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22,500 | Let $M = 36 \cdot 36 \cdot 85 \cdot 128$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. | \frac{1}{4094} | 28.125 |
22,501 | Given an ellipse $C$: $\frac{x^2}{16}+\frac{y^2}{4}=1$, with the left and right foci being $F_{1}$ and $F_{2}$, respectively. Line $l$ intersects the ellipse at points $A$ and $B$, where the chord $AB$ is bisected by the point $(\sqrt{3},\frac{\sqrt{3}}{2})$.
$(1)$ Find the equation of line $l$;
$(2)$ Find the area of $\triangle F_{1}AB$. | 2\sqrt{15} | 5.46875 |
22,502 | For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $500$ that are neither $3$-nice nor $5$-nice. | 266 | 0 |
22,503 | Prisha writes down one integer three times and another integer two times, with their sum being $105$, and one of the numbers is $15$. Calculate the other number. | 30 | 26.5625 |
22,504 | In the geometric sequence $\{a_n\}$, $S_4=1$, $S_8=3$, then the value of $a_{17}+a_{18}+a_{19}+a_{20}$ is. | 16 | 57.03125 |
22,505 | Calculate $[(6^{6} \div 6^{5})^3 \cdot 8^3] \div 4^3$. | 1728 | 84.375 |
22,506 | How many multiples of 5 are there between 105 and 500? | 79 | 4.6875 |
22,507 | A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number? | 99 | 4.6875 |
22,508 | For real numbers $t,$ consider the point of intersection of the triplet of lines $3x - 2y = 8t - 5$, $2x + 3y = 6t + 9$, and $x + y = 2t + 1$. All the plotted points lie on a line. Find the slope of this line. | -\frac{1}{6} | 18.75 |
22,509 | Alice's favorite number is between $90$ and $150$. It is a multiple of $13$, but not a multiple of $4$. The sum of its digits should be a multiple of $4$. What is Alice's favorite number? | 143 | 40.625 |
22,510 | $ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ .
| 30 | 11.71875 |
22,511 | Eleven positive integers from a list of fifteen positive integers are $3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23$. What is the largest possible value of the median of this list of fifteen positive integers? | 17 | 30.46875 |
22,512 | Let the coefficient of $x^{-4}$ in the expansion of $\left(1- \frac {1}{x^{2}}\right)^{n}$ (where $n\in\mathbb{N}_{+}$) be denoted as $a_{n}$. Calculate the value of $$\frac {1}{a_{2}}+ \frac {1}{a_{3}}+…+ \frac {1}{a_{2015}}$$. | \frac {4028}{2015} | 98.4375 |
22,513 | The area of the base of a hemisphere is $144\pi$. A cylinder of the same radius as the hemisphere and height equal to the radius of the hemisphere is attached to its base. What is the total surface area of the combined solid (hemisphere + cylinder)? | 576\pi | 34.375 |
22,514 | Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction. | \frac{1}{12} | 87.5 |
22,515 | Assume that the scores $X$ of 400,000 students in a math mock exam in Yunnan Province approximately follow a normal distribution $N(98,100)$. It is known that a student's score ranks among the top 9100 in the province. Then, the student's math score will not be less than ______ points. (Reference data: $P(\mu -\sigma\ \ \lt X \lt \mu +\sigma )=0.6827, P(\mu -2\sigma\ \ \lt X \lt \mu +2\sigma )=0.9545$) | 118 | 75 |
22,516 | Given the circle $x^{2}-2x+y^{2}-2y+1=0$, find the cosine value of the angle between the two tangents drawn from the point $P(3,2)$. | \frac{3}{5} | 46.09375 |
22,517 | A classroom has 9 desks arranged in a row. Two students, Alex and Jessie, choose their desks at random. What is the probability that they do not sit next to each other? | \frac{7}{9} | 76.5625 |
22,518 | Calculate the surface integrals of the first kind:
a) \(\iint_{\sigma}|x| dS\), where \(\sigma\) is defined by \(x^2 + y^2 + z^2 = 1\), \(z \geqslant 0\).
b) \(\iint_{\sigma} (x^2 + y^2) dS\), where \(\sigma\) is defined by \(x^2 + y^2 = 2z\), \(z = 1\).
c) \(\iint_{\sigma} (x^2 + y^2 + z^2) dS\), where \(\sigma\) is the part of the cone defined by \(z^2 - x^2 - y^2 = 0\), \(z \geqslant 0\), truncated by the cylinder \(x^2 + y^2 - 2x = 0\). | 3\sqrt{2} \pi | 9.375 |
22,519 | During the Qingming Festival, a certain school, in order to commemorate the revolutionary martyrs, requires students to participate in the "Qingming Sacrifice to the Martyrs" activity by either visiting the Revolutionary Martyrs Memorial Hall or participating online. Students can only choose one way to participate. It is known that the ratio of the number of students in the three grades of the middle school, Grade 7, Grade 8, and Grade 9, is $4:5:6$. In order to understand the way students participate in the "Qingming Sacrifice to the Martyrs" activity, a stratified sampling method is used for investigation, and the following data is obtained:
| Grade and Participation Method | Grade 7 | Grade 8 | Grade 9 |
|-------------------------------|---------|---------|---------|
| Visit Memorial Hall | $2a-1$ | $8$ | $10$ |
| Online Participation | $a$ | $b$ | $2$ |
$(1)$ Find the values of $a$ and $b$;
$(2)$ From the students surveyed in each grade of the school who chose to participate in the "Qingming Sacrifice to the Martyrs" activity online, randomly select two students. Find the probability that these two students are from the same grade. | \frac{5}{21} | 44.53125 |
22,520 | A regular octagon's perimeter is given as $P=16\sqrt{2}$. If $R_i$ denotes the midpoint of side $V_iV_{i+1}$ (with $V_8V_1$ as the last side), calculate the area of the quadrilateral $R_1R_3R_5R_7$.
A) $8$
B) $4\sqrt{2}$
C) $10$
D) $8 + 4\sqrt{2}$
E) $12$ | 8 + 4\sqrt{2} | 12.5 |
22,521 | If the line $l_1: x + ay + 6 = 0$ is parallel to the line $l_2: (a-2)x + 3y + 2a = 0$, calculate the distance between lines $l_1$ and $l_2$. | \frac{8\sqrt{2}}{3} | 85.9375 |
22,522 | Solve the equations:
(1) $(x-3)^2+2x(x-3)=0$
(2) $x^2-4x+1=0$. | 2-\sqrt{3} | 0 |
22,523 | In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta = 4$.
$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line segment $OM$, and satisfies $|OP| \cdot |OM| = 16$. Find the rectangular coordinate equation of the locus $C_{2}$ of point $P$.
$(2)$ Suppose the polar coordinates of point $A$ are $({2, \frac{π}{3}})$, point $B$ lies on the curve $C_{2}$. Find the maximum value of the area of $\triangle OAB$. | 2 + \sqrt{3} | 34.375 |
22,524 | In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$ respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at point $R$, with $BR = 8$ and $PR = 9$. If $\triangle BRP$ is a right triangle with $\angle BRP = 90^\circ$, what is the area of the square $ABCD$?
A) 144
B) 169
C) 225
D) 256
E) 289 | 225 | 18.75 |
22,525 | Compute: $104 \times 96$. | 9984 | 100 |
22,526 | Suppose there are 15 dogs including Rex and Daisy. We need to divide them into three groups of sizes 6, 5, and 4. How many ways can we form the groups such that Rex is in the 6-dog group and Daisy is in the 4-dog group? | 72072 | 0 |
22,527 | Let \(A\) and \(B\) be two moving points on the ellipse \(x^2 + 3y^2 = 1\), and \(OA \perp OB\) (where \(O\) is the origin). Find the product of the maximum and minimum values of \( |AB| \). | \frac{2 \sqrt{3}}{3} | 32.8125 |
22,528 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively. The radius of the circumcircle of $\triangle ABC$ is $1$, and $b = acosC - \frac{{\sqrt{3}}}{6}ac$.
$(Ⅰ)$ Find the value of $a$;
$(Ⅱ)$ If $b = 1$, find the area of $\triangle ABC$. | \frac{\sqrt{3}}{4} | 69.53125 |
22,529 | Given that the function $f(x)$ defined on $\mathbb{R}$ satisfies $f(4)=2-\sqrt{3}$, and for any $x$, $f(x+2)=\frac{1}{-f(x)}$, find $f(2018)$. | -2-\sqrt{3} | 44.53125 |
22,530 | Let O be the center of the square ABCD. If 3 points are chosen from O, A, B, C, and D at random, find the probability that the 3 points are collinear. | \frac{1}{5} | 53.125 |
22,531 | An experimenter is conducting an experiment that involves implementing five procedures in sequence. Procedure A must only occur either as the first or the last step, and procedures C and D must be implemented consecutively. The number of possible arrangements for the sequence of these procedures is _______. | 24 | 67.1875 |
22,532 | Given an ellipse $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left focus $F_{1}$ and right focus $F_{2}$, and the focal distance $F_{1}F_{2}$ is $2$. A line passing through $F_{1}$ intersects the ellipse $E$ at points $A$ and $B$, and the perimeter of $\triangle ABF_{2}$ is $4\sqrt{3}$.
$(1)$ Find the equation of the ellipse $E$;
$(2)$ If the slope of line $AB$ is $2$, find the area of $\triangle ABF_{2}$. | \frac{4\sqrt{15}}{7} | 39.84375 |
22,533 | Given a geometric sequence $\{a_{n}\}$, the sum of the first n terms is $S_{n}$, $S_{2}=7$ and $S_{6}=91$. Calculate $S_{4}$. | 28 | 57.03125 |
22,534 | Given the lines $l_1: ax+2y-1=0$ and $l_2: 8x+ay+2-a=0$, if $l_1 \parallel l_2$, find the value of the real number $a$. | -4 | 50.78125 |
22,535 | A traffic light follows a cycle of green for 45 seconds, yellow for 5 seconds, and red for 40 seconds. Sam observes the light for a random five-second interval. What is the probability that the light changes from one color to another during his observation? | \frac{1}{6} | 56.25 |
22,536 | The number 2015 can be represented as a sum of consecutive integers in several ways, for example, $2015 = 1007 + 1008$ or $2015 = 401 + 402 + 403 + 404 + 405$. How many ways can this be done? | 16 | 42.1875 |
22,537 | Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (average value) of $6^D$ ? | 10071 | 10.15625 |
22,538 | For a positive integer $n>1$ , let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$ . For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$ . Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$ . Find the largest integer not exceeding $\sqrt{N}$ | 19 | 45.3125 |
22,539 | A pen costs $\mathrm{Rs.}\, 13$ and a note book costs $\mathrm{Rs.}\, 17$ . A school spends exactly $\mathrm{Rs.}\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ note books such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018-19$ , the school spends a little more than $\mathrm{Rs.}\, 10000$ and buys $y$ pens and $x$ note books. How much **more** did the school pay? | 40 | 21.875 |
22,540 | The shape shown is made up of three similar right-angled triangles. The smallest triangle has two sides of side-length 2, as shown. What is the area of the shape? | 14 | 38.28125 |
22,541 | A projectile is launched with an initial velocity of $u$ at an angle of $\phi$ from the horizontal. The trajectory of the projectile is given by the parametric equations:
\[
x = ut \cos \phi,
\]
\[
y = ut \sin \phi - \frac{1}{2} gt^2,
\]
where $t$ is time and $g$ is the acceleration due to gravity. Suppose $u$ is constant but $\phi$ varies from $0^\circ$ to $180^\circ$. As $\phi$ changes, the highest points of the trajectories trace a closed curve. The area enclosed by this curve can be expressed as $d \cdot \frac{u^4}{g^2}$. Find the value of $d$. | \frac{\pi}{8} | 46.09375 |
22,542 | Tio Mané has two boxes, one with seven distinct balls numbered from 1 to 7 and another with eight distinct balls numbered with all prime numbers less than 20. He draws one ball from each box.
Calculate the probability that the product is odd. What is the probability that the product of the numbers on the drawn balls is even? | \frac{1}{2} | 83.59375 |
22,543 | What is the result of subtracting $7.305$ from $-3.219$? | -10.524 | 100 |
22,544 | A person rolled a fair six-sided die $100$ times and obtained a $6$ $19$ times. What is the approximate probability of rolling a $6$? | 0.19 | 28.125 |
22,545 | Determine the fifth-largest divisor of 2,500,000,000. | 156,250,000 | 0 |
22,546 | What is the total volume and the total surface area in square feet of three cubic boxes if their edge lengths are 3 feet, 5 feet, and 6 feet, respectively? | 420 | 27.34375 |
22,547 | Given $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$, where $\alpha \in (0, \pi)$, calculate the value of $\sin\left(\alpha + \frac{\pi}{12}\right)$. | \frac{2\sqrt{2} + \sqrt{3}}{6} | 0 |
22,548 | Given the parametric equation of line $l$ as $$\begin{cases} x= \sqrt {3}+t \\ y=7+ \sqrt {3}t\end{cases}$$ ($t$ is the parameter), a coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho \sqrt {a^{2}\sin^{2}\theta+4\cos^{2}\theta}=2a$ ($a>0$).
1. Find the Cartesian equation of curve $C$.
2. Given point $P(0,4)$, line $l$ intersects curve $C$ at points $M$ and $N$. If $|PM|\cdot|PN|=14$, find the value of $a$. | \frac{2\sqrt{21}}{3} | 1.5625 |
22,549 | Given that $| \overrightarrow{a}|=5$, $| \overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ on vector $\overrightarrow{b}$. | -4 | 35.9375 |
22,550 | Compute \[ \left\lfloor \dfrac {1007^3}{1005 \cdot 1006} - \dfrac {1005^3}{1006 \cdot 1007} + 5 \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$ | 12 | 0 |
22,551 | Given an isosceles triangle $ABC$ with $AB = AC = 40$ units and $BC = 24$ units, let $CX$ be the angle bisector of $\angle BCA$. Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$. Provide your answer as a simplified fraction. | \frac{3}{5} | 59.375 |
22,552 | Given $\sqrt[3]{2.37} \approx 1.333$ and $\sqrt[3]{23.7} \approx 2.872$, determine the approximate value of $\sqrt[3]{2370}$. | 13.33 | 56.25 |
22,553 | For all positive reals $ a$ , $ b$ , and $ c$ , what is the value of positive constant $ k$ satisfies the following inequality?
$ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ . | 6020 | 94.53125 |
22,554 | The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$ | 105 | 6.25 |
22,555 | If the function $f(x)=\frac{1}{3}x^{3}-\frac{3}{2}x^{2}+ax+4$ is strictly decreasing on the interval $[-1,4]$, then the value of the real number $a$ is ______. | -4 | 65.625 |
22,556 | In the expression $10 \square 10 \square 10 \square 10 \square 10$, fill in the four spaces with each of the operators "+", "-", "×", and "÷" exactly once. The maximum possible value of the resulting expression is: | 109 | 5.46875 |
22,557 | Given that Josie jogs parallel to a canal along which a boat is moving at a constant speed in the same direction and counts 130 steps to reach the front of the boat from behind it, and 70 steps from the front to the back, find the length of the boat in terms of Josie's steps. | 91 | 40.625 |
22,558 | In a sequence of positive integers starting from 1, certain numbers are painted red according to the following rules: First paint 1, then the next 2 even numbers $2, 4$; then the next 3 consecutive odd numbers after 4, which are $5, 7, 9$; then the next 4 consecutive even numbers after 9, which are $10, 12, 14, 16$; then the next 5 consecutive odd numbers after 16, which are $17, 19, 21, 23, 25$. Following this pattern, we get a red subsequence $1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, \cdots$. What is the 2003rd number in this red subsequence? | 3943 | 0.78125 |
22,559 | Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order). | 2: 3: 4 | 65.625 |
22,560 | A community plans to organize three activities, "Book Club," "Fun Sports," and "Environmental Theme Painting," to enrich the lives of residents. A total of 120 people have signed up for the activities, with each resident participating in at most two activities. It is known that 80 people participate in the "Book Club," 50 people in "Fun Sports," and 40 people in "Environmental Theme Painting." Additionally, 20 people participate in both the "Book Club" and "Fun Sports," and 10 people participate in both "Fun Sports" and "Environmental Theme Painting." Find the number of people who participate in both the "Book Club" and "Environmental Theme Painting." | 20 | 89.0625 |
22,561 | Given the parameter equation of line $l$ and the equation of circle $C$ in the polar coordinate system, find the rectangular coordinate equation of circle $C$ and the minimum value of $\frac{1}{|PA|} + \frac{1}{|PB|}$, where $P(1, 2)$ and $A$, $B$ are the intersection points of line $l$ and circle $C$.
The parameter equation of line $l$ in the rectangular coordinate system is $\begin{cases} x = 1 + t\cos\alpha\\ y = 2 + t\sin\alpha \end{cases}$ ($t$ is the parameter), and the equation of circle $C$ in the polar coordinate system (with the same unit length and origin as the rectangular coordinate system, and the positive $x$-axis as the polar axis) is $\rho = 6\sin\theta$. | \frac{2\sqrt{7}}{7} | 74.21875 |
22,562 | In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b\tan A = (2c-b)\tan B$.
$(1)$ Find angle $A$;
$(2)$ If $\overrightarrow{m}=(0,-1)$ and $\overrightarrow{n}=(\cos B, 2\cos^2\frac{C}{2})$, find the minimum value of $|\overrightarrow{m}+\overrightarrow{n}|$. | \frac{\sqrt{2}}{2} | 36.71875 |
22,563 | During the November monthly exam at our school, approximately 1,000 science students participated, with mathematics scores distributed normally as $\xi \sim N(100, a^2), (a > 0)$, and a full score of 150. The statistics showed that about 60% of the students scored between 80 and 120 points. Therefore, approximately \_\_\_\_\_\_ students scored no less than 120 points in this monthly exam. | 200 | 93.75 |
22,564 | Given that $A$ is an interior angle of $\triangle ABC$, when $x= \frac {5\pi}{12}$, the function $f(x)=2\cos x\sin (x-A)+\sin A$ attains its maximum value. The sides opposite to the angles $A$, $B$, $C$ of $\triangle ABC$ are $a$, $b$, $c$ respectively.
$(1)$ Find the angle $A$;
$(2)$ If $a=7$ and $\sin B + \sin C = \frac {13 \sqrt {3}}{14}$, find the area of $\triangle ABC$. | 10\sqrt{3} | 50 |
22,565 | Let point P be a fixed point inside a circle ⊙O with a radius of 5, and OP=4. The sum of all possible integer values of the chord lengths passing through point P is. | 40 | 57.8125 |
22,566 | A school wishes to understand the psychological state of learning among its senior students and adopts a systematic sampling method to select 40 students out of 800 for a test. The students are randomly assigned numbers from 1 to 800 and then grouped. In the first group, number 18 is selected through simple random sampling. Among the 40 selected students, those with numbers in the range [1, 200] take test paper A, numbers in the range [201, 560] take test paper B, and the remaining students take test paper C. Calculate the number of students who take test paper C. | 12 | 46.09375 |
22,567 | Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her third wave? | 31 | 28.90625 |
22,568 | If a sequence's sum of the first $n$ terms, $S_n = 1 - 5 + 9 - \ldots + (-1)^{n+1}(4n - 3)$ for $n \in \mathbb{N}^*$, find the value of $S_{15} - S_{22} + S_{31}$. | 134 | 47.65625 |
22,569 | Vasya has 9 different books by Arkady and Boris Strugatsky, each containing a single work by the authors. Vasya wants to arrange these books on a shelf in such a way that:
(a) The novels "Beetle in the Anthill" and "Waves Extinguish the Wind" are next to each other (in any order).
(b) The stories "Restlessness" and "A Story About Friendship and Non-friendship" are next to each other (in any order).
In how many ways can Vasya do this?
Choose the correct answer:
a) \(4 \cdot 7!\);
b) \(9!\);
c) \(\frac{9!}{4!}\);
d) \(4! \cdot 7!\);
e) another answer. | 4 \cdot 7! | 99.21875 |
22,570 | Find the result of $46_8 - 63_8$ and express your answer in base 10. | -13 | 91.40625 |
22,571 | Given that $F$ is the focus of the parabola $x^{2}=8y$, $P$ is a moving point on the parabola, and the coordinates of $A$ are $(0,-2)$, find the minimum value of $\frac{|PF|}{|PA|}$. | \frac{\sqrt{2}}{2} | 14.84375 |
22,572 | Let \( x \) be a real number such that \( x + \frac{1}{x} = 5 \). Define \( S_m = x^m + \frac{1}{x^m} \). Determine the value of \( S_6 \). | 12098 | 63.28125 |
22,573 | Inside the square \(ABCD\), points \(K\) and \(M\) are marked (point \(M\) is inside triangle \(ABD\), point \(K\) is inside \(BMC\)) such that triangles \(BAM\) and \(DKM\) are congruent \((AM = KM, BM = MD, AB = KD)\). Find \(\angle KCM\) if \(\angle AMB = 100^\circ\). | 35 | 2.34375 |
22,574 | In a class of 120 students, the teacher recorded the following scores for an exam. Calculate the average score for the class.
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{Score (\%)}&\textbf{Number of Students}\\\hline
95&12\\\hline
85&24\\\hline
75&30\\\hline
65&20\\\hline
55&18\\\hline
45&10\\\hline
35&6\\\hline
\end{tabular} | 69.83 | 0.78125 |
22,575 | Given the equations $z^2 = 1 + 3\sqrt{10}i$ and $z^2 = 2 - 2\sqrt{2}i$, where $i = \sqrt{-1}$, find the vertices formed by the solutions of these equations on the complex plane and compute the area of the quadrilateral they form.
A) $17\sqrt{6} - 2\sqrt{2}$
B) $18\sqrt{6} - 3\sqrt{2}$
C) $19\sqrt{6} - 2\sqrt{2}$
D) $20\sqrt{6} - 2\sqrt{2}$ | 19\sqrt{6} - 2\sqrt{2} | 4.6875 |
22,576 | For real numbers \( x \) and \( y \), simplify the equation \(\cfrac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} + 2\frac{1}{y}} = 4\) and express it as \(\frac{x+y}{x+2y}\). | \frac{4}{11} | 57.03125 |
22,577 | Determine the minimum number of digits to the right of the decimal point required to express the fraction $\frac{987654321}{2^{30} \cdot 5^5}$ as a decimal. | 30 | 53.125 |
22,578 |
Out of 100 externally identical marbles, one is radioactive, but I don't know which one it is. A friend of mine would buy only non-radioactive marbles from me, at a price of 1 forint each. Another friend of mine has an instrument that can determine whether or not there is a radioactive marble among any number of marbles. He charges 1 forint per measurement, but if there is a radioactive marble among those being measured, all of the marbles in the measurement will become radioactive.
What is the maximum profit I can absolutely achieve? | 92 | 55.46875 |
22,579 | Given the equation of line $l$ is $ax+by+c=0$, where $a$, $b$, and $c$ form an arithmetic sequence, the maximum distance from the origin $O$ to the line $l$ is ______. | \sqrt{5} | 21.875 |
22,580 | In the diagram, \( Z \) lies on \( XY \) and the three circles have diameters \( XZ \), \( ZY \), and \( XY \). If \( XZ = 12 \) and \( ZY = 8 \), calculate the ratio of the area of the shaded region to the area of the unshaded region. | \frac{12}{13} | 92.96875 |
22,581 | Let $p$, $q$, and $r$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Find the value of $(1+p)(1+q)(1+r)$. | 51 | 94.53125 |
22,582 | Consider finding the result when we compute the series $$1^3 + 2^3 + 3^3 + \dots + 49^3 + 50^3$$ and the series $$(-1)^3 + (-2)^3 + (-3)^3 + \dots + (-49)^3 + (-50)^3,$$ then subtract the second series' result from the first series' result. What is the sum? | 3251250 | 83.59375 |
22,583 | Calculate the probability of selecting the letter "$s$" in the word "statistics". | \frac{3}{10} | 10.15625 |
22,584 | A rectangle in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Compute the radius $d$ to the nearest tenth such that the probability the point is within $d$ units from any lattice point is $\tfrac{1}{4}$. | 0.3 | 28.90625 |
22,585 | Let $T = TNFTPP$ . As $n$ ranges over the integers, the expression $n^4 - 898n^2 + T - 2160$ evaluates to just one prime number. Find this prime.
[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question. For those who wanted to try the problem separately, <details><summary>here's the value of T</summary>$T=2161$</details>. | 1801 | 53.90625 |
22,586 | Given that the sum of the first $n$ terms ($S_n$) of the sequence $\{a_n\}$ satisfies $S_n = 2a_n - 1$ ($n \in \mathbb{N}^*$).
(1) Find the general term formula of the sequence $\{a_n\}$;
(2) If the sequence $\{b_n\}$ satisfies $b_n = 1 + \log_2 a_n$,
(I) Find the sum of the first $n$ terms ($T_n$) of the sequence $\{a_n b_n\}$;
(II) Find the minimum value of $\frac{b_n^2 + 9}{(\log_2 a_n) + 2}$. | \frac{13}{3} | 40.625 |
22,587 | Find an integer $n$ such that the decimal representation of the number $5^{n}$ contains at least 1968 consecutive zeros. | 1968 | 75.78125 |
22,588 | $F$ is the right focus of the hyperbola $C: \dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 \left(a > 0, b > 0\right)$. A perpendicular line is drawn from point $F$ to asymptote $C$, with the foot of the perpendicular denoted as $A$, intersecting another asymptote at point $B$. If $2\overrightarrow {AF} = \overrightarrow {FB}$, then find the eccentricity of $C$. | \dfrac{2\sqrt{3}}{3} | 30.46875 |
22,589 | The increasing sequence $1, 3, 4, 9, 10, 12, 13, \dots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $150^{\text{th}}$ term of this sequence. | 2280 | 18.75 |
22,590 | Joe has exactly enough paint to paint the surface (excluding the bases) of a cylinder with radius 3 and height 4. It turns out this is also exactly enough paint to paint the entire surface of a cube. The volume of this cube is \( \frac{48}{\sqrt{K}} \). What is \( K \)? | \frac{36}{\pi^3} | 60.15625 |
22,591 | Given real numbers $x$ and $y$ satisfying $x^{2}+2y^{2}-2xy=4$, find the maximum value of $xy$. | 2\sqrt{2} + 2 | 0.78125 |
22,592 | Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = -a_n - \left(\frac{1}{2}\right)^{n-1} + 2$, and $(1) b_n = 2^n a_n$, find the general term formula for $\{b_n\}$. Also, $(2)$ find the maximum term of $\{a_n\}$. | \frac{1}{2} | 70.3125 |
22,593 | What is the greatest integer less than or equal to \[\frac{5^{80} + 3^{80}}{5^{75} + 3^{75}}?\] | 3124 | 51.5625 |
22,594 | Given that one of the children is a boy and the probability of having a boy or a girl is equal, calculate the probability that the other child is a girl. | \frac{2}{3} | 96.875 |
22,595 | Given $x \gt 0$, $y \gt 0$, and $x+y=1$, find the minimum value of $\frac{2{x}^{2}-x+1}{xy}$. | 2\sqrt{2}+1 | 0 |
22,596 | Given the arithmetic sequence $\{a_n\}$, $S_n$ denotes the sum of its first $n$ terms. Given that $a_4 + a_8 = 4$, find the value of $S_{11} + a_6$. | 24 | 92.1875 |
22,597 | In Dr. Strange's laboratory, there are some bacteria. Each day, 11 bacteria are eliminated, and each night, 5 bacteria are added. If there are 50 bacteria on the morning of the first day, on which day will all the bacteria be eliminated? | 10 | 5.46875 |
22,598 | Solve the system of equations:
\begin{cases}
\frac{m}{3} + \frac{n}{2} = 1 \\
m - 2n = 2
\end{cases} | \frac{2}{7} | 0 |
22,599 | Given the function \(\frac{(4^t - 2t)t}{16^t}\), find the maximum value for real values of \(t\). | \frac{1}{8} | 4.6875 |
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