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28,700 | Given $F(x) = \int_{0}^{x} (t^{2} + 2t - 8) \, dt$, where $x > 0$.
1. Determine the intervals of monotonicity for $F(x)$.
2. Find the maximum and minimum values of the function $F(x)$ on the interval $[1, 3]$. | -\frac{28}{3} | 61.71875 |
28,701 | $AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$ . A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$ . Find the area of the square. | 36 | 0.78125 |
28,702 | The real sequence \( x_0, x_1, x_2, \ldots \) is defined by \( x_0 = 1 \), \( x_1 = 2 \), and \( n(n+1) x_{n+1} = n(n-1) x_n - (n-2) x_{n-1} \). Find \( \frac{x_0}{x_1} + x_1 x_2 + \cdots + \frac{x_{50}}{x_{51}} \). | 1326 | 11.71875 |
28,703 | How many positive four-digit integers of the form $\_\_35$ are divisible by 35? | 13 | 78.90625 |
28,704 | Choose one of the three conditions: ①$ac=\sqrt{3}$, ②$c\sin A=3$, ③$c=\sqrt{3}b$, and supplement it in the following question. If the triangle in the question exists, find the value of $c$; if the triangle in the question does not exist, explain the reason.<br/>Question: Does there exist a $\triangle ABC$ where the internal angles $A$, $B$, $C$ have opposite sides $a$, $b$, $c$, and $\sin A=\sqrt{3}\sin B$, $C=\frac{π}{6}$, _______ $?$<br/>Note: If multiple conditions are chosen to answer separately, the first answer will be scored. | 2\sqrt{3} | 13.28125 |
28,705 | 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} | 1.5625 |
28,706 | In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE=8$, and $AF=4$. Find the area of $ABCD$. | 192\sqrt{3}-96 | 1.5625 |
28,707 | Given $f(x)=2x^3-6x^2+a$ (where $a$ is a constant), the function has a maximum value of 3 on the interval $[-2, 2]$. Find the minimum value of $f(x)$ on the interval $[-2, 2]$. | -29 | 0.78125 |
28,708 | In the diagram, semicircles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, such that they are mutually tangent. Point $C$ is at one third the distance from $A$ to $B$, so $\overline{AC} = \frac{1}{3} \overline{AB}$ and $\overline{CB} = \frac{2}{3} \overline{AB}$. If $\overline{CD} \perp \overline{AB}$, find the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius. | 1:4 | 0 |
28,709 | A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$ . Heracle defeats a hydra by cutting it into two parts which are no joined. Find the minimum $N$ for which Heracle can defeat any hydra with $100$ necks by no more than $N$ hits. | 10 | 1.5625 |
28,710 | Given in $\triangle ABC$, $\tan A$ and $\tan B$ are the two real roots of the equation $x^2 + ax + 4 = 0$:
(1) If $a = -8$, find the value of $\tan C$;
(2) Find the minimum value of $\tan C$, and specify the corresponding values of $\tan A$ and $\tan B$. | \frac{4}{3} | 7.8125 |
28,711 | Eight distinct integers are picked at random from $\{1,2,3,\ldots,15\}$. What is the probability that, among those selected, the third smallest is $5$? | \frac{4}{17} | 0 |
28,712 | Fill in the four boxes with the operations "+", "-", "*", and "$\div$" each exactly once in the expression 10 □ 10 □ 10 □ 10 □ 10 to maximize the value. What is the maximum value? | 109 | 0 |
28,713 | Given the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{2} = 1$ with three non-collinear points $A$, $B$, $C$ on it. The midpoints of $AB$, $BC$, $AC$ are $D$, $E$, $F$ respectively. If the sum of the slopes of $OD$, $OE$, $OF$ is $-1$, find the value of $\frac{1}{k_{AB}} + \frac{1}{k_{BC}} + \frac{1}{k_{AC}}$. | -2 | 27.34375 |
28,714 | Given that point $A(-2,3)$ lies on the axis of parabola $C$: $y^{2}=2px$, and the line passing through point $A$ is tangent to $C$ at point $B$ in the first quadrant. Let $F$ be the focus of $C$. Then, $|BF|=$ _____ . | 10 | 5.46875 |
28,715 | Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to definitely get inside?
b) On average, how much time will Petya need?
c) What is the probability that Petya will get inside in less than a minute? | \frac{29}{120} | 3.90625 |
28,716 | 6 small circles of equal radius and 1 large circle are arranged as shown in the diagram. The area of the large circle is 120. What is the area of one of the small circles? | 40 | 0 |
28,717 | In the line $8x + 5y + c = 0$, find the value of $c$ if the product of the $x$- and $y$- intercepts is $24$. | -8\sqrt{15} | 18.75 |
28,718 | Given a square region with a side length of 1 meter, and a total of 5120 beans within the square with 4009 beans within the inscribed circle, determine the approximate value of pi rounded to three decimal places. | 3.13 | 0 |
28,719 | Between A and B, there are 6 parallel network cables, with their maximum information capacities being 1, 1, 2, 2, 3, and 4, respectively. Now, if we randomly select 3 of these network cables, in how many ways can we ensure that the sum of the maximum information capacities of these 3 cables is not less than 6? | 15 | 33.59375 |
28,720 | Given a regular quadrangular pyramid \( S-ABCD \) with a side edge length of 4, and \( \angle ASB = 30^\circ \). Points \( E \), \( F \), and \( G \) are taken on side edges \( SB \), \( SC \), and \( SD \), respectively. Find the minimum value of \( AE + EF + FG + GA \). | 4\sqrt{3} | 3.90625 |
28,721 | In a diagram, there is an equilateral triangle with a side length of $10$ m. Calculate both the perimeter and the height of the triangle. | 5\sqrt{3} | 78.125 |
28,722 | Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$ . If $\angle DBC = 15^{\circ}$ , then find $AD^2$ .
*Proposed by Anand Iyer* | 60000 | 14.84375 |
28,723 | Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$ with $g(x) \neq 0$, $f'(x)g(x) > f(x)g'(x)$. Let $a$ and $b$ be the points obtained from two consecutive throws of the same die. If $f(x) - a^x g(x) = 0$ and $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} \geq \frac{10}{3}$, calculate the probability that the equation $abx^2 + 8x + 1 = 0$ has two distinct real roots. | \frac{13}{36} | 4.6875 |
28,724 | In how many distinct ways can I arrange my six keys on a keychain, if I want to put my house key next to my car key and additionally ensure my bike key is next to my mailbox key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order by reflection or rotation). | 24 | 10.9375 |
28,725 | Let \( A B C D \) be an isosceles trapezoid with \( [A B] \) as the larger base. It is given that the diagonals intersect at a point \( O \) such that \(\frac{O A}{O C}=2\). Given that the area of triangle \( B O C \) is 10, what is the area of the trapezoid \( A B C D \)? | 45 | 7.03125 |
28,726 | Two cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ miles along the track in some direction. The other car travels $3$ miles along the track in some direction. Then the two cars are once again at the same point along the track. If $n$ is a positive integer, find the sum of all possible values of $n.$ | 89 | 0 |
28,727 | Compute the number of five-digit positive integers whose digits have exactly $30$ distinct permutations (the permutations do not necessarily have to be valid five-digit integers).
*Proposed by David Sun* | 9720 | 6.25 |
28,728 | Let $g$ be the function defined by $g(x) = -3 \sin(2\pi x)$. How many values of $x$ such that $-3 \le x \le 3$ satisfy the equation $g(g(g(x))) = g(x)$? | 48 | 0 |
28,729 | Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$ . Let
\[AO = 5, BO =6, CO = 7, DO = 8.\]
If $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$ , determine $\frac{OM}{ON}$ . | 35/48 | 12.5 |
28,730 | The table below shows the Gross Domestic Product (GDP) of China from 2012 to 2018 in trillion US dollars:
| Year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 |
|------|------|------|------|------|------|------|------|
| Year Code $x$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| GDP $y$ (trillion US dollars) | 8.5 | 9.6 | 10.4 | 11 | 11.1 | 12.1 | 13.6 |
$(1)$ From the data in the table, it is known that there is a strong linear correlation between $x$ and $y$. Find the linear regression equation with $x$ as the explanatory variable and $y$ as the predicted variable.
$(2)$ Given that the GDP of the United States in 2018 was approximately 20.5 trillion US dollars, using the conclusion from part (1), determine in which year China could surpass the GDP of the United States in 2018.
Reference data: $\sum_{i=1}^7{y_i}=76.3$, $\sum_{i=1}^7{y_i}{x_i}=326.2$
Reference formulas: The least squares estimates for the slope and intercept in the regression equation $\hat{y}=\hat{b}x+\hat{a}$ are:
$\hat{b}=\frac{\sum_{i=1}^n{(y_i-\overline{y})(x_i-\overline{x})}}{\sum_{i=1}^n{(x_i-\overline{x})^2}}=\frac{\sum_{i=1}^n{y_ix_i-n\overline{y}\overline{x}}}{\sum_{i=1}^n{x_i^2-n\overline{x}^2}}$
$\hat{a}=\overline{y}-\hat{b}\overline{x}$. | 2028 | 25.78125 |
28,731 | Circle $C$ has its center at $C(5, 5)$ and has a radius of 3 units. Circle $D$ has its center at $D(14, 5)$ and has a radius of 3 units. What is the area of the gray region bound by the circles and the $x$-axis?
```asy
import olympiad; size(150); defaultpen(linewidth(0.8));
xaxis(0,18,Ticks("%",1.0));
yaxis(0,9,Ticks("%",1.0));
fill((5,5)--(14,5)--(14,0)--(5,0)--cycle,gray(0.7));
filldraw(circle((5,5),3),fillpen=white);
filldraw(circle((14,5),3),fillpen=white);
dot("$C$",(5,5),S); dot("$D$",(14,5),S);
``` | 45 - \frac{9\pi}{2} | 21.875 |
28,732 | Find the shortest distance between the point $(5,10)$ and the parabola given by the equation $x = \frac{y^2}{3}.$ | \sqrt{53} | 7.03125 |
28,733 | Let $d_1 = a^2 + 3^a + a \cdot 3^{(a+1)/2}$ and $d_2 = a^2 + 3^a - a \cdot 3^{(a+1)/2}$. If $1 \le a \le 300$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $7$? | 43 | 1.5625 |
28,734 | If the function $$f(x)=(2m+3)x^{m^2-3}$$ is a power function, determine the value of $m$. | -1 | 1.5625 |
28,735 | Determine the periodicity of the following functions. If the function is periodic, find its smallest positive period:
(1) \( y = \tan x - \cot x \);
(2) \( y = \sin (\cos x) \);
(3) \( y = \sin x^{2} \). | 2\pi | 20.3125 |
28,736 | You can arrange 15 balls in the shape of a triangle, but you cannot arrange 96 balls in the shape of a square (missing one ball). Out of how many balls, not exceeding 50, can you arrange them both in the shape of a triangle and a square? | 36 | 15.625 |
28,737 |
Write the expression
$$
K=\frac{\frac{1}{a+b}-\frac{2}{b+c}+\frac{1}{c+a}}{\frac{1}{b-a}-\frac{2}{b+c}+\frac{1}{c-a}}+\frac{\frac{1}{b+c}-\frac{2}{c+a}+\frac{1}{a+b}}{\frac{1}{c-b}-\frac{2}{c+a}+\frac{1}{a-b}}+\frac{\frac{1}{c+a}-\frac{2}{a+b}+\frac{1}{b+c}}{\frac{1}{a-c}-\frac{2}{a+b}+\frac{1}{b-c}}
$$
in a simpler form. Calculate its value if \( a=5, b=7, c=9 \). Determine the number of operations (i.e., the total number of additions, subtractions, multiplications, and divisions) required to compute \( K \) from the simplified expression and from the original form. Also, examine the case when \( a=5, b=7, c=1 \). What are the benefits observed from algebraic simplifications in this context? | 0.0625 | 0 |
28,738 | Let point $P$ lie on the curve $y= \frac {1}{2}e^{x}$, and point $Q$ lie on the curve $y=\ln (2x)$. Find the minimum value of $|PQ|$. | \sqrt {2}(1-\ln 2) | 0 |
28,739 | Let $a,b$ be real numbers such that $|a| \neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6.$ Find the value of the expression $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}.$ | 18/7 | 46.09375 |
28,740 | The number of students in Jakob's graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Jakob's graduating class? | 122 | 43.75 |
28,741 | Two trucks are transporting identical sacks of flour from France to Spain. The first truck carries 118 sacks, and the second one carries only 40. Since the drivers of these trucks lack the pesetas to pay the customs duty, the first driver leaves 10 sacks with the customs officers, after which they only need to pay 800 pesetas. The second driver does similarly, but he leaves only 4 sacks and the customs officer pays him an additional 800 pesetas.
How much does each sack of flour cost, given that the customs officers take exactly the amount of flour needed to pay the customs duty in full? | 1600 | 0 |
28,742 | If $x_1=5, x_2=401$ , and
\[
x_n=x_{n-2}-\frac 1{x_{n-1}}
\]
for every $3\leq n \leq m$ , what is the largest value of $m$ ? | 2007 | 7.8125 |
28,743 | What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2003}+1\right)^{11/8}$? | 375 | 4.6875 |
28,744 | Calculate the volume of the solid of revolution obtained by rotating a right triangle with sides 3, 4, and 5 around one of its legs that form the right angle. | 12 \pi | 38.28125 |
28,745 | Given a geometric sequence $\{a_n\}$ with a common ratio $q$, and it satisfies $|q| > 1$, $a_2 + a_7 = 2$, and $a_4 \cdot a_5 = -15$, find the value of $a_{12}$. | -\frac{25}{3} | 7.03125 |
28,746 | Jacqueline has 40% less sugar than Liliane, and Bob has 30% less sugar than Liliane. Express the relationship between the amounts of sugar that Jacqueline and Bob have as a percentage. | 14.29\% | 0 |
28,747 | Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{4}{9} | 0.78125 |
28,748 | If one of the 13 provinces or territories is chosen at random, what is the probability that it joined Canadian Confederation between 1890 and 1969? | $\frac{4}{13}$ | 0 |
28,749 | Find the sum of all real solutions to the equation \[\frac{x-3}{x^2+3x+2} = \frac{x-7}{x^2-7x+12}.\] | \frac{26}{3} | 63.28125 |
28,750 | How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x, y)$ satisfying $1 \leq x \leq 5$ and $1 \leq y \leq 5$? | 2164 | 0 |
28,751 | The sequence is $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{3}$, $\frac{1}{4}$, $\frac{2}{4}$, $\frac{3}{4}$, ... , $\frac{1}{m+1}$, $\frac{2}{m+1}$, ... , $\frac{m}{m+1}$, ... Find the $20^{th}$ term. | \frac{6}{7} | 1.5625 |
28,752 | What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base- $5$ representation? | 755 | 0 |
28,753 | A bullet with a mass of \( m = 10 \) g, flying horizontally with a speed of \( v_{1} = 400 \) m/s, passes through a massive board and emerges from it with a speed of \( v_{2} = 100 \) m/s. Find the amount of work done on the bullet by the resistive force of the board. | 750 | 64.84375 |
28,754 | Triangle $PQR$ has sides $\overline{PQ}$, $\overline{QR}$, and $\overline{RP}$ of length 47, 14, and 50, respectively. Let $\omega$ be the circle circumscribed around $\triangle PQR$ and let $S$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{RP}$ that is not on the same side of $\overline{RP}$ as $Q$. The length of $\overline{PS}$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find the greatest integer less than or equal to $m + \sqrt{n}$. | 14 | 0 |
28,755 | Chandra now has five bowls and five glasses, and each expands to a new set of colors: red, blue, yellow, green, and purple. However, she dislikes pairing the same colors; thus, a bowl and glass of the same color cannot be paired together like a red bowl with a red glass. How many acceptable combinations can Chandra make when choosing a bowl and a glass? | 44 | 60.15625 |
28,756 | 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 \( A_{1} A_{2} A_{3} \).
\( A_{1}(7, 2, 4) \)
\( A_{2}(7, -1, -2) \)
\( A_{3}(3, 3, 1) \)
\( A_{4}(-4, 2, 1) \) | 21.5 | 0 |
28,757 | For a three-digit number \(\overline{a b c}\) that satisfies \(\overline{a b c} = 37(a + b + c)\), how many such three-digit numbers are there? | 15 | 52.34375 |
28,758 | From the set $\left\{ \frac{1}{3}, \frac{1}{2}, 2, 3 \right\}$, select a number and denote it as $a$. From the set $\{-2, -1, 1, 2\}$, select another number and denote it as $b$. Then, the probability that the graph of the function $y=a^{x}+b$ passes through the third quadrant is ______. | \frac{3}{8} | 2.34375 |
28,759 | "The Nine Chapters on the Mathematical Art" is the first mathematical monograph in China, which includes the following problem: "There is a gold rod, 5 feet in length. Cutting 1 foot from the base, it weighs 4 jin. Cutting 1 foot from the tip, it weighs 2 jin. How much does the gold rod weigh?" This means: "There is a gold rod (uniformly varying in thickness) 5 feet long. Cutting 1 foot from the base end, it weighs 4 jin. Cutting 1 foot from the tip end, it weighs 2 jin. How much does the gold rod weigh?" The answer is \_\_\_\_\_\_. | 15 | 50.78125 |
28,760 | What is the largest four-digit negative integer congruent to $1 \pmod{17}$? | -1002 | 25 |
28,761 | Given $2014$ points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points? | 2013 | 15.625 |
28,762 | Given that the graph of a power function passes through the points $(2,16)$ and $(\frac{1}{2},m)$, find the value of $m$. | \frac{1}{16} | 68.75 |
28,763 | During the past summer, 100 graduates from the city of $N$ applied to 5 different universities in our country. It turned out that during the first and second waves, each university was unable to reach exactly half of the applicants to that university. In addition, representatives of at least three universities were unable to reach those graduates. What is the maximum number of graduates from the city of $N$ who could have been of interest to the military recruitment office? | 83 | 0 |
28,764 | 12. If $p$ is the smallest positive prime number such that there exists an integer $n$ for which $p$ divides $n^{2}+5n+23$, then $p=$ ______ | 13 | 0 |
28,765 | Find $x$ such that $\lceil x \rceil \cdot x = 156$. Express $x$ as a decimal. | 12 | 3.90625 |
28,766 | Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger completes one lap of the track in a time, \(t\) seconds, where \(t\) is a positive integer. The product of the smallest and largest possible integer values of \(t\) is: | 3705 | 17.96875 |
28,767 | Three numbers $a, b,$ and $c$ were written on a board. They were erased and replaced with the numbers $a-1, b+1,$ and $c^2$. After this, it turned out that the same numbers were on the board as initially (possibly in a different order). What values can the number $a$ take, given that the sum of the numbers $a, b,$ and $c$ is 2008? If necessary, round your answer to the nearest hundredth. | 1004 | 28.125 |
28,768 | In the Cartesian coordinate plane $(xOy)$, if the line $ax + y - 2 = 0$ intersects the circle centered at $C$ with the equation $(x - 1)^2 + (y - a)^2 = 16$ at points $A$ and $B$, and $\triangle ABC$ is a right triangle, then the value of the real number $a$ is _____. | -1 | 27.34375 |
28,769 | Given $x-y=1$ and $x^3-y^3=2$, find the values of $x^4+y^4$ and $x^5-y^5$. | \frac{29}{9} | 71.875 |
28,770 | Petya wants to color some cells of a $6 \times 6$ square so that there are as many vertices as possible that belong to exactly three colored squares. What is the maximum number of such vertices he can achieve? | 25 | 5.46875 |
28,771 | Rectangles \( A B C D, D E F G, C E I H \) have equal areas and integer sides. Find \( D G \) if \( B C = 19 \). | 380 | 0 |
28,772 | Find the smallest natural number ending in the digit 2 such that it doubles when this digit is moved to the beginning. | 105263157894736842 | 94.53125 |
28,773 | Add $-45.367$, $108.2$, and $23.7654$, then round your answer to the nearest tenth. | 86.6 | 97.65625 |
28,774 | Rectangle $EFGH$ has area $4032$. An ellipse with area $4032\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle? | 8\sqrt{2016} | 0 |
28,775 | Randomly color the four vertices of a tetrahedron with two colors, red and yellow. The probability that "three vertices on the same face are of the same color" is ______. | \dfrac{5}{8} | 6.25 |
28,776 | What is the largest integer that must divide the product of any $5$ consecutive integers? | 60 | 0.78125 |
28,777 | Find the number of six-digit palindromes. | 9000 | 7.03125 |
28,778 | How many real \( x \) satisfy the equation \( x = \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor + \left\lfloor \frac{x}{5} \right\rfloor \)? | 30 | 57.03125 |
28,779 | In the complex plane, the graph of \( |z - 5| = 3|z + 5| \) intersects the graph of \( |z| = k \) in exactly one point. Find all possible values of \( k \). | 12.5 | 0 |
28,780 | Let $$\overset{ .}{a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}}|_{m}$$ be defined as $a_{0}+a_{1}\times m+\ldots+a_{n-1}\times m^{n-1}+a_{n}\times m^{n}$, where $n\leq m$, $m$ and $n$ are positive integers, $a_{k}\in\{0,1,2,\ldots,m-1\}$ ($k=0,1,2,\ldots,n$) and $a_{n}\neq 0$;
(1) Calculate $$\overset{ .}{2016}|_{7}$$\= \_\_\_\_\_\_ ;
(2) Let the set $A(m,n)=$$\{x|x= \overset{ .}{a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}}|_{m}\}$, then the sum of all elements in $A(m,n)$ is \_\_\_\_\_\_ . | 699 | 1.5625 |
28,781 | There are two ribbons, one is 28 cm long and the other is 16 cm long. Now, we want to cut them into shorter ribbons of the same length without any leftovers. What is the maximum length of each short ribbon in centimeters? And how many such short ribbons can be cut? | 11 | 81.25 |
28,782 | Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive? | 36 | 75.78125 |
28,783 | For some positive integers $a$ and $b$, the product \[\log_a(a+1) \cdot \log_{a+1} (a+2) \dotsm \log_{b-2} (b-1) \cdot\log_{b-1} b\]contains exactly $1000$ terms, and its value is $3.$ Compute $a+b.$ | 1010 | 62.5 |
28,784 | Given that \(a_{1}, a_{2}, \cdots, a_{n}\) are \(n\) people corresponding to \(A_{1}, A_{2}, \cdots, A_{n}\) cards (\(n \geq 2\), \(a_{i}\) corresponds to \(A_{i}\)). Now \(a_{1}\) picks a card from the deck randomly, and then each person in sequence picks a card. If their corresponding card is still in the deck, they take it, otherwise they pick a card randomly. What is the probability that \(a_{n}\) gets \(A_{n}\)? | \frac{1}{2} | 20.3125 |
28,785 | In a bus, there are single and double seats. In the morning, 13 people were sitting in the bus, and there were 9 completely free seats. In the evening, 10 people were sitting in the bus, and there were 6 completely free seats. How many seats are there in the bus? | 16 | 3.125 |
28,786 | A traffic light runs repeatedly through the following cycle: green for 45 seconds, then yellow for 5 seconds, then blue for 10 seconds, and finally red for 40 seconds. Peter picks a random five-second time interval to observe the light. What is the probability that the color changes while he is watching? | 0.15 | 0 |
28,787 | Given a right triangle $ABC$. On the extension of the hypotenuse $BC$, a point $D$ is chosen such that the line $AD$ is tangent to the circumcircle $\omega$ of triangle $ABC$. The line $AC$ intersects the circumcircle of triangle $ABD$ at point $E$. It turns out that the angle bisector of $\angle ADE$ is tangent to the circle $\omega$. In what ratio does point $C$ divide segment $AE$? | 1:2 | 31.25 |
28,788 | Let \( a \), \( b \), and \( c \) be the roots of the polynomial \( x^3 - 15x^2 + 25x - 10 = 0 \). Compute
\[
(a-b)^2 + (b-c)^2 + (c-a)^2.
\] | 125 | 0 |
28,789 | If the width of a rectangle is increased by 3 cm and the height is decreased by 3 cm, its area does not change. What would happen to the area if, instead, the width of the original rectangle is decreased by 4 cm and the height is increased by 4 cm? | 28 | 23.4375 |
28,790 | For a nonnegative integer $n$, let $r_{11}(n)$ represent the remainder when $n$ is divided by $11.$ What is the $15^{\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_{11}(7n)\le 3~?$$ | 41 | 1.5625 |
28,791 | In a right triangle JKL, where angle J is the right angle, KL measures 20 units, and JL measures 12 units. Calculate $\tan K$. | \frac{4}{3} | 7.03125 |
28,792 | How many 3-term geometric sequences $a$ , $b$ , $c$ are there where $a$ , $b$ , and $c$ are positive integers with $a < b < c$ and $c = 8000$ ? | 39 | 1.5625 |
28,793 | Given the function $f(x)=x^{3}+3x^{2}-9x+3.$ Find:
(I) The interval(s) where $f(x)$ is increasing;
(II) The extreme values of $f(x)$. | -2 | 41.40625 |
28,794 | Find the product of the roots of the equation \[(3x^4 + 2x^3 - 9x + 30)(4x^3 - 20x^2 + 24x - 36) = 0.\] | -90 | 6.25 |
28,795 | Given an ellipse $E: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and upper vertex at B. Point P is on E, point D is at (0, -2b), and the maximum area of △PBD is $\frac{3\sqrt{2}}{2}$.
(I) Find the equation of E;
(II) If line DP intersects E at another point Q, and lines BP and BQ intersect the x-axis at points M and N, respectively, determine whether $|OM|\cdot|ON|$ is a constant value. | \frac{2}{3} | 7.03125 |
28,796 | The U.S. produces about 8 million tons of apples each year. Initially, $30\%$ of the apples are mixed with other products. If the production increases by 1 million tons, the percentage mixed with other products increases by $5\%$ for each additional million tons. Of the remaining apples, $60\%$ is used to make apple juice and $40\%$ is sold fresh. Calculate how many million tons of apples are sold fresh. | 2.24 | 11.71875 |
28,797 | The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$ . | -24 | 12.5 |
28,798 | The orchestra has more than 150 members but fewer than 300 members. When they line up in rows of 6 there are two extra people; when they line up in rows of 8 there are four extra people; and when they line up in rows of 9 there are six extra people. How many members are in the orchestra? | 212 | 0.78125 |
28,799 | A cube with an edge length of 2 Chinese feet is cut 4 times horizontally and then 5 times vertically. What is the total surface area of all the small blocks after cutting, in square Chinese feet? | 96 | 7.8125 |
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