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29,100 | Given the function \( f(x) = x^3 + ax^2 + bx + c \) where \( a, b, \) and \( c \) are nonzero integers, if \( f(a) = a^3 \) and \( f(b) = b^3 \), what is the value of \( c \)? | 16 | 13.28125 |
29,101 | Inside a right triangle \(ABC\) with hypotenuse \(AC\), a point \(M\) is chosen such that the areas of triangles \(ABM\) and \(BCM\) are one-third and one-quarter of the area of triangle \(ABC\) respectively. Find \(BM\) if \(AM = 60\) and \(CM = 70\). If the answer is not an integer, round it to the nearest whole number. | 38 | 0.78125 |
29,102 | If $|x| + x + y = 14$ and $x + |y| - y = 16,$ find $x + y.$ | -2 | 0 |
29,103 | Given \( 0 \leq m-n \leq 1 \) and \( 2 \leq m+n \leq 4 \), when \( m - 2n \) reaches its maximum value, what is the value of \( 2019m + 2020n \)? | 2019 | 2.34375 |
29,104 | In a modified game similar to Deal or No Deal, participants choose a box at random from a set of 30 boxes, each containing one of the following values:
\begin{tabular}{|c|c|}
\hline
\$0.50 & \$2,000 \\
\hline
\$2 & \$10,000 \\
\hline
\$10 & \$20,000 \\
\hline
\$20 & \$40,000 \\
\hline
\$50 & \$100,000 \\
\hline
\$100 & \$200,000 \\
\hline
\$500 & \$400,000 \\
\hline
\$1,000 & \$800,000 \\
\hline
\$1,500 & \$1,000,000 \\
\hline
\end{tabular}
After choosing a box, participants eliminate other boxes by opening them. What is the minimum number of boxes a participant needs to eliminate to have at least a 50% chance of holding a box containing at least \$200,000? | 20 | 17.1875 |
29,105 | In the ancient Chinese mathematical text "The Mathematical Classic of Sunzi", there is a problem stated as follows: "Today, a hundred deer enter the city. Each family takes one deer, but not all are taken. Then, three families together take one deer, and all deer are taken. The question is: how many families are there in the city?" In this problem, the number of families in the city is ______. | 75 | 35.15625 |
29,106 | Katie writes a different positive integer on the top face of each of the fourteen cubes in the pyramid shown. The sum of the nine integers written on the cubes in the bottom layer is 50. The integer written on each of the cubes in the middle and top layers of the pyramid is equal to the sum of the integers on the four cubes underneath it. What is the greatest possible integer that she can write on the top cube? | 118 | 0 |
29,107 | Given the equations:
\[
\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 4 \quad \text{and} \quad \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 3,
\]
find the value of \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\). | 16 | 0.78125 |
29,108 | Determine the value of
\[2023 + \frac{1}{2} \left( 2022 + \frac{1}{2} \left( 2021 + \dots + \frac{1}{2} \left( 4 + \frac{1}{2} \cdot (3 + 1) \right) \right) \dotsb \right).\] | 4044 | 42.1875 |
29,109 | A bridge needs to be constructed over a river with a required elevation of 800 feet from one side to the other. Determine the additional bridge length required if the gradient is reduced from 2% to 1.5%. | 13333.33 | 13.28125 |
29,110 | It is known that the vertex of angle $\theta$ is at the origin of coordinates, its initial side coincides with the positive half-axis of the x-axis, and its terminal side falls on the ray $y= \frac {1}{2}x$ ($x\leq0$).
(Ⅰ) Find the value of $\cos\left( \frac {\pi}{2}+\theta \right)$;
(Ⅱ) If $\cos\left( \alpha+ \frac {\pi}{4} \right)=\sin\theta$, find the value of $\sin\left(2\alpha+ \frac {\pi}{4}\right)$. | - \frac { \sqrt {2}}{10} | 0 |
29,111 | A number like 45132 is called a "wave number," which means the tens and thousands digits are both larger than their respective neighboring digits. What is the probability of forming a non-repeating five-digit "wave number" using the digits 1, 2, 3, 4, 5? | \frac{1}{15} | 0.78125 |
29,112 | Medians $\overline{DP}$ and $\overline{EQ}$ of isosceles $\triangle DEF$, where $DE=EF$, are perpendicular. If $DP= 21$ and $EQ = 28$, then what is ${DE}$? | \frac{70}{3} | 46.09375 |
29,113 | There are $5$ people arranged in a row. Among them, persons A and B must be adjacent, and neither of them can be adjacent to person D. How many different arrangements are there? | 36 | 1.5625 |
29,114 | In the center of a square, there is a police officer, and in one of the vertices, there is a gangster. The police officer can run throughout the whole square, while the gangster can only run along its sides. It is known that the ratio of the maximum speed of the police officer to the maximum speed of the gangster is: 0.5; 0.49; 0.34; 1/3. Can the police officer run in such a way that at some point he will be on the same side of the square as the gangster? | 1/3 | 0 |
29,115 |
A phone number \( d_{1} d_{2} d_{3}-d_{4} d_{5} d_{6} d_{7} \) is called "legal" if the number \( d_{1} d_{2} d_{3} \) is equal to \( d_{4} d_{5} d_{6} \) or to \( d_{5} d_{6} d_{7} \). For example, \( 234-2347 \) is a legal phone number. Assume each \( d_{i} \) can be any digit from 0 to 9. How many legal phone numbers exist? | 19990 | 6.25 |
29,116 | In the Cartesian coordinate system $(xOy)$, let the line $l: \begin{cases} x=2-t \\ y=2t \end{cases} (t \text{ is a parameter})$, and the curve $C_{1}: \begin{cases} x=2+2\cos \theta \\ y=2\sin \theta \end{cases} (\theta \text{ is a parameter})$. In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis:
(1) Find the polar equations of $C_{1}$ and $l$:
(2) Let curve $C_{2}: \rho=4\sin\theta$. The curve $\theta=\alpha(\rho > 0, \frac{\pi}{4} < \alpha < \frac{\pi}{2})$ intersects with $C_{1}$ and $C_{2}$ at points $A$ and $B$, respectively. If the midpoint of segment $AB$ lies on line $l$, find $|AB|$. | \frac{4\sqrt{10}}{5} | 3.90625 |
29,117 |
On a circular track with a perimeter of 360 meters, three individuals A, B, and C start from the same point: A starts first, running counterclockwise. Before A completes one lap, B and C start simultaneously, running clockwise. When A and B meet for the first time, C is exactly halfway between them. After some time, when A and C meet for the first time, B is also exactly halfway between them. If B's speed is four times that of A's, how many meters has A run when B and C started? | 90 | 1.5625 |
29,118 | Given the parametric equation of line $l$ is $\begin{cases} & x=1+3t \\ & y=2-4t \end{cases}$ (where $t$ is the parameter), calculate the cosine of the inclination angle of line $l$. | -\frac{3}{5} | 14.84375 |
29,119 | Given the letters a, b, c, d, e arranged in a row, find the number of arrangements where both a and b are not adjacent to c. | 36 | 18.75 |
29,120 | How many different 4-edge trips are there from $A$ to $B$ in a cube, where the trip can visit one vertex twice (excluding start and end vertices)? | 36 | 13.28125 |
29,121 | Find the number of pairs of natural numbers \((x, y)\) such that \(1 \leq x, y \leq 1000\) and \(x^2 + y^2\) is divisible by 5. | 200000 | 8.59375 |
29,122 | Given the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \dfrac{2\sqrt{3}}{3})$, with the left and right foci being $F_1$ and $F_2$ respectively. The circle $x^2 + y^2 = 2$ intersects with the line $x + y + b = 0$ at a chord of length $2$.
(Ⅰ) Find the standard equation of the ellipse $C$;
(Ⅱ) Let $Q$ be a moving point on the ellipse $C$ that is not on the $x$-axis, $O$ be the origin. A line parallel to $OQ$ passing through $F_2$ intersects the ellipse $C$ at two distinct points $M$ and $N$. Investigate whether the value of $\dfrac{|MN|}{|OQ|^2}$ is a constant. If it is, find this constant; if not, please explain why. | \dfrac{2\sqrt{3}}{3} | 5.46875 |
29,123 | Given the sequence $\{a_k\}_{k=1}^{11}$ of real numbers defined by $a_1=0.5$, $a_2=(0.51)^{a_1}$, $a_3=(0.501)^{a_2}$, $a_4=(0.511)^{a_3}$, and in general,
$a_k=\begin{cases}
(0.\underbrace{501\cdots 01}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is odd,} \\
(0.\underbrace{501\cdots 011}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is even.}
\end{cases}$
Rearrange the numbers in the sequence $\{a_k\}_{k=1}^{11}$ in decreasing order to produce a new sequence $\{b_k\}_{k=1}^{11}$. Find the sum of all integers $k$, $1\le k \le 11$, such that $a_k = b_k$. | 30 | 0 |
29,124 | Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, satisfying $2c+b-2a\cos B=0$.
$(1)$ Find angle $A$;
$(2)$ If $a=2\sqrt{3}$, $\overrightarrow{BA}\cdot \overrightarrow{AC}=\frac{3}{2}$, and $AD$ is the median of $\triangle ABC$, find the length of $AD$. | \frac{\sqrt{6}}{2} | 71.09375 |
29,125 | Find the number of different possible rational roots of the polynomial:
\[6x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 10 = 0.\] | 22 | 0 |
29,126 | The number of minutes in a week is closest to: | 10000 | 10.15625 |
29,127 | We define the polynomial $$ P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x. $$ Find the largest prime divisor of $P (2)$ . | 61 | 1.5625 |
29,128 | What is the least positive integer value of $x$ such that $(3x)^2 + 3 \cdot 29 \cdot 3x + 29^2$ is a multiple of 43? | 19 | 6.25 |
29,129 | Given:
$$ \frac{ \left( \frac{1}{3} \right)^2 + \left( \frac{1}{4} \right)^2 }{ \left( \frac{1}{5} \right)^2 + \left( \frac{1}{6} \right)^2} = \frac{37x}{73y} $$
Express $\sqrt{x} \div \sqrt{y}$ as a common fraction. | \frac{75 \sqrt{73}}{6 \sqrt{61} \sqrt{37}} | 0 |
29,130 | Consider real numbers $A$ , $B$ , \dots, $Z$ such that \[
EVIL = \frac{5}{31}, \;
LOVE = \frac{6}{29}, \text{ and }
IMO = \frac{7}{3}.
\] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$ , find the value of $m+n$ .
*Proposed by Evan Chen* | 579 | 0.78125 |
29,131 | Compute the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 2 = 0,\] given that all of the roots are real and nonnegative. | 46 | 14.0625 |
29,132 | Juca is a scout exploring the vicinity of his camp. After collecting fruits and wood, he needs to fetch water from the river and return to his tent. Represent Juca by the letter $J$, the river by the letter $r$, and his tent by the letter $B$. The distance from the feet of the perpendiculars $C$ and $E$ on $r$ from points $J$ and $B$ is $180 m$. What is the shortest distance Juca can travel to return to his tent, passing through the river? | 180\sqrt{2} | 29.6875 |
29,133 | What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le2$ and $|x|+|y|+|z-2|\le2$? | \frac{8}{3} | 3.125 |
29,134 | Given the ellipse $C\_1: \frac{x^2}{8} + \frac{y^2}{4} = 1$ with left and right foci $F\_1$ and $F\_2$, a line $l\_1$ is drawn through point $F\_1$ perpendicular to the $x$-axis. Line $l\_2$ is perpendicular to $l\_1$ at point $P$, and the perpendicular bisector of segment $PF\_2$ intersects $l\_2$ at point $M$.
(I) Find the equation of the trajectory $C\_2$ of point $M$;
(II) Two perpendicular lines $AC$ and $BD$ are drawn through point $F\_2$, intersecting ellipse $C\_1$ at points $A$, $C$ and $B$, $D$ respectively. Find the minimum value of the area of quadrilateral $ABCD$. | \frac{64}{9} | 0.78125 |
29,135 | A cuckoo clock chimes "cuckoo" on the hour, with the number of "cuckoo" calls equal to the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand with his finger until he moved the clock forward by 7 hours. How many times did the clock chime "cuckoo" during this period? | 43 | 1.5625 |
29,136 | Given that the angle between the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute, and for any $(x,y)$ that satisfies $|x\overrightarrow{a}+y\overrightarrow{b}|=1$ and $xy\geqslant 0$, the inequality $|x+2y|\leqslant \frac{8}{\sqrt{15}}$ holds. Find the minimum value of $\overrightarrow{a}\cdot\overrightarrow{b}$. | \frac{1}{4} | 3.125 |
29,137 | We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $ . | 25 | 39.0625 |
29,138 | Given that the terminal side of angle $\alpha$ ($0 < \alpha < \frac{\pi}{2}$) passes through the point $(\cos 2\beta, 1+\sin 3\beta \cos \beta - \cos 3\beta \sin \beta)$, where $\frac{\pi}{2} < \beta < \pi$ and $\beta \neq \frac{3\pi}{4}$, calculate $\alpha - \beta$. | -\frac{3\pi}{4} | 10.15625 |
29,139 | Vasya and Petya are participating in a school sports-entertainment game. They only have one pair of roller skates between the two of them, and need to traverse a distance of 3 km as quickly as possible. They start simultaneously, with one running and the other roller-skating. At any moment, the one on roller skates can leave them for the other and continue running without them. This exchange can occur as many times as desired. Find the minimum time for both friends to complete the distance (which is determined by who finishes last), given that Vasya's running and roller skating speeds are 4 km/h and 8 km/h respectively, while Petya's speeds are 5 km/h and 10 km/h. Assume that no time is lost during the exchange of roller skates. | 0.5 | 21.875 |
29,140 | Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | 20\sqrt{5} | 0.78125 |
29,141 | The maximum and minimum values of the function y=2x^3-3x^2-12x+5 on the interval [0,3] need to be determined. | -15 | 98.4375 |
29,142 | Given the sequence 2, $\frac{5}{3}$, $\frac{3}{2}$, $\frac{7}{5}$, $\frac{4}{3}$, ..., then $\frac{17}{15}$ is the \_\_\_\_\_ term in this sequence. | 14 | 74.21875 |
29,143 | A six-digit palindrome is a positive integer with respective digits $abcdcba$, where $a$ is non-zero. Let $T$ be the sum of all six-digit palindromes. Calculate the sum of the digits of $T$. | 20 | 0 |
29,144 | Given that the polynomial \(x^2 - kx + 24\) has only positive integer roots, find the average of all distinct possibilities for \(k\). | 15 | 98.4375 |
29,145 | If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 30$ and \[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{240}{pqr} = 1,\] compute $pqr$. | 1080 | 2.34375 |
29,146 | In a certain sequence, the first term is $a_1 = 101$ and the second term is $a_2 = 102$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = n + 2$ for all $n \geq 1$. Determine $a_{50}$. | 117 | 0.78125 |
29,147 | Given a polynomial with integer coefficients,
\[16x^5 + b_4x^4 + b_3x^3 + b_2x^2 + b_1x + 24 = 0,\]
find the number of different possible rational roots of this polynomial. | 32 | 68.75 |
29,148 | Find the values of the real number \( a \) such that all the roots of the polynomial in the variable \( x \),
\[ x^{3}-2x^{2}-25x+a \]
are integers. | -50 | 7.03125 |
29,149 | Consider a unit square $ABCD$ whose bottom left vertex is at the origin. A circle $\omega$ with radius $\frac{1}{3}$ is inscribed such that it touches the square's bottom side at point $M$. If $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M$, where $A$ is at the top left corner of the square, find the length of $AP$. | \frac{1}{3} | 7.8125 |
29,150 | In triangle \(ABC\), \(BK\) is the median, \(BE\) is the angle bisector, and \(AD\) is the altitude. Find the length of side \(AC\) if it is known that lines \(BK\) and \(BE\) divide segment \(AD\) into three equal parts and the length of \(AB\) is 4. | 2\sqrt{3} | 0 |
29,151 | A power locomotive's hourly electricity consumption cost is directly proportional to the cube of its speed. It is known that when the speed is $20$ km/h, the hourly electricity consumption cost is $40$ yuan. Other costs amount to $400$ yuan per hour. The maximum speed of the locomotive is $100$ km/h. At what speed should the locomotive travel to minimize the total cost of traveling from city A to city B? | 20 \sqrt[3]{5} | 0 |
29,152 | In the three-dimensional Cartesian coordinate system, the equation of the plane passing through $P(x_{0}, y_{0}, z_{0})$ with normal vector $\overrightarrow{m}=(a, b, c)$ is $a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$, and the equation of the line passing through $P(x_{0}, y_{0}, z_{0})$ with direction vector $\overrightarrow{n}=(A, B, C)$ is $\frac{x-x_{0}}{A}=\frac{y-y_{0}}{B}=\frac{z-z_{0}}{C}$. Read the above information and solve the following problem: Given the equation of the plane $\alpha$ as $3x+y-z-5=0$, and the equation of the line $l$ passing through the point $P(0,0,0)$ as $x=\frac{y}{2}=-z$, then one direction vector of the line $l$ is ______, and the cosine value of the angle between the line $l$ and the plane $\alpha$ is ______. | \frac{\sqrt{55}}{11} | 8.59375 |
29,153 | Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$ . | 18 | 0.78125 |
29,154 | Given that the radius of a sphere is 24cm, and the height of a cone is equal to the diameter of this sphere, and the surface area of the sphere is equal to the surface area of the cone, then the volume of this cone is \_\_\_\_\_\_ cm<sup>3</sup>. | 12288\pi | 8.59375 |
29,155 | How many whole numbers between 1 and 2000 do not contain the digits 1 or 2? | 1535 | 0 |
29,156 | Right triangle $ABC$ has one leg of length 9 cm, another leg of length 12 cm, and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express your answer as a common fraction. | \frac{45}{8} | 7.03125 |
29,157 | Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits. | 972538 | 94.53125 |
29,158 | There are 100 chips numbered from 1 to 100 placed in the vertices of a regular 100-gon in such a way that they follow a clockwise order. In each move, it is allowed to swap two chips located at neighboring vertices if their numbers differ by at most $k$. What is the smallest value of $k$ for which, by a series of such moves, it is possible to achieve a configuration where each chip is shifted one position clockwise relative to its initial position? | 50 | 81.25 |
29,159 | Define a sequence by \( a_0 = \frac{1}{3} \) and \( a_n = 1 + (a_{n-1} - 1)^3 \). Compute the infinite product \( a_0 a_1 a_2 \dotsm \). | \frac{3}{5} | 1.5625 |
29,160 | A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is at least half of the number of beetles in the tenth jar. How many beetles are in the sixth jar? | 16 | 9.375 |
29,161 | Let $\\((2-x)^5 = a_0 + a_1x + a_2x^2 + \ldots + a_5x^5\\)$. Evaluate the value of $\dfrac{a_0 + a_2 + a_4}{a_1 + a_3}$. | -\dfrac{122}{121} | 3.125 |
29,162 | Given the function $f(x)=(\sin x+\cos x)^{2}+2\cos ^{2}x$,
(1) Find the smallest positive period and the monotonically decreasing interval of the function $f(x)$;
(2) When $x\in[0, \frac{\pi}{2}]$, find the maximum and minimum values of $f(x)$. | 2+\sqrt{2} | 0 |
29,163 | Find the least positive integer $n$ such that the prime factorizations of $n$ , $n + 1$ , and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not). | 33 | 99.21875 |
29,164 | Li Qiang rented a piece of land from Uncle Zhang, for which he has to pay Uncle Zhang 800 yuan and a certain amount of wheat every year. One day, he did some calculations: at that time, the price of wheat was 1.2 yuan per kilogram, which amounted to 70 yuan per mu of land; but now the price of wheat has risen to 1.6 yuan per kilogram, so what he pays is equivalent to 80 yuan per mu of land. Through Li Qiang's calculations, you can find out how many mu of land this is. | 20 | 9.375 |
29,165 | Let $n$ and $k$ be integers satisfying $\binom{2k}{2} + n = 60$ . It is known that $n$ days before Evan's 16th birthday, something happened. Compute $60-n$ .
*Proposed by Evan Chen* | 45 | 7.8125 |
29,166 | Rectangle ABCD and right triangle AEF share side AD and have the same area. Side AD = 8, and side AB = 7. If EF, which is perpendicular to AD, is denoted as x, determine the length of hypotenuse AF. | 2\sqrt{65} | 92.1875 |
29,167 | An acute isosceles triangle, \( ABC \), is inscribed in a circle. Through \( B \) and \( C \), tangents to the circle are drawn, meeting at point \( D \). If \( \angle ABC = \angle ACB = 3 \angle D \) and \( \angle BAC = k \pi \) in radians, then find \( k \). | \frac{5}{11} | 7.03125 |
29,168 | Given a pyramid with a vertex and base ABCD, each vertex is painted with one color, ensuring that two vertices on the same edge are of different colors. There are 5 different colors available. Calculate the total number of distinct coloring methods. (Answer with a number) | 420 | 3.125 |
29,169 | Given $f(x)=x^{2}-ax$, $g(x)=\ln x$, $h(x)=f(x)+g(x)$,
(1) Find the range of values for the real number $a$ such that $f(x) \geq g(x)$ holds true for any $x$ within their common domain;
(2) Suppose $h(x)$ has two critical points $x_{1}$, $x_{2}$, with $x_{1} \in (0, \frac{1}{2})$, and if $h(x_{1}) - h(x_{2}) > m$ holds true, find the maximum value of the real number $m$. | \frac{3}{4} - \ln 2 | 1.5625 |
29,170 | Given an isosceles trapezoid \(ABCD\), where \(AD \parallel BC\), \(BC = 2AD = 4\), \(\angle ABC = 60^\circ\), and \(\overrightarrow{CE} = \frac{1}{3} \overrightarrow{CD}\), calculate the value of \(\overrightarrow{CA} \cdot \overrightarrow{BE}\). | -10 | 0.78125 |
29,171 | Compute the value of the following expression:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))))). \] | 2046 | 32.8125 |
29,172 | Points \(A\) and \(B\) are connected by two arcs of circles, convex in opposite directions: \(\cup A C B = 117^\circ 23'\) and \(\cup A D B = 42^\circ 37'\). The midpoints \(C\) and \(D\) of these arcs are connected to point \(A\). Find the angle \(C A D\). | 80 | 0 |
29,173 | Let \(a,\) \(b,\) \(c,\) \(d,\) \(e,\) \(f,\) \(g,\) and \(h\) be real numbers such that \(abcd = 8\) and \(efgh = 16.\) Find the minimum value of
\[
(ae)^2 + (bf)^2 + (cg)^2 + (dh)^2.
\] | 32 | 42.1875 |
29,174 | Determine the value of $l$ for which
\[\frac{9}{x + y + 1} = \frac{l}{x + z - 1} = \frac{13}{z - y + 2}.\] | 22 | 1.5625 |
29,175 | Let $x_{0}$ be a zero of the function $f(x)=\sin \pi x$, and suppose it satisfies $|x_{0}| + f\left(x_{0}+ \frac{1}{2}\right) < 11$. Calculate the number of such zeros. | 21 | 46.09375 |
29,176 | A regular tetrahedron has two spheres, one inscribed within it and the other circumscribed around it. Between each face of the tetrahedron and the circumscribed sphere, there are four smaller spheres centered on each face. Given a point \( P \) within the circumscribed sphere, the probability that \( P \) lies within one of the five smaller spheres is closest to: | 0.2 | 6.25 |
29,177 | An influenza outbreak occurred in regions $A$, $B$, and $C$. The percentages of people with influenza in these three regions are $6\%$, $5\%$, and $4\%$ respectively. It is assumed that the populations of these three regions are in the ratio of $3:5:2$. Now, a person is randomly selected from these three regions.
$(1)$ Find the probability that this person has influenza.
$(2)$ If this person has influenza, find the probability that this person is from region $A$. | \frac{18}{51} | 0 |
29,178 | Teacher Tan awarded a stack of exercise books to the students who were named "Outstanding Students" in the math Olympiad class. If each student is awarded 3 books, there are 7 books left over; if each student is awarded 5 books, there are 9 books short. How many students received the award? How many exercise books are there in total? | 31 | 66.40625 |
29,179 | Semicircles of diameter 4 inches are aligned in a linear pattern, with a second row staggered under the first such that the flat edges of the semicircles in the second row touch the midpoints of the arcs in the first row. What is the area, in square inches, of the shaded region in an 18-inch length of this pattern? Express your answer in terms of $\pi$. | 16\pi | 49.21875 |
29,180 | For every integers $ a,b,c$ whose greatest common divisor is $n$ , if
\[ \begin{array}{l} {x \plus{} 2y \plus{} 3z \equal{} a}
{2x \plus{} y \minus{} 2z \equal{} b}
{3x \plus{} y \plus{} 5z \equal{} c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number $ n$ ? | 28 | 5.46875 |
29,181 | In the quadrilateral pyramid \( P-ABCD \), it is given that \( AB \parallel CD \), \( AB \perp AD \), \( AB = 4 \), \( AD = 2\sqrt{2} \), \( CD = 2 \), and \( PA \perp \) plane \( ABCD \) with \( PA = 4 \). Let \( Q \) be a point on the segment \( PB \), and the sine of the angle between the line \( QC \) and the plane \( PAC \) is \( \frac{\sqrt{3}}{3} \). Find the value of \( \frac{PQ}{PB} \). | 1/2 | 22.65625 |
29,182 | Let \( A, B, C, D \) be four points in space that are not coplanar. Each pair of points is connected by an edge with a probability of \( \frac{1}{2} \), and whether or not there is an edge between any two pairs of points is independent of the others. Determine the probability that \( A \) and \( B \) can be connected by a path (composed of one or multiple edges). | \frac{3}{4} | 7.03125 |
29,183 |
A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is $810 \mathrm{~cm}^{2}$. | 36 | 6.25 |
29,184 | The integers $a$ , $b$ , $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$ . Determine the largest possible value of $d$ , | 2016 | 0.78125 |
29,185 | Given real numbers \( x \) and \( y \) that satisfy \( x^{2} + y^{2} \leq 5 \), find the maximum and minimum values of the function \( f(x, y) = 3|x+y| + |4y+9| + |7y-3x-18| \). | 27 + 6\sqrt{5} | 0 |
29,186 | In a right circular cone ($S-ABC$), $SA =2$, the midpoints of $SC$ and $BC$ are $M$ and $N$ respectively, and $MN \perp AM$. Determine the surface area of the sphere that circumscribes the right circular cone ($S-ABC$). | 12\pi | 3.90625 |
29,187 | Given the ellipse $C$: $mx^{2}+3my^{2}=1$ ($m > 0$) with a major axis length of $2\sqrt{6}$, and $O$ is the origin.
$(1)$ Find the equation of the ellipse $C$.
$(2)$ Let point $A(3,0)$, point $B$ be on the $y$-axis, and point $P$ be on the ellipse $C$ and to the right of the $y$-axis. If $BA=BP$, find the minimum value of the area of quadrilateral $OPAB$. | 3\sqrt{3} | 1.5625 |
29,188 | In a cylinder with a base radius of 6, there are two spheres, each with a radius of 6, and the distance between their centers is 13. If a plane is tangent to these two spheres and intersects the cylindrical surface forming an ellipse, what is the sum of the lengths of the major axis and minor axis of this ellipse? | 25 | 19.53125 |
29,189 | Evaluate \(\lim_{n \to \infty} \frac{1}{n^5} \sum (5r^4 - 18r^2s^2 + 5s^4)\), where the sum is over all \(r, s\) satisfying \(0 < r, s \leq n\). | -1 | 21.09375 |
29,190 | Print 90,000 five-digit numbers
$$
10000, 10001, \cdots, 99999
$$
on cards, with each card displaying one five-digit number. Some numbers printed on the cards (e.g., 19806 when reversed reads 90861) can be read in two different ways and may cause confusion. How many cards will display numbers that do not cause confusion? | 89100 | 55.46875 |
29,191 | Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is *monotonically bounded* if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$ . We say that a term $a_k$ in the sequence with $2\leq k\leq 2^{16}-1$ is a *mountain* if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$ . Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$ .
*Proposed by Michael Ren* | 49153 | 0 |
29,192 | Given $A(-1,\cos \theta)$, $B(\sin \theta,1)$, if $|\overrightarrow{OA}+\overrightarrow{OB}|=|\overrightarrow{OA}-\overrightarrow{OB}|$, then find the value of the acute angle $\theta$. | \frac{\pi}{4} | 74.21875 |
29,193 | A three-digit number has digits a, b, and c in the hundreds, tens, and units place respectively. If a < b and b > c, then the number is called a "convex number". If you randomly select three digits from 1, 2, 3, and 4 to form a three-digit number, what is the probability that it is a "convex number"? | \frac{1}{3} | 53.90625 |
29,194 | Given School A and School B each have 3 teachers signing up for volunteer teaching, with School A having 2 males and 1 female, and School B having 1 male and 2 females, calculate the probability that the selected 2 teachers have the same gender. | \frac{4}{9} | 14.0625 |
29,195 | Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color, where there are $7$ colors to choose from. Find the number of possible different colorings such that both the ends of each diagonal and each pair of adjacent vertices have different colors. | 5040 | 7.03125 |
29,196 | Given the letters in the word $SUCCESS$, determine the number of distinguishable rearrangements where all the vowels are at the end. | 20 | 6.25 |
29,197 | In this Number Wall, you add the numbers next to each other and write the sum in the block directly above the two numbers. Which number will be in the block labeled '$m$'? [asy]
draw((0,0)--(8,0)--(8,2)--(0,2)--cycle);
draw((2,0)--(2,2));
draw((4,0)--(4,2));
draw((6,0)--(6,2));
draw((1,2)--(7,2)--(7,4)--(1,4)--cycle);
draw((3,2)--(3,4));
draw((5,2)--(5,4));
draw((2,4)--(2,6)--(6,6)--(6,4)--cycle);
draw((4,4)--(4,6));
draw((3,6)--(3,8)--(5,8)--(5,6));
label("$m$",(1,1));
label("3",(3,1));
label("9",(5,1));
label("6",(7,1));
label("16",(6,3));
label("54",(4,7));
[/asy] | 12 | 39.84375 |
29,198 | Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers. | 17 | 8.59375 |
29,199 | A tetrahedron is formed using the vertices of a cube. How many such distinct tetrahedrons can be formed? | 58 | 67.1875 |
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