn, it can ensure getting balls of at least three different colors. | 88 |
80,076 | In an acute triangle $ABC$, if $\sin A = 2\sin B\sin C$, then the minimum value of $\tan A\tan B\tan C$ is ______. | 8 |
80,145 | Uncle Chernomor assigns 9 or 10 of his thirty-three bogatyrs to duty every evening. What is the minimum number of days required so that all the bogatyrs have been on duty the same number of times? | 7 |
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