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83,552 | From 2005 numbers, 20 numbers are to be sampled. If the systematic sampling method is used, what is the sampling interval? | 100 |
83,593 | Given that $P$ is a point on the parabola $y^2=4x$, let the distance from $P$ to the directrix be $d_1$, and the distance from $P$ to point $A(1, 4)$ be $d_2$. Find the minimum value of $d_1+d_2$. | 4 |
83,618 | A company has a total of 150 employees, among which there are 15 senior titles, 45 intermediate titles, and 90 junior titles. Now, stratified sampling is used to select 30 people. The number of people with senior, intermediate, and junior titles selected are respectively \_\_\_\_\_\_\_\_, \_\_\_\_\_\_\_\_, \_\_\_\_\_\_\_\_. | 18 |
83,659 | There is a house at the center of a circular field. From it, 6 straight roads radiate, dividing the field into 6 equal sectors. Two geologists start their journey from the house, each choosing a road at random and traveling at a speed of 4 km/h. Determine the probability that the distance between them after an hour is at least 6 km. | 0.5 |
83,716 | For a natural number $N$, if at least eight out of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called an "Eight Immortals Number." What is the smallest "Eight Immortals Number" greater than $2000$? | 2016 |
83,721 | 7. Determine the area of the polygon formed by the ordered pairs $(x, y)$ where $x$ and $y$ are positive integers which satisfy the equation
$$
\frac{1}{x}+\frac{1}{y}=\frac{1}{13} .
$$ | 12096 |
83,726 | A school's emergency telephone tree operates as follows: The principal calls two students. In the first level, there is only the principal. In the second level, two students are contacted. For each subsequent level, each student contacts two new students who have not yet been contacted. This process continues with each student from the previous level contacting two new students. After the 8th level, how many students in total have been contacted? | 254 |
83,751 | If the line $x + my + 6 = 0$ is parallel to the line $(m - 2)x + 3y + 2m = 0$, then the value of $m$ is $\_\_\_\_\_\_\_\_$. | -1 |
83,789 | Given the function
$$
f(x)=
\begin{cases}
x^2 - 1, & x \leq 0 \\
3x, & x > 0
\end{cases},
$$
if $f(x) = 15$, then $x = \_\_\_\_\_\_$. | 5 |
83,794 | Distribute 16 identical books among 4 classes, with each class receiving at least one book and each class receiving a different number of books. How many different distribution methods are there? (Answer in digits) | 216 |
83,852 | Example 30 (1995 National Training Team Selection Exam Question) Find the smallest prime $p$ that cannot be expressed as $\left|3^{a}-2^{b}\right|$, where $a$ and $b$ are non-negative integers. | 41 |
83,854 | How many triangles exist in which the measures of the angles, measured in degrees, are whole numbers? | 2700 |
83,855 | How many (infinite) arithmetic sequences of natural numbers are there, in which the ratio of the sum of the first $2 n$ terms to the sum of the first $n$ terms does not depend on the choice of $n$, and furthermore, one of the terms of the sequence is the number 1971? | 9 |
83,881 | Five students (including A, B, C) are arranged in a row. A must be adjacent to B, and A must not be adjacent to C. The number of different ways to arrange them is _____. (Provide your answer in numerical form) | 36 |
83,899 | At most, how many interior angles greater than $180^\circ$ can a 2006-sided polygon have? | 2003 |
83,957 | Given that the sum of the first $n$ terms ($S_n$) of an arithmetic sequence {$a_n$} has a maximum value, and $\frac{a_{15}}{a_{14}} < -1$, determine the maximum value of $n$ that makes $S_n > 0$. | 27 |
83,959 | Let \( S = \{1, 2, \cdots, 2005\} \). Find the minimum value of \( n \) such that any set of \( n \) pairwise coprime elements from \( S \) contains at least one prime number. | 16 |
83,962 | 10.196. Given a triangle $A B C$, in which $2 h_{c}=A B$ and $\angle A=75^{\circ}$. Find the measure of angle $C$. | 75 |
83,964 | A state issues license plates consisting of 6 digits (each digit ranging from 0 to 9) and stipulates that any two license plates must differ by at least two digits (thus, plate numbers 027592 and 020592 cannot be used simultaneously). Determine the maximum number of license plates that can be issued. | 100000 |
83,971 | I1.2 Given that $f(x)=-x^{2}+10 x+9$, and $2 \leq x \leq \frac{a}{9}$. If $b$ is the difference of the maximum and minimum values of $f$, find the value of $b$. | 9 |
84,012 | # Problem 10.
Set $A$ on the plane $O x y$ is defined by the equation $x^{2}+y^{2}=2 x+2 y+23$. Set $B$ on the same plane is defined by the equation $|x-1|+|y-1|=5$. Set $C$ is the intersection of sets $A$ and $B$. What is the maximum value that the product of the lengths of $n$ segments $X Y_{1} \cdot X Y_{2} \cdot X Y_{3} \cdot \ldots \cdot X Y_{n}$ can take, where point $X$ is an arbitrarily chosen point from set $A$, and points $Y_{1}, Y_{2}, Y_{3}, \ldots, Y_{n}$ are all elements of set $C$? | 1250 |
84,036 | The number of distinct tetrahedra that can be formed using the vertices of a triangular prism is ___. | 12 |
84,037 | ## Task 1
A bear can live to be 50 years old, a fox can live to be one fifth of that; a wolf can live 5 years longer than a fox.
How old can a wolf, how old can a fox become? | 10 |
84,043 | Given four positive integers \(a, b, c,\) and \(d\) satisfying the equations \(a^2 = c(d + 20)\) and \(b^2 = c(d - 18)\). Find the value of \(d\). | 180 |
84,044 | The solution to the fractional equation $\frac{a-2}{x+3}+1=\frac{3x}{3+x}$ in terms of $x$ is an integer, and the inequality system in terms of $y$ $\left\{\begin{array}{l}{\frac{1}{3}y+1≥\frac{y+3}{2}}\\{\frac{a+y}{2}＜y-1}\end{array}\right.$ has a solution with at most six integer solutions. The sum of all integer values of $a$ that satisfy the conditions is ______. | -20 |
84,130 | In a football tournament, seven teams played: each team played once with each other team. In the next round, teams that scored twelve or more points advance. A win gives 3 points, a draw gives 1 point, and a loss gives 0 points. What is the maximum number of teams that can advance to the next round? | 5 |
84,180 | Given a population consisting of 20 individuals labeled as $01$, $02$, $...$, $19$, $20$. Using the following random number table, select 5 individuals. The selection method is to start from the 5th and 6th columns of the 1st row in the random number table and sequentially select two digits from left to right. What is the label of the 5th selected individual?
$7816\ 6572\ 0802\ 6314\ 0702\ 4369\ 9728\ 0198$
$3204\ 9234\ 4935\ 8200\ 3623\ 4869\ 6938\ 7481$ | 01 |
84,188 | Convert the octal number $127_8$ to a binary number. | 1010111_2 |
84,205 | The Gnollish language consists of 3 words, ``splargh,'' ``glumph,'' and ``amr.'' In a sentence, ``splargh'' cannot come directly before ``glumph''; all other sentences are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Gnollish? | 21 |
84,208 | The famous Italian mathematician Fibonacci, while studying the problem of rabbit population growth, discovered a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ..., where starting from the third number, each number is equal to the sum of the two numbers preceding it. This sequence of numbers $\{a_n\}$ is known as the "Fibonacci sequence". Determine which term in the Fibonacci sequence is represented by $$\frac { a_{ 1 }^{ 2 }+ a_{ 2 }^{ 2 }+ a_{ 3 }^{ 2 }+…+ a_{ 2015 }^{ 2 }}{a_{2015}}$$. | 2016 |
84,245 | Given that the equation concerning $x$, $\frac{m-1}{x-1} - \frac{x}{x-1} = 0$, has a repeated root, find the value of $m$. | 2 |
84,270 | 21. [11] Let $f(n)$ be the number of distinct prime divisors of $n$ less than 6. Compute
$$
\sum_{n=1}^{2020} f(n)^{2} .
$$ | 3431 |
84,278 | Given that $a+b=-2$ and $b < 0$, find the value of $a$ that minimizes the expression $\frac{1}{2|a|} - \frac{|a|}{b}$. | 2 |
84,304 | In 1937, the German mathematician Collatz proposed a famous conjecture: for any positive integer $n$, if $n$ is even, divide it by $2$ (i.e., $\frac{n}{2}$); if $n$ is odd, multiply it by $3$ and add $1$ (i.e., $3n+1$). By repeating this operation, after a finite number of steps, you will always reach $1$. Currently, the Collatz conjecture cannot be proven or disproven. Now, please investigate: if the $8$th term after applying the rule to a positive integer $n$ (initial term) is $1$, then the number of different values of $n$ is ____. | 6 |
84,369 | A bag of fresh milk costs 3 yuan, and the supermarket has a buy 5 get 1 free offer. With 20 yuan, mom can buy at most bags of fresh milk and still have yuan left. | 2 |
84,555 | Let point $P$ be a point on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, and let $F_1$ and $F_2$ be the foci. If $\angle F_1PF_2$ is a right angle, then the area of $\triangle F_1PF_2$ is ______. | 16 |
84,565 | Add 3 digits after 325 to make a six-digit number such that it is divisible by 3, 4, and 5, and make this number as small as possible. What is the new six-digit number? | 325020 |
84,581 | Let $\left\{a_{n}\right\}$ be a sequence of positive integers that is monotonically increasing, i.e.,
$$
1 \leqq a_{1} \leqq a_{2} \leqq \ldots \leqq a_{n} \leqq \ldots
$$
For any integer $m \geqq 1$, let $b_{m}$ be the index of the first element in the sequence $\left\{a_{n}\right\}$ that is at least $m$.
If $a_{19}=88$, then within what range does the sum
$$
a_{1}+a_{2}+\ldots+a_{19}+b_{1}+b_{2}+\ldots+b_{88}
$$
lie? | 1760 |
84,604 | Four distinct natural numbers, one of which is an even prime number, have the following properties:
- The sum of any two numbers is a multiple of 2.
- The sum of any three numbers is a multiple of 3.
- The sum of all four numbers is a multiple of 4.
Find the smallest possible sum of these four numbers. | 44 |
84,661 | There are four individuals, A, B, C, and D, who need to cross a bridge from the left side to the right side at night. The bridge can only accommodate two people at a time, and there's only one flashlight available, which must be used to cross the bridge. The fastest times each individual takes to cross the bridge are as follows: A takes 2 minutes; B takes 3 minutes; C takes 8 minutes; D takes 10 minutes. As the quicker person must wait for the slower one, the task is to get all four across the bridge in 21 minutes or less. The first to cross the bridge are A and B together, then ____ returns alone. After coming back, the flashlight is passed to ____ and ____, who cross together. Once they reach the other side, they hand the flashlight to ____, who brings it back. Finally, A and B cross the bridge together again. | 21 |
84,753 | The product of all elements in a finite set $S$ is referred to as the "product number" of set $S$. Given the set $M=$ $\left\{\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \cdots, \frac{1}{100}\right\}$, determine the sum of the "product numbers" of all even-numbered subsets (2 elements, 4 elements, $\cdots$, 98 elements) of $M$. | 24.255 |
84,769 | Given 9 points in space, with no 4 points being coplanar, connect some of these points with line segments. Determine the maximum number of triangles that can exist in the graph without forming a tetrahedron. | 27 |
84,815 | 1. Find the bases $z$ of all number systems in which the four-digit number $(1001)_{z}$ is divisible by the two-digit number $(41)_{z}$. | 5 |
84,816 | The number of triangles with vertices' coordinates $(x, y)$ that satisfy $1 \leqslant x \leqslant 4, 1 \leqslant y \leqslant 4$, and where $x$ and $y$ are integers is $\qquad$ . | 516 |
84,840 | The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19a, \frac{124b}{13}, \frac{c}{13}$ is an arithmetic progression. Find the common ratio of the geometric progression. | 247 |
84,855 | Let \( f(a, b, c) = \frac{1}{\sqrt{1+2a}} + \frac{1}{\sqrt{1+2b}} + \frac{1}{\sqrt{1+2c}} \), where \( a, b, c > 0 \) and \( abc = 1 \). Find the minimum value of the constant \( \lambda \) such that \( f(a, b, c) < \lambda \) always holds. | 2 |
84,900 | A facility has 7 consecutive parking spaces, and there are 3 different models of cars to be parked. If it is required that among the remaining 4 parking spaces, exactly 3 are consecutive, then the number of different parking methods is \_\_\_\_\_\_. | 72 |
84,985 | Xiao Ming places several chess pieces into a $3 \times 3$ grid of square cells. Each cell can have no pieces, one piece, or more than one piece. After calculating the total number of pieces in each row and each column, we obtain 6 different numbers. What is the minimum number of chess pieces required to achieve this? | 8 |
85,044 | The set $A=\{a, \frac{b}{a}, 1\}$ and the set $B=\{a^2, a+b, 0\}$. If $A=B$, find the value of $a^{2013}+b^{2014}$. | -1 |
85,062 | Given that the arithmetic square root of $a$ is equal to itself, find the value of $a^2+1$. | 2 |
85,073 | Given a triangular prism $ABC-A_1B_1C_1$ with the area of one of its lateral faces $ABB_1A_1$ being 4, and the distance from the lateral edge $CC_1$ to the lateral face $ABB_1A_1$ being 2, find the volume of the triangular prism $ABC-A_1B_1C_1$. | 4 |
85,122 | A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere, with its base \( A B C \) and lateral edges \( A A_{1}, B B_{1}, C C_{1} \). Segment \( C D \) is the diameter of this sphere, and points \( K \) and \( L \) are the midpoints of edges \( A A_{1} \) and \( A B \) respectively. Find the volume of the prism if \( D L = \sqrt{2} \) and \( D K = \sqrt{3} \). | 4 |
85,133 | Given 6000 cards, each with a unique natural number from 1 to 6000 written on it. It is required to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done? | 179940 |
85,199 | Given that $c > 0$, find the minimum value of $\frac{3}{a} - \frac{4}{b} + \frac{5}{c}$ when non-zero real numbers $a$ and $b$ satisfy the equation $4a^{2} - 2ab + 4b^{2} - c = 0$ and make $|2a + b|$ maximum. | -2 |
85,226 | 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 |
85,231 | 14-22 Solve the equation $\left[x^{3}\right]+\left[x^{2}\right]+[x]=\{x\}-1$.
(Kiev Mathematical Olympiad, 1972) | -1 |
85,264 | Compute: $$\left\lfloor\frac{2005^{3}}{2003 \cdot 2004}-\frac{2003^{3}}{2004 \cdot 2005}\right\rfloor$$ | 8 |
85,275 | Let \( a \) and \( b \) be two known positive constants such that \( a > b \). Points \( P \) and \( Q \) are on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\). If the line connecting point \( A(-a, 0) \) to point \( Q \) is parallel to the line \( OP \) and intersects the \( y \)-axis at point \( R \), then find the value of \(\frac{|AQ| \cdot AR}{OP^{2}}\), where \( O \) is the origin of the coordinate system. | 2 |
85,278 | In the isosceles $\triangle ABC$, $AB = AC = \sqrt{5}$, $D$ is a point on side $BC$ that is not the midpoint, and $E$ is the symmetric point of $C$ with respect to line $AD$. The extension of $EB$ intersects the extension of $AD$ at point $F$. Find the value of $AD \cdot AF$. | 5 |
85,325 | Find the greatest number \( A \) for which the following statement is true.
No matter how we pick seven real numbers between 1 and \( A \), there will always be two numbers among them whose ratio \( h \) satisfies \( \frac{1}{2} \leq h \leq 2 \). | 64 |
85,403 | At a fun fair, coupons can be used to purchase food. Each coupon is worth $5, $8, or $12. For example, for a $15 purchase you can use three coupons of $5, or use one coupon of $5 and one coupon of $8 and pay $2 by cash. Suppose the prices in the fun fair are all whole dollars. What is the largest amount that you cannot purchase using only coupons? | 19 |
85,441 | After persistent searching, Andrey found a two-digit number which does not end in 0 and has a property such that, when inserting a 0 between its tens and units digits, it becomes a three-digit number divisible by the original two-digit number without a remainder. When he showed his result to the teacher, the teacher praised the boy and noted that not only the number Andrey found has this property. Find all such two-digit numbers. How many are there? | 3 |
85,464 | Append the same digit to the left and right of the number 10 so that the resulting four-digit number is divisible by 12. | 4104 |
85,500 | Given $f(x) = 2 + \log_3x$, where $x \in [1,9]$, find the maximum value of the function $y = f(x)^2 + f(x^2)$, and the value of $x$ for which $y$ attains this maximum value. | 3 |
85,524 | Let \(a\) and \(b\) be real numbers. For any real number \(x\) satisfying \(0 \leqslant x \leqslant 1\), it holds that \(|ax + b| \leqslant 1\). Find the maximum value of \(|20a + 14b| + |20a - 14b|\). | 80 |
85,548 | Using the digits 1, 3, and 5, Monica forms three-digit numbers that are greater than 150. How many numbers can Monica form? | 21 |
85,565 | Given two sets $\{a^2, 0, -1\} = \{a, b, 0\}$, find the value of $a^{2014} + b^{2014}$. | 2 |
85,580 | How many real numbers $x$ are solutions to the following equation? $$2003^{x}+2004^{x}=2005^{x}$$ | 1 |
85,607 | Find the maximum value of the function $f(x) = \frac{4x - 4x^3}{1 + 2x^2 + x^4}$ on the set of real numbers $R$. | 1 |
85,611 | Find all integer values of \(a\), not exceeding 15 in absolute value, for which the inequality
$$
\frac{4x - a - 4}{6x + a - 12} \leq 0
$$
holds for all \(x\) in the interval \([2, 3]\). In the answer, indicate the sum of all such \(a\). | -7 |
85,623 | Problem 3. On a river, two identical sightseeing boats departed from one pier in opposite directions at 13:00. At the same time, a raft also set off from the pier. After an hour, one of the boats turned around and headed in the opposite direction. At 15:00, the second boat did the same. What is the speed of the current if, at the moment the boats met, the raft had drifted 7.5 km from the pier? | 2.5 |
85,626 | A $4 \times 4 \times 4$ cube is cut into 64 smaller $1 \times 1 \times 1$ cubes. Then, 16 of the $1 \times 1 \times 1$ small cubes are painted red. The requirement is that, in any group of 4 small cubes parallel to any edge, exactly 1 cube is painted red. How many different ways are there to paint the cubes? (Painting methods that are the same after rotation are also considered different methods.) | 576 |
85,670 | Write the process of using the Horner's algorithm to find the value of the function $\_(f)\_()=1+\_x+0.5x^2+0.16667x^3+0.04167x^4+0.00833x^5$ at $x=-0.2$. | 0.81873 |
85,697 | 6. On the coordinate plane, the area of the plane region bounded by the conditions $\left\{\begin{array}{l}y \geqslant-|x|-1, \\ y \leqslant-2|x|+3\end{array}\right.$ is | 16 |
85,755 | Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a}|=2$, $| \overrightarrow {b}|=1$, and $\overrightarrow {b} \perp ( \overrightarrow {a}+ \overrightarrow {b})$, determine the projection of vector $\overrightarrow {a}$ onto vector $\overrightarrow {b}$. | -1 |
85,788 | Given the circle $O: x^2 + y^2 = 4$, and the line $l: mx - y + 1 = 0$ intersects circle $O$ at points $A$ and $C$, the line $n: x + my - m = 0$ intersects circle $O$ at points $B$ and $D$. The maximum area of quadrilateral $ABCD$ is \_\_\_\_\_\_. | 7 |
85,801 | Given an geometric sequence $\{a\_n\}$ with a common ratio $q \neq 1$, $a\_1 = \frac{1}{2}$, and the sum of the first $n$ terms is $S\_n$. It is also known that $a\_2 + S\_2$, $a\_3 + S\_3$, $a\_4 + S\_4$ form an arithmetic sequence. Find $a\_n + S\_n=\_\_\_\_\_\_\_.$ | 1 |
85,817 | Dora was rowing a boat upstream. At 12:00 PM, she decided to take a break and realized that her adventure map had fallen into the water. She immediately turned the boat around and went downstream to look for the map. She found the floating map at 12:21 PM. If the speed of the water current and the speed of the boat in still water are both constant, how many minutes did the map float in the water? | 21 |
85,834 | Given the circle $(x+1)^{2}+y^{2}=4$ and the parabola $y^{2}=mx(m\neq 0)$ intersect the directrix at points $A$ and $B$, and $|AB|=2 \sqrt {3}$, then the value of $m$ is \_\_\_\_\_\_. | 8 |
85,873 | Calculate the value of the polynomial $f(x) = 2x^7 + x^6 - 3x^3 + 2x + 1$ using Horner's method (also known as Qin Jiushao algorithm) when $x = 2$. Determine the number of multiplication and addition operations required to compute the value. | 7 |
85,878 | Let \( x \) and \( y \) be positive real numbers and \( \theta \) an angle such that \( \theta \neq \frac{\pi}{2} n \) for any integer \( n \). Suppose
$$
\frac{\sin \theta}{x}=\frac{\cos \theta}{y}
$$
and
$$
\frac{\cos ^{4} \theta}{x^{4}}+\frac{\sin ^{4} \theta}{y^{4}}=\frac{97 \sin 2 \theta}{x^{3} y+y^{3} x} .
$$
Compute \( \frac{x}{y}+\frac{y}{x} \). | 4 |
85,882 | In a certain survey, a stratified random sampling with sample size proportion allocation was used. Some of the data are shown in the table below:
| Sample Category | Sample Size | Mean | Variance |
|-----------------|-------------|------|----------|
| $A$ | $10$ | $3.5$| $2$ |
| $B$ | $30$ | $5.5$| $1$ |
Based on this data, the variance of the total sample is ______. | 2 |
85,898 | Klepitsyn V.A.
Vasya multiplied a certain number by 10 and got a prime number. And Petya multiplied the same number by 15, but still got a prime number.
Could it be that neither of them made a mistake? | 0.2 |
85,907 | How many five-digit numbers can be formed by selecting 3 different digits from $\{0, 1, 2, 3\}$ for the first three positions and 2 different digits from $\{5, 6, 7, 8\}$ for the last two positions, under the condition that 0 and 5 cannot be adjacent? | 198 |
85,919 | Problem 7.1. Jerry has nine cards with digits from 1 to 9. He lays them out in a row, forming a nine-digit number. Tom writes down all 8 two-digit numbers formed by adjacent digits (for example, for the number 789456123, these numbers are $78, 89, 94, 45$, $56, 61, 12, 23$). For each two-digit number divisible by 9, Tom gives Jerry a piece of cheese. What is the maximum number of pieces of cheese Jerry can get? | 4 |
85,925 | When the base-$b$ number $11011_b$ is multiplied by $b-1$, then $1001_b$ is added, what is the result (written in base $b$)? | 100100 |
85,929 | Let $l$, $m$, $n$ be three different lines in space, and $\alpha$, $\beta$ be two non-coincident planes in space. The following four propositions are given:
① If $l$ and $m$ are skew lines, and $m \parallel n$, then $l$ and $n$ are skew lines;
② If $l \parallel \alpha$, and $\alpha \parallel \beta$, then $l \parallel \beta$;
③ If $\alpha \perp \beta$, $l \perp \alpha$, and $m \perp \beta$, then $l \perp m$;
④ If $m \parallel \alpha$, and $m \parallel n$, then $n \parallel \alpha$.
Among these propositions, the correct ones are. (Please fill in the numbers of all propositions you think are correct) | 3 |
85,947 | On a \(10 \times 10\) grid, there are 11 horizontal grid lines and 11 vertical grid lines. The line segments connecting adjacent nodes on the same line are called "links." What is the minimum number of links that must be removed so that at each node, there are at most 3 remaining links? | 41 |
85,955 | From the numbers $1, 2, \cdots, 2004$, select $k$ numbers such that among the selected $k$ numbers, there are three numbers that can form the side lengths of a triangle (with the condition that the three side lengths are pairwise distinct). Find the smallest value of $k$ that satisfies this condition. | 17 |
86,038 | What is the smallest three-digit positive integer which can be written in the form \( p q^{2} r \), where \( p, q \), and \( r \) are distinct primes? | 126 |
86,073 | Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, getting a result of \( V \) cubic centimeters. Ritmo, on the other hand, measured each dimension to the nearest centimeter and multiplied the rounded measurements, getting a result of 2017 cubic centimeters. Find the positive difference between the least and greatest possible positive values for \( V \). | 4035 |
86,077 | What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 9? | 4096 |
86,086 | Given $A=5\sqrt{2x+1}$, $B=3\sqrt{x+3}$, $C=\sqrt{10x+3y}$, where $A$ and $B$ are the simplest quadratic surds, and $A+B=C$, find the value of $\sqrt{2y-x^2}$. | 14 |
86,117 | Divide the natural numbers from 1 to 2010 into groups, such that the greatest common divisor of any three numbers within each group is 1. What is the minimum number of groups required? | 503 |
86,237 | There is a strip of paper with three types of scale lines that divide the strip into 6 parts, 10 parts, and 12 parts along its length. If the strip is cut along all the scale lines, into how many parts is the strip divided? | 20 |
86,276 | Given that $f(x)$ is an even function defined on $\mathbb{R}$ and satisfies $f(x+2)=-\frac{1}{f(x)}$, when $1\leq x\leq 2$, $f(x)=x-2$. Find the value of $f(6.5)$. | -0.5 |
86,358 | In triangle \( ABC \), \( \angle A = 40^\circ \), \( \angle B = 20^\circ \), and \( AB - BC = 4 \). Find the length of the angle bisector of angle \( C \). | 4 |
86,370 | In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midsegment is 5. Find the height of the trapezoid. | 4.8 |
86,488 | A person named Jia and their four colleagues each own a car with license plates ending in 9, 0, 2, 1, and 5, respectively. To comply with the local traffic restriction rules from the 5th to the 9th day of a certain month (allowing cars with odd-ending numbers on odd days and even-ending numbers on even days), they agreed to carpool. Each day they can pick any car that meets the restriction, but Jia’s car can be used for one day at most. The number of different carpooling arrangements is __________. | 80 |
86,555 | Given the set $A={1,m+2,m^{2}+4}$, and $5\in A$, find the value of $m$. | 3 |
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