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30,000 | In the rectangular coordinate system, the symmetric point of point $A(-2,1,3)$ with respect to the $x$-axis is point $B$. It is also known that $C(x,0,-2)$, and $|BC|=3 \sqrt{2}$. Find the value of $x$. | -6 | 19.53125 |
30,001 | A right triangle $ABC$ is inscribed in a circle. From the vertex $C$ of the right angle, a chord $CM$ is drawn, intersecting the hypotenuse at point $K$. Find the area of triangle $ABM$ if $BK: AB = 3:4$, $BC=2\sqrt{2}$, $AC=4$. | \frac{36}{19} \sqrt{2} | 0 |
30,002 | Given $f(x) = \begin{cases} x+2 & (x\leq-1) \\ x^{2} & (-1<x<2) \\ 2x & (x\geq2) \end{cases}$, determine the value of $x$ if $f(x)=3$. | \sqrt{3} | 55.46875 |
30,003 | What is the nearest integer to $(3+\sqrt2)^6$? | 7414 | 1.5625 |
30,004 | A sports conference has 12 teams in two divisions of 6 each. How many games are in a complete season for the conference if each team must play every other team in its own division three times and every team in the other division twice? | 162 | 78.90625 |
30,005 | Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as $\frac{a}{b}$ for positive integers $a,b$ with $\gcd (a,b) = 1$ , then find $a + b$ .
*Proposed by Vismay Sharan* | 831 | 3.125 |
30,006 | Read the following material and complete the corresponding tasks.
平衡多项式
定义:对于一组多项式$x+a$,$x+b$,$x+c$,$x+d(a$,$b$,$c$,$d$是常数）,当其中两个多项式的乘积与另外两个多项式乘积的差是一个常数$p$时,称这样的四个多项式是一组平衡多项式,$p$的绝对值是这组平衡多项式的平衡因子.
例如:对于多项式$x+1$,$x+2$,$x+5$,$x+6$,因为$(x+1)(x+6)-(x+2)(x+5)=(x^{2}+7x+6)-(x^{2}+7x+10)=-4$,所以多项式$x+1$,$x+2$,$x+5$,$x+6$是一组平衡多项式,其平衡因子为$|-4|=4$.
任务:
$(1)$小明发现多项式$x+3$,$x+4$,$x+6$,$x+7$是一组平衡多项式,在求其平衡因子时,列式如下:$(x+3)(x+7)-(x+4)(x+6)$,根据他的思路求该组平衡多项式的平衡因子；
$A$.判断多项式$x-1$,$x-2$,$x-4$,$x-5$是否为一组平衡多项式,若是,求出其平衡因子；若不是,说明理由.
$B$.若多项式$x+2$,$x-4$,$x+1$,$x+m(m$是常数）是一组平衡多项式,求$m$的值. | -5 | 57.03125 |
30,007 | Let $\triangle DEF$ be an isosceles triangle with $DE = DF$. Three circles are defined as follows: the circle $\Omega$ with its center at the centroid of $\triangle DEF$, and two circles $\Omega_1$ and $\Omega_2$, where $\Omega_1$ is tangent to $\overline{EF}$ and externally tangent to the other sides extended, while $\Omega_2$ is tangent to $\overline{DE}$ and $\overline{DF}$ while internally tangent to $\Omega$. Determine the smallest integer value for the perimeter of $\triangle DEF$ if $EF$ is the smallest integer side, and $\Omega_1$ and $\Omega_2$ intersect only once. | 12 | 0.78125 |
30,008 | Determine the product of the solutions to the quadratic equation $49 = -2x^2 - 8x$. | \frac{49}{2} | 55.46875 |
30,009 | The sides of rectangle $ABCD$ have lengths $12$ and $5$. A right triangle is drawn so that no point of the triangle lies outside $ABCD$, and one of its angles is $30^\circ$. The maximum possible area of such a triangle can be written in the form $p \sqrt{q} - r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r$. | 28 | 39.0625 |
30,010 | Consider a list of numbers \[8, 3, x, 3, 7, 3, y\]. When the mean, median, and mode of this list are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible values of \(y\) if \(x=6\)?
A) $\frac{51}{13}$
B) $33$
C) $\frac{480}{13}$
D) $40$
E) $34$ | \frac{480}{13} | 35.9375 |
30,011 | Given $$|\vec{a}|=3, |\vec{b}|=2$$. If $$\vec{a} \cdot \vec{b} = -3$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ is \_\_\_\_\_\_. | \frac{2}{3}\pi | 0 |
30,012 | Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{11}{14} | 64.0625 |
30,013 | There exist $s$ unique nonnegative integers $m_1 > m_2 > \cdots > m_s$ and $s$ integers $b_k$ ($1\le k\le s$), with each $b_k$ either $1$ or $-1$, such that \[b_13^{m_1} + b_23^{m_2} + \cdots + b_s3^{m_s} = 1007.\] Find $m_1 + m_2 + \cdots + m_s$. | 15 | 14.84375 |
30,014 | The necessary and sufficient condition for the lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel is "$a=$ ______". | -2 | 54.6875 |
30,015 | How many integers between 10000 and 100000 include the block of digits 178? | 280 | 22.65625 |
30,016 | Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face. | 16 | 98.4375 |
30,017 | There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from the left of the remaining candies, and then ate every ninth one of them, also moving to the right. How many candies are left after this? | 761 | 0 |
30,018 | Calculate the sum of the sequence $2 - 6 + 10 - 14 + 18 - \cdots - 98 + 102$. | -52 | 17.96875 |
30,019 | Let \( n \geq 1 \) be a positive integer. We say that an integer \( k \) is a fan of \( n \) if \( 0 \leq k \leq n-1 \) and there exist integers \( x, y, z \in \mathbb{Z} \) such that
\[
\begin{aligned}
x^2 + y^2 + z^2 &\equiv 0 \pmod{n}; \\
xyz &\equiv k \pmod{n}.
\end{aligned}
\]
Let \( f(n) \) be the number of fans of \( n \). Determine \( f(2020) \). | 101 | 1.5625 |
30,020 | The sequence $\{a\_n\}$ satisfies $(a_{n+1}-1)(1-a_{n})=a_{n}$, $a_{8}=2$, then $S_{2017}=$ _____ . | \frac {2017}{2} | 26.5625 |
30,021 | In a WeChat group, members A, B, C, D, and E simultaneously grab 4 red envelopes. Each person can grab at most one red envelope, and all red envelopes are grabbed. Among the 4 red envelopes, there are two worth 2 yuan and two worth 3 yuan (red envelopes with the same amount are considered the same). The number of situations where both A and B grab a red envelope is \_\_\_\_\_\_ (Answer in digits). | 18 | 35.15625 |
30,022 | In a circle, parallel chords of lengths 8, 15, and 17 determine central angles of $\gamma$, $\delta$, and $\gamma + \delta$ radians, respectively, where $\gamma + \delta < \pi$. If $\cos \gamma$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator? | 32 | 0.78125 |
30,023 | Two circles of radius 3 and 4 are internally tangent to a larger circle. The larger circle circumscribes both the smaller circles. Find the area of the shaded region surrounding the two smaller circles within the larger circle. Express your answer in terms of \(\pi\). | 24\pi | 33.59375 |
30,024 | At a family outing to a theme park, the Thomas family, comprising three generations, plans to purchase tickets. The two youngest members, categorized as children, get a 40% discount. The two oldest members, recognized as seniors, enjoy a 30% discount. The middle generation no longer enjoys any discount. Grandmother Thomas, whose senior ticket costs \$7.50, has taken the responsibility to pay for everyone. Calculate the total amount Grandmother Thomas must pay.
A) $46.00$
B) $47.88$
C) $49.27$
D) $51.36$
E) $53.14$ | 49.27 | 10.15625 |
30,025 | Let $A$ , $B$ , and $C$ be distinct points on a line with $AB=AC=1$ . Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$ . What is the degree measure of the acute angle formed by lines $EC$ and $BF$ ?
*Ray Li* | 75 | 51.5625 |
30,026 | Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 2028 | 46.875 |
30,027 | Given the function $f(x)=2x^{3}-ax^{2}+1$ $(a\in\mathbb{R})$ has exactly one zero in the interval $(0,+\infty)$, find the sum of the maximum and minimum values of $f(x)$ on the interval $[-1,1]$. | -3 | 67.96875 |
30,028 | Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy? | 406 | 82.8125 |
30,029 | Given a moving circle P that is internally tangent to the circle M: (x+1)²+y²=8 at the fixed point N(1,0).
(1) Find the trajectory equation of the moving circle P's center.
(2) Suppose the trajectory of the moving circle P's center is curve C. A and B are two points on curve C. The perpendicular bisector of line segment AB passes through point D(0, 1/2). Find the maximum area of △OAB (O is the coordinate origin). | \frac{\sqrt{2}}{2} | 16.40625 |
30,030 | Let \(n = 2^{20}3^{25}\). How many positive integer divisors of \(n^2\) are less than \(n\) but do not divide \(n\)? | 499 | 89.84375 |
30,031 | Li Fang has 4 shirts of different colors, 3 skirts of different patterns, and 2 dresses of different styles. Calculate the total number of different choices she has for the May Day celebration. | 14 | 22.65625 |
30,032 | Let $(p_n)$ and $(q_n)$ be sequences of real numbers defined by the equation
\[ (3+i)^n = p_n + q_n i\] for all integers $n \geq 0$, where $i = \sqrt{-1}$. Determine the value of
\[ \sum_{n=0}^\infty \frac{p_n q_n}{9^n}.\]
A) $\frac{5}{8}$
B) $\frac{7}{8}$
C) $\frac{3}{4}$
D) $\frac{15}{16}$
E) $\frac{9}{20}$ | \frac{3}{4} | 2.34375 |
30,033 | Positive integers $a$, $b$, and $c$ are such that $a<b<c$. Consider the system of equations
\[
2x + y = 2022 \quad \text{and} \quad y = |x-a| + |x-b| + |x-c|
\]
Determine the minimum value of $c$ such that the system has exactly one solution. | 1012 | 21.09375 |
30,034 | A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of $2\sqrt{2}$ from its original location?
*Proposed by Connor Gordon* | 3/8 | 10.9375 |
30,035 | Given quadrilateral $\Box FRDS$ with $\triangle FDR$ being a right-angled triangle at point $D$, with side lengths $FD = 3$ inches, $DR = 4$ inches, $FR = 5$ inches, and $FS = 8$ inches, and $\angle RFS = \angle FDR$, find the length of RS. | \sqrt{89} | 18.75 |
30,036 | The stem and leaf plot represents the heights, in inches, of the players on the Pine Ridge Middle School boys' basketball team. Calculate the mean height of the players on the team. (Note: $5|3$ represents 53 inches.)
Height of the Players on the Basketball Team (inches)
$4|8$
$5|0\;1\;4\;6\;7\;7\;9$
$6|0\;3\;4\;5\;7\;9\;9$
$7|1\;2\;4$ | 61.44 | 0 |
30,037 | Given two fuses, each of which burns for exactly one minute if lit from one end (but may burn non-uniformly), how can you measure 45 seconds using these fuses? (You can light the fuse from either of its two ends.) | 45 | 89.84375 |
30,038 | Ms. Garcia weighed the packages in three different pairings and obtained weights of 162, 164, and 168 pounds. Find the total weight of all four packages. | 247 | 3.90625 |
30,039 | Determine the volume of the original cube given that one dimension is increased by $3$, another is decreased by $2$, and the third is left unchanged, and the volume of the resulting rectangular solid is $6$ more than that of the original cube. | (3 + \sqrt{15})^3 | 0.78125 |
30,040 | A point $(x, y)$ is randomly selected such that $0 \leq x \leq 4$ and $0 \leq y \leq 5$. What is the probability that $x + y \leq 5$? Express your answer as a common fraction. | \frac{3}{5} | 17.1875 |
30,041 | 29 boys and 15 girls came to the ball. Some of the boys danced with some of the girls (at most once with each person in the pair). After the ball, each individual told their parents how many times they danced. What is the maximum number of different numbers that the children could mention? | 29 | 0.78125 |
30,042 | A class has 50 students, and their scores in a math test $\xi$ follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \leq \xi \leq 100) = 0.3$. Estimate the number of students scoring above 110. | 10 | 42.1875 |
30,043 | For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of
\[\frac{abc(a + b + c)}{(a + b)^2 (b + c)^2}.\] | \frac{1}{4} | 0 |
30,044 | Let $a_n$ be the number obtained by writing the integers 1 to $n$ from left to right. For instance, $a_4 = 1234$ and $a_{12} = 123456789101112$. For $1 \le k \le 150$, how many $a_k$ are divisible by both 3 and 5? | 10 | 0 |
30,045 | A certain college student had the night of February 23 to work on a chemistry problem set and a math problem set (both due on February 24, 2006). If the student worked on his problem sets in the math library, the probability of him finishing his math problem set that night is 95% and the probability of him finishing his chemistry problem set that night is 75%. If the student worked on his problem sets in the the chemistry library, the probability of him finishing his chemistry problem set that night is 90% and the probability of him finishing his math problem set that night is 80%. Since he had no bicycle, he could only work in one of the libraries on February 23rd. He works in the math library with a probability of 60%. Given that he finished both problem sets that night, what is the probability that he worked on the problem sets in the math library? | 95/159 | 0 |
30,046 | 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 | 0 |
30,047 | Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{50}{x - 50} = x + 5.
\] | 51 | 72.65625 |
30,048 | All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | \frac{2}{3} | 3.90625 |
30,049 | A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside this cylinder? | 2\sqrt{61} | 82.03125 |
30,050 | Given $|a|=3$, $|b-2|=9$, and $a+b > 0$, find the value of $ab$. | -33 | 47.65625 |
30,051 |
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height dropped from vertex $A_{4}$ onto the face $A_{1} A_{2} A_{3}$.
$A_{1}(1, 1, 2)$
$A_{2}(-1, 1, 3)$
$A_{3}(2, -2, 4)$
$A_{4}(-1, 0, -2)$ | \sqrt{\frac{35}{2}} | 0 |
30,052 | Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_i$ for every $i=\overline{0,4}$ ?
<details><summary>thanks </summary>Thanks to the user Vlados021 for translating the problem.</details> | \frac{1 + \sqrt{5}}{2} | 41.40625 |
30,053 | Given 5 different letters from the word "equation", find the total number of different arrangements that contain "qu" where "qu" are consecutive and in the same order. | 480 | 25.78125 |
30,054 | Team A has a probability of $$\frac{2}{3}$$ of winning each set in a best-of-five set match, and Team B leads 2:0 after the first two sets. Calculate the probability of Team B winning the match. | \frac{19}{27} | 24.21875 |
30,055 | Given the hyperbola $\frac {x^{2}}{16}- \frac {y^{2}}{9}=1$, and a chord AB with a length of 6 connected to the left focus F₁, calculate the perimeter of △ABF₂ (F₂ being the right focus). | 28 | 38.28125 |
30,056 | You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$ , the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$ .) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$ . What is $\lfloor E\rfloor,$ the greatest integer less than or equal to $E$ ?
*Author: Alex Zhu* | 695 | 18.75 |
30,057 | Within a cube with edge length 6, there is a regular tetrahedron with edge length \( x \) that can rotate freely inside the cube. What is the maximum value of \( x \)? | 2\sqrt{6} | 5.46875 |
30,058 | Compute the expression: $\left( \pi - 1 \right)^{0} + \left( \frac{1}{2} \right)^{-1} + \left| 5 - \sqrt{27} \right| - 2 \sqrt{3}$. | 8 - 5 \sqrt{3} | 12.5 |
30,059 | Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes. | 27 | 25 |
30,060 | How many rectangles can be formed where each of the four vertices are points on a 4x4 grid with points spaced evenly along the grid lines? | 36 | 13.28125 |
30,061 | \(AB\) is the diameter of a circle; \(BC\) is a tangent; \(D\) is the point where line \(AC\) intersects the circle. It is known that \(AD = 32\) and \(DC = 18\). Find the radius of the circle. | 20 | 28.90625 |
30,062 | Given that $a$ and $b \in R$, the function $f(x) = \ln(x + 1) - 2$ is tangent to the line $y = ax + b - \ln2$ at $x = -\frac{1}{2}$. Let $g(x) = e^x + bx^2 + a$. If the inequality $m \leqslant g(x) \leqslant m^2 - 2$ holds true in the interval $[1, 2]$, determine the real number $m$. | e + 1 | 7.03125 |
30,063 | As shown in the diagram, squares \( a \), \( b \), \( c \), \( d \), and \( e \) are used to form a rectangle that is 30 cm long and 22 cm wide. Find the area of square \( e \). | 36 | 7.8125 |
30,064 | Given that \( a \) is a positive real number and \( b \) is an integer between \( 2 \) and \( 500 \), inclusive, find the number of ordered pairs \( (a,b) \) that satisfy the equation \( (\log_b a)^{1001}=\log_b(a^{1001}) \). | 1497 | 19.53125 |
30,065 | In parallelogram $ABCD$ , the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$ . If the area of $XYZW$ is $10$ , find the area of $ABCD$ | 40 | 17.96875 |
30,066 | Let $x$ be a positive real number. Define
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] Given that $A^3+B^3+C^3 + 8ABC = 2014$ , compute $ABC$ .
*Proposed by Evan Chen* | 183 | 27.34375 |
30,067 | Jillian drives along a straight road that goes directly from her house $(J)$ to her Grandfather's house $(G)$. Some of this road is on flat ground and some is downhill or uphill. Her car travels downhill at $99 \mathrm{~km} / \mathrm{h}$, on flat ground at $77 \mathrm{~km} / \mathrm{h}$, and uphill at $63 \mathrm{~km} / \mathrm{h}$. It takes Jillian 3 hours and 40 minutes to drive from $J$ to $G$. It takes her 4 hours and 20 minutes to drive from $G$ to $J$. The distance between $J$ and $G$, in $\mathrm{km}$, is: | 308 | 1.5625 |
30,068 | Pete has some trouble slicing a 20-inch (diameter) pizza. His first two cuts (from center to circumference of the pizza) make a 30º slice. He continues making cuts until he has gone around the whole pizza, each time trying to copy the angle of the previous slice but in fact adding 2º each time. That is, he makes adjacent slices of 30º, 32º, 34º, and so on. What is the area of the smallest slice? | 5\pi | 1.5625 |
30,069 | Given the sets $A=\{1, 3, 2m-1\}$ and $B=\{3, m^2\}$; if $B \subseteq A$, find the value of the real number $m$. | -1 | 16.40625 |
30,070 | Given the sets of consecutive integers where each set starts with one more element than the preceding one and the first element of each set is one more than the last element of the preceding set, find the sum of the elements in the 21st set. | 4641 | 87.5 |
30,071 | Cirlce $\Omega$ is inscribed in triangle $ABC$ with $\angle BAC=40$ . Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$ . Tangent form $D$ touches $\Omega$ in $E$ . FInd $\angle BEC$ . | 110 | 10.15625 |
30,072 | How many natural numbers between 200 and 400 are divisible by 8? | 25 | 50.78125 |
30,073 | Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$ . Suppose the area of $\vartriangle DOC$ is $2S/9$ . Find the value of $a/b$ . | \frac{2 + 3\sqrt{2}}{7} | 3.90625 |
30,074 | Calculate:<br/>$(1)4.7+\left(-2.5\right)-\left(-5.3\right)-7.5$;<br/>$(2)18+48\div \left(-2\right)^{2}-\left(-4\right)^{2}\times 5$;<br/>$(3)-1^{4}+\left(-2\right)^{2}\div 4\times [5-\left(-3\right)^{2}]$;<br/>$(4)(-19\frac{15}{16})×8$ (Solve using a simple method). | -159\frac{1}{2} | 0 |
30,075 | $A$ and $B$ are $46$ kilometers apart. Person A rides a bicycle from point $A$ to point $B$ at a speed of $15$ kilometers per hour. One hour later, person B rides a motorcycle along the same route from point $A$ to point $B$ at a speed of $40$ kilometers per hour.
$(1)$ After how many hours can person B catch up to person A?
$(2)$ If person B immediately returns to point $A after reaching point $B, how many kilometers away from point $B will they meet person A on the return journey? | 10 | 14.0625 |
30,076 | Distribute 5 students into 3 groups: Group A, Group B, and Group C, with the condition that Group A must have at least two people, and Groups B and C must have at least one person each. How many different distribution schemes exist? | 80 | 10.15625 |
30,077 | The lock opens only if a specific three-digit number is entered. An attempt consists of randomly selecting three digits from a given set of five. The code was guessed correctly only on the last of all attempts. How many attempts preceded the successful one? | 124 | 12.5 |
30,078 | How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? | 60 | 0 |
30,079 | In how many ways can 7 people sit around a round table, considering that two seatings are the same if one is a rotation of the other, and additionally, one specific person must sit between two particular individuals? | 240 | 13.28125 |
30,080 | Let $(b_1,b_2,b_3,\ldots,b_{14})$ be a permutation of $(1,2,3,\ldots,14)$ for which
$b_1>b_2>b_3>b_4>b_5>b_6>b_7>b_8 \mathrm{\ and \ } b_8<b_9<b_{10}<b_{11}<b_{12}<b_{13}<b_{14}.$
Find the number of such permutations. | 1716 | 10.9375 |
30,081 | Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\gcd(k, n)=1$ is a prime. | 105 | 90.625 |
30,082 | In a convex polygon, all its diagonals are drawn. These diagonals divide the polygon into several smaller polygons. What is the maximum number of sides that a polygon in the subdivision can have if the original polygon has:
a) 13 sides;
b) 1950 sides? | 1950 | 50 |
30,083 | An employee arrives at the unit randomly between 7:50 and 8:30. Calculate the probability that he can clock in on time. | \frac{3}{8} | 0.78125 |
30,084 | Let $x_1, x_2, x_3, \dots, x_{50}$ be positive real numbers such that $x_1^2 + x_2^2 + x_3^2 + \dots + x_{50}^2 = 1.$ Find the maximum value of
\[
\frac{x_1}{1 + x_1^2} + \frac{x_2}{1 + x_2^2} + \frac{x_3}{1 + x_3^2} + \dots + \frac{x_{50}}{1 + x_{50}^2}.
\] | \frac{1}{2} | 3.90625 |
30,085 | Given that the eccentricities of a confocal ellipse and a hyperbola are \( e_1 \) and \( e_2 \), respectively, and the length of the minor axis of the ellipse is twice the length of the imaginary axis of the hyperbola, find the maximum value of \( \frac{1}{e_1} + \frac{1}{e_2} \). | 5/2 | 1.5625 |
30,086 | There is a string of lights with a recurrent pattern of three blue lights followed by four yellow lights, spaced 7 inches apart. Determine the distance in feet between the 4th blue light and the 25th blue light, given that 1 foot equals 12 inches. | 28 | 0 |
30,087 | Let \( x, y, z \in [0, 1] \). The maximum value of \( M = \sqrt{|x-y|} + \sqrt{|y-z|} + \sqrt{|z-x|} \) is ______ | \sqrt{2} + 1 | 3.90625 |
30,088 | The maximum and minimum values of the function $f(x) = -x^2 + 2x + 3$ in the interval $[-2, 3]$ are ___ and ___, respectively. | -5 | 85.9375 |
30,089 | Bees, in processing flower nectar into honey, remove a significant amount of water. Research has shown that nectar usually contains about $70\%$ water, while the honey produced from it contains only $17\%$ water. How many kilograms of nectar must bees process to obtain 1 kilogram of honey? | 2.77 | 61.71875 |
30,090 | The general formula of the sequence \\(\{a_n\}\) is \\(a_n=n\cos \frac{n\pi}{2}\\), and the sum of its first \\(n\\) terms is \\(S_n\\). Find \\(S_{2019}\\). | -1010 | 37.5 |
30,091 | Three people, including one girl, are to be selected from a group of $3$ boys and $2$ girls, determine the probability that the remaining two selected individuals are boys. | \frac{2}{3} | 0 |
30,092 | Two real numbers are selected independently at random from the interval $[-15, 15]$. The product of those numbers is considered only if both numbers are outside the interval $[-5, 5]$. What is the probability that the product of those numbers, when considered, is greater than zero? | \frac{2}{9} | 26.5625 |
30,093 | The power function $f(x)=(m^{2}+2m-2)x^{m}$ is a decreasing function on $(0,+\infty)$. Find the value of the real number $m$. | -3 | 7.8125 |
30,094 | Calculate: $(128)^{\frac{7}{3}}$ | 65536 \cdot \sqrt[3]{2} | 28.125 |
30,095 | When the two diagonals of an $8 \times 10$ grid are drawn, calculate the number of the $1 \times 1$ squares that are not intersected by either diagonal. | 48 | 9.375 |
30,096 | Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{24}/R_4$. | 15 | 45.3125 |
30,097 | The value of $\log_5{3125}$ is between which consecutive integers $c$ and $d$? Find $c+d$. | 11 | 13.28125 |
30,098 | Simplify first, then evaluate: $(1+\frac{4}{a-1})÷\frac{a^2+6a+9}{a^2-a}$, where $a=2$. | \frac{2}{5} | 79.6875 |
30,099 | For a nonnegative integer \(n\), let \(r_{11}(7n)\) stand for the remainder left when \(n\) is divided by \(11.\) For example, \(r_{11}(7 \cdot 3) = 10.\)
What is the \(15^{\text{th}}\) entry in an ordered list of all nonnegative integers \(n\) that satisfy $$r_{11}(7n) \leq 5~?$$ (Note that the first entry in this list is \(0\).) | 29 | 3.90625 |
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