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40.3k
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5.15k
| ground_truth
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100
|
|---|---|---|---|
10,400 |
Let triangle $ABC$ be an equilateral triangle. A point $M$ in the plane of the triangle satisfies $\overrightarrow{AM}=\frac{1}{3}\overrightarrow{AB}+\frac{2}{3}\overrightarrow{AC}$. The cosine value of the angle between the vectors $\overrightarrow{AM}$ and $\overrightarrow{BC}$ is ______.
|
\frac{\sqrt{7}}{14}
| 82.8125 |
10,401 |
Scatterbrained Scientist had a sore knee. The doctor prescribed 10 pills for the knee, to be taken one pill daily. These pills help in $90 \%$ of cases, but in $2 \%$ of cases, there is a side effect—it eliminates scatterbrainedness, if present.
Another doctor prescribed the Scientist pills for scatterbrainedness, also to be taken one per day for 10 consecutive days. These pills cure scatterbrainedness in $80 \%$ of cases, but in $5 \%$ of cases, there is a side effect—the knee pain stops.
The two bottles of pills look similar, and when the Scientist went on a ten-day business trip, he took one bottle with him but paid no attention to which one. He took one pill daily for ten days and returned completely healthy: the scatterbrainedness was gone and the knee pain was no more. Find the probability that the Scientist took the pills for scatterbrainedness.
|
0.69
| 0 |
10,402 |
Determine how many integers are in the list containing the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them.
|
871
| 63.28125 |
10,403 |
Xiaoming and Xiaojun start simultaneously from locations A and B, heading towards each other. If both proceed at their original speeds, they meet after 5 hours. If both increase their speeds by 2 km/h, they meet after 3 hours. The distance between locations A and B is 30 km.
|
30
| 21.09375 |
10,404 |
Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{4}} \left( x^{2} + 17.5 \right) \sin 2x \, dx
$$
|
\frac{68 + \pi}{8}
| 0.78125 |
10,405 |
On an island, there are knights, liars, and followers; each one knows who is who among them. All 2018 island inhabitants were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The inhabitants answered one by one in such a way that the others could hear. Knights told the truth, liars lied. Each follower answered the same as the majority of those who had answered before them, and if the number of "Yes" and "No" answers was equal, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island inhabitants?
|
1009
| 51.5625 |
10,406 |
Find the largest integer \( a \) such that the expression
\[
a^2 - 15a - (\tan x - 1)(\tan x + 2)(\tan x + 5)(\tan x + 8)
\]
is less than 35 for all values of \( x \in (-\pi/2, \pi/2) \).
|
10
| 14.84375 |
10,407 |
Given a parallelogram \(ABCD\) with sides \(AB=2\) and \(BC=3\), find the area of this parallelogram, given that the diagonal \(AC\) is perpendicular to the segment \(BE\), where \(E\) is the midpoint of side \(AD\).
|
\sqrt{35}
| 10.9375 |
10,408 |
Let **v** be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of $\|\mathbf{v}\|$.
|
10 - 2\sqrt{5}
| 86.71875 |
10,409 |
Find a ten-digit number where the first digit indicates how many times the digit 0 appears in the number, the second digit indicates how many times the digit 1 appears, and so forth, with the tenth digit indicating how many times the digit 9 appears in the number.
Generalize and solve the problem for a number system with base $n$.
|
6210001000
| 41.40625 |
10,410 |
A number contains only two kinds of digits: 3 or 4, and both 3 and 4 appear at least once. The number is a multiple of both 3 and 4. What is the smallest such number?
|
3444
| 57.8125 |
10,411 |
In the complex plane, the points \( 0, z, \frac{1}{z}, z+\frac{1}{z} \) form a parallelogram with an area of \( \frac{35}{37} \). If the real part of \( z \) is greater than 0, find the minimum value of \( \left| z + \frac{1}{z} \right| \).
|
\frac{5 \sqrt{74}}{37}
| 0 |
10,412 |
In triangle $ABC$, if $A = \frac{\pi}{3}$, $\tan B = \frac{1}{2}$, and $AB = 2\sqrt{3} + 1$, then find the length of $BC$.
|
\sqrt{15}
| 46.09375 |
10,413 |
Let \(A B C D\) be a square of side length 13. Let \(E\) and \(F\) be points on rays \(A B\) and \(A D\), respectively, so that the area of square \(A B C D\) equals the area of triangle \(A E F\). If \(E F\) intersects \(B C\) at \(X\) and \(B X = 6\), determine \(D F\).
|
\sqrt{13}
| 0.78125 |
10,414 |
Mathematician Wiener, the founder of cybernetics, was asked about his age during his Ph.D. awarding ceremony at Harvard University because he looked very young. Wiener's interesting response was: "The cube of my age is a four-digit number, and the fourth power of my age is a six-digit number. These two numbers together use all the digits from 0 to 9 exactly once, with no repetition or omission." What is Wiener's age that year? (Note: The cube of a number \(a\) is equal to \(a \times a \times a\), and the fourth power of a number \(a\) is equal to \(a \times a \times a \times a\)).
|
18
| 83.59375 |
10,415 |
In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel with lengths of 10 and 24 units respectively, and the altitude is 15 units. Points $G$ and $H$ are the midpoints of sides $AD$ and $BC$, respectively. Determine the area of quadrilateral $GHCD$.
|
153.75
| 68.75 |
10,416 |
Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n= \frac{b_{n-1}}{2}$ if $b_{n-1}$ is even and $b_n=3b_{n-1}+1$ if $b_{n-1}$ is odd. For how many positive integers $b_1 \le 1000$ is it true that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$?
|
250
| 28.125 |
10,417 |
The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
*Proposed by Aaron Lin*
|
1006
| 31.25 |
10,418 |
What is the ratio of the sides of a triangle in which the sum of the lengths of the altitudes taken two at a time corresponds to the ratio 5:7:8?
|
10:15:6
| 48.4375 |
10,419 |
The number $7.21\times 10^{11}$ has how many digits in the original number.
|
12
| 6.25 |
10,420 |
Using the seven digits $1, 2, 3, 4, 5, 6, 7$ to appropriately arrange them into a 7-digit number so that it is a multiple of 11, how many such numbers can be formed?
|
576
| 42.1875 |
10,421 |
Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity?
|
225
| 53.125 |
10,422 |
Simplify first, then evaluate: $\dfrac{x^{2}-4x+4}{2x}\div \dfrac{x^{2}-2x}{x^{2}}+1$. Choose a suitable number from $0$, $1$, $2$, substitute it in and evaluate.
|
\dfrac{1}{2}
| 64.84375 |
10,423 |
Given that a 1-step requires 4 toothpicks, a 2-step requires 10 toothpicks, a 3-step requires 18 toothpicks, and a 4-step requires 28 toothpicks, determine the number of additional toothpicks needed to build a 6-step staircase.
|
26
| 25 |
10,424 |
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0, 2, 4, 7, 8, 9\) (digits can repeat) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?
|
1728
| 37.5 |
10,425 |
Lily is riding her bicycle at a constant rate of 15 miles per hour and Leo jogs at a constant rate of 9 miles per hour. If Lily initially sees Leo 0.75 miles in front of her and later sees him 0.75 miles behind her, determine the duration of time, in minutes, that she can see Leo.
|
15
| 91.40625 |
10,426 |
Given the function \( f(x) \):
\[ f(x) = \begin{cases}
\ln x & \text{if } x > 1, \\
\frac{1}{2} x + \frac{1}{2} & \text{if } x \leq 1
\end{cases} \]
If \( m < n \) and \( f(m) = f(n) \), what is the minimum value of \( n - m \)?
|
3 - 2\ln 2
| 67.1875 |
10,427 |
There are three types of snacks for the kitten. It eats a stick of cat food every 1 day, an egg yolk every 2 days, and nutritional cream every 3 days. The kitten ate cat stick and nutritional cream on March 23, and ate cat stick and egg yolk on March 25. Which day in March does the kitten eat all three types of snacks for the first time?
|
29
| 49.21875 |
10,428 |
Find the number of four-digit numbers in which all digits are different, the first digit is divisible by 2, and the sum of the first and last digits is divisible by 3.
|
672
| 82.8125 |
10,429 |
Given a triangle \( ABC \) with the condition:
\[ \cos (\angle A - \angle B) + \sin (\angle A + \angle B) = 2 \]
Find the side \( BC \) if \( AB = 4 \).
|
2\sqrt{2}
| 74.21875 |
10,430 |
In triangle ABC, the lengths of the three sides are three consecutive natural numbers, and the largest angle is twice the smallest angle. Calculate the area of this triangle.
|
\frac {15 \sqrt {7}}{4}
| 0 |
10,431 |
Given real numbers $x$ and $y$ that satisfy the system of inequalities $\begin{cases} x - 2y - 2 \leqslant 0 \\ x + y - 2 \leqslant 0 \\ 2x - y + 2 \geqslant 0 \end{cases}$, if the minimum value of the objective function $z = ax + by + 5 (a > 0, b > 0)$ is $2$, determine the minimum value of $\frac{2}{a} + \frac{3}{b}$.
|
\frac{10 + 4\sqrt{6}}{3}
| 7.03125 |
10,432 |
Reading material: In general, the equation $\frac{1}{x}+\frac{1}{y}=1$ is not valid, but some special real numbers can make it valid. For example, when $x=2$ and $y=2$, $\frac{1}{2}+\frac{1}{2}=1$ is valid. We call $(2,2)$ a "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid. Please complete the following questions:
$(1)$ Among the pairs $(\frac{4}{3},4), (1,1)$, the "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid is ______;
$(2)$ If $(5-t,5+t)$ is a "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid, find the value of $t$;
$(3)$ If $(m,n)$ is a "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid, and $a=b+m$, $b=c+n$, find the minimum value of the algebraic expression $\left(a-c\right)^{2}-12\left(a-b\right)\left(b-c\right)$.
|
-36
| 57.03125 |
10,433 |
The polynomial \( x^8 - 4x^7 + 7x^6 + \cdots + a_0 \) has all its roots positive and real numbers. Find the possible values for \( a_0 \).
|
\frac{1}{256}
| 57.03125 |
10,434 |
The sequence is defined recursively:
\[ x_{0} = 0, \quad x_{n+1} = \frac{(n^2 + n + 1) x_{n} + 1}{n^2 + n + 1 - x_{n}}. \]
Find \( x_{8453} \).
|
8453
| 86.71875 |
10,435 |
The population size (in number of animals) of a certain animal species is given by the equation $y=a\log_{2}(x+1)$. Suppose that the population size of this animal species in the first year was 100 animals. What will be the population size in the 15th year.
|
400
| 82.03125 |
10,436 |
Find the angle $D A C$ given that $A B = B C$ and $A C = C D$, and the lines on which points $A, B, C, D$ lie are parallel with equal distances between adjacent lines. Point $A$ is to the left of $B$, $C$ is to the left of $B$, and $D$ is to the right of $C$.
|
30
| 18.75 |
10,437 |
A high school is holding a speech contest with 10 participants. There are 3 students from Class 1, 2 students from Class 2, and 5 students from other classes. Using a draw to determine the speaking order, what is the probability that the 3 students from Class 1 are placed consecutively (in consecutive speaking slots) and the 2 students from Class 2 are not placed consecutively?
|
$\frac{1}{20}$
| 0 |
10,438 |
Given $ \tan \left( \alpha + \frac{\pi}{4} \right) = -\frac{1}{2} $ and $\frac{\pi}{2} < \alpha < \pi$, calculate the value of $ \frac{\sin 2\alpha - 2\cos^2 \alpha}{\sin \left(\alpha - \frac{\pi}{4} \right)} $.
|
-\frac{2 \sqrt{5}}{5}
| 92.1875 |
10,439 |
Express $\frac{214_8}{32_5} + \frac{343_9}{133_4}$ in base 10.
|
\frac{9134}{527}
| 2.34375 |
10,440 |
For a positive integer $n$ , let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$ . For example, $f(96)=1+2+3+6+9=21$ . Find the largest positive integer $n$ such that for all positive integers $k$ , there is some positive integer $a$ such that $f^k(a)=n$ , where $f^k(a)$ denotes $f$ applied $k$ times to $a$ .
*2021 CCA Math Bonanza Lightning Round #4.3*
|
15
| 25 |
10,441 |
If \( a^3 + b^3 + c^3 = 3abc = 6 \) and \( a^2 + b^2 + c^2 = 8 \), find the value of \( \frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a} \).
|
-8
| 12.5 |
10,442 |
Using one each of the coins and bills of 1 jiao, 2 jiao, 5 jiao, 1 yuan, 2 yuan, and 5 yuan, how many different monetary values can be formed?
|
63
| 56.25 |
10,443 |
Calculate: \((56 \times 0.57 \times 0.85) \div(2.8 \times 19 \times 1.7) =\)
|
0.3
| 96.09375 |
10,444 |
Rectangle $PQRS$ is inscribed in a semicircle with diameter $\overline{GH}$, such that $PR=20$, and $PG=SH=12$. Determine the area of rectangle $PQRS$.
A) $120\sqrt{6}$
B) $150\sqrt{6}$
C) $160\sqrt{6}$
D) $180\sqrt{6}$
E) $200\sqrt{6}$
|
160\sqrt{6}
| 28.125 |
10,445 |
Find the infinite sum of \(\frac{1^{3}}{3^{1}}+\frac{2^{3}}{3^{2}}+\frac{3^{3}}{3^{3}}+\frac{4^{3}}{3^{4}}+\cdots\).
求 \(\frac{1^{3}}{3^{1}}+\frac{2^{3}}{3^{2}}+\frac{3^{3}}{3^{3}}+\frac{4^{3}}{3^{4}}+\cdots\) 無限項之和。
|
\frac{33}{8}
| 57.8125 |
10,446 |
Find the smallest natural number that consists of identical digits and is divisible by 18.
|
666
| 45.3125 |
10,447 |
The function $f(x) = (m^2 - m - 1)x^m$ is a power function, and it is a decreasing function on $x \in (0, +\infty)$. The value of the real number $m$ is
|
-1
| 31.25 |
10,448 |
Three of the four vertices of a rectangle are $(3, 7)$, $(12, 7)$, and $(12, -4)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 3)^2 + (y + 4)^2 = 16$?
|
4\pi
| 80.46875 |
10,449 |
Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?
*Author: Ray Li*
|
270
| 89.84375 |
10,450 |
$8[x]$ represents the greatest integer not exceeding the real number $x$. Then, $\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots+\left[\log _{2} 2012\right]=$ $\qquad$ .
|
18084
| 90.625 |
10,451 |
Solve the equations:
(1) $x(x+4)=-5(x+4)$
(2) $(x+2)^2=(2x-1)^2$
|
-\frac{1}{3}
| 0 |
10,452 |
In the rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C:ρ\sin^2θ=2a\cos θ (a > 0)$, the line $l:\begin{cases}x=-2+\frac{\sqrt{2}}{2}t\\y=-4+\frac{\sqrt{2}}{2}t\end{cases} (t$ is the parameter$) $ passes through point $P(-2,-4)$ and intersects curve $C$ at points $M$ and $N$.
(I) Find the ordinary equations of curve $C$ and line $l$;
(II) If $|PM|, |MN|, |PN|$ form a geometric sequence, find the value of real number $a$.
|
a=1
| 79.6875 |
10,453 |
All natural numbers from 1 to 1000 inclusive are divided into two groups: even and odd. In which group is the sum of all the digits used to write the numbers greater and by how much?
|
499
| 60.15625 |
10,454 |
Rotate an equilateral triangle with side length $2$ around one of its sides to form a solid of revolution. The surface area of this solid is ______.
|
4\sqrt{3}\pi
| 52.34375 |
10,455 |
A cylindrical tank with radius 6 feet and height 7 feet is lying on its side. The tank is filled with water to a depth of 3 feet. Find the volume of water in the tank, in cubic feet.
|
84\pi - 63\sqrt{3}
| 19.53125 |
10,456 |
Given that 2 students exercised 0 days, 4 students exercised 1 day, 5 students exercised 2 days, 3 students exercised 4 days, 7 students exercised 5 days, and 2 students exercised 6 days, calculate the average number of days exercised last week by the students in Ms. Brown's class.
|
3.17
| 3.90625 |
10,457 |
At a meeting of cactus enthusiasts, 80 cactophiles presented their collections, each consisting of cacti of different species. It turned out that no single species of cactus is found in all collections simultaneously, but any 15 people have cacti of the same species. What is the minimum total number of cactus species that can be in all collections?
|
16
| 37.5 |
10,458 |
If a 31-day month is taken at random, find \( c \), the probability that there are 5 Sundays in the month.
|
3/7
| 25 |
10,459 |
In a circular arrangement of 101 natural numbers, it is known that among any 5 consecutive numbers, there are at least two even numbers. What is the minimum number of even numbers that can be among the listed numbers?
|
41
| 43.75 |
10,460 |
A "clearance game" has the following rules: in the $n$-th round, a die is rolled $n$ times. If the sum of the points from these $n$ rolls is greater than $2^n$, then the player clears the round. Questions:
(1) What is the maximum number of rounds a player can clear in this game?
(2) What is the probability of clearing the first three rounds consecutively?
(Note: The die is a uniform cube with the numbers $1, 2, 3, 4, 5, 6$ on its faces. After rolling, the number on the top face is the result of the roll.)
|
\frac{100}{243}
| 2.34375 |
10,461 |
Adam and Bettie are playing a game. They take turns generating a random number between $0$ and $127$ inclusive. The numbers they generate are scored as follows: $\bullet$ If the number is zero, it receives no points. $\bullet$ If the number is odd, it receives one more point than the number one less than it. $\bullet$ If the number is even, it receives the same score as the number with half its value.
if Adam and Bettie both generate one number, the probability that they receive the same score is $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$ . Find $p$ .
|
429
| 85.9375 |
10,462 |
Given that $\overrightarrow{AB} \perp \overrightarrow{AC}$, $|\overrightarrow{AB}|= \frac{1}{t}$, $|\overrightarrow{AC}|=t$, and point $P$ is a point on the plane of $\triangle ABC$ such that $\overrightarrow{AP}= \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} + \frac{4\overrightarrow{AC}}{|\overrightarrow{AC}|}$. Find the real value(s) of $t$ that satisfy $\overrightarrow{AP} \perp \overrightarrow{BC}$.
|
\frac{1}{2}
| 82.8125 |
10,463 |
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{1}{4}} \frac{\sqrt[3]{\frac{x}{16}}-\frac{1}{4}}{\sqrt{\frac{1}{4}+x}-\sqrt{2x}}$
|
-\frac{2\sqrt{2}}{6}
| 0 |
10,464 |
In the tetrahedron \( A B C D \),
$$
\begin{array}{l}
AB=1, BC=2\sqrt{6}, CD=5, \\
DA=7, AC=5, BD=7.
\end{array}
$$
Find its volume.
|
\frac{\sqrt{66}}{2}
| 0 |
10,465 |
If a class of 30 students is seated in a movie theater, then in any case at least two classmates will be in the same row. If the same is done with a class of 26 students, then at least three rows will be empty. How many rows are in the theater?
|
29
| 53.90625 |
10,466 |
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30} \cdot 5^3}$ as a decimal?
|
30
| 65.625 |
10,467 |
For a four-digit natural number $M$, let the digit in the thousands place be $a$, in the hundreds place be $b$, in the tens place be $c$, and in the units place be $d$. The two-digit number formed by the thousands and units digits of $M$ is $A=10a+d$, and the two-digit number formed by the tens and hundreds digits of $M$ is $B=10c+b$. If the difference between $A$ and $B$ is equal to the negative of the sum of the thousands and hundreds digits of $M$, then $M$ is called an "open number." Determine whether $1029$ is an "open number" (fill in "yes" or "no"). If $M$ is an "open number," let $G(M)=\frac{b+13}{c-a-d}$. Find the maximum value of $M$ that satisfies the condition when $G(M)$ is divisible by $7$.
|
8892
| 0 |
10,468 |
In the triangular prism \(A-BCD\), the side edges \(AB, AC, AD\) are mutually perpendicular. The areas of triangles \(\triangle ABC\), \(\triangle ACD\), and \(\triangle ADB\) are \(\frac{\sqrt{2}}{2}\), \(\frac{\sqrt{3}}{2}\), and \(\frac{\sqrt{6}}{2}\) respectively. Find the volume of the circumscribed sphere of the triangular prism \(A-BCD\).
|
\sqrt{6}\pi
| 7.8125 |
10,469 |
A woman wants freshly baked cookies delivered exactly at 18:00 for an event. Delivery trucks, upon finishing baking, travel with varying speeds due to potential traffic conditions:
- If there is moderate traffic, the trucks travel at an average speed of 60 km/h and would arrive at 17:45.
- If there are traffic jams, the trucks travel at an average speed of 20 km/h and would arrive at 18:15.
Determine the average speed the delivery truck must maintain to arrive exactly at 18:00.
|
30
| 34.375 |
10,470 |
In acute triangle $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $4\sin ^{2} \frac {B+C}{2}-\cos 2A= \frac {7}{2}$.
1. Find the measure of angle $A$.
2. If the altitude on side $BC$ is $1$, find the minimum area of $\triangle ABC$.
|
\frac { \sqrt {3}}{3}
| 0 |
10,471 |
Arrange for 7 staff members to be on duty from May 1st to May 7th. Each person is on duty for one day, with both members A and B not being scheduled on May 1st and 2nd. The total number of different scheduling methods is $\_\_\_\_\_\_\_$.
|
2400
| 74.21875 |
10,472 |
Peter has three times as many sisters as brothers. His sister Louise has twice as many sisters as brothers. How many children are there in the family?
|
13
| 50 |
10,473 |
If \( N \) is a multiple of 84 and \( N \) contains only the digits 6 and 7, what is the smallest \( N \) that meets these conditions?
|
76776
| 51.5625 |
10,474 |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{\begin{array}{l}{x=2+\sqrt{5}\cos\theta,}\\{y=\sqrt{5}\sin\theta}\end{array}\right.$ ($\theta$ is the parameter). Line $l$ passes through point $P(1,-1)$ with a slope of $60^{\circ}$ and intersects curve $C$ at points $A$ and $B$. <br/>$(1)$ Find the general equation of curve $C$ and a parametric equation of line $l$;<br/>$(2)$ Find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.
|
\frac{\sqrt{16+2\sqrt{3}}}{3}
| 6.25 |
10,475 |
Calculate the area of one petal of the curve $\rho = \sin^2 \varphi$.
|
\frac{3\pi}{16}
| 12.5 |
10,476 |
The cost of 60 copies of the first volume and 75 copies of the second volume is 2700 rubles. In reality, the total payment for all these books was only 2370 rubles because a discount was applied: 15% off the first volume and 10% off the second volume. Find the original price of these books.
|
20
| 57.03125 |
10,477 |
Write $\mathbf{2012}$ as the sum of $N$ distinct positive integers, where $N$ is at its maximum. What is the maximum value of $N$?
|
62
| 77.34375 |
10,478 |
Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $\frac{x^{2}}{20} + \frac{y^{2}}{4} = 1$, a line passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. If $|F_{1}A| + |F_{1}B| = 5\sqrt{5}$, then $|AB| = $ ______.
|
3\sqrt{5}
| 85.15625 |
10,479 |
Let $a$ , $b$ , $c$ , $d$ , $e$ be positive reals satisfying \begin{align*} a + b &= c a + b + c &= d a + b + c + d &= e.\end{align*} If $c=5$ , compute $a+b+c+d+e$ .
*Proposed by Evan Chen*
|
40
| 81.25 |
10,480 |
When Xiaohong was 3 years old, her mother's age was the same as Xiaohong's age this year. When her mother is 78 years old, Xiaohong's age is the same as her mother’s age this year. How old is Xiaohong’s mother this year?
|
53
| 68.75 |
10,481 |
Two right triangles $\triangle ABC$ and $\triangle ABD$ share a side $AB$ in such way that $AB=8$, $BC=12$, and $BD=10$. Let $E$ be a point on $BC$ such that $BE=9$. Determine the area of $\triangle ABE$.
|
36
| 76.5625 |
10,482 |
In the plane rectangular coordinate system $(xOy)$, establish a polar coordinate system $(.$ with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. If the polar coordinate equation of the line $l$ is $\sqrt {2}ρ\cos (θ- \dfrac {π}{4})-2=0$, and the polar coordinate equation of the curve $C$ is: $ρ\sin ^{2}θ=\cos θ$, shrink the abscissa of all points on the curve $C$ to half of the original, and the ordinate remains unchanged. Then, translate it to the right by one unit to obtain the curve $C_{1}$.
(I) Find the rectangular coordinate equation of the curve $C_{1}$;
(II) It is known that the line $l$ intersects with the curve $C_{1}$ at points $A$ and $B$. Point $P(2,0)$, find the value of $(|PA|+|PB|)$.
|
2 \sqrt {6}
| 0 |
10,483 |
In the triangular pyramid $P-ABC$, $PA\bot $ plane $ABC$, $\triangle ABC$ is an isosceles triangle, where $AB=BC=2$, $\angle ABC=120{}^\circ $, and $PA=4$. The surface area of the circumscribed sphere of the triangular pyramid $P-ABC$ is __________.
|
32\pi
| 46.875 |
10,484 |
Given a circle $C: (x-3)^{2}+y^{2}=25$ and a line $l: (m+1)x+(m-1)y-2=0$ (where $m$ is a parameter), the minimum length of the chord intercepted by the circle $C$ and the line $l$ is ______.
|
4\sqrt{5}
| 7.03125 |
10,485 |
\[
1.047. \left(\frac{\sqrt{561^{2} - 459^{2}}}{4 \frac{2}{7} \cdot 0.15 + 4 \frac{2}{7} : \frac{20}{3}} + 4 \sqrt{10}\right) : \frac{1}{3} \sqrt{40}
\]
|
125
| 14.84375 |
10,486 |
Let \( A \) be a subset of \(\{1, 2, 3, \ldots, 2019\}\) having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements in \( A \)?
|
505
| 37.5 |
10,487 |
There is a board of size 7×12 cells and a cube, the side of which is equal to a cell. One face of the cube is painted with non-drying paint. The cube can be placed in a certain cell of the board and rolled over an edge to an adjacent face. The cube cannot be placed twice in the same cell. What is the maximum number of cells that the cube can visit without soiling the board with paint?
|
84
| 68.75 |
10,488 |
When young fishermen were asked how many fish each of them caught, the first one replied, "I caught half the number of fish that my friend caught, plus 10 fish." The second one said, "And I caught as many as my friend, plus 20 fish." How many fish did the fishermen catch?
|
100
| 97.65625 |
10,489 |
It is known that there are a total of $n$ students in the first grade of Shuren High School, with $550$ male students. They are divided into layers based on gender, and $\frac{n}{10}$ students are selected to participate in a wetland conservation knowledge competition. It is given that there are $10$ more male students than female students among the participants. Find the value of $n$.
|
1000
| 71.875 |
10,490 |
Given Joy has 50 thin rods, one each of every integer length from 1 cm through 50 cm, and rods with lengths 8 cm, 12 cm, and 25 cm are already placed on a table, determine the number of the remaining rods that can be chosen as the fourth rod to form a quadrilateral with positive area.
|
36
| 40.625 |
10,491 |
If the graph of the linear function $y=-3x+m$ (where $m$ is a constant) passes through the second, third, and fourth quadrants, then the possible values of $m$ are _____. (Write down one possible value)
|
-1
| 71.09375 |
10,492 |
Three boxes each contain an equal number of hockey pucks. Each puck is either black or gold. All 40 of the black pucks and exactly $\frac{1}{7}$ of the gold pucks are contained in one of the three boxes. Determine the total number of gold hockey pucks.
|
140
| 49.21875 |
10,493 |
We roll five dice, each a different color. In how many ways can the sum of the rolls be 11?
|
205
| 8.59375 |
10,494 |
Calculate the square of 1007 without using a calculator.
|
1014049
| 100 |
10,495 |
The product of two consecutive page numbers is $20{,}412$. What is the sum of these two page numbers?
|
285
| 57.8125 |
10,496 |
Let $ABC$ be a right-angled triangle with $\angle ABC=90^\circ$ , and let $D$ be on $AB$ such that $AD=2DB$ . What is the maximum possible value of $\angle ACD$ ?
|
30
| 69.53125 |
10,497 |
Given that the sequence $\{a_n\}$ forms a geometric sequence, and $a_n > 0$.
(1) If $a_2 - a_1 = 8$, $a_3 = m$. ① When $m = 48$, find the general formula for the sequence $\{a_n\}$. ② If the sequence $\{a_n\}$ is unique, find the value of $m$.
(2) If $a_{2k} + a_{2k-1} + \ldots + a_{k+1} - (a_k + a_{k-1} + \ldots + a_1) = 8$, where $k \in \mathbb{N}^*$, find the minimum value of $a_{2k+1} + a_{2k+2} + \ldots + a_{3k}$.
|
32
| 30.46875 |
10,498 |
A pedestrian walked 5.5 kilometers in 1 hour but did not reach point \( B \) (short by \(2 \pi - 5.5\) km). Therefore, the third option is longer than the first and can be excluded.
In the first case, they need to cover a distance of 5.5 km along the alley. If they move towards each other, the required time is \(\frac{5.5}{20 + 5.5}\) hours.
In the second case, moving towards each other, after \(\frac{2 \pi - 5.5}{5.5}\) hours the pedestrian will reach point \( B \), while the cyclist will still be riding on the highway (since \(\frac{4}{15} > \frac{2 \pi - 5.5}{5.5}\)). Thus, the cyclist will always be on the highway and the closing speed of the pedestrian and the cyclist will be \(15 + 5.5 = 20.5\) km/h. They will meet in \(\frac{2 \pi - 1.5}{20.5}\) hours.
Compare the numbers obtained in cases 1 and 2:
$$
\frac{5.5}{25.5} = \frac{11}{51} < 0.22 < 0.23 < \frac{4 \cdot 3.14 - 3}{41} < \frac{2 \pi - 1.5}{20.5}
$$
Therefore, the answer is given by the first case.
**Note**: Some participants interpreted the condition differently, considering that the pedestrian and the cyclist met after an hour, and at that time, the pedestrian asked the cyclist to bring him the keys. Since the condition does not specify where exactly the neighbor was, and with a different interpretation of the condition, a similar problem arises (although with slightly more cumbersome calculations), the jury decided to accept both interpretations of the condition. The plan for solving the second interpretation is provided.
|
11/51
| 24.21875 |
10,499 |
There is a card game called "Twelve Months" that is played only during the Chinese New Year. The rules are as follows:
Step 1: Take a brand new deck of playing cards, remove the two jokers and the four Kings, leaving 48 cards. Shuffle the remaining cards.
Step 2: Lay out the shuffled cards face down into 12 columns, each column consisting of 4 cards.
Step 3: Start by turning over the first card in the first column. If the card is numbered \(N \ (N=1,2, \cdots, 12\), where J and Q correspond to 11 and 12 respectively, regardless of suit, place the card face up at the end of the \(N\)th column.
Step 4: Continue by turning over the first face-down card in the \(N\)th column and follow the same process as in step 3.
Step 5: Repeat this process until you cannot continue. If all 12 columns are fully turned over, it signifies that the next 12 months will be smooth and prosperous. Conversely, if some columns still have face-down cards remaining at the end, it indicates that there will be some difficulties in the corresponding months.
Calculate the probability that all columns are fully turned over.
|
1/12
| 0 |
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