Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
18,900 |
What is the greatest common factor of the numbers 2750 and 9450?
|
50
| 100 |
18,901 |
The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$ .
|
\frac{\sqrt{3}}{2}
| 25 |
18,902 |
Liam needs to add 125 to 96 and then round the result to the nearest hundred. He plans to round the numbers before adding them. He rounds 125 to 100 and 96 to 100, resulting in an answer of $100 + 100 = 200$. Is this answer correct? What is the correct answer?
|
200
| 98.4375 |
18,903 |
Let $a$ and $b$ be the numbers obtained by rolling a pair of dice twice. The probability that the equation $x^{2}-ax+2b=0$ has two distinct real roots is $\_\_\_\_\_\_$.
|
\frac{1}{4}
| 0 |
18,904 |
A positive integer divisor of $10!$ is chosen at random. Calculate the probability that the divisor chosen is a perfect square, expressed as a simplified fraction $\frac{m}{n}$, and find the sum of the numerator and denominator.
|
10
| 59.375 |
18,905 |
For a positive integer $n$ , let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$ . Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$ .
|
18
| 92.1875 |
18,906 |
A curious archaeologist is holding a competition where participants must guess the age of a unique fossil. The age of the fossil is formed from the six digits 2, 2, 5, 5, 7, and 9, and the fossil's age must begin with a prime number.
|
90
| 1.5625 |
18,907 |
Simplify first, then evaluate: $3a^{2}+\left[a^{2}+\left(5a^{2}-2a\right)-3\left(a^{2}-3a\right)\right]$, where $a=-2$.
|
10
| 76.5625 |
18,908 |
The volume of a box is 360 cubic units where $a, b,$ and $c$ are integers such that $1 < c < b < a$ with $c$ being a prime number. Determine the largest possible value of $b$.
|
12
| 5.46875 |
18,909 |
Given the new operation $n\heartsuit m=n^{3+m}m^{2+n}$, evaluate $\frac{2\heartsuit 4}{4\heartsuit 2}$.
|
\frac{1}{2}
| 95.3125 |
18,910 |
In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}=1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half of the $x$-axis as the polar axis, a polar coordinate system is established with the same unit length. It is known that the polar equation of line $l$ is $\rho(2\cos \theta - \sin \theta) = 6$.
$(1)$ After extending the x-coordinate of all points on curve $C_{1}$ by $\sqrt {3}$ times and the y-coordinate by $2$ times to obtain curve $C_{2}$, write down the Cartesian equation of line $l$ and the parametric equation of curve $C_{2}$;
$(2)$ Let $P$ be any point on curve $C_{2}$. Find the maximum distance from point $P$ to line $l$.
|
2 \sqrt {5}
| 0 |
18,911 |
$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$. Given that $f(6) = 6$, determine $f(2012)$.
|
-2000
| 10.15625 |
18,912 |
The math teacher wants to arrange 6 multiple-choice questions into a large practice exam paper, requiring that the two easy questions, $A$ and $B$, must be adjacent, and the two difficult questions, $E$ and $F$, cannot be adjacent. There are a total of $\_\_\_\_\_\_\_\_\_$ different arrangements. (Answer with a number)
|
144
| 34.375 |
18,913 |
Given that the mean score of the students in the first section is 92 and the mean score of the students in the second section is 78, and the ratio of the number of students in the first section to the number of students in the second section is 5:7, calculate the combined mean score of all the students in both sections.
|
\frac{1006}{12}
| 0 |
18,914 |
Detached calculation.
327 + 46 - 135
1000 - 582 - 128
(124 - 62) × 6
500 - 400 ÷ 5
|
420
| 92.96875 |
18,915 |
Evaluate the infinite geometric series:
$$\frac{5}{3} - \frac{5}{4} + \frac{25}{48} - \frac{125}{384} + \dots$$
|
\frac{20}{21}
| 71.875 |
18,916 |
Let \(ABCD\) be a convex quadrilateral, and let \(M_A,\) \(M_B,\) \(M_C,\) \(M_D\) denote the midpoints of sides \(BC,\) \(CA,\) \(AD,\) and \(DB,\) respectively. Find the ratio \(\frac{[M_A M_B M_C M_D]}{[ABCD]}.\)
|
\frac{1}{4}
| 2.34375 |
18,917 |
Twelve million added to twelve thousand equals what number?
|
12012000
| 98.4375 |
18,918 |
If $f\left(x\right)=\ln |a+\frac{1}{{1-x}}|+b$ is an odd function, then $a=$____, $b=$____.
|
\ln 2
| 1.5625 |
18,919 |
Given the polynomial $f(x)=x^{6}-5x^{5}+6x^{4}+x^{2}+0.3x+2$, use Horner's method to calculate $f(-2)$ and find the value of $v_{1}$.
|
-7
| 57.8125 |
18,920 |
Given a quadratic polynomial \( P(x) \). It is known that the equations \( P(x) = x - 2 \) and \( P(x) = 1 - x / 2 \) each have exactly one root. What is the discriminant of \( P(x) \)?
|
-\frac{1}{2}
| 13.28125 |
18,921 |
If real numbers \(a\), \(b\), and \(c\) satisfy \(a^{2} + b^{2} + c^{2} = 9\), then what is the maximum value of the algebraic expression \((a - b)^{2} + (b - c)^{2} + (c - a)^{2}\)?
|
27
| 42.1875 |
18,922 |
Given sets A = {x | $x^2 - ax + a^2 - 19 = 0$}, B = {x | $x^2 - 5x + 6 = 0$}, C = {x | $x^2 + 2x - 8 = 0$} such that $A \cap B \neq \emptyset$ and $A \cap C = \emptyset$. Find the value of the real number $a$.
|
-2
| 64.0625 |
18,923 |
The sum of three numbers $x$, $y$, and $z$ is 150. If we increase $x$ by 7, decrease $y$ by 12 and multiply $z$ by 4, the three resulting numbers are equal. What is the value of $y$?
|
\frac{688}{9}
| 89.0625 |
18,924 |
In Perfectville, the streets are all $30$ feet wide and the blocks they enclose are all squares of side length $500$ feet. Jane runs around the block on the $500$-foot side of the street, while John runs on the opposite side of the street. How many more feet than Jane does John run for every lap around the block?
|
240
| 65.625 |
18,925 |
Given that the solution set for the inequality $ax^2+ax+2>0$ is $\mathbb{R}$ (the set of all real numbers), let the set of all numerical values of the real number $a$ be denoted as $M$.
(1) Find the set $M$.
(2) If $t>0$, for all $a \in M$, it holds that $(a^2-2a)t \leq t^2 + 3t - 46$. Find the minimum value of $t$.
|
46
| 53.125 |
18,926 |
Given $\cos\alpha + \cos\beta = \frac{1}{2}$, $\sin\alpha + \sin\beta = \frac{\sqrt{3}}{2}$, then $\cos(\alpha - \beta) =$ \_\_\_\_\_\_.
|
-\frac{1}{2}
| 96.875 |
18,927 |
Given $\sqrt{15129}=123$ and $\sqrt{x}=0.123$, calculate the value of $x$.
|
0.015129
| 98.4375 |
18,928 |
In a jar, there are 5 salted duck eggs of the same size and shape, among which 3 have green shells, and 2 have white shells. If two eggs are taken out one after another without replacement, calculate:
(1) The probability of the first egg taken out being green-shelled;
(2) The probability that both the first and the second eggs taken out are green-shelled;
(3) Given that the first egg taken out is green-shelled, the probability that the second egg is also green-shelled.
|
\frac{1}{2}
| 92.1875 |
18,929 |
If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.
The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.
a) Find the probability that under this condition, event \( B \) did not occur at all.
b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \).
|
\frac{16}{3}
| 0 |
18,930 |
Simplify first, then find the value of $\frac{{{a^2}-{b^2}}}{{{a^2}b-a{b^2}}}÷(1+\frac{{{a^2}+{b^2}}}{2ab})$, where $a=\sqrt{3}-\sqrt{11}$ and $b=\sqrt{3}+\sqrt{11}$.
|
\frac{\sqrt{3}}{3}
| 76.5625 |
18,931 |
There is infinite sequence of composite numbers $a_1,a_2,...,$ where $a_{n+1}=a_n-p_n+\frac{a_n}{p_n}$ ; $p_n$ is smallest prime divisor of $a_n$ . It is known, that $37|a_n$ for every $n$ .
Find possible values of $a_1$
|
37^2
| 0 |
18,932 |
A pair of dice is rolled twice. What is the probability that the sum of the numbers facing up is 5?
A) $\frac{1}{9}$
B) $\frac{1}{4}$
C) $\frac{1}{36}$
D) 97
|
\frac{1}{9}
| 77.34375 |
18,933 |
In the mystical mountain, there are only two types of legendary creatures: Nine-Headed Birds and Nine-Tailed Foxes. A Nine-Headed Bird has nine heads and one tail, while a Nine-Tailed Fox has nine tails and one head.
A Nine-Headed Bird discovers that, excluding itself, the total number of tails of the other creatures on the mountain is 4 times the number of heads. A Nine-Tailed Fox discovers that, excluding itself, the total number of tails of the other creatures on the mountain is 3 times the number of heads. How many Nine-Tailed Foxes are there on the mountain?
|
14
| 8.59375 |
18,934 |
For what value of $n$ does $|6 + ni| = 6\sqrt{5}$?
|
12
| 81.25 |
18,935 |
Let the complex numbers \( z_1 \) and \( z_2 \) correspond to the points \( A \) and \( B \) on the complex plane respectively, and suppose \( \left|z_1\right| = 4 \) and \( 4z_1^2 - 2z_1z_2 + z_2^2 = 0 \). Let \( O \) be the origin. Calculate the area of triangle \( \triangle OAB \).
|
8\sqrt{3}
| 82.03125 |
18,936 |
Given that $F\_1$ and $F\_2$ are two foci of the hyperbola $x^2-y^2=1$, and $P$ is a point on the hyperbola such that $\angle F\_1PF\_2=60^{\circ}$, determine the area of $\triangle F\_1PF\_2$.
|
\sqrt{3}
| 97.65625 |
18,937 |
The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $5$. Another die is marked with $2$, $3$, $4$, $6$, $7$, and $9$. What is the probability that the sum of the top two numbers will be $6$, $8$, or $10$?
A) $\frac{8}{36}$
B) $\frac{11}{36}$
C) $\frac{15}{36}$
D) $\frac{18}{36}$
|
\frac{11}{36}
| 63.28125 |
18,938 |
The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Determine the $150^{\mbox{th}}$ term of this sequence.
|
2280
| 23.4375 |
18,939 |
Given that the focus of the parabola $y^{2}=2px\left(p \gt 0\right)$ is $F\left(4,0\right)$, and $O$ is the origin.
$(1)$ Find the equation of the parabola.
$(2)$ A line with a slope of $1$ passes through point $F$ and intersects the parabola at points $A$ and $B$. Find the area of $\triangle AOB$.
|
32\sqrt{2}
| 20.3125 |
18,940 |
There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"?
|
12
| 41.40625 |
18,941 |
Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ with a common difference of $d$. If $a_{1}=190$, $S_{20} \gt 0$, and $S_{24} \lt 0$, then one possible value for the integer $d$ is ______.
|
-17
| 32.8125 |
18,942 |
Let the triangle $ABC$ have area $1$ . The interior bisectors of the angles $\angle BAC,\angle ABC, \angle BCA$ intersect the sides $(BC), (AC), (AB) $ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $E$ . The lines $EF, FG,GE$ intersect the bisectors $(AL), (CQ) ,(BN)$ respectively at points $P, M, R$ . Determine the area of the hexagon $LMNPR$ .
|
1/2
| 53.90625 |
18,943 |
Given that $2ax^3 - 3bx + 8 = 18$ when $x = -1$, determine the value of $9b - 6a + 2$.
|
32
| 98.4375 |
18,944 |
The line RS does not intersect with △ABC. Perpendiculars are drawn from the vertices A, B, and C of △ABC to line RS, and the corresponding feet of the perpendiculars are D, E, and F respectively. It is given that AD = 10, BE = 6, and CF = 24. Let H be the foot of the perpendicular drawn from the centroid G of △ABC to line RS. If x denotes the length of the segment GH, find x.
|
\frac{40}{3}
| 92.96875 |
18,945 |
Let positive integers \( a, b, c, d \) satisfy \( a > b > c > d \) and \( a+b+c+d=2004 \), \( a^2 - b^2 + c^2 - d^2 = 2004 \). Find the minimum value of \( a \).
|
503
| 92.96875 |
18,946 |
What is the largest divisor of 540 that is less than 80 and also a factor of 180?
|
60
| 77.34375 |
18,947 |
The lengths of the three sides of a triangle are 11, 15, and \( k \) \((k \in Z^{+})\). The number of values of \( k \) that make the triangle an obtuse triangle is:
|
13
| 74.21875 |
18,948 |
Given that the random variable $\xi$ follows the normal distribution $N(1, \sigma^2)$, and $P(\xi \leq 4) = 0.84$, find the probability $P(\xi \leq -2)$.
|
0.16
| 97.65625 |
18,949 |
What is the minimum value of $w$ if $w=3x^2 + 4y^2 - 12x + 8y + 15$?
|
-1
| 85.15625 |
18,950 |
On a 6 by 6 grid of points, what fraction of the larger square's area is inside the shaded square if the shaded square is rotated 45 degrees with vertices at points (2,2), (3,3), (2,4), and (1,3)? Express your answer as a common fraction.
|
\frac{1}{18}
| 43.75 |
18,951 |
If $\overrightarrow{a} = (2, 3)$, $\overrightarrow{b} = (-4, 7)$, and $\overrightarrow{a} + \overrightarrow{c} = 0$, then the projection of $\overrightarrow{c}$ in the direction of $\overrightarrow{b}$ is \_\_\_\_\_\_.
|
-\frac{\sqrt{65}}{5}
| 2.34375 |
18,952 |
Given points $A(-2,-2)$, $B(-2,6)$, $C(4,-2)$, and point $P$ moving on the circle $x^{2}+y^{2}=4$, find the maximum value of $|PA|^{2}+|PB|^{2}+|PC|^{2}$.
|
88
| 9.375 |
18,953 |
Given a set of $n$ positive integers in which the difference between any two elements is either divisible by 5 or divisible by 25, find the maximum value of $n$.
|
25
| 42.1875 |
18,954 |
A student, Alex, is required to do a specified number of homework assignments to earn homework points using a different system: for the first four points, each point requires one homework assignment; for the next four points (points 5-8), each requires two homework assignments; then, for points 9-12, each requires three assignments, etc. Calculate the smallest number of homework assignments necessary for Alex to earn a total of 20 homework points.
|
60
| 14.0625 |
18,955 |
Given the lines $x-y-1=0$ and $x-y-5=0$ both intersect circle $C$ creating chords of length 10, find the area of circle $C$.
|
27\pi
| 56.25 |
18,956 |
In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Charlie, and Dana, refuse to sit in four consecutive seats?
|
3507840
| 92.96875 |
18,957 |
Find the smallest positive integer $Y$ such that a number $U$, made only of digits 0s and 1s, is divisible by 15, and $U = 15Y$.
|
74
| 38.28125 |
18,958 |
Given $m+n=2$ and $mn=-2$. Find the value of:
1. $2^{m}\cdot 2^{n}-(2^{m})^{n}$
2. $(m-4)(n-4)$
3. $(m-n)^{2}$.
|
12
| 96.875 |
18,959 |
What is the sum of the digits of the square of the number 22222?
|
46
| 41.40625 |
18,960 |
Mrs. Johnson recorded the following scores for a test taken by her 120 students. Calculate the average percent score for these students.
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
95&10\\\hline
85&20\\\hline
75&40\\\hline
65&30\\\hline
55&15\\\hline
45&3\\\hline
0&2\\\hline
\end{tabular}
|
71.33
| 1.5625 |
18,961 |
What is $3.57 - 1.14 - 0.23$?
|
2.20
| 17.1875 |
18,962 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $c=2$, $b=\sqrt{2}a$. The maximum area of $\triangle ABC$ is ______.
|
2\sqrt{2}
| 28.125 |
18,963 |
Given an arithmetic sequence $\{a_{n}\}$, $\{b_{n}\}$, with their sums of the first $n$ terms being $S_{n}$ and $T_{n}$ respectively, and satisfying $\frac{S_n}{T_n}=\frac{n+3}{2n-1}$, find $\frac{a_5}{T_9}$.
|
\frac{4}{51}
| 25 |
18,964 |
If the function $f(x)=\frac{1}{2}(2-m)x^{2}+(n-8)x+1$ $(m > 2)$ is monotonically decreasing in the interval $[-2,-1]$, find the maximum value of $mn$.
|
18
| 10.15625 |
18,965 |
Given an arithmetic sequence $\{a_n\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\{b_n\}$.
|
\frac {175}{264}
| 32.03125 |
18,966 |
Given a sequence of distinct positive integers $(i\_1, i\_2, \ldots, i\_n)$ where $n$ is a positive integer greater than or equal to 2, if $i\_p > i\_q$ when $p > q$, then $i\_p$ and $i\_q$ are called a "good order" of the sequence. The number of "good orders" in a sequence is called the "good order number" of that sequence. For example, the sequence (1, 3, 4, 2) has "good orders" "1, 3", "1, 4", "1, 2", "3, 4", so its "good order number" is 4. If the "good order number" of the sequence of distinct positive integers $(a\_1, a\_2, a\_3, a\_4, a\_5, a\_6, a\_7)$ is 3, then the "good order number" of $(a\_7, a\_6, a\_5, a\_4, a\_3, a\_2, a\_1)$ is $\_\_\_\_\_\_\_.$
|
18
| 85.9375 |
18,967 |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(1000,0),(1000,1005),$ and $(0,1005)$. What is the probability that $x > 5y$? Express your answer as a common fraction.
|
\frac{20}{201}
| 59.375 |
18,968 |
For a given point $P$ on the curve $x^2 - y - \ln x = 0$, what is the minimum distance from point $P$ to the line $y = x - 2$?
|
\sqrt{2}
| 78.90625 |
18,969 |
The number of right-angled triangles with integer side lengths $a$ and $b$ (where $b < 2011$), and hypotenuse length $b + 1$ is .
|
31
| 18.75 |
18,970 |
What is the value of $102^{4} - 4 \cdot 102^{3} + 6 \cdot 102^2 - 4 \cdot 102 + 1$?
|
104060401
| 96.09375 |
18,971 |
A parallelogram in the coordinate plane has vertices at points (2,1), (7,1), (5,6), and (10,6). Calculate the sum of the perimeter and the area of the parallelogram.
|
35 + 2\sqrt{34}
| 39.84375 |
18,972 |
In how many ways can a President, a Vice-President, and a Secretary be chosen from a group of 6 people, given the following constraints:
- The President must be one of the first three members of the group.
- The Vice-President must be one of the last four members of the group.
- No person can hold more than one office.
|
36
| 6.25 |
18,973 |
If a person A is taller or heavier than another person B, then we note that A is *not worse than* B. In 100 persons, if someone is *not worse than* other 99 people, we call him *excellent boy*. What's the maximum value of the number of *excellent boys*?
|
100
| 78.90625 |
18,974 |
In triangle $PQR$, angle $R$ is a right angle, $PR=15$ units, and $Q$ is on a circle centered at $R$. Squares $PQRS$ and $PRUT$ are formed on sides $PQ$ and $PR$, respectively. What is the number of square units in the sum of the areas of the two squares $PQRS$ and $PRUT$?
|
450
| 31.25 |
18,975 |
Given a function $f(x)$ that always satisfies the following conditions on its domain $\mathbb{R}$:
① $f(x) = f(-x)$, ② $f(2+x) = f(2-x)$, when $x \in [0, 4)$, $f(x) = -x^2 + 4x$.
(1) Find $f(8)$.
(2) Find the number of zeros of $f(x)$ in $[0, 2015]$.
|
504
| 57.03125 |
18,976 |
A malfunctioning thermometer shows a temperature of $+1^{\circ}$ in freezing water and $+105^{\circ}$ in the steam of boiling water. Currently, this thermometer shows $+17^{\circ}$; what is the true temperature?
|
15.38
| 41.40625 |
18,977 |
A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. How many different flags can be made?
|
260
| 17.1875 |
18,978 |
Given $a \gt 0$, $b\in R$, if the inequality $\left(ax-2\right)(-x^{2}-bx+4)\leqslant 0$ holds for all $x \gt 0$, then the minimum value of $b+\frac{3}{a}$ is ______.
|
2\sqrt{2}
| 42.96875 |
18,979 |
Given that the median of the numbers $3, 5, 7, 23,$ and $x$ is equal to the mean of those five numbers, calculate the sum of all real numbers $x$.
|
-13
| 42.96875 |
18,980 |
A random sample of $10$ households was taken from a residential area, and the monthly income data ${x_i}$ (in units of thousand yuan) of the $i$-th household was obtained. The following statistical values were calculated: $\sum_{i=1}^{10} x_i = 80$, $\sum_{i=1}^{10} y_i = 20$, $\sum_{i=1}^{10} x_i y_i = 184$, and $\sum_{i=1}^{10} x_i^2 = 720$.
(1) Find the linear regression equation $\hat{y} = \hat{b}x + \hat{a}$ of the monthly savings $y$ with respect to the monthly income $x$, and determine whether the variables $x$ and $y$ are positively or negatively correlated.
(2) Predict the monthly savings of a household in this residential area with a monthly income of $7$ thousand yuan.
|
1.7
| 94.53125 |
18,981 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $c=a\cos B+b\sin A$.
(1) Find $A$;
(2) If $a=2$ and $b=c$, find the area of $\triangle ABC$.
|
\sqrt{2}+1
| 33.59375 |
18,982 |
What is the greatest integer less than 200 for which the greatest common factor of that integer and 72 is 9?
|
189
| 44.53125 |
18,983 |
Let $ABC$ be a triangle with $\angle BAC=40^\circ $ , $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$ . If $AD\parallel OG$ , find $\angle ACB$ .
|
70
| 48.4375 |
18,984 |
Mena listed the numbers from 1 to 30 one by one. Emily copied these numbers and substituted every digit 2 with digit 1. Both calculated the sum of the numbers they wrote. By how much is the sum that Mena calculated greater than the sum that Emily calculated?
|
103
| 22.65625 |
18,985 |
How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits?
|
90
| 89.84375 |
18,986 |
Real numbers $a$ , $b$ , $c$ which are differ from $1$ satisfies the following conditions;
(1) $abc =1$ (2) $a^2+b^2+c^2 - \left( \dfrac{1}{a^2} + \dfrac{1}{b^2} + \dfrac{1}{c^2} \right) = 8(a+b+c) - 8 (ab+bc+ca)$ Find all possible values of expression $\dfrac{1}{a-1} + \dfrac{1}{b-1} + \dfrac{1}{c-1}$ .
|
-\frac{3}{2}
| 11.71875 |
18,987 |
Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=\ln (1+ \frac {1}{n})$, find the value of $e^{a_7+a_8+a_9}$.
|
\frac {20}{21}
| 19.53125 |
18,988 |
One night, 21 people exchanged phone calls $n$ times. It is known that among these people, there are $m$ people $a_{1}, a_{2}, \cdots, a_{m}$ such that $a_{i}$ called $a_{i+1}$ (for $i=1,2, \cdots, m$ and $a_{m+1}=a_{1}$), and $m$ is an odd number. If no three people among these 21 people have all exchanged calls with each other, determine the maximum value of $n$.
|
101
| 0 |
18,989 |
Calculate $7 \cdot 9\frac{2}{5}$.
|
65 \frac{4}{5}
| 82.8125 |
18,990 |
A bag contains 4 tan, 3 pink, 5 violet, and 2 green chips. If all 14 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the 4 tan chips, the 3 pink chips, and the 5 violet chips are each drawn consecutively, and there is at least one green chip placed between any two groups of these chips of other colors? Express your answer as a common fraction.
|
\frac{1440}{14!}
| 0 |
18,991 |
Given that the angles A, B, C of triangle ABC correspond to the sides a, b, c respectively, and vectors $\overrightarrow {m}$ = (a, $- \sqrt {3}b$) and $\overrightarrow {n}$ = (cosA, sinB), and $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$.
(1) Find angle A.
(2) If $a = \sqrt{39}$ and $c = 5$, find the area of triangle ABC.
|
\frac{5\sqrt{3}}{2}
| 92.1875 |
18,992 |
Given an ellipse $C$ with one focus at $F_{1}(2,0)$ and the corresponding directrix $x=8$, and eccentricity $e=\frac{1}{2}$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ Find the length of the chord cut from the ellipse $C$ by a line passing through the other focus and having a slope of $45^{\circ}$.
|
\frac{48}{7}
| 45.3125 |
18,993 |
A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. Find the x-coordinate of the point on the $x$-axis where a line, tangent to both circles, intersects. The line should intersect the x-axis to the right of the origin.
|
4.5
| 15.625 |
18,994 |
Point $O$ is the center of the regular octagon $ABCDEFGH$, and $Y$ is the midpoint of side $\overline{CD}$. Calculate the fraction of the area of the octagon that is shaded, where the shaded region includes triangles $\triangle DEO$, $\triangle EFO$, $\triangle FGO$, and half of $\triangle DCO$.
|
\frac{7}{16}
| 74.21875 |
18,995 |
Given points A(1, 2, 1), B(-2, $$\frac{7}{2}$$, 4), and D(1, 1, 1), if $$\vec{AP} = 2\vec{PB}$$, then the value of |$$\vec{PD}$$| is ______.
|
2\sqrt{3}
| 91.40625 |
18,996 |
All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the tenth number in that list?
|
147
| 2.34375 |
18,997 |
Given complex numbers $z_{1}=m+i$ and $z_{2}=2+mi$, where $i$ is the imaginary unit and $m\in R$.
$(1)$ If $z_{1}\cdot z_{2}$ is a pure imaginary number, find the value of $m$.
$(2)$ If ${z}_{1}^{2}-2{z}_{1}+2=0$, find the imaginary part of $\frac{{z}_{2}}{{z}_{1}}$.
|
-\frac{1}{2}
| 64.0625 |
18,998 |
Let \(\mathcal{S}\) be a set of 16 points in the plane, no three collinear. Let \(\chi(\mathcal{S})\) denote the number of ways to draw 8 line segments 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 \(\mathcal{S}\).
|
1430
| 91.40625 |
18,999 |
Given the function $f(x)=2x-\sin x$, if the positive real numbers $a$ and $b$ satisfy $f(a)+f(2b-1)=0$, then the minimum value of $\dfrac {1}{a}+ \dfrac {4}{b}$ is ______.
|
9+4 \sqrt {2}
| 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.