Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
20,400 |
Olympus Corporation has released a new version of its popular Vowel Soup. In this version, each vowel (A, E, I, O, U) appears 7 times in each bowl. How many six-letter "words" can be formed from this Olympus Vowel Soup? Note: Words do not need to be actual words in the English language.
|
15625
| 25.78125 |
20,401 |
Given the equations
$$
z^{2}=4+4 \sqrt{15} i \text { and } z^{2}=2+2 \sqrt{3} i,
$$
the roots are the coordinates of the vertices of a parallelogram in the complex plane. If the area $S$ of the parallelogram can be expressed as $p \sqrt{q} - r \sqrt{s}$ (where $p, q, r, s \in \mathbf{Z}_{+}$, and $r$ and $s$ are not perfect squares), find the value of $p+q+r+s$.
|
20
| 14.0625 |
20,402 |
Given the definition: $min\{a,b\} = \begin{cases} a,\quad a\leqslant b \\ b,\quad a > b \end{cases}$. If a point $P(x,y)$ is randomly selected within the region defined by $\begin{cases} 0\leqslant x\leqslant 2 \\ 0\leqslant y\leqslant 6 \end{cases}$, find the probability that $x$ and $y$ satisfy $min\{3x-2y+6,x-y+4\}=x-y+4$.
|
\frac{2}{3}
| 76.5625 |
20,403 |
Given the function $f(x) = f'(1)e^{x-1} - f(0)x + \frac{1}{2}x^2$ (where $f'(x)$ is the derivative of $f(x)$, and $e$ is the base of the natural logarithm), and $g(x) = \frac{1}{2}x^2 + ax + b$ ($a \in \mathbb{R}, b \in \mathbb{R}$):
(Ⅰ) Find the explicit formula for $f(x)$ and its extremum;
(Ⅱ) If $f(x) \geq g(x)$, find the maximum value of $\frac{b(a+1)}{2}$.
|
\frac{e}{4}
| 32.03125 |
20,404 |
Given an ellipse $\dfrac {x^{2}}{4}+ \dfrac {y^{2}}{3}=1$ with its left and right foci denoted as $F_{1}$ and $F_{2}$ respectively, and a point $P$ on the ellipse. If $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}= \dfrac {5}{2}$, calculate $| \overrightarrow{PF_{1}}|\cdot| \overrightarrow{PF_{2}}|$.
|
\dfrac{7}{2}
| 57.8125 |
20,405 |
Given $f(n) = n^2 \cos(n\pi)$ and $a_n = f(n) + f(n+1)$, find the sum of $a_1 + a_2 + a_3 + \cdots + a_{100}$.
|
-100
| 92.1875 |
20,406 |
Compute the following expressions:
(1) $2 \sqrt{12} -6 \sqrt{ \frac{1}{3}} + \sqrt{48}$
(2) $(\sqrt{3}-\pi)^{0}-\frac{\sqrt{20}-\sqrt{15}}{\sqrt{5}}+(-1)^{2017}$
|
\sqrt{3} - 2
| 40.625 |
20,407 |
In a triangle \( \triangle ABC \), \(a\), \(b\), and \(c\) are the sides opposite to angles \(A\), \(B\), and \(C\) respectively, with \(B= \dfrac {2\pi}{3}\). If \(a^{2}+c^{2}=4ac\), then find the value of \( \dfrac {\sin (A+C)}{\sin A\sin C} \).
|
\dfrac{10\sqrt{3}}{3}
| 67.1875 |
20,408 |
There are 552 weights with masses of 1g, 2g, 3g, ..., 552g. Divide them into three equal weight piles.
|
50876
| 0 |
20,409 |
A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $16$, $25$, and $36$, respectively. Furthermore, a circle centered at $Q$ inside $\triangle DEF$ cuts off a segment from $u_3$ with area $9$. Find the area of $\triangle DEF$.
|
225
| 3.90625 |
20,410 |
Given that the terminal side of angle $\alpha$ is in the second quadrant and intersects the unit circle at point $P(m, \frac{\sqrt{15}}{4})$.
$(1)$ Find the value of the real number $m$;
$(2)$ Let $f(\alpha) = \frac{\cos(2\pi - \alpha) + \tan(3\pi + \alpha)}{\sin(\pi - \alpha) \cdot \cos(\alpha + \frac{3\pi}{2})}$. Find the value of $f(\alpha)$.
|
-\frac{4 + 16\sqrt{15}}{15}
| 67.96875 |
20,411 |
Square $XYZW$ has area $144$. Point $P$ lies on side $\overline{XW}$, such that $XP = 2WP$. Points $Q$ and $R$ are the midpoints of $\overline{ZP}$ and $\overline{YP}$, respectively. Quadrilateral $XQRW$ has an area of $20$. Calculate the area of triangle $RWP$.
|
12
| 55.46875 |
20,412 |
The energy stored by a pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges are initially placed at the corners of a square with each side length $d$. This configuration stores a total of $20$ Joules of energy. How much energy, in Joules, would be stored if two of these charges are moved such that they form a new square with each side doubled (i.e., side length $2d$)?
|
10
| 73.4375 |
20,413 |
The diagram shows a rhombus and two sizes of regular hexagon. What is the ratio of the area of the smaller hexagon to the area of the larger hexagon?
|
1:4
| 0.78125 |
20,414 |
Given that five boys, A, B, C, D, and E, are randomly assigned to stay in 3 standard rooms (with at most two people per room), calculate the probability that A and B stay in the same standard room.
|
\frac{1}{5}
| 23.4375 |
20,415 |
Use the Horner's method to calculate the value of the polynomial $f(x) = 7x^7 + 6x^6 + 5x^5 + 4x^4 + 3x^3 + 2x^2 + x$ when $x = 3$, and find the value of $v_3$.
|
262
| 45.3125 |
20,416 |
The eccentricity of the ellipse $\frac {x^{2}}{9}+ \frac {y^{2}}{4+k}=1$ is $\frac {4}{5}$. Find the value of $k$.
|
21
| 32.8125 |
20,417 |
$A_{2n}^{n+3} + A_{4}^{n+1} = \boxed{\text{\_\_\_\_\_\_\_\_}}$.
|
744
| 10.15625 |
20,418 |
How many positive integers less than $800$ are either a perfect cube or a perfect square?
|
35
| 7.03125 |
20,419 |
How many three-digit numbers are increased by 99 when their digits are reversed?
|
80
| 85.9375 |
20,420 |
In the sequence $\{a_{n}\}$, the sum of the first $n$ terms is $S_{n}=2^{n}+2$. Find the sum $T_{20}$ of the first 20 terms in the sequence $\{\log _{2}a_{n}\}$.
|
192
| 89.0625 |
20,421 |
In the polar coordinate system, the distance from the center of the circle $\rho=4\cos\theta$ ($\rho\in\mathbb{R}$) to the line $\theta= \frac {\pi}{3}$ can be found using the formula for the distance between a point and a line in polar coordinates.
|
\sqrt {3}
| 0 |
20,422 |
The sum of the digits in the product of $\overline{A A A A A A A A A} \times \overline{B B B B B B B B B}$.
|
81
| 53.90625 |
20,423 |
Given that Sofia has a $5 \times 7$ index card, if she shortens the length of one side by $2$ inches and the card has an area of $21$ square inches, find the area of the card in square inches if instead she shortens the length of the other side by $1$ inch.
|
30
| 35.15625 |
20,424 |
Solve the equations:<br/>$(1)x^{2}-5x+1=0$;<br/>$(2) 2\left(x-5\right)^{2}+x\left(x-5\right)=0$.
|
\frac{10}{3}
| 0 |
20,425 |
What is the probability that when two numbers are randomly selected from the set {1, 2, 3, 4}, one number is twice the other?
|
\frac{1}{3}
| 90.625 |
20,426 |
Given positive integers $a$ and $b$ are members of a set where $a \in \{2, 3, 5, 7\}$ and $b \in \{2, 4, 6, 8\}$, and the sum of $a$ and $b$ must be even, determine the smallest possible value for the expression $2 \cdot a - a \cdot b$.
|
-12
| 42.96875 |
20,427 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=\sqrt{2}$, and $\overrightarrow{a}\perp(\overrightarrow{a}+2\overrightarrow{b})$, calculate the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$.
|
-1
| 3.125 |
20,428 |
The parabolas $y = (x - 2)^2$ and $x + 6 = (y - 2)^2$ intersect at four points $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$. Find
\[
x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.
\]
|
16
| 68.75 |
20,429 |
For all real numbers $x$ and $y$, define the mathematical operation $\diamond$ such that $x \diamond 0 = 2x, x \diamond y = y \diamond x$, and $(x + 1) \diamond y = (x \diamond y) \cdot (y + 2)$. What is the value of $6 \diamond 3$?
|
93750
| 29.6875 |
20,430 |
Let $\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\mathcal S$ , such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal S)$ across all such $\mathcal S$ .
*Ankan Bhattacharya*
|
1430
| 89.0625 |
20,431 |
Let $ y_0$ be chosen randomly from $ \{0, 50\}$ , let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$ , let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$ , and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$ . (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$ , $ P(1) \equal{} y_1$ , $ P(2) \equal{} y_2$ , and $ P(3) \equal{} y_3$ . What is the expected value of $ P(4)$ ?
|
107
| 1.5625 |
20,432 |
In $\triangle ABC$, $\cos A= \frac{\sqrt{3}}{3}$, $c=\sqrt{3}$, and $a=3\sqrt{2}$. Find the value of $\sin C$ and the area of $\triangle ABC$.
|
\frac{5\sqrt{2}}{2}
| 35.15625 |
20,433 |
Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make $?$
|
52
| 10.9375 |
20,434 |
Two circles of radius \( r \) are externally tangent to each other and internally tangent to the ellipse \( x^2 + 4y^2 = 8 \). Find \( r \).
|
\frac{\sqrt{6}}{2}
| 1.5625 |
20,435 |
Find the product of all positive integral values of $x$ such that $x^2 - 40x + 399 = q$ for some prime number $q$. Note that there must be at least one such $x$.
|
396
| 67.96875 |
20,436 |
A particle with charge $8.0 \, \mu\text{C}$ and mass $17 \, \text{g}$ enters a magnetic field of magnitude $\text{7.8 mT}$ perpendicular to its non-zero velocity. After 30 seconds, let the absolute value of the angle between its initial velocity and its current velocity, in radians, be $\theta$ . Find $100\theta$ .
*(B. Dejean, 5 points)*
|
1.101
| 3.125 |
20,437 |
Given the line $y=ax$ intersects the circle $C:x^2+y^2-2ax-2y+2=0$ at points $A$ and $B$, and $\Delta ABC$ is an equilateral triangle, then the area of circle $C$ is __________.
|
6\pi
| 34.375 |
20,438 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=1$, and $2\cos C - 2a - c = 0$.
(Ⅰ) Find the magnitude of angle $B$;
(Ⅱ) Find the distance from the circumcenter of $\triangle ABC$ to side $AC$.
|
\frac{\sqrt{3}}{6}
| 60.9375 |
20,439 |
In Mr. Johnson's class, 12 out of 20 students received an 'A' grade and the rest received a 'B' grade. Mrs. Smith, teaching a different class, observed that the proportion of students getting 'A' was the same. If Mrs. Smith has 30 students total, how many students received an 'A' grade? Moreover, if the same proportion received 'B' as in Mr. Johnson’s class, how many students in Mrs. Smith’s class received 'B'?
|
12
| 85.9375 |
20,440 |
The sizes of circular pizzas are determined by their diameter. If Lana's initial pizza was 14 inches in diameter and she decides to order a larger pizza with a diameter of 18 inches instead, what is the percent increase in the area of her pizza?
|
65.31\%
| 60.15625 |
20,441 |
Given the function $f(x) = x^3 - ax^2 + 3x$, and $x=3$ is an extremum of $f(x)$.
(Ⅰ) Determine the value of the real number $a$;
(Ⅱ) Find the equation of the tangent line $l$ to the graph of $y=f(x)$ at point $P(1, f(1))$;
(Ⅲ) Find the minimum and maximum values of $f(x)$ on the interval $[1, 5]$.
|
15
| 65.625 |
20,442 |
Given $x \gt -1$, $y \gt 0$, and $x+2y=1$, find the minimum value of $\frac{1}{x+1}+\frac{1}{y}$.
|
\frac{3+2\sqrt{2}}{2}
| 3.90625 |
20,443 |
In a right triangle $\triangle PQR$, we know that $\tan Q = 0.5$ and the length of $QP = 16$. What is the length of $QR$?
|
8 \sqrt{5}
| 21.875 |
20,444 |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is
$$
\begin{cases}
x=2+t\cos \alpha \\
y=1+t\sin \alpha
\end{cases}
(t \text{ is the parameter}),
$$
In the polar coordinate system (which uses the same unit length as the Cartesian coordinate system $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis), the equation of circle $C$ is $\rho = 6\cos \theta$.
(1) Find the Cartesian coordinate equation of circle $C$.
(2) Suppose circle $C$ intersects line $l$ at points $A$ and $B$. If point $P$ has coordinates $(2,1)$, find the minimum value of $|PA|+|PB|$.
|
2\sqrt{7}
| 59.375 |
20,445 |
There was a bonus fund in a certain institution. It was planned to distribute the fund such that each employee of the institution would receive $50. However, it turned out that the last employee on the list would receive only $45. Then, in order to ensure fairness, it was decided to give each employee $45, leaving $95 undistributed, which would be carried over to the fund for the next year. What was the amount of the initial fund?
|
950
| 40.625 |
20,446 |
One interior angle in a triangle measures $50^{\circ}$. What is the angle between the bisectors of the remaining two interior angles?
|
65
| 27.34375 |
20,447 |
Evaluate the argument $\theta$ of the complex number
\[
e^{11\pi i/60} + e^{31\pi i/60} + e^{51 \pi i/60} + e^{71\pi i /60} + e^{91 \pi i /60}
\]
expressed in the form $r e^{i \theta}$ with $0 \leq \theta < 2\pi$.
|
\frac{17\pi}{20}
| 3.125 |
20,448 |
Let set $\mathcal{A}$ be a 70-element subset of $\{1,2,3,\ldots,120\}$, and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$.
|
3501
| 92.96875 |
20,449 |
Let $d$ be a positive number such that when $144$ is divided by $d$, the remainder is $9$. Compute the sum of all possible two-digit values of $d$.
|
87
| 80.46875 |
20,450 |
Let $ABC$ be an equilateral triangle . Let point $D$ lie on side $AB,E$ lie on side $AC, D_1$ and $E_1$ lie on side BC such that $AB=DB+BD_1$ and $AC=CE+CE_1$ . Calculate the smallest angle between the lines $DE_1$ and $ED_1$ .
|
60
| 72.65625 |
20,451 |
The sides of a triangle have lengths \( 13, 17, \) and \( k, \) where \( k \) is a positive integer. For how many values of \( k \) is the triangle obtuse?
|
14
| 52.34375 |
20,452 |
An integer $N$ is selected at random in the range $1 \leq N \leq 2030$. Calculate the probability that the remainder when $N^{12}$ is divided by $7$ is $1$.
|
\frac{6}{7}
| 6.25 |
20,453 |
A chessboard’s squares are labeled with numbers as follows:
[asy]
unitsize(0.8 cm);
int i, j;
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
label("$\frac{1}{" + string(9 - i + j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]
Eight of the squares are chosen such that each row and each column has exactly one selected square. Find the maximum sum of the labels of these eight chosen squares.
|
\frac{8}{9}
| 11.71875 |
20,454 |
If $2\tan\alpha=3\tan \frac{\pi}{8}$, then $\tan\left(\alpha- \frac{\pi}{8}\right)=$ ______.
|
\frac{5\sqrt{2}+1}{49}
| 16.40625 |
20,455 |
After reading the following solution, answer the question: Xiaofang found in the simplification of $\sqrt{7+4\sqrt{3}}$ that first, $\sqrt{7+4\sqrt{3}}$ can be simplified to $\sqrt{7+2\sqrt{12}}$. Since $4+3=7$ and $4\times 3=12$, that is, ${(\sqrt{4})^2}+{(\sqrt{3})^2}=7$, $\sqrt{4}×\sqrt{3}=\sqrt{12}$, so $\sqrt{7+4\sqrt{3}}=\sqrt{7+2\sqrt{12}}=\sqrt{{{(\sqrt{4})}^2}+2\sqrt{4×3}+{{(\sqrt{3})}^2}}=\sqrt{{{(\sqrt{4}+\sqrt{3})}^2}}=2+\sqrt{3}$. The question is:<br/>$(1)$ Fill in the blanks: $\sqrt{4+2\sqrt{3}}=$______, $\sqrt{5-2\sqrt{6}}=$______; <br/>$(2)$ Further research reveals that the simplification of expressions in the form of $\sqrt{m±2\sqrt{n}}$ can be done by finding two positive numbers $a$ and $b\left(a \gt b\right)$ such that $a+b=m$, $ab=n$, that is, ${(\sqrt{a})^2}+{(\sqrt{b})^2}=m$, $\sqrt{a}×\sqrt{b}=\sqrt{n}$, then we have: $\sqrt{m±2\sqrt{n}}=\_\_\_\_\_\_.$<br/>$(3)$ Simplify: $\sqrt{4-\sqrt{15}$ (Please write down the simplification process).
|
\frac{\sqrt{10}}{2}-\frac{\sqrt{6}}{2}
| 1.5625 |
20,456 |
Calculate $3.6 \times 0.3$.
|
1.08
| 100 |
20,457 |
If $\tan \alpha= \sqrt {2}$, then $2\sin ^{2}\alpha-\sin \alpha\cos \alpha+\cos ^{2}\alpha=$ \_\_\_\_\_\_ .
|
\frac {5- \sqrt {2}}{3}
| 0 |
20,458 |
There are two targets, A and B. A shooter shoots at target A once, with a probability of $\frac{3}{4}$ of hitting it and scoring $1$ point, or missing it and scoring $-1$ point. The shooter shoots at target B twice, with a probability of $\frac{2}{3}$ of hitting it and scoring $2$ points each time, or missing it and scoring $0$ points. The results of each shot are independent. Assuming the shooter completes all three shots, what is the probability of scoring $3$ points?
|
\frac{4}{9}
| 39.0625 |
20,459 |
For an arithmetic sequence $b_1, b_2, b_3, \dots,$ let
\[S_n = b_1 + b_2 + b_3 + \dots + b_n,\]and let
\[T_n = S_1 + S_2 + S_3 + \dots + S_n.\]Given the value of $S_{2023},$ then you can uniquely determine the value of $T_n$ for some integer $n.$ What is this integer $n$?
|
3034
| 7.8125 |
20,460 |
Compute the following sum:
\[
\frac{1}{2^{2024}} \sum_{n = 0}^{1011} (-3)^n \binom{2024}{2n}.
\]
|
-\frac{1}{2}
| 41.40625 |
20,461 |
Given the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\cdot b^{t}$, where $a$ and $b$ are non-zero constants, and $v=5\%$ after $6$ months and $v=10\%$ after $12$ months, determine the time needed for this type of garbage to completely decompose.
|
32
| 3.125 |
20,462 |
If the line $y=ax+b$ is a tangent line of the graph of the function $f(x)=\ln{x}-\frac{1}{x}$, then the minimum value of $a+b$ is $\_\_\_\_\_\_\_\_$.
|
-1
| 71.09375 |
20,463 |
Given the function $y=\left[x\right]$, which is called the Gaussian function and represents the greatest integer not exceeding $x$, such as $\left[\pi \right]=3$, $\left[-2.5\right]=-3$. The solution set of the inequality $\frac{[x]}{[x]-4}＜0$ is ______; when $x \gt 0$, the maximum value of $\frac{[x]}{[x]^2+4}$ is ______.
|
\frac{1}{4}
| 93.75 |
20,464 |
Given the line $l: y = kx + m$ intersects the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ at points $A$ and $P$, and intersects the $x$-axis and $y$-axis at points $N$ and $M$ respectively, with $PM = MN$. Point $Q$ is the reflection of point $P$ across the $x$-axis, and the extension of line $QM$ intersects the ellipse at point $B$. Perpendicular lines to the $x$-axis are drawn from points $A$ and $B$, with the feet of the perpendiculars being $A_1$ and $B_1$ respectively.
(Ⅰ) If the left and right foci of the ellipse $C$ and one endpoint of its minor axis form the vertices of an equilateral triangle, and point $D(1,\frac{3}{2})$ lies on the ellipse $C$, find the equation of the ellipse $C$;
(Ⅱ) When $k = \frac{1}{2}$, if point $N$ bisects the line segment $A_1B_1$, find the eccentricity of the ellipse $C$.
|
\frac{1}{2}
| 42.1875 |
20,465 |
Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$ . What is the length of $AD$ .
|
19
| 3.90625 |
20,466 |
In a race, all runners must start at point $A$, touch any part of a 1500-meter wall, and then stop at point $B$. Given that the distance from $A$ directly to the wall is 400 meters and from the wall directly to $B$ is 600 meters, calculate the minimum distance a participant must run to complete this. Express your answer to the nearest meter.
|
1803
| 21.875 |
20,467 |
Kelvin the Frog is playing the game of Survival. He starts with two fair coins. Every minute, he flips all his coins one by one, and throws a coin away if it shows tails. The game ends when he has no coins left, and Kelvin's score is the *square* of the number of minutes elapsed. What is the expected value of Kelvin's score? For example, if Kelvin flips two tails in the first minute, the game ends and his score is 1.
|
\frac{64}{9}
| 12.5 |
20,468 |
Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$.
|
\frac{3-\sqrt{3}}{3}
| 18.75 |
20,469 |
In $\triangle ABC$, it is known that $\cos C + (\cos A - \sqrt{3} \sin A) \cos B = 0$.
(1) Find the measure of angle $B$.
(2) If $\sin (A - \frac{\pi}{3}) = \frac{3}{5}$, find $\sin 2C$.
|
\frac{24 + 7\sqrt{3}}{50}
| 5.46875 |
20,470 |
James has 6 ounces of tea in a ten-ounce mug and 6 ounces of milk in a separate ten-ounce mug. He first pours one-third of the tea from the first mug into the second mug and stirs well. Then he pours one-fourth of the mixture from the second mug back into the first. What fraction of the liquid in the first mug is now milk?
A) $\frac{1}{6}$
B) $\frac{1}{5}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$
|
\frac{1}{4}
| 81.25 |
20,471 |
A number is composed of 6 millions, 3 tens of thousands, and 4 thousands. This number is written as ____, and when rewritten in terms of "ten thousands" as the unit, it becomes ____ ten thousands.
|
603.4
| 21.875 |
20,472 |
Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2\sin A + \sin B = 2\sin C\cos B$ and the area of $\triangle ABC$ is $S = \frac{\sqrt{3}}{2}c$, find the minimum value of $ab$.
|
12
| 11.71875 |
20,473 |
What is the total number of digits used when the first 3003 positive even integers are written?
|
11460
| 35.15625 |
20,474 |
Calculate $52103_{8} - 1452_{9}$ in base 10.
|
20471
| 21.875 |
20,475 |
There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?
|
1024
| 37.5 |
20,476 |
A class has $50$ students. The math scores $\xi$ of an exam follow a normal distribution $N(100, 10^{2})$. Given that $P(90 \leqslant \xi \leqslant 100)=0.3$, estimate the number of students with scores of $110$ or higher.
|
10
| 53.125 |
20,477 |
Evaluate the value of $\frac{1}{4}\cdot\frac{8}{1}\cdot\frac{1}{32}\cdot\frac{64}{1} \dotsm \frac{1}{1024}\cdot\frac{2048}{1}$.
|
32
| 21.09375 |
20,478 |
Given an arithmetic sequence $\{a_n\}$, where $a_n \in \mathbb{N}^*$, and $S_n = \frac{1}{8}(a_n+2)^2$. If $b_n = \frac{1}{2}a_n - 30$, find the minimum value of the sum of the first \_\_\_\_\_\_ terms of the sequence $\{b_n\}$.
|
15
| 37.5 |
20,479 |
Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2g(a^2 + b^2) = [g(a)]^2 + [g(b)]^2\] for all nonnegative integers $a$ and $b.$
Let $n$ be the number of possible values of $g(16),$ and let $s$ be the sum of the possible values of $g(16).$ Find $n \times s.$
|
99
| 40.625 |
20,480 |
The diagonal lengths of a rhombus are 18 units and 26 units. Calculate both the area and the perimeter of the rhombus.
|
20\sqrt{10}
| 59.375 |
20,481 |
A parallelogram has side lengths of 10, $12x-2$, $5y+5$, and $4$. What is the value of $x+y$?
[asy]draw((0,0)--(24,0)--(30,20)--(6,20)--cycle);
label("$12x-2$",(15,0),S);
label("10",(3,10),W);
label("$5y+5$",(18,20),N);
label("4",(27,10),E);
[/asy]
|
\frac{4}{5}
| 1.5625 |
20,482 |
The difference between two perfect squares is 221. What is the smallest possible sum of the two perfect squares?
|
24421
| 0 |
20,483 |
Given a line segment $\overline{AB}=10$ cm, a point $C$ is placed on $\overline{AB}$ such that $\overline{AC} = 6$ cm and $\overline{CB} = 4$ cm. Three semi-circles are drawn with diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, external to the segment. If a line $\overline{CD}$ is drawn perpendicular to $\overline{AB}$ from point $C$ to the boundary of the smallest semi-circle, find the ratio of the shaded area to the area of a circle taking $\overline{CD}$ as its radius.
|
\frac{3}{2}
| 54.6875 |
20,484 |
In $\triangle ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, find the length of side $AB$.
|
\frac{50}{3}
| 74.21875 |
20,485 |
Sarah subscribes to a virtual fitness class platform that charges a monthly membership fee plus a per-class fee. If Sarah paid a total of $30.72 in February for 4 classes, and $54.72 in March for 8 classes, with the monthly membership fee increasing by 10% from February to March, calculate the fixed monthly membership fee.
|
7.47
| 75.78125 |
20,486 |
Calculate the volumes of solids formed by rotating the region bounded by the function graphs about the \( O y \) (y-axis).
$$
y = x^{2} + 1, \quad y = x, \quad x = 0, \quad x = 1
$$
|
\frac{5\pi}{6}
| 76.5625 |
20,487 |
For a modified toothpick pattern, the first stage is constructed using 5 toothpicks. If each subsequent stage is formed by adding three more toothpicks than the previous stage, what is the total number of toothpicks needed for the $15^{th}$ stage?
|
47
| 86.71875 |
20,488 |
Using systematic sampling to select a sample of size 20 from 180 students, the students are randomly numbered from 1 to 180. They are then divided into 20 groups in order of their number (group 1: numbers 1-9, group 2: numbers 10-18, ..., group 20: numbers 172-180). If the number drawn from group 20 is 176, what is the number drawn from group 3?
|
23
| 45.3125 |
20,489 |
How many integers $n$ (with $1 \le n \le 2021$ ) have the property that $8n + 1$ is a perfect square?
|
63
| 82.03125 |
20,490 |
Given that point $A(1,\sqrt{5})$ lies on the parabola $C:y^{2}=2px$, the distance from $A$ to the focus of $C$ is ______.
|
\frac{9}{4}
| 88.28125 |
20,491 |
What is the base five sum of the numbers $212_{5}$ and $12_{5}$?
|
224_5
| 80.46875 |
20,492 |
Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.
a) Write down any five-digit palindromic number that is divisible by 5.
b) How many five-digit palindromic numbers are there that are divisible by 5?
|
100
| 88.28125 |
20,493 |
Consider the function $f(x)=\cos^2x+a\sin x- \frac{a}{4}- \frac{1}{2}$, where $0 \leq x \leq \frac{\pi}{2}$ and $a > 0$.
(1) Express the maximum value $M(a)$ of $f(x)$ in terms of $a$.
(2) Find the value of $a$ when $M(a)=2$.
|
\frac{10}{3}
| 69.53125 |
20,494 |
If each side of a regular hexagon consists of 6 toothpicks, and there are 6 sides, calculate the total number of toothpicks used to build the hexagonal grid.
|
36
| 32.8125 |
20,495 |
Five volunteers and two elderly people are taking a photo, and they need to be arranged in a row. The two elderly people must stand next to each other but cannot be at either end of the row. How many different arrangements are possible?
|
960
| 44.53125 |
20,496 |
Given the parabola $y^2 = 4x$, a line passing through point $P(4, 0)$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. Find the minimum value of $y_1^2 + y_2^2$.
|
32
| 94.53125 |
20,497 |
Given the function $y=\cos({2x+\frac{π}{3}})$, determine the horizontal shift of the graph of the function $y=\sin 2x$.
|
\frac{5\pi}{12}
| 13.28125 |
20,498 |
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
|
16
| 33.59375 |
20,499 |
In the line $5x + 8y + c = 0$, the sum of the $x$- and $y$-intercepts is $26$. Find $c$.
|
-80
| 74.21875 |
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