Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
9,200 |
Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journey, and halved its speed again for the next eighth part of the journey, and so on. After the eighth decrease in speed, the car did not change its speed until the end of the trip. By how many times did the second car take more time to complete the entire journey than the first car?
|
5/3
| 10.9375 |
9,201 |
Two concentric circles have radii of $12$ meters and $24$ meters respectively. A rabbit starts at point X and runs along a sixth of the circumference of the larger circle, followed by a straight line to the smaller circle, then along a third of the circumference of the smaller circle, and finally across the diameter of the smaller circle back to the closest point on the larger circle's circumference. Calculate the total distance the rabbit runs.
|
16\pi + 36
| 47.65625 |
9,202 |
Let \( z_{1} \) and \( z_{2} \) be complex numbers such that \( \left|z_{1}\right|=3 \), \( \left|z_{2}\right|=5 \), and \( \left|z_{1} + z_{2}\right|=7 \). Find the value of \( \arg \left(\left( \frac{z_{2}}{z_{1}} \right)^{3}\right) \).
|
\pi
| 78.125 |
9,203 |
If you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example, $3333$ ), two values (for example, $2666$ ), three values (for example, $5215$ ), or four values (for example, $4236$ ). The mean number of values that show is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .
|
887
| 42.96875 |
9,204 |
Let \( A = \{1, 2, \cdots, 10\} \). If the equation \( x^2 - bx - c = 0 \) satisfies \( b, c \in A \) and the equation has at least one root \( a \in A \), then the equation is called a "beautiful equation". Find the number of "beautiful equations".
|
12
| 100 |
9,205 |
Given the function $f(x)=e^{ax}-x-1$, where $a\neq 0$. If $f(x)\geqslant 0$ holds true for all $x\in R$, then the set of possible values for $a$ is \_\_\_\_\_\_.
|
\{1\}
| 10.15625 |
9,206 |
Usually, I go up the escalator in the subway. I have calculated that when I walk up the moving escalator, I ascend 20 steps, and it takes me exactly 60 seconds. My wife walks up the stairs more slowly and only ascends 16 steps; therefore, her total time to ascend the escalator is longer - it is 72 seconds.
How many steps would I have to climb if the escalator suddenly breaks down?
|
40
| 81.25 |
9,207 |
Walnuts and hazelnuts were delivered to a store in $1 \mathrm{~kg}$ packages. The delivery note only mentioned that the shipment's value is $1978 \mathrm{Ft}$, and its weight is $55 \mathrm{~kg}$. The deliverers remembered the following:
- Walnuts are more expensive;
- The prices per kilogram are two-digit numbers, and one can be obtained by swapping the digits of the other;
- The price of walnuts consists of consecutive digits.
How much does $1 \mathrm{~kg}$ of walnuts cost?
|
43
| 42.96875 |
9,208 |
If a pot can hold 2 cakes at a time and it takes 5 minutes to cook both sides of a cake, calculate the minimum time it will take to cook 3 cakes thoroughly.
|
15
| 79.6875 |
9,209 |
In $\triangle ABC$, with $AB=3$, $AC=4$, $BC=5$, let $I$ be the incenter of $\triangle ABC$ and $P$ be a point inside $\triangle IBC$ (including the boundary). If $\overrightarrow{AP}=\lambda \overrightarrow{AB} + \mu \overrightarrow{AC}$ (where $\lambda, \mu \in \mathbf{R}$), find the minimum value of $\lambda + \mu$.
|
7/12
| 25.78125 |
9,210 |
Given the parametric equations of curve $C_1$ are $$\begin{cases} x=2\cos\theta \\ y=\sin\theta\end{cases}(\theta \text{ is the parameter}),$$ and the parametric equations of curve $C_2$ are $$\begin{cases} x=-3+t \\ y= \frac {3+3t}{4}\end{cases}(t \text{ is the parameter}).$$
(1) Convert the parametric equations of curves $C_1$ and $C_2$ into standard equations;
(2) Find the maximum and minimum distances from a point on curve $C_1$ to curve $C_2$.
|
\frac {12-2 \sqrt {13}}{5}
| 0 |
9,211 |
Given the inequality about $x$, $|x-1|+|x+a|\leqslant 8$, the minimum value of $a$ is ________.
|
-9
| 87.5 |
9,212 |
Given \( x = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{10^{6}}} \), calculate the value of \([x]\).
|
1998
| 99.21875 |
9,213 |
(2014•Shanghai) In a certain game, the scores are 1, 2, 3, 4, 5. The random variable $\xi$ represents Xiao Bai's score in this game. If $E(\xi) = 4.2$, then the probability that Xiao Bai scores 5 points is at least ___.
|
0.2
| 28.125 |
9,214 |
The entire surface of a cube with dimensions $13 \times 13 \times 13$ was painted red, and then this cube was cut into $1 \times 1 \times 1$ cubes. All faces of the $1 \times 1 \times 1$ cubes that were not painted red were painted blue. By what factor is the total area of the blue faces greater than the total area of the red faces?
|
12
| 60.9375 |
9,215 |
Ostap Bender organized an elephant distribution for the residents in the city of Fuks. 28 members of a union and 37 non-union members came for the distribution. Ostap distributed the elephants equally among all union members and also equally among all non-union members.
It turned out that there was only one possible way to distribute the elephants (such that all elephants were distributed). What is the largest number of elephants that Ostap Bender could have? (It is assumed that each attendee received at least one elephant.)
|
1036
| 57.03125 |
9,216 |
A flock of geese was flying over several lakes. On each lake, half of the geese and an additional half goose landed, while the rest continued flying. All the geese landed after seven lakes. How many geese were in the flock?
|
127
| 67.1875 |
9,217 |
There are 4 white tiles, 4 yellow tiles, and 8 green tiles, all of which are square tiles with a side length of 1. These tiles are used to create square patterns with a side length of 4. How many different patterns can be created?
|
900900
| 97.65625 |
9,218 |
In $\triangle ABC$, angle bisectors $BD$ and $CE$ intersect at $I$, with $D$ and $E$ located on $AC$ and $AB$ respectively. A perpendicular from $I$ to $DE$ intersects $DE$ at $P$, and the extension of $PI$ intersects $BC$ at $Q$. If $IQ = 2 IP$, find $\angle A$.
|
60
| 82.03125 |
9,219 |
To monitor the skier's movements, the coach divided the track into three sections of equal length. It was found that the skier's average speeds on these three separate sections were different. The time required for the skier to cover the first and second sections together was 40.5 minutes, and for the second and third sections - 37.5 minutes. Additionally, it was determined that the skier's average speed on the second section was the same as the average speed for the first and third sections combined. How long did it take the skier to reach the finish?
|
58.5
| 3.90625 |
9,220 |
For the quadrilateral $ABCD$, it is known that $\angle BAC = \angle CAD = 60^{\circ}$, and $AB + AD = AC$. Additionally, it is known that $\angle ACD = 23^{\circ}$. How many degrees is the angle $\angle ABC$?
|
83
| 71.09375 |
9,221 |
There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$ , $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$ . What is the most probable sum he can get?
|
127
| 53.90625 |
9,222 |
Given the line $l$: $2mx - y - 8m - 3 = 0$ and the circle $C$: $x^2 + y^2 - 6x + 12y + 20 = 0$, find the shortest length of the chord that line $l$ cuts on circle $C$.
|
2\sqrt{15}
| 3.125 |
9,223 |
Let \(x_{1}, x_{2}, \ldots, x_{200}\) be natural numbers greater than 2 (not necessarily distinct). In a \(200 \times 200\) table, the numbers are arranged as follows: at the intersection of the \(i\)-th row and the \(k\)-th column, the number \(\log _{x_{k}} \frac{x_{i}}{9}\) is written. Find the smallest possible value of the sum of all the numbers in the table.
|
-40000
| 82.03125 |
9,224 |
There are some identical square pieces of paper. If a part of them is paired up to form rectangles with a length twice their width, the total perimeter of all the newly formed rectangles is equal to the total perimeter of the remaining squares. Additionally, the total perimeter of all shapes after pairing is 40 centimeters less than the initial total perimeter. What is the initial total perimeter of all square pieces of paper in centimeters?
|
280
| 64.0625 |
9,225 |
A four-digit number $2\Box\Box5$ is divisible by $45$. How many such four-digit numbers are there?
|
11
| 32.8125 |
9,226 |
At the New-Vasyuki currency exchange, 11 tugriks are traded for 14 dinars, 22 rupees for 21 dinars, 10 rupees for 3 thalers, and 5 crowns for 2 thalers. How many tugriks can be exchanged for 13 crowns?
|
13
| 58.59375 |
9,227 |
In triangle $XYZ,$ $\angle Y = 45^\circ,$ $\angle Z = 90^\circ,$ and $XZ = 6.$ Find $YZ.$
|
3\sqrt{2}
| 92.96875 |
9,228 |
\(A, B, C, D\) are consecutive vertices of a parallelogram. Points \(E, F, P, H\) lie on sides \(AB\), \(BC\), \(CD\), and \(AD\) respectively. Segment \(AE\) is \(\frac{1}{3}\) of side \(AB\), segment \(BF\) is \(\frac{1}{3}\) of side \(BC\), and points \(P\) and \(H\) bisect the sides they lie on. Find the ratio of the area of quadrilateral \(EFPH\) to the area of parallelogram \(ABCD\).
|
37/72
| 22.65625 |
9,229 |
If \( 9210 - 9124 = 210 - \square \), the value represented by the \( \square \) is:
|
124
| 71.09375 |
9,230 |
Find the GCD (Greatest Common Divisor) of all the numbers of the form \( n^{13} - n \).
|
2730
| 81.25 |
9,231 |
Given the mean of the data $x_1, x_2, \ldots, x_n$ is 2, and the variance is 3, calculate the mean and variance of the data $3x_1+5, 3x_2+5, \ldots, 3x_n+5$.
|
27
| 76.5625 |
9,232 |
Compute the sum $i^{-103} + i^{-102} + \cdots + i^{-1} + i^0 + i^1 + \cdots + i^{102} + i^{103} + \sum_{n=1}^{103} n$.
|
5355
| 3.90625 |
9,233 |
In triangle $ABC$, we have $\angle A = 90^\circ$ and $\sin B = \frac{3}{5}$. Find $\cos C$.
|
\frac{3}{5}
| 93.75 |
9,234 |
Elective 4-5: Selected Topics on Inequalities. Given the function $f(x) = |2x-1| + |2x+3|$.
$(1)$ Solve the inequality $f(x) \geqslant 6$;
$(2)$ Let the minimum value of $f(x)$ be $m$, and let the positive real numbers $a, b$ satisfy $2ab + a + 2b = m$. Find the minimum value of $a + 2b$.
|
2\sqrt{5} - 2
| 12.5 |
9,235 |
Given a sequence $\left\{a_{n}\right\}$, where $a_{1}=a_{2}=1$, $a_{3}=-1$, and $a_{n}=a_{n-1} a_{n-3}$, find $a_{1964}$.
|
-1
| 17.1875 |
9,236 |
Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$ .
Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible.
Justify your answer
|
3/7
| 89.84375 |
9,237 |
Define: \( a \oplus b = a \times b \), \( c \bigcirc d = d \times d \times d \times \cdots \times d \) (d multiplied c times). Find \( (5 \oplus 8) \oplus (3 \bigcirc 7) \).
|
13720
| 87.5 |
9,238 |
Given \( f(u) = u^{2} + au + (b-2) \), where \( u = x + \frac{1}{x} \) (with \( x \in \mathbb{R} \) and \( x \neq 0 \)). If \( a \) and \( b \) are real numbers such that the equation \( f(u) = 0 \) has at least one real root, find the minimum value of \( a^{2} + b^{2} \).
|
4/5
| 21.875 |
9,239 |
In triangle $ABC$, $AB = AC = 17$, $BC = 16$, and point $G$ is the centroid of the triangle. Points $A'$, $B'$, and $C'$ are the images of $A$, $B$, and $C$, respectively, after a $90^\circ$ clockwise rotation about $G$. Determine the area of the union of the two regions enclosed by triangles $ABC$ and $A'B'C'$.
|
240
| 26.5625 |
9,240 |
A student is using the given data in the problem statement to find an approximate solution (accurate to 0.1) for the equation $\lg x = 2 - x$. He sets $f(x) = \lg x + x - 2$, finds that $f(1) < 0$ and $f(2) > 0$, and uses the "bisection method" to obtain 4 values of $x$, calculates the sign of their function values, and concludes that the approximate solution of the equation is $x \approx 1.8$. Among the 4 values he obtained, what is the second value?
|
1.75
| 64.84375 |
9,241 |
A right-angled triangle has an area of \( 36 \mathrm{~m}^2 \). A square is placed inside the triangle such that two sides of the square are on two sides of the triangle, and one vertex of the square is at one-third of the longest side.
Determine the area of this square.
|
16
| 21.875 |
9,242 |
Given that \(\sin (x + \sin x) = \cos (x - \cos x)\), where \(x \in [0, \pi]\). Find \(x\).
|
\frac{\pi}{4}
| 92.96875 |
9,243 |
Let \(ABCD\) be a square of side length 13. Let \(E\) and \(F\) be points on rays \(AB\) and \(AD\), respectively, so that the area of square \(ABCD\) equals the area of triangle \(AEF\). If \(EF\) intersects \(BC\) at \(X\) and \(BX=6\), determine \(DF\).
|
\sqrt{13}
| 0 |
9,244 |
Find the smallest integer \( n > 1 \) such that \(\frac{1^2 + 2^2 + 3^2 + \ldots + n^2}{n}\) is a square.
|
337
| 100 |
9,245 |
Calculate the lengths of the arcs of curves defined by the equations in polar coordinates.
$$
\rho=5(1-\cos \varphi),-\frac{\pi}{3} \leq \varphi \leq 0
$$
|
20 \left(1 - \frac{\sqrt{3}}{2}\right)
| 0 |
9,246 |
Person A can only be in the first or second position, and person B can only be in the second or third position. Find the total number of different possible arrangements of five people in a row.
|
18
| 39.0625 |
9,247 |
In triangle $\triangle ABC$, $A(7,8)$, $B(10,4)$, $C(2,-4)$, then $S_{\triangle ABC}$ is ______.
|
28
| 93.75 |
9,248 |
What is the probability that in a random sequence of 8 ones and 2 zeros, there are exactly three ones between the two zeros?
|
2/15
| 27.34375 |
9,249 |
The ratio of the dividend to the divisor is 9:2, and the ratio of the divisor to the quotient is ____.
|
\frac{2}{9}
| 13.28125 |
9,250 |
The value of the expression $\frac{\sin 10^{\circ}}{1-\sqrt{3}\tan 10^{\circ}}$ can be simplified using trigonometric identities and calculated exactly.
|
\frac{1}{4}
| 77.34375 |
9,251 |
Given the function $y= \sqrt {x^{2}-ax+4}$, find the set of all possible values of $a$ such that the function is monotonically decreasing on the interval $[1,2]$.
|
\{4\}
| 14.0625 |
9,252 |
Given that construction teams A and B each have a certain number of workers. If team A lends 90 workers to team B, then team B's total number of workers will be twice that of team A. If team B lends a certain number of workers to team A, then team A's total number of workers will be 6 times that of team B. How many workers did team A originally have at least?
|
153
| 75.78125 |
9,253 |
The forecast predicts an 80 percent chance of rain for each day of a three-day festival. If it doesn't rain, there is a 50% chance it will be sunny and a 50% chance it will be cloudy. Mina and John want exactly one sunny day during the festival for their outdoor activities. What is the probability that they will get exactly one sunny day?
|
0.243
| 20.3125 |
9,254 |
Let \( g(x) = \log_{\frac{1}{3}}\left(\log_9\left(\log_{\frac{1}{9}}\left(\log_{81}\left(\log_{\frac{1}{81}}x\right)\right)\right)\right) \). Determine the length of the interval that forms the domain of \( g(x) \), and express it in the form \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers. What is \( p+q \)?
A) 80
B) 82
C) 84
D) 86
E) 88
|
82
| 38.28125 |
9,255 |
Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list?
|
871
| 46.875 |
9,256 |
Given the sequence \( S_{1} = 1, S_{2} = 1 - 2, S_{3} = 1 - 2 + 3, S_{4} = 1 - 2 + 3 - 4, S_{5} = 1 - 2 + 3 - 4 + 5, \cdots \), find the value of \( S_{1} + S_{2} + S_{3} + \cdots + S_{299} \).
|
150
| 89.84375 |
9,257 |
In triangle \(ABC\) on side \(AB\) points \(E\) and \(F\) lie. The area of triangle \(AEC\) is \(1 \text{ cm}^2\), the area of triangle \(EFC\) is \(3 \text{ cm}^2\), and the area of triangle \(FBC\) is \(2 \text{ cm}^2\). Point \(T\) is the centroid of triangle \(AFC\), and point \(G\) is the intersection of lines \(CT\) and \(AB\). Point \(R\) is the centroid of triangle \(EBC\), and point \(H\) is the intersection of lines \(CR\) and \(AB\).
Determine the area of triangle \(GHC\).
(E. Semerádová)
|
1.5
| 0 |
9,258 |
Suppose you have two bank cards for making purchases: a debit card and a credit card. Today you decided to buy airline tickets worth 20,000 rubles. If you pay for the purchase with the credit card (the credit limit allows it), you will have to repay the bank within $\mathrm{N}$ days to stay within the grace period in which the credit can be repaid without extra charges. Additionally, in this case, the bank will pay cashback of $0.5 \%$ of the purchase amount after 1 month. If you pay for the purchase with the debit card (with sufficient funds available), you will receive a cashback of $1 \%$ of the purchase amount after 1 month. It is known that the annual interest rate on the average monthly balance of funds on the debit card is $6 \%$ per year (Assume for simplicity that each month has 30 days, the interest is credited to the card at the end of each month, and the interest accrued on the balance is not compounded). Determine the minimum number of days $\mathrm{N}$, under which all other conditions being equal, it is more profitable to pay for the airline tickets with the credit card. (15 points)
|
31
| 26.5625 |
9,259 |
Given a set of data: $10.1$, $9.8$, $10$, $x$, $10.2$, the average of these data is $10$. Calculate the variance of this set of data.
|
0.02
| 13.28125 |
9,260 |
Natural numbers \( a, b, c \) are such that \( 1 \leqslant a < b < c \leqslant 3000 \). Find the largest possible value of the quantity
$$
\gcd(a, b) + \gcd(b, c) + \gcd(c, a)
$$
|
3000
| 59.375 |
9,261 |
In the polar coordinate system, the length of the chord cut by the ray $θ= \dfrac {π}{4}$ on the circle $ρ=4\sin θ$ is __________.
|
2\sqrt {2}
| 0 |
9,262 |
How many ten-digit numbers exist in which there are at least two identical digits?
|
8996734080
| 99.21875 |
9,263 |
Find the area of a triangle with side lengths 13, 14, and 14.
|
6.5\sqrt{153.75}
| 0 |
9,264 |
An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$ . Now, she pulls her arms in and her moment of inertia is now $\frac{7}{10}I$ . What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)?
|
$\dfrac{10}{7}$
| 0 |
9,265 |
There are 8 identical balls in a box, consisting of three balls numbered 1, three balls numbered 2, and two balls numbered 3. A ball is randomly drawn from the box, returned, and then another ball is randomly drawn. The product of the numbers on the balls drawn first and second is denoted by $\xi$. Find the expected value $E(\xi)$.
|
225/64
| 42.1875 |
9,266 |
Evaluate or simplify:
\\((1)\\dfrac{\\sqrt{1-2\\sin {15}^{\\circ}\\cos {15}^{\\circ}}}{\\cos {15}^{\\circ}-\\sqrt{1-\\cos^2 {165}^{\\circ}}}\\)；
\\((2)\\)Given \\(| \\vec{a} |=4\\), \\(| \\vec{b} |=2\\), and the angle between \\(\\vec{a}\\) and \\(\\vec{b}\\) is \\(\\dfrac{2\\pi }{3}\\), find the value of \\(| \\vec{a} + \\vec{b} |\\).
|
2\\sqrt{3}
| 94.53125 |
9,267 |
For how many positive integers \( n \) is the sum
\[
(-n)^{3} + (-n+1)^{3} + \cdots + (n-2)^{3} + (n-1)^{3} + n^{3} + (n+1)^{3}
\]
less than \( 3129 \)?
|
13
| 78.125 |
9,268 |
A chemistry student conducted an experiment: starting with a bottle filled with syrup solution, the student poured out one liter of liquid, refilled the bottle with water, then poured out one liter of liquid again, and refilled the bottle with water once more. As a result, the syrup concentration decreased from 36% to 1%. Determine the volume of the bottle in liters.
|
1.2
| 39.0625 |
9,269 |
Given a triangle \(ABC\) with sides opposite to the angles \(A\), \(B\), and \(C\) being \(a\), \(b\), and \(c\) respectively, and it is known that \( \sqrt {3}\sin A-\cos (B+C)=1\) and \( \sin B+\sin C= \dfrac {8}{7}\sin A\) with \(a=7\):
(Ⅰ) Find the value of angle \(A\);
(Ⅱ) Calculate the area of \( \triangle ABC\).
|
\dfrac {15 \sqrt {3}}{4}
| 0 |
9,270 |
There are 32 ones written on the board. Each minute, Carlsson erases any two numbers, writes their sum on the board, and then eats an amount of candy equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 32 minutes?
|
496
| 88.28125 |
9,271 |
Let \(a,\) \(b,\) \(c\) be distinct real numbers such that
\[\frac{a}{1 + b} = \frac{b}{1 + c} = \frac{c}{1 + a} = k.\] Find the product of all possible values of \(k.\)
|
-1
| 19.53125 |
9,272 |
For any real number \( x \), let \( f(x) \) be the minimum of the values \( 4x + 1 \), \( x + 2 \), and \( -2x + 4 \). What is the maximum value of \( f(x) \)?
|
\frac{8}{3}
| 50.78125 |
9,273 |
How many positive three-digit integers less than 700 have at least two digits that are the same and none of the digits can be zero?
|
150
| 3.125 |
9,274 |
Given \( f(x)=a \sin ((x+1) \pi)+b \sqrt[3]{x-1}+2 \), where \( a \) and \( b \) are real numbers and \( f(\lg 5) = 5 \), find \( f(\lg 20) \).
|
-1
| 65.625 |
9,275 |
In the parallelepiped $ABCD A_1B_1C_1D_1$, the face $ABCD$ is a square with side length 5, the edge $AA_1$ is also equal to 5, and this edge forms angles of $60^\circ$ with the edges $AB$ and $AD$. Find the length of the diagonal $BD_1$.
|
5\sqrt{3}
| 56.25 |
9,276 |
Let $T=TNFTPP$ . Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$ , $CD = 5$ , and $BD = 6$ . Determine the area of quadrilateral $ACDB$ .
[asy]
draw(circle((0,0),10));
draw((0,10)--(0,0)--(10,0)--(0,10));
draw((0,3)--(4,0));
label("O",(0,0),SW);
label("C",(0,3),W);
label("A",(0,10),N);
label("D",(4,0),S);
label("B",(10,0),E);
[/asy]
[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question. For those who wanted to try the problem separately, <details><summary>here's the value of T</summary>$T=10$</details>.
|
44
| 48.4375 |
9,277 |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and it is known that $a_2=1$, $S_4=8$. Find $a_5$ and $S_{10}$.
|
80
| 43.75 |
9,278 |
When the base-16 number $B1234_{16}$ is written in base 2, how many base-2 digits (bits) does it have?
|
20
| 44.53125 |
9,279 |
Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 4. The point $K$ is defined such that $\overrightarrow{L K}=3 \overrightarrow{F A}-\overrightarrow{F B}$. Determine whether $K$ lies inside, on the boundary, or outside of $A B C D E F$, and find the length of the segment $K A$.
|
\frac{4 \sqrt{3}}{3}
| 3.125 |
9,280 |
Given the function $f(x)=x^3-3x-a$, find the value of $(M-N)$, where $M$ and $-N$ are the maximum and minimum values of $f(x)$ on the interval $[0, 3]$.
|
20
| 33.59375 |
9,281 |
There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there were 2013 non-empty boxes. How many boxes were there in total? Answer: 28182.
|
28182
| 39.0625 |
9,282 |
Evaluate the infinite geometric series: $$\frac{4}{3} - \frac{5}{12} + \frac{25}{144} - \frac{125}{1728} + \dots$$
|
\frac{64}{63}
| 74.21875 |
9,283 |
The diagram shows a square and a regular decagon that share an edge. One side of the square is extended to meet an extended edge of the decagon. What is the value of \( x \)?
A) 15
B) 18
C) 21
D) 24
E) 27
|
18
| 13.28125 |
9,284 |
Let $f(x) = |x-1| + |x+1|$, where $x \in \mathbb{R}$.
(Ⅰ) Solve the inequality $f(x) \leq 4$;
(Ⅱ) If there exists a non-zero real number $b$ such that the inequality $f(x) \geq \frac{|2b+1| + |1-b|}{|b|}$ holds, find the maximum value of $x$ when $x$ is a negative number.
|
-1.5
| 0 |
9,285 |
Given the function $f(x)=\sin x+\lambda\cos x (\lambda\in\mathbb{R})$ is symmetric about $x=-\frac{\pi}{4}$, find the equation of one of the axes of symmetry of function $g(x)$ obtained by expanding the horizontal coordinate of each point of the graph of $f(x)$ by a factor of $2$ and then shifting it right by $\frac{\pi}{3}$.
|
\frac{11\pi}{6}
| 6.25 |
9,286 |
The number $2.29^{\star \star} N$ is an integer. Its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a perfect fourth power.
|
18
| 67.96875 |
9,287 |
While moving down an escalator, Walker counted 50 steps. Trotman, who moved three times faster than Walker, counted 75 steps. If the escalator were to stop, how many steps could be counted on its visible part? It is assumed that both individuals moved at a constant speed and that the escalator's speed was constant.
|
100
| 92.1875 |
9,288 |
A balloon that inflates into the shape of a perfect cube is being blown up at a rate such that at time \( t \) (in fortnights), it has a surface area of \( 6t \) square furlongs. At what rate, in cubic furlongs per fortnight, is the air being pumped in when the surface area is 144 square furlongs?
|
3\sqrt{6}
| 59.375 |
9,289 |
Given $\cos (\alpha - \pi) = -\frac{5}{13}$, and $\alpha$ is an angle in the fourth quadrant, find the value of $\sin (-2\pi + \alpha)$.
|
-\frac{12}{13}
| 99.21875 |
9,290 |
Given a company needs to select 8 engineering and technical personnel from its 6 subsidiaries to form a task force, with each subsidiary contributing at least one person, calculate the total number of ways to allocate these 8 positions.
|
21
| 85.15625 |
9,291 |
Given the two-variable function
$$
f(a, b)=\max _{x \in[-1,1]}\left\{\left|x^{2}-a x-b\right|\right\},
$$
find the minimum value of \( f(a, b) \).
|
\frac{1}{2}
| 78.125 |
9,292 |
Find \( k \) such that for all real numbers \( a, b, \) and \( c \):
\[
(a+b)(b+c)(c+a) = (a+b+c)(ab + bc + ca) + k \, abc
\]
|
-1
| 90.625 |
9,293 |
Let $S_{n}$ and $T_{n}$ denote the sum of the first $n$ terms of the arithmetic sequences ${ a_{n} }$ and ${ b_{n} }$, respectively. Given that $\frac{S_{n}}{T_{n}} = \frac{7n}{n+3}$, find the value of $\frac{a_{5}}{b_{5}}$.
|
\frac{21}{4}
| 67.96875 |
9,294 |
30 beads (blue and green) were arranged in a circle. 26 beads had a neighboring blue bead, and 20 beads had a neighboring green bead. How many blue beads were there?
|
18
| 8.59375 |
9,295 |
Peter's most listened-to CD contains eleven tracks. His favorite is the eighth track. When he inserts the CD into the player and presses one button, the first track starts, and by pressing the button seven more times, he reaches his favorite song. If the device is in "random" mode, he can listen to the 11 tracks in a randomly shuffled order. What are the chances that he will reach his favorite track with fewer button presses in this way?
|
7/11
| 90.625 |
9,296 |
Find the value of the expression \(\cos ^{4} \frac{7 \pi}{24}+\sin ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{17 \pi}{24}+\cos ^{4} \frac{13 \pi}{24}\).
|
\frac{3}{2}
| 55.46875 |
9,297 |
Given $\cos\alpha = \frac{5}{13}$ and $\cos(\alpha - \beta) = \frac{4}{5}$, with $0 < \beta < \alpha < \frac{\pi}{2}$,
$(1)$ Find the value of $\tan 2\alpha$;
$(2)$ Find the value of $\cos\beta$.
|
\frac{56}{65}
| 18.75 |
9,298 |
Given a point P in the plane satisfying $|PM| - |PN| = 2\sqrt{2}$, with $M(-2,0)$, $N(2,0)$, and $O(0,0)$,
(1) Find the locus S of point P;
(2) A straight line passing through the point $(2,0)$ intersects with S at points A and B. Find the minimum value of the area of triangle $\triangle OAB$.
|
2\sqrt{2}
| 31.25 |
9,299 |
In a cabinet, there are 3 pairs of different shoes. If 2 shoes are randomly taken out, let event A denote "the taken out shoes do not form a pair"; event B denote "both taken out shoes are for the same foot"; event C denote "one shoe is for the left foot and the other is for the right foot, but they do not form a pair".
(Ⅰ) Please list all the basic events;
(Ⅱ) Calculate the probabilities of events A, B, and C respectively.
|
\dfrac{2}{5}
| 57.8125 |
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