Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
10,200 |
To open the safe, you need to enter a code — a number consisting of seven digits: twos and threes. The safe will open if there are more twos than threes, and the code is divisible by both 3 and 4. Create a code that opens the safe.
|
2222232
| 35.15625 |
10,201 |
Given a regular 2017-sided polygon \(A_{1} A_{2} \cdots A_{2017}\) inscribed in a unit circle \(\odot O\), choose any two different vertices \(A_{i}, A_{j}\). Find the probability that \(\overrightarrow{O A_{i}} \cdot \overrightarrow{O A_{j}} > \frac{1}{2}\).
|
1/3
| 42.1875 |
10,202 |
In the product
\[
24^{a} \cdot 25^{b} \cdot 26^{c} \cdot 27^{d} \cdot 28^{e} \cdot 29^{f} \cdot 30^{g}
\]
seven numbers \(1, 2, 3, 5, 8, 10, 11\) were assigned to the exponents \(a, b, c, d, e, f, g\) in some order. Find the maximum number of zeros that can appear at the end of the decimal representation of this product.
|
32
| 31.25 |
10,203 |
Given $f(\alpha) = \frac{\sin(\frac{\pi}{2} + \alpha) + 3\sin(-\pi - \alpha)}{2\cos(\frac{11\pi}{2} - \alpha) - \cos(5\pi - \alpha)}$.
(I) Simplify $f(\alpha)$;
(II) If $\tan \alpha = 3$, find the value of $f(\alpha)$.
|
-2
| 69.53125 |
10,204 |
There is a box containing 3 red balls and 3 white balls, which are identical in size and shape. A fair die is rolled, and the number rolled determines the number of balls drawn from the box. What is the probability that the number of red balls drawn is greater than the number of white balls drawn?
|
19/60
| 6.25 |
10,205 |
There are four points that are $7$ units from the line $y = 20$ and $10$ units from the point $(10, 20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?
|
120
| 42.96875 |
10,206 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\sin C + \sin(B - A) = \sqrt{2} \sin 2A$, and $A \neq \frac{\pi}{2}$.
(I) Find the range of values for angle $A$;
(II) If $a = 1$, the area of $\triangle ABC$ is $S = \frac{\sqrt{3} + 1}{4}$, and $C$ is an obtuse angle, find the measure of angle $A$.
|
\frac{\pi}{6}
| 43.75 |
10,207 |
Given an arithmetic sequence $\left\{a_{n}\right\}$ with the sum of the first 12 terms being 60, find the minimum value of $\left|a_{1}\right| + \left|a_{2}\right| + \cdots + \left|a_{12}\right|$.
|
60
| 66.40625 |
10,208 |
Convert the base 2 number \(1011111010_2\) to its base 4 representation.
|
23322_4
| 94.53125 |
10,209 |
The school choir originally had 36 members, and 9 more people joined. Calculate the total number of people in the choir now.
|
45
| 100 |
10,210 |
Calculate: $(10 \times 19 \times 20 \times 53 \times 100 + 601) \div 13 = \ ?$
|
1549277
| 33.59375 |
10,211 |
What is the largest integer \( k \) whose square \( k^2 \) is a factor of \( 10! \)?
|
720
| 88.28125 |
10,212 |
The solutions to the equation $(z-4)^6 = 64$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labelled $D, E,$ and $F$. What is the least possible area of triangle $DEF$?
|
\sqrt{3}
| 71.09375 |
10,213 |
Given $$\alpha \in \left( \frac{5}{4}\pi, \frac{3}{2}\pi \right)$$ and it satisfies $$\tan\alpha + \frac{1}{\tan\alpha} = 8$$, then $\sin\alpha\cos\alpha = \_\_\_\_\_\_$; $\sin\alpha - \cos\alpha = \_\_\_\_\_\_$.
|
-\frac{\sqrt{3}}{2}
| 14.0625 |
10,214 |
Let $0=x_0<x_1<\cdots<x_n=1$ .Find the largest real number $ C$ such that for any positive integer $ n $ , we have $$ \sum_{k=1}^n x^2_k (x_k - x_{k-1})>C $$
|
\frac{1}{3}
| 98.4375 |
10,215 |
Nathan and his two younger twin sisters' ages multiply to 72. Find the sum of their three ages.
|
14
| 41.40625 |
10,216 |
A herd of elephants. Springs are bubbling at the bottom of the lake. A herd of 183 elephants could drink it dry in one day, and a herd of 37 elephants could do so in 5 days. How many days will it take for 1 elephant to drink the lake dry?
|
365
| 50.78125 |
10,217 |
In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle?
|
65
| 73.4375 |
10,218 |
Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.
|
48
| 2.34375 |
10,219 |
Seven members of the family are each to pass through one of seven doors to complete a challenge. The first person can choose any door to activate. After completing the challenge, the adjacent left and right doors will be activated. The next person can choose any unchallenged door among the activated ones to complete their challenge. Upon completion, the adjacent left and right doors to the chosen one, if not yet activated, will also be activated. This process continues until all seven members have completed the challenge. The order in which the seven doors are challenged forms a seven-digit number. How many different possible seven-digit numbers are there?
|
64
| 30.46875 |
10,220 |
Given real numbers $a$ and $b \gt 0$, if $a+2b=1$, then the minimum value of $\frac{3}{b}+\frac{1}{a}$ is ______.
|
7 + 2\sqrt{6}
| 47.65625 |
10,221 |
Mary and James each sit in a row of 10 chairs. They choose their seats at random. Two of the chairs (chair number 4 and chair number 7) are broken and cannot be chosen. What is the probability that they do not sit next to each other?
|
\frac{3}{4}
| 28.90625 |
10,222 |
Given the prime factorization of $215^7$, $p^7 \cdot q^6 \cdot r^6$, where $p$, $q$, and $r$ are prime numbers, determine the number of positive integer divisors of $215^7$ that are perfect squares or perfect cubes (or both).
|
21
| 69.53125 |
10,223 |
The monkey has 100 bananas and its home is 50 meters away. The monkey can carry at most 50 bananas at a time and eats one banana for every meter walked. Calculate the maximum number of bananas the monkey can bring home.
|
25
| 45.3125 |
10,224 |
The base of a triangle is 20; the medians drawn to the lateral sides are 18 and 24. Find the area of the triangle.
|
288
| 11.71875 |
10,225 |
In the Cartesian coordinate system xOy, the polar equation of circle C is $\rho=4$. The parametric equation of line l, which passes through point P(1, 2), is given by $$\begin{cases} x=1+ \sqrt {3}t \\ y=2+t \end{cases}$$ (where t is a parameter).
(I) Write the standard equation of circle C and the general equation of line l;
(II) Suppose line l intersects circle C at points A and B, find the value of $|PA| \cdot |PB|$.
|
11
| 14.0625 |
10,226 |
Given a deck of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning pair.
|
\frac{1}{2}
| 17.1875 |
10,227 |
In the addition sum shown, \(J\), \(K\), and \(L\) stand for different digits. What is the value of \(J + K + L\)?
\[
\begin{array}{r}
J K L \\
J L L \\
+J K L \\
\hline 479
\end{array}
\]
|
11
| 7.03125 |
10,228 |
On the coordinate plane (\( x; y \)), a circle with radius 4 and center at the origin is drawn. A line given by the equation \( y = 4 - (2 - \sqrt{3}) x \) intersects the circle at points \( A \) and \( B \). Find the sum of the length of segment \( A B \) and the length of the shorter arc \( A B \).
|
4\sqrt{2 - \sqrt{3}} + \frac{2\pi}{3}
| 14.84375 |
10,229 |
A regular $n$-gon has $n$ diagonals, its perimeter is $p$, and the sum of the lengths of all the diagonals is $q$. What is $\frac{p}{q} + \frac{q}{p}$?
|
\sqrt{5}
| 0 |
10,230 |
Tim's quiz scores were 85, 87, 92, 94, 78, and 96. Calculate his mean score and find the range of his scores.
|
18
| 92.1875 |
10,231 |
Given that \( x_{1}, x_{2}, x_{3}, x_{4} \) are all positive numbers and \( x_{1} + x_{2} + x_{3} + x_{4} = \pi \), find the minimum value of the expression \(\left(2 \sin^2 x_{1}+\frac{1}{\sin^2 x_{1}}\right)\left(2 \sin^2 x_{2}+\frac{1}{\sin^2 x_{2}}\right)\left(2 \sin^2 x_{3}+\frac{1}{\sin^2 x_{3}}\right)\left(2 \sin^2 x_{4}+\frac{1}{\sin^2 x_{4}}\right)\).
|
81
| 80.46875 |
10,232 |
The real number \( a \) is such that \( 2a - \frac{1}{a} = 3 \). What is \( 16a^{4} + \frac{1}{a^{4}} \)?
|
161
| 12.5 |
10,233 |
Given that $\cos(\frac{\pi}{6} - \alpha) = \frac{3}{5}$, find the value of $\cos(\frac{5\pi}{6} + \alpha)$:
A) $\frac{3}{5}$
B) $-\frac{3}{5}$
C) $\frac{4}{5}$
D) $-\frac{4}{5}$
|
-\frac{3}{5}
| 97.65625 |
10,234 |
If for any $x \in D$, the inequality $f_1(x) \leq f(x) \leq f_2(x)$ holds, then the function $f(x)$ is called a "compromise function" of the functions $f_1(x)$ to $f_2(x)$ over the interval $D$. It is known that the function $f(x) = (k-1)x - 1$, $g(x) = 0$, $h(x) = (x+1)\ln x$, and $f(x)$ is a "compromise function" of $g(x)$ to $h(x)$ over the interval $[1, 2e]$, then the set of values of the real number $k$ is \_\_\_\_\_\_.
|
\{2\}
| 0.78125 |
10,235 |
Given a segment \( AB \) of fixed length 3 with endpoints moving on the parabola \( y^2 = x \), find the shortest distance from the midpoint \( M \) of segment \( AB \) to the y-axis.
|
\frac{5}{4}
| 14.84375 |
10,236 |
How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?
|
42
| 26.5625 |
10,237 |
$AL$ and $BM$ are the angle bisectors of triangle $ABC$. The circumcircles of triangles $ALC$ and $BMC$ intersect again at point $K$, which lies on side $AB$. Find the measure of angle $ACB$.
|
60
| 29.6875 |
10,238 |
What is the base \(2\) representation of \(125_{10}\)?
|
1111101_2
| 98.4375 |
10,239 |
A rectangle with a perimeter of 100 cm was divided into 70 identical smaller rectangles by six vertical cuts and nine horizontal cuts. What is the perimeter of each smaller rectangle if the total length of all cuts equals 405 cm?
|
13
| 35.9375 |
10,240 |
On the base \(AC\) of an isosceles triangle \(ABC (AB = BC)\), point \(M\) is marked. It is known that \(AM = 7\), \(MB = 3\), \(\angle BMC = 60^\circ\). Find the length of segment \(AC\).
|
17
| 60.9375 |
10,241 |
In a convex 13-gon, all the diagonals are drawn. They divide it into polygons. Consider a polygon among them with the largest number of sides. What is the maximum number of sides it can have?
|
13
| 10.15625 |
10,242 |
How many ways are there to arrange 5 identical red balls and 5 identical blue balls in a line if there cannot be three or more consecutive blue balls in the arrangement?
|
126
| 61.71875 |
10,243 |
The six-digit number $M=\overline{abc321}$, where $a, b, c$ are three different numbers, and all are greater than 3. If $M$ is a multiple of 7, what is the smallest value of $M$?
|
468321
| 81.25 |
10,244 |
In triangle $ABC$, $AB = AC = 15$ and $BC = 14$. Points $D, E, F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE} \parallel \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. What is the perimeter of parallelogram $ADEF$?
|
30
| 53.90625 |
10,245 |
On the sides $AB$ and $CD$ of rectangle $ABCD$, points $E$ and $F$ are marked such that $AFCE$ forms a rhombus. It is known that $AB = 16$ and $BC = 12$. Find $EF$.
|
15
| 1.5625 |
10,246 |
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
| 40.625 |
10,247 |
Two cars, Car A and Car B, travel towards each other from cities A and B, which are 330 kilometers apart. Car A starts from city A first. After some time, Car B starts from city B. The speed of Car A is $\frac{5}{6}$ of the speed of Car B. When the two cars meet, Car A has traveled 30 kilometers more than Car B. Determine how many kilometers Car A had traveled before Car B started.
|
55
| 54.6875 |
10,248 |
The graph of the quadratic $y = ax^2 + bx + c$ has these properties: (1) The maximum value of $y = ax^2 + bx + c$ is 4, which occurs at $x = 2$. (2) The graph passes through the point $(0,-16)$. If the graph also passes through the point $(5,n)$, what is the value of $n$?
|
-41
| 96.875 |
10,249 |
Given that the domain of the function $f(x)$ is $\mathbf{R}$, and $f(x+2) - 2$ is an odd function, while $f(2x+1)$ is an even function. If $f(1) = 0$, determine the value of $f(1) + f(2) + \cdots + f(2023)$.
|
4046
| 28.90625 |
10,250 |
On the sides $AB, AC, BC$ of an equilateral triangle $ABC$, with a side length of 2, points $C_{1}, B_{1}, A_{1}$ are chosen respectively.
What is the maximum possible value of the sum of the radii of the circles inscribed in the triangles $AB_{1}C_{1}$, $A_{1}BC_{1}$, and $A_{1}B_{1}C$?
|
\frac{\sqrt{3}}{2}
| 60.9375 |
10,251 |
From the set \( \{1, 2, 3, \ldots, 999, 1000\} \), select \( k \) numbers. If among the selected numbers, there are always three numbers that can form the side lengths of a triangle, what is the smallest value of \( k \)? Explain why.
|
16
| 48.4375 |
10,252 |
Given that the determinant of a $2 \times 2$ matrix $A = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is 3, find the values of $\begin{vmatrix} 3a & 3b \\ 3c & 3d \end{vmatrix}$ and $\begin{vmatrix} 4a & 2b \\ 4c & 2d \end{vmatrix}$.
|
24
| 89.84375 |
10,253 |
In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=4$ and $\frac{\cos B}{\cos C} = \frac{4}{2a - c}$.
(1) Find the measure of angle $B$;
(2) Find the maximum area of $\triangle ABC$.
|
4\sqrt{3}
| 78.125 |
10,254 |
Calculate the sum of $213_4 + 132_4 + 321_4$ and express your answer in base $4$.
|
1332_4
| 93.75 |
10,255 |
In an isosceles triangle \(ABC \) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\).
In what ratio does the line \(AD\) divide the height \(BE\) of the triangle \(ABC\), counted from the vertex \(B\)?
|
1:2
| 9.375 |
10,256 |
Find the product of all possible real values for $k$ such that the system of equations $$ x^2+y^2= 80 $$ $$ x^2+y^2= k+2x-8y $$ has exactly one real solution $(x,y)$ .
*Proposed by Nathan Xiong*
|
960
| 71.09375 |
10,257 |
In a given triangle, for $\angle P$ to be the largest angle of the triangle, it must be that $a < y < b$. The side lengths are given by $y+6$, $2y+1$, and $5y-10$. What is the least possible value of $b-a$, expressed as a common fraction?
|
4.5
| 0 |
10,258 |
On a road, there are three locations $A$, $O$, and $B$. $O$ is between $A$ and $B$, and $A$ is 1360 meters away from $O$. Two individuals, Jia and Yi, start simultaneously from points $A$ and $O$ towards point $B$. At the 10th minute after departure, both Jia and Yi are equidistant from point $O$. In the 40th minute, Jia and Yi meet at point $B$. What is the distance between points $O$ and $B$ in meters?
|
2040
| 53.90625 |
10,259 |
Given the hyperbola $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1 (m < 0 < n)$ with asymptote equations $y = \pm \sqrt{2}x$, calculate the hyperbola's eccentricity.
|
\sqrt{3}
| 42.96875 |
10,260 |
In a particular state, the design of vehicle license plates was changed from an old format to a new one. Under the old scheme, each license plate consisted of two letters followed by three digits (e.g., AB123). The new scheme is made up of four letters followed by two digits (e.g., ABCD12). Calculate by how many times has the number of possible license plates increased.
A) $\frac{26}{10}$
B) $\frac{26^2}{10^2}$
C) $\frac{26^2}{10}$
D) $\frac{26^3}{10^3}$
E) $\frac{26^3}{10^2}$
|
\frac{26^2}{10}
| 98.4375 |
10,261 |
There are 7 parking spaces in a row in a parking lot, and now 4 cars need to be parked. If 3 empty spaces need to be together, calculate the number of different parking methods.
|
120
| 71.875 |
10,262 |
(1) Point $P$ is any point on the curve $y=x^{2}-\ln x$. The minimum distance from point $P$ to the line $x-y-4=0$ is ______.
(2) If the tangent line to the curve $y=g(x)$ at the point $(1,g(1))$ is $y=2x+1$, then the equation of the tangent line to the curve $f(x)=g(x)+\ln x$ at the point $(1,f(1))$ is ______.
(3) Given that the distance from point $P(1,0)$ to one of the asymptotes of the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ is $\frac{1}{2}$, the eccentricity of the hyperbola $C$ is ______.
(4) A line passing through point $M(1,1)$ with a slope of $-\frac{1}{2}$ intersects the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ at points $A$ and $B$. If $M$ is the midpoint of segment $AB$, then the eccentricity of the ellipse $C$ is ______.
|
\frac{\sqrt{2}}{2}
| 71.09375 |
10,263 |
What is the distance between the two (non-intersecting) face diagonals on adjacent faces of a unit cube?
|
\frac{\sqrt{3}}{3}
| 14.84375 |
10,264 |
Petya and his three classmates started a 100-meter race simultaneously, and Petya finished first. Twelve seconds after the race began, no one had finished yet, and all four participants had collectively run a total of 288 meters. When Petya finished the race, the other three participants had a combined distance of 40 meters left to the finish line. How many meters did Petya run in the first 12 seconds? Justify your answer. It is assumed that each participant ran with a constant speed.
|
80
| 43.75 |
10,265 |
Let $A$ be the set of all real numbers $a$ that satisfy $\left(a-2\right)x^{2}+2\left(a-2\right)x-4 \lt 0$ for any $x\in R$. Let $B$ be the set of all real numbers $x$ that satisfy $\left(a-2\right)x^{2}+2\left(a-2\right)x-4 \lt 0$ for any $a\in \left[-2,2\right]$. Find $A\cap (\complement _{R}B)$.
|
\{-1\}
| 71.09375 |
10,266 |
A numerical sequence is defined by the conditions: \( a_{1}=1 \), \( a_{n+1}=a_{n}+\left \lfloor \sqrt{a_{n}} \right \rfloor \).
How many perfect squares are among the first terms of this sequence that do not exceed 1,000,000?
|
10
| 64.84375 |
10,267 |
In a six-digit decimal number $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$, each digit $a_{i}(1 \leqslant i \leqslant 6)$ is an odd number, and the digit 1 is not allowed to appear consecutively (for example, 135131 and 577797 satisfy the conditions, while 311533 does not satisfy the conditions). Find the total number of such six-digit numbers. $\qquad$ .
|
13056
| 39.84375 |
10,268 |
Solve the equation \(\frac{15}{x\left(\sqrt[3]{35-8 x^{3}}\right)}=2x+\sqrt[3]{35-8 x^{3}}\). Write the sum of all obtained solutions as the answer.
|
2.5
| 0.78125 |
10,269 |
A particle moves in a straight line inside a square of side 1. It is reflected from the sides, but absorbed by the four corners. It starts from an arbitrary point \( P \) inside the square. Let \( c(k) \) be the number of possible starting directions from which it reaches a corner after traveling a distance \( k \) or less. Find the smallest constant \( a_2 \), such that for some constants \( a_1 \) and \( a_0 \), \( c(k) \leq a_2 k^2 + a_1 k + a_0 \) for all \( P \) and all \( k \).
|
\pi
| 36.71875 |
10,270 |
Teacher Zhang led the students of class 6 (1) to plant trees. The students can be divided into 5 equal groups. It is known that each teacher and student plants the same number of trees, with a total of 527 trees planted. How many students are there in class 6 (1)?
|
30
| 32.8125 |
10,271 |
Let \( x_{i} \in \mathbf{R} \), \( x_{i} \geqslant 0 \) for \( i=1,2,3,4,5 \), and \( \sum_{i=1}^{5} x_{i} = 1 \). Find the minimum value of \( \max \left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{4}, x_{4} + x_{5}\right\} \).
|
\frac{1}{3}
| 26.5625 |
10,272 |
In parallelogram \( A B C D \), the height drawn from vertex \( B \) of the obtuse angle to side \( DA \) divides it in a ratio of 5:3, starting from vertex \( D \). Find the ratio \( AC:BD \) if \( AD:AB=2 \).
|
2:1
| 3.90625 |
10,273 |
For positive integers $n$, let $h(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, and $k$ is divisible by 3. How many positive integer divisors does $h(2010)$ have?
|
4022
| 27.34375 |
10,274 |
Let $y = ax^{2} + x - b$ where $a \in \mathbb{R}$ and $b \in \mathbb{R}$.
$(1)$ If $b = 1$ and the set $\{x | y = 0\}$ has exactly one element, find the set of possible values for the real number $a$.
$(2)$ Solve the inequality with respect to $x$: $y < (a-1)x^{2} + (b+2)x - 2b$.
$(3)$ When $a > 0$ and $b > 1$, let $P$ be the solution set of the inequality $y > 0$, and $Q = \{x | -2-t < x < -2+t\}$. If for any positive number $t$, $P \cap Q \neq \varnothing$, find the maximum value of $\frac{1}{a} - \frac{1}{b}$.
|
\frac{1}{2}
| 27.34375 |
10,275 |
A positive integer \( m \) has the property that when multiplied by 12, the result is a four-digit number \( n \) of the form \( 20A2 \) for some digit \( A \). What is the four-digit number \( n \)?
|
2052
| 46.875 |
10,276 |
It is known that none of the digits of a three-digit number is zero, and the sum of all possible two-digit numbers composed of the digits of this number is equal to the number itself. Find the largest such three-digit number.
|
396
| 82.03125 |
10,277 |
Let $f(x) = x^3 - 9x^2 + 27x - 25$ and let $g(f(x)) = 3x + 4$. What is the sum of all possible values of $g(7)$?
|
39
| 34.375 |
10,278 |
On a line passing through the center $O$ of a circle with radius 12, points $A$ and $B$ are chosen such that $OA=15$, $AB=5$, and $A$ lies between $O$ and $B$. Tangents are drawn from points $A$ and $B$ to the circle, with the points of tangency lying on the same side of the line $OB$. Find the area of triangle $ABC$, where $C$ is the point of intersection of these tangents.
|
150/7
| 2.34375 |
10,279 |
Given the positive sequence $\{a_n\}$, where $a_1=2$, $a_2=1$, and $\frac {a_{n-1}-a_{n}}{a_{n}a_{n-1}}= \frac {a_{n}-a_{n+1}}{a_{n}a_{n+1}}(n\geqslant 2)$, find the value of the 2016th term of this sequence.
|
\frac{1}{1008}
| 89.0625 |
10,280 |
Let \( f(x) = x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1 \). When \( f(x^{10}) \) is divided by \( f(x) \), the remainder is \( b \). Find the value of \( b \).
|
10
| 94.53125 |
10,281 |
Given $\{a_{n}\}\left(n\in N*\right)$ is an arithmetic sequence with a common difference of $-2$, and $a_{6}$ is the geometric mean of $a_{2}$ and $a_{8}$. Let $S_{n}$ be the sum of the first $n$ terms of $\{a_{n}\}$. Find the value of $S_{10}$.
|
90
| 96.09375 |
10,282 |
Given that \(a - b = 2 + \sqrt{3}\) and \(b - c = 2 - \sqrt{3}\), find the value of \(a^2 + b^2 + c^2 - ab - bc - ca\).
|
15
| 50.78125 |
10,283 |
A pedestrian reported to a traffic officer the number of a car whose driver grossly violated traffic rules. This number is expressed as a four-digit number, where the unit digit is the same as the tens digit, and the hundreds digit is the same as the thousands digit. Moreover, this number is a perfect square. What is this number?
|
7744
| 19.53125 |
10,284 |
In a round-robin tournament among $8$ chess players (each pair plays one match), the scoring rules are: the winner of a match earns $2$ points, a draw results in $1$ point for each player, and the loser scores $0$ points. The final scores of the players are all different, and the score of the player in second place equals the sum of the scores of the last four players. What is the score of the second-place player?
|
12
| 23.4375 |
10,285 |
Given a real coefficient fourth-degree polynomial with a leading coefficient of 1 that has four imaginary roots, where the product of two of the roots is \(32+\mathrm{i}\) and the sum of the other two roots is \(7+\mathrm{i}\), determine the coefficient of the quadratic term.
|
114
| 1.5625 |
10,286 |
Compute the smallest base-10 positive integer greater than 7 that is a palindrome when written in both base 3 and 5.
|
26
| 3.125 |
10,287 |
On the lateral side \( CD \) of trapezoid \( ABCD \) (\( AD \parallel BC \)), a point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is drawn to segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).
|
18
| 8.59375 |
10,288 |
Given $0\leqslant x\_0 < 1$, for all integers $n > 0$, let $x\_n= \begin{cases} 2x_{n-1}, & 2x_{n-1} < 1 \\ 2x_{n-1}-1, & 2x_{n-1} \geqslant 1 \end{cases}$. Find the number of $x\_0$ that makes $x\_0=x\_6$ true.
|
64
| 16.40625 |
10,289 |
Evaluate the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^4}{9^4 - 1} + \frac{3^8}{9^8 - 1} + \cdots.$$
|
\frac{1}{2}
| 71.875 |
10,290 |
A traffic light runs repeatedly through the following cycle: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark picks a random five-second time interval to watch the light. What is the probability that the color changes while he is watching?
|
\frac{3}{20}
| 43.75 |
10,291 |
Using five nines (9), arithmetic operations, and exponentiation, create the numbers from 1 to 13.
|
13
| 1.5625 |
10,292 |
Let \(a, b, c\) be arbitrary real numbers such that \(a > b > c\) and \((a - b)(b - c)(c - a) = -16\). Find the minimum value of \(\frac{1}{a - b} + \frac{1}{b - c} - \frac{1}{c - a}\).
|
\frac{5}{4}
| 32.8125 |
10,293 |
Elena intends to buy 7 binders priced at $\textdollar 3$ each. Coincidentally, a store offers a 25% discount the next day and an additional $\textdollar 5$ rebate for purchases over $\textdollar 20$. Calculate the amount Elena could save by making her purchase on the day of the discount.
|
10.25
| 43.75 |
10,294 |
A shopkeeper purchases 2000 pens at a cost of $0.15 each. If the shopkeeper wants to sell them for $0.30 each, calculate the number of pens that need to be sold to make a profit of exactly $120.00.
|
1400
| 23.4375 |
10,295 |
If the function \( f(x) = (x^2 - 1)(x^2 + ax + b) \) satisfies \( f(x) = f(4 - x) \) for any \( x \in \mathbb{R} \), what is the minimum value of \( f(x) \)?
|
-16
| 32.03125 |
10,296 |
For $\{1, 2, 3, \ldots, 10\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all such alternating sums for $n=10$.
|
5120
| 4.6875 |
10,297 |
This century will mark the 200th anniversary of the birth of the famous Russian mathematician Pafnuty Lvovich Chebyshev, a native of Kaluga province. The sum of the digits in the hundreds and thousands places of the year he was born is 3 times the sum of the digits in the units and tens places, and the digit in the tens place is greater than the digit in the units place. Determine the year of birth of P.L. Chebyshev, given that he was born and died in the same century and lived for 73 years.
|
1821
| 82.03125 |
10,298 |
Given the function $f(x)=|x-2|+|x-a^{2}|$.
$(1)$ If the inequality $f(x)\leqslant a$ has solutions for $x$, find the range of the real number $a$;
$(2)$ If the positive real numbers $m$, $n$ satisfy $m+2n=a$, when $a$ takes the maximum value from $(1)$, find the minimum value of $\left( \dfrac {1}{m}+ \dfrac {1}{n}\right)$.
|
\dfrac {3}{2}+ \sqrt {2}
| 0 |
10,299 |
Eight numbers \( a_{1}, a_{2}, a_{3}, a_{4} \) and \( b_{1}, b_{2}, b_{3}, b_{4} \) satisfy the following equations:
$$
\left\{\begin{array}{c}
a_{1} b_{1}+a_{2} b_{3}=1 \\
a_{1} b_{2}+a_{2} b_{4}=0 \\
a_{3} b_{1}+a_{4} b_{3}=0 \\
a_{3} b_{2}+a_{4} b_{4}=1
\end{array}\right.
$$
It is known that \( a_{2} b_{3}=7 \). Find \( a_{4} b_{4} \).
|
-6
| 26.5625 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.