Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
18,600 |
In 2019, our county built 4 million square meters of new housing, of which 2.5 million square meters are mid-to-low-priced houses. It is expected that in the coming years, the average annual increase in the area of new housing in our county will be 8% higher than the previous year. In addition, the area of mid-to-low-priced houses built each year will increase by 500,000 square meters compared to the previous year. So, by the end of which year:<br/>
$(1)$ The cumulative area of mid-to-low-priced houses built in our county over the years (with 2019 as the first cumulative year) will first exceed 22.5 million square meters?<br/>
$(2)$ The proportion of the area of mid-to-low-priced houses built in that year to the total area of housing built in that year will first exceed 85%? (Reference data: $1.08^{4}\approx 1.36$, $1.08^{5}\approx 1.47$)
|
2024
| 23.4375 |
18,601 |
The lateral surface area of a circular truncated cone is given by the formula, find the value for the lateral surface area of the cone where the radii of the upper and lower bases are $r=1$ and $R=4$ and the height is $4$.
|
25\pi
| 90.625 |
18,602 |
Two different numbers are selected from 1, 2, 3, 4, 5 to form a point (x, y). Find the probability that this point lies above the line x+y-5=0.
|
\frac{3}{5}
| 35.9375 |
18,603 |
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^{2}\) and \(13^{2}\)?
|
165
| 0.78125 |
18,604 |
In trapezoid \(ABCD\), the side \(AD\) is perpendicular to the bases and is equal to 9. \(CD\) is 12, and the segment \(AO\), where \(O\) is the point of intersection of the diagonals of the trapezoid, is equal to 6. Find the area of triangle \(BOC\).
|
\frac{108}{5}
| 1.5625 |
18,605 |
Given the sequence 1, $\frac{1}{2}$, $\frac{2}{1}$, $\frac{1}{3}$, $\frac{2}{2}$, $\frac{3}{1}$, $\frac{1}{4}$, $\frac{2}{3}$, $\frac{3}{2}$, $\frac{4}{1}$, ..., then $\frac{3}{5}$ is the \_\_\_\_\_\_ term of this sequence.
|
24
| 44.53125 |
18,606 |
What is the greatest integer less than or equal to \[\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}?\]
|
3124
| 55.46875 |
18,607 |
If person A has either a height or weight greater than person B, then person A is considered not inferior to person B. Among 100 young boys, if a person is not inferior to the other 99, he is called an outstanding boy. What is the maximum number of outstanding boys among the 100 boys?
|
100
| 71.09375 |
18,608 |
Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2?
|
\frac{1}{4}
| 92.1875 |
18,609 |
Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \in [-3, -1]$, $f(x) = -(x+2)^2$, and when $x \in [-1, 3)$, $f(x) = x$. Calculate the value of $f(1) + f(2) + f(3) + \ldots + f(2015)$.
|
336
| 46.09375 |
18,610 |
Our school's basketball team has won the national middle school basketball championship multiple times! In one competition, including our school's basketball team, 7 basketball teams need to be randomly divided into two groups (one group with 3 teams and the other with 4 teams) for the group preliminaries. The probability that our school's basketball team and the strongest team among the other 6 teams end up in the same group is ______.
|
\frac{3}{7}
| 61.71875 |
18,611 |
Given that the radius of the base of a cone is $2$ and the area of the unfolded side of the cone is $8\pi$, find the volume of the inscribed sphere in the cone.
|
\frac{32\sqrt{3}}{27}\pi
| 57.03125 |
18,612 |
Using the numbers $1$, $2$, $3$, $4$ to form a four-digit number without repeating digits, the number of four-digit numbers larger than $2134$ is _____. (Answer in digits)
|
17
| 17.1875 |
18,613 |
Define a new function $\$N$ such that $\$N = 0.75N + 2$. Calculate $\$(\$(\$30))$.
|
17.28125
| 0.78125 |
18,614 |
What is the smallest positive value of $m$ so that the equation $18x^2 - mx + 252 = 0$ has integral solutions?
|
162
| 13.28125 |
18,615 |
When Lisa squares her favorite $2$ -digit number, she gets the same result as when she cubes the sum of the digits of her favorite $2$ -digit number. What is Lisa's favorite $2$ -digit number?
|
27
| 95.3125 |
18,616 |
The perimeter of triangle \(ABC\) is 1. A circle \(\omega\) touches side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the length of segment \(XY\).
|
\frac{1}{2}
| 74.21875 |
18,617 |
Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$ th time, for any nonnegative integer $n$ , he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\tfrac ab$ for coprime positive integers $a$ and $b$ . Find $a + b$ .
*Proposed by Isaac Chen*
|
201
| 53.125 |
18,618 |
Six chairs are placed in a row. Find the number of ways 3 people can sit randomly in these chairs such that no two people sit next to each other.
|
24
| 92.96875 |
18,619 |
Determine the value of $k$ such that the equation
\[\frac{x + 3}{kx - 2} = x\] has exactly one solution.
|
-\frac{3}{4}
| 83.59375 |
18,620 |
What is the product of the digits in the base 8 representation of $9876_{10}$?
|
96
| 13.28125 |
18,621 |
Pirate Bob shares his treasure with Pirate Sam in a peculiar manner. Bob first declares, ``One for me, one for you,'' keeping one coin for himself and starting Sam's pile with one coin. Then Bob says, ``Two for me, and two for you,'' adding two more coins to his pile but updating Sam's total to two coins. This continues until Bob says, ``$x$ for me, $x$ for you,'' at which he takes $x$ more coins and makes Sam's total $x$ coins in total. After all coins are distributed, Pirate Bob has exactly three times as many coins as Pirate Sam. Find out how many gold coins they have between them?
|
20
| 13.28125 |
18,622 |
Given triangle $\triangle ABC$ with $\cos C = \frac{2}{3}$, $AC = 4$, and $BC = 3$, calculate the value of $\tan B$.
|
4\sqrt{5}
| 85.9375 |
18,623 |
While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$ .
|
35
| 93.75 |
18,624 |
Given triangle $ ABC$ . Point $ O$ is the center of the excircle touching the side $ BC$ . Point $ O_1$ is the reflection of $ O$ in $ BC$ . Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$ .
|
60
| 75 |
18,625 |
Convert the binary number $1101100_{(2)}$ to a decimal number.
|
108
| 78.90625 |
18,626 |
The taxi fare in Metropolis City is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles you can ride for $15.
|
3.75
| 49.21875 |
18,627 |
Shift the graph of the function $y = \sin\left(\frac{\pi}{3} - x\right)$ to obtain the graph of the function $y = \cos\left(x + \frac{2\pi}{3}\right)$.
|
\frac{\pi}{2}
| 47.65625 |
18,628 |
Find the number of solutions to:
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval $(0,50 \pi)$.
|
50
| 70.3125 |
18,629 |
Read the following material: The overall idea is a common thinking method in mathematical problem solving: Here is a process of a student factorizing the polynomial $(x^{2}+2x)(x^{2}+2x+2)+1$. Regard "$x^{2}+2x$" as a whole, let $x^{2}+2x=y$, then the original expression $=y^{2}+2y+1=\left(y+1\right)^{2}$, and then restore "$y$".
**Question:**
(1) ① The student's factorization result is incorrect, please write down the correct result directly ______;
② According to material $1$, please try to imitate the above method to factorize the polynomial $(x^{2}-6x+8)(x^{2}-6x+10)+1$;
(2) According to material $1$, please try to imitate the above method to calculate:
$(1-2-3-\ldots -2020)\times \left(2+3+\ldots +2021\right)-\left(1-2-3-\ldots -2021\right)\times \left(2+3+\ldots +2020\right)$.
|
2021
| 45.3125 |
18,630 |
The sum of the heights on the two equal sides of an isosceles triangle is equal to the height on the base. Find the sine of the base angle.
|
$\frac{\sqrt{15}}{4}$
| 0 |
18,631 |
Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has a diameter of 8 cm. In Crate A, the pipes are packed directly on top of each other in 25 rows of 8 pipes each across the width of the crate. In Crate B, pipes are packed in a staggered (hexagonal) pattern that results in 24 rows, with the rows alternating between 7 and 8 pipes.
After the crates have been packed with an equal number of 200 pipes each, what is the positive difference in the total heights (in cm) of the two packings?
|
200 - 96\sqrt{3}
| 5.46875 |
18,632 |
A three-digit natural number $abc$ is termed a "convex number" if and only if the digits $a$, $b$, and $c$ (representing the hundreds, tens, and units place, respectively) satisfy $a < b$ and $c < b$. Given that $a$, $b$, and $c$ belong to the set $\{5, 6, 7, 8, 9\}$ and are distinct from one another, find the probability that a randomly chosen three-digit number $abc$ is a "convex number".
|
\frac {1}{3}
| 75.78125 |
18,633 |
A rectangular garden needs to be enclosed on three sides using a 70-meter rock wall as one of the sides. Fence posts are placed every 10 meters along the fence, including at the ends where the fence meets the rock wall. If the area of the garden is 2100 square meters, calculate the fewest number of posts required.
|
14
| 8.59375 |
18,634 |
Which of the following is equal to $2017 - \frac{1}{2017}$?
|
$\frac{2018 \times 2016}{2017}$
| 0 |
18,635 |
In how many ways can the number 5 be expressed as the sum of one or more positive integers?
|
16
| 10.9375 |
18,636 |
A line passes through the distinct vectors $\mathbf{u}$ and $\mathbf{v}.$ For some value of $k$, the vector
\[k \mathbf{u} + \frac{5}{8} \mathbf{v}\] must also lie on the line. Find $k$.
|
\frac{3}{8}
| 97.65625 |
18,637 |
Given the random variable $X$ follows a normal distribution $N(-1, \sigma^2)$, and $P(-3 \leq X \leq -1) = 0.4$, calculate the probability of $X$ being greater than or equal to $1$.
|
0.1
| 89.0625 |
18,638 |
Using the digits $1$, $2$, $3$, $5$, and $6$ exactly once, the five-digit positive integers are formed and arranged in ascending order. What is the $60^{\text{th}}$ integer in this list?
|
32651
| 5.46875 |
18,639 |
Given the set $\{a,b,c\}=\{1,2,3\}$, and three relations: $①a\neq 3$, $②b=3$, $③c\neq 1$. Only one of these relations is correct. Find the value of $100a+10b+c=\_\_\_\_\_\_.$
|
312
| 47.65625 |
18,640 |
Given that the right focus of ellipse $I$: $\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0)$ is $(2 \sqrt{2},0)$, and ellipse $I$ passes through point $(3,1)$.
(1) Find the equation of ellipse $I$;
(2) Let line $l$ with slope $1$ intersect ellipse $I$ at two distinct points $A$ and $B$. Construct an isosceles triangle $PAB$ with base $AB$ and vertex $P$ at coordinates $(-3,2)$. Find the area of $∆PAB$.
|
\frac {9}{2}
| 7.8125 |
18,641 |
A sphere is inside a cube with an edge length of $3$, and it touches all $12$ edges of the cube. Find the volume of the sphere.
|
9\sqrt{2}\pi
| 21.09375 |
18,642 |
What is the total area of two overlapping circles where circle A has center at point $A(2, -1)$ and passes through point $B(5, 4)$, and circle B has center at point $C(3, 3)$ and passes through point $D(5, 8)$? Express your answer in terms of $\pi$.
|
63\pi
| 15.625 |
18,643 |
The sum of the numerical coefficients in the binomial $(2a+2b)^7$ is $\boxed{32768}$.
|
16384
| 39.84375 |
18,644 |
Given a hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. One of its asymptotes is $x+\sqrt{2}y=0$. Point $M$ lies on the hyperbola, and $MF\_1 \perp x$-axis. If $F\_2$ is also a focus of the parabola $y^{2}=12x$, find the distance from $F\_1$ to line $F\_2M$.
|
\frac{6}{5}
| 69.53125 |
18,645 |
The congruent sides of an isosceles triangle are each 8 cm long, and the perimeter is 26 cm. In centimeters, what is the length of the base? Also, find the area of the triangle.
|
5\sqrt{39}
| 89.0625 |
18,646 |
Given the geometric sequence with 8 inserted numbers between 1 and 3, find the product of these 8 inserted numbers.
|
81
| 85.15625 |
18,647 |
During a journey, the distance read on the odometer was 450 miles. On the return trip, using snow tires for the same distance, the reading was 440 miles. If the original wheel radius was 15 inches, find the increase in the wheel radius, correct to the nearest hundredth of an inch.
|
0.34
| 8.59375 |
18,648 |
A round-robin tennis tournament is organized where each player is supposed to play every other player exactly once. However, the tournament is scheduled to have one rest day during which no matches will be played. If there are 10 players in the tournament, and the tournament was originally scheduled for 9 days, but one day is now a rest day, how many matches will be completed?
|
40
| 43.75 |
18,649 |
The number of different arrangements of $6$ rescue teams to $3$ disaster sites, where site $A$ has at least $2$ teams and each site is assigned at least $1$ team.
|
360
| 0.78125 |
18,650 |
Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if $CX$ bisects $\angle ACB$. In the diagram, $AB = 34$, $BC = 35$, and $AC = 39$. Let the coordinates be:
- $A = (-17,0)$,
- $B = (17,0)$,
- $C = (0,30)$.
Express your answer as a common fraction.
|
\frac{35}{39}
| 93.75 |
18,651 |
A triangular pyramid stack of apples is formed with a base consisting of 6 apples on each side. Each higher layer is formed by 3 apples per row, with the top layer consisting of a single apple. Calculate the total number of apples used in the entire stack.
|
56
| 54.6875 |
18,652 |
Given the vectors $\overrightarrow{a}=(\cos 40^{\circ},\sin 40^{\circ})$, $\overrightarrow{b}=(\sin 20^{\circ},\cos 20^{\circ})$, and $\overrightarrow{u}= \sqrt {3} \overrightarrow{a}+λ \overrightarrow{b}$ (where $λ∈R$), find the minimum value of $|\overrightarrow{u}|$.
|
\frac {\sqrt {3}}{2}
| 0 |
18,653 |
Adnan is trying to remember his four-digit PIN. He is sure it contains the digits 5, 3, 7, and 0 but can't recall the order in which they appear. How many different arrangements are possible for his PIN?
|
24
| 85.15625 |
18,654 |
What is the largest five-digit number whose digits add up to 20?
|
99200
| 85.15625 |
18,655 |
The angle that has the same terminal side as $- \frac{\pi}{3}$ is $\frac{\pi}{3}$.
|
\frac{5\pi}{3}
| 42.1875 |
18,656 |
A die with faces showing the numbers $0,1,2,3,4,5$ is rolled until the total sum of the rolled numbers exceeds 12. What is the most likely value of this sum?
|
13
| 17.1875 |
18,657 |
Given a sequence $\{a_n\}$, let $S_n$ denote the sum of its first $n$ terms. Define $T_n = \frac{S_1 + S_2 + \dots + S_n}{n}$ as the "ideal number" of the sequence $a_1, a_2, \dots, a_n$. If the "ideal number" of the sequence $a_1, a_2, \dots, a_{502}$ is $2012$, calculate the "ideal number" of the sequence $2, a_1, a_2, \dots, a_{502}$.
|
2010
| 60.9375 |
18,658 |
Given two vectors in the plane, $\mathbf{a} = (2m+1, 3)$ and $\mathbf{b} = (2, m)$, and $\mathbf{a}$ is in the opposite direction to $\mathbf{b}$, calculate the magnitude of $\mathbf{a} + \mathbf{b}$.
|
\sqrt{2}
| 91.40625 |
18,659 |
All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list?
|
156
| 5.46875 |
18,660 |
Given the sequence 1, 1+2, 2+3+4, 3+4+5+6, ..., the value of the 8th term in this sequence is: ______.
|
84
| 8.59375 |
18,661 |
What time is it 2017 minutes after 20:17?
|
05:54
| 21.875 |
18,662 |
$g(x):\mathbb{Z}\rightarrow\mathbb{Z}$ is a function that satisfies $$ g(x)+g(y)=g(x+y)-xy. $$ If $g(23)=0$ , what is the sum of all possible values of $g(35)$ ?
|
210
| 78.125 |
18,663 |
A note contains three two-digit numbers that are said to form a sequence with a fourth number under a cryptic condition. The numbers provided are 46, 19, and 63, but the fourth number is unreadable. You know that the sum of the digits of all four numbers is $\frac{1}{4}$ of the total sum of these four numbers. What is the fourth number?
|
28
| 14.0625 |
18,664 |
Given $f(\alpha)=\frac{2\sin(2\pi-\alpha)\cos(2\pi+\alpha)-\cos(-\alpha)}{1+\sin^{2}\alpha+\sin(2\pi+\alpha)-\cos^{2}(4\pi-\alpha)}$, find the value of $f\left(-\frac{23}{6}\pi \right)$.
|
-\sqrt{3}
| 84.375 |
18,665 |
Let $P$ be the centroid of triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. The area of triangle $ABC$ is 24. Find the area of triangle $G_1 G_2 G_3$.
|
\frac{8}{3}
| 93.75 |
18,666 |
Given a revised graph for Lambda Corp., the number of employees at different tenure periods is represented with the following marks:
- Less than 1 year: 3 marks
- 1 to less than 2 years: 6 marks
- 2 to less than 3 years: 5 marks
- 3 to less than 4 years: 4 marks
- 4 to less than 5 years: 2 marks
- 5 to less than 6 years: 2 marks
- 6 to less than 7 years: 3 marks
- 7 to less than 8 years: 2 marks
- 8 to less than 9 years: 1 mark
- 9 to less than 10 years: 1 mark
Determine what percent of the employees have worked there for $6$ years or more.
|
24.14\%
| 1.5625 |
18,667 |
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle?
*Author: Ray Li*
|
256
| 16.40625 |
18,668 |
Given a geometric sequence $\left\{a_{n}\right\}$ with real terms, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then $S_{40}$ is equal to:
|
150
| 82.03125 |
18,669 |
John has saved up $5235_9$ dollars for a trip to Japan. A round-trip airline ticket costs $1250_8$ dollars. In base ten, how many dollars will he have left for lodging and food?
|
3159
| 4.6875 |
18,670 |
What is the product of all real numbers that are tripled when added to their reciprocals?
|
-\frac{1}{2}
| 82.8125 |
18,671 |
Calculate: $|\sqrt{8}-2|+(\pi -2023)^{0}+(-\frac{1}{2})^{-2}-2\cos 60^{\circ}$.
|
2\sqrt{2}+2
| 69.53125 |
18,672 |
Given the function $y=\sin (3x+ \frac {\pi}{3})\cos (x- \frac {\pi}{6})+\cos (3x+ \frac {\pi}{3})\sin (x- \frac {\pi}{6})$, find the equation of one of the axes of symmetry.
|
\frac {\pi}{12}
| 80.46875 |
18,673 |
Given that $F$ is the left focus of the ellipse $C:\frac{{x}^{2}}{3}+\frac{{y}^{2}}{2}=1$, $M$ is a moving point on the ellipse $C$, and point $N(5,3)$, then the minimum value of $|MN|-|MF|$ is ______.
|
5 - 2\sqrt{3}
| 60.9375 |
18,674 |
Find the square root of $\dfrac{10!}{210}$.
|
24\sqrt{30}
| 11.71875 |
18,675 |
In the arithmetic sequence $\left\{a_{n}\right\}$, if $\frac{a_{11}}{a_{10}} < -1$ and the sum of its first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\quad$ ).
|
19
| 54.6875 |
18,676 |
In the multiplication shown, $P, Q,$ and $R$ are all different digits such that
$$
\begin{array}{r}
P P Q \\
\times \quad Q \\
\hline R Q 5 Q
\end{array}
$$
What is the value of $P + Q + R$?
|
17
| 0 |
18,677 |
Among all the four-digit numbers without repeated digits, how many numbers have the digit in the thousandth place 2 greater than the digit in the unit place?
|
448
| 10.9375 |
18,678 |
Encrypt integers by the following method: the digit of each number becomes the units digit of its product with 7, then replace each digit _a_ with $10 - _a_$. If a number is encrypted by the above method and becomes 473392, then the original number is ______.
|
891134
| 53.90625 |
18,679 |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?
|
115
| 28.90625 |
18,680 |
A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$ , $DP = 2$ , $CP = 3$ . Determine the length of the segment $AP$ .
|
\sqrt{7}
| 33.59375 |
18,681 |
Each segment with endpoints at the vertices of a regular 100-gon is colored red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers were placed at the vertices such that the sum of their squares equals 1, and at the segments, the products of the numbers at the endpoints were placed. Then, the sum of the numbers on the red segments was subtracted by the sum of the numbers on the blue segments. What is the largest possible value that could be obtained?
I. Bogdanov
|
1/2
| 14.84375 |
18,682 |
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom and sides.
[asy]/* AMC8 2002 #22 Problem */
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
draw((0,1)--(0.5,1.5)--(1.5,1.5)--(1,1));
draw((1,0)--(1.5,0.5)--(1.5,1.5));
draw((0.5,1.5)--(1,2)--(1.5,2));
draw((1.5,1.5)--(1.5,3.5)--(2,4)--(3,4)--(2.5,3.5)--(2.5,0.5)--(1.5,.5));
draw((1.5,3.5)--(2.5,3.5));
draw((1.5,1.5)--(3.5,1.5)--(3.5,2.5)--(1.5,2.5));
draw((3,4)--(3,3)--(2.5,2.5));
draw((3,3)--(4,3)--(4,2)--(3.5,1.5));
draw((4,3)--(3.5,2.5));
draw((2.5,.5)--(3,1)--(3,1.5));[/asy]
|
26
| 28.90625 |
18,683 |
Given the function f(x) = $\sqrt {2}$sin $\frac {x}{2}$cos $\frac {x}{2}$ - $\sqrt {2}$sin<sup>2</sup> $\frac {x}{2}$,
(1) Find the smallest positive period of f(x);
(2) Find the minimum value of f(x) in the interval [-π, 0].
|
-1 - \frac { \sqrt {2}}{2}
| 0 |
18,684 |
Given the function $y=4\cos (2x+\frac{\pi}{4})$, determine the direction and magnitude of horizontal shift required to obtain the graph of the function $y=4\cos 2x$.
|
\frac{\pi}{8}
| 46.875 |
18,685 |
Evaluate the value of $3^2 \times 4 \times 6^3 \times 7!$.
|
39191040
| 12.5 |
18,686 |
Given that $\tan α$ and $\tan β$ are the roots of the equation $x^{2}+3 \sqrt {3}x+4=0$, and $\(- \frac {π}{2} < α < \frac {π}{2}\)$, $\(- \frac {π}{2} < β < \frac {π}{2}\)$, find $α+β$.
|
- \frac {2\pi}{3}
| 39.84375 |
18,687 |
In the Cartesian coordinate system $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. If $M$ is a moving point on the parabola, determine the maximum value of $\frac{|MO|}{|MF|}$.
|
\frac{2\sqrt{3}}{3}
| 37.5 |
18,688 |
If the line passing through the point $P(2,1)$ intersects the graph of the function $f(x)= \frac{2x+3}{2x-4}$ at points $A$ and $B$, and $O$ is the origin, calculate the value of $( \overrightarrow{OA}+ \overrightarrow{OB})\cdot \overrightarrow{OP}$.
|
10
| 69.53125 |
18,689 |
Among all triangles $ABC,$ find the maximum value of $\cos A + \cos B \cos C.$
|
\frac{5}{2}
| 0 |
18,690 |
Given that Square $ABCD$ has side length $5$, point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$, calculate the degree measure of $\angle AMD$.
|
45
| 34.375 |
18,691 |
Plane M is parallel to plane N. There are 3 different points on plane M and 4 different points on plane N. The maximum number of tetrahedrons with different volumes that can be determined by these 7 points is ____.
|
34
| 20.3125 |
18,692 |
Let $ABC$ be an acute triangle. Let $H$ and $D$ be points on $[AC]$ and $[BC]$ , respectively, such that $BH \perp AC$ and $HD \perp BC$ . Let $O_1$ be the circumcenter of $\triangle ABH$ , and $O_2$ be the circumcenter of $\triangle BHD$ , and $O_3$ be the circumcenter of $\triangle HDC$ . Find the ratio of area of $\triangle O_1O_2O_3$ and $\triangle ABH$ .
|
1/4
| 84.375 |
18,693 |
The volume of the box is 360 cubic units where $a, b,$ and $c$ are integers with $1<c<b<a$. What is the largest possible value of $b$?
|
12
| 2.34375 |
18,694 |
Given that \\(AB\\) is a chord passing through the focus of the parabola \\(y^{2} = 4\sqrt{3}x\\), and the midpoint \\(M\\) of \\(AB\\) has an x-coordinate of \\(2\\), calculate the length of \\(AB\\.
|
4 + 2\sqrt{3}
| 6.25 |
18,695 |
In triangle $ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, and the height from $C$ to line $AB$ is $12$, find the length of side $AB$.
|
\frac{50}{3}
| 48.4375 |
18,696 |
A hyperbola with its center shifted to $(1,1)$ passes through point $(4, 2)$. The hyperbola opens horizontally, with one of its vertices at $(3, 1)$. Determine $t^2$ if the hyperbola also passes through point $(t, 4)$.
|
36
| 8.59375 |
18,697 |
A square with a side length of one unit has one of its vertices separated from the other three by a line \( e \). The products of the distances of opposite vertex pairs from \( e \) are equal. What is the distance of the center of the square from \( e \)?
|
\frac{1}{2}
| 28.90625 |
18,698 |
Given that the leftmost position can be occupied by student A or B, and the rightmost position cannot be occupied by student A, find the number of different arrangements of the six high-performing students from Class 1, Grade 12.
|
216
| 28.90625 |
18,699 |
A belt is placed without slack around two non-crossing circular pulleys which have radii of $15$ inches and $5$ inches respectively. The distance between the points where the belt contacts the pulleys is $30$ inches. Determine the distance between the centers of the two pulleys.
A) $20$ inches
B) $10\sqrt{10}$ inches
C) $40$ inches
D) $50$ inches
|
10\sqrt{10}
| 69.53125 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.