Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
11,200 |
Each term of a geometric sequence $\left\{a_{n}\right\}$ is a real number, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then find $S_{4n}$.
|
150
| 74.21875 |
11,201 |
Given the scores of the other five students are $83$, $86$, $88$, $91$, $93$, and Xiaoming's score is both the mode and the median among these six scores, find Xiaoming's score.
|
88
| 88.28125 |
11,202 |
A firecracker was thrown vertically upwards with a speed of $20 \mathrm{~m/s}$. Three seconds after the start of the flight, it exploded into two fragments of equal mass. The first fragment flew horizontally immediately after the explosion with a speed of $48 \mathrm{~m/s}$. Find the speed of the second fragment (in m/s) right after the explosion. Assume the acceleration due to gravity is $10 \mathrm{~m/s}^2$.
|
52
| 31.25 |
11,203 |
Alice and Bob are playing a game where Alice declares, "My number is 36." Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in Alice's. What is the smallest possible number Bob can choose?
|
72
| 80.46875 |
11,204 |
Given that $sinα + cosα = \frac{\sqrt{2}}{3}, α ∈ (0, π)$, find the value of $sin(α + \frac{π}{12})$.
|
\frac{2\sqrt{2} + \sqrt{3}}{6}
| 3.90625 |
11,205 |
Three different numbers are chosen at random from the list \(1, 3, 5, 7, 9, 11, 13, 15, 17, 19\). The probability that one of them is the mean of the other two is \(p\). What is the value of \(\frac{120}{p}\) ?
|
720
| 28.90625 |
11,206 |
Convert the binary number \(11111011111_2\) to its decimal representation.
|
2015
| 96.875 |
11,207 |
$ABCD$ is a trapezium such that $\angle ADC=\angle BCD=60^{\circ}$ and $AB=BC=AD=\frac{1}{2}CD$. If this trapezium is divided into $P$ equal portions $(P>1)$ and each portion is similar to trapezium $ABCD$ itself, find the minimum value of $P$.
The sum of tens and unit digits of $(P+1)^{2001}$ is $Q$. Find the value of $Q$.
If $\sin 30^{\circ}+\sin ^{2} 30^{\circ}+\ldots+\sin Q 30^{\circ}=1-\cos ^{R} 45^{\circ}$, find the value of $R$.
Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-8x+(R+1)=0$. If $\frac{1}{\alpha^{2}}$ and $\frac{1}{\beta^{2}}$ are the roots of the equation $225x^{2}-Sx+1=0$, find the value of $S$.
|
34
| 19.53125 |
11,208 |
What is the largest divisor by which $29 \cdot 14$ leaves the same remainder when divided by $13511, 13903,$ and $14589$?
|
98
| 19.53125 |
11,209 |
The six-digit number \( 2PQRST \) is multiplied by 3, and the result is the six-digit number \( PQRST2 \). What is the sum of the digits of the original number?
|
27
| 92.1875 |
11,210 |
Suppose the curve C has the polar coordinate equation $ρ\sin^2θ - 8\cos θ = 0$. Establish a rectangular coordinate system $xoy$ with the pole as the origin and the non-negative semi-axis of the polar axis as the $x$-axis. A line $l$, with an inclination angle of $α$, passes through point $P(2, 0)$.
(1) Write the rectangular coordinate equation of curve C and the parametric equation of line $l$.
(2) Suppose points $Q$ and $G$ have polar coordinates $\left(2, \dfrac{3π}{2}\right)$ and $\left(2, π\right)$, respectively. If line $l$ passes through point $Q$ and intersects curve $C$ at points $A$ and $B$, find the area of triangle $GAB$.
|
16\sqrt{2}
| 21.875 |
11,211 |
Positive numbers \( \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \) satisfy \( a+b+c+d=100 \) and \( \frac{a}{b+c+d}+\frac{b}{a+c+d}+\frac{c}{a+b+d}+\frac{d}{a+b+c}=95 \). Then, \( \frac{1}{b+c+d}+\frac{1}{a+c+d}+\frac{1}{a+b+d}+\frac{1}{a+b+c} = \quad \)
|
\frac{99}{100}
| 60.15625 |
11,212 |
How many natural numbers not exceeding 500 are not divisible by 2, 3, or 5?
|
134
| 96.875 |
11,213 |
A company plans to invest in 3 different projects among 5 candidate cities around the Bohai Economic Rim, which are Dalian, Yingkou, Panjin, Jinzhou, and Huludao. The number of projects invested in the same city cannot exceed 2. How many different investment plans can the company have? (Answer with a number).
|
120
| 70.3125 |
11,214 |
On a shelf, there are 4 different comic books, 5 different fairy tale books, and 3 different story books, all lined up in a row. If the fairy tale books cannot be separated from each other, and the comic books also cannot be separated from each other, how many different arrangements are there?
|
345600
| 32.03125 |
11,215 |
Suppose an integer $x$ , a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$ . Find the largest value of $p$ .
|
47
| 43.75 |
11,216 |
A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors, starting and ending with the same color?
|
\frac{1}{126}
| 36.71875 |
11,217 |
Pigsy picked a bag of wild fruits from the mountain to bring back to Tang Monk. As he walked, he got hungry and took out half of them, hesitated, put back two, and then ate the rest. This happened four times: taking out half, putting back two, and then eating. By the time he reached Tang Monk, there were 5 wild fruits left in the bag. How many wild fruits did Pigsy originally pick?
|
20
| 28.90625 |
11,218 |
Using the digits 3, 4, 7, and 8, form two two-digit numbers (each digit can only be used once and must be used) such that their product is maximized. What is the maximum product?
|
6142
| 53.125 |
11,219 |
In acute triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and $\sqrt{3}b=2asinBcosC+2csinBcosA$.
$(1)$ Find the measure of angle $B$;
$(2)$ Given $a=3$, $c=4$,
① Find $b$,
② Find the value of $\cos \left(2A+B\right)$.
|
-\frac{23}{26}
| 56.25 |
11,220 |
For his birthday, Piglet baked a big cake weighing 10 kg and invited 100 guests. Among them was Winnie-the-Pooh, who has a weakness for sweets. The birthday celebrant announced the cake-cutting rule: the first guest cuts themselves a piece of cake equal to \(1\%\) of the remaining cake, the second guest cuts themselves a piece of cake equal to \(2\%\) of the remaining cake, the third guest cuts themselves a piece of cake equal to \(3\%\) of the remaining cake, and so on. Which position in the queue should Winnie-the-Pooh take to get the largest piece of cake?
|
10
| 48.4375 |
11,221 |
Given that $M$ and $N$ are the common points of circle $A$: $x^{2}+y^{2}-2x=0$ and circle $B$: $x^{2}+y^{2}+2x-4y=0$, find the area of $\triangle BMN$.
|
\frac{3}{2}
| 92.96875 |
11,222 |
A quadrilateral is divided by its diagonals into four triangles. The areas of three of them are 10, 20, and 30, each being less than the area of the fourth triangle. Find the area of the given quadrilateral.
|
120
| 53.125 |
11,223 |
There is a three-digit number \( A \). By placing a decimal point in front of one of its digits, we get a number \( B \). If \( A - B = 478.8 \), find \( A \).
|
532
| 77.34375 |
11,224 |
Real numbers \(a, b, c\) are such that \(a + \frac{1}{b} = 9\), \(b + \frac{1}{c} = 10\), \(c + \frac{1}{a} = 11\). Find the value of the expression \(abc + \frac{1}{abc}\).
|
960
| 2.34375 |
11,225 |
Let $$\overrightarrow {a} = (sinx, \frac {3}{4})$$, $$\overrightarrow {b} = (\frac {1}{3}, \frac {1}{2}cosx)$$, and $$\overrightarrow {a}$$ is parallel to $$\overrightarrow {b}$$. Find the acute angle $x$.
|
\frac {\pi}{4}
| 96.875 |
11,226 |
The cube of $a$ and the fourth root of $b$ vary inversely. If $a=3$ when $b=256$, then find $b$ when $ab=81$.
|
16
| 13.28125 |
11,227 |
We have learned methods to factorize a polynomial, such as the method of common factors and the application of formulas. In fact, there are other methods for factorization, such as grouping method and splitting method.<br/>① Grouping method:<br/>For example, $x^{2}-2xy+y^{2}-25=(x^{2}-2xy+y^{2})-25=(x-y)^{2}-5^{2}=(x-y-5)(x-y+5)$.<br/>② Splitting method:<br/>For example, $x^{2}+2x-3=x^{2}+2x+1-4=(x+1)^{2}-2^{2}=(x+1-2)(x+1+2)=(x-1)(x+3)$.<br/>$(1)$ Following the above methods, factorize as required:<br/>① Using grouping method: $9x^{2}+6x-y^{2}+1$;<br/>② Using splitting method: $x^{2}-6x+8$;<br/>$(2)$ Given: $a$, $b$, $c$ are the three sides of $\triangle ABC$, $a^{2}+5b^{2}+c^{2}-4ab-6b-10c+34=0$, find the perimeter of $\triangle ABC$.
|
14
| 96.875 |
11,228 |
A line passing through the vertex \( A \) of triangle \( ABC \) perpendicular to its median \( BD \) bisects this median.
Find the ratio of the sides \( AB \) and \( AC \).
|
\frac{1}{2}
| 24.21875 |
11,229 |
If $11 = x^6 + \frac{1}{x^6}$, find the value of $x^3 + \frac{1}{x^3}$.
|
\sqrt{13}
| 2.34375 |
11,230 |
Let \(A, B, C\), and \(D\) be four points that are not coplanar. A plane passes through the centroid of triangle \(ABC\) that is parallel to the lines \(AB\) and \(CD\). In what ratio does this plane divide the median drawn to the side \(CD\) of triangle \(ACD\)?
|
1:2
| 44.53125 |
11,231 |
What is the least natural number that can be added to 71,382 to create a palindrome?
|
35
| 0 |
11,232 |
The height of a trapezoid, whose diagonals are mutually perpendicular, is 4. Find the area of the trapezoid if one of its diagonals is 5.
|
\frac{50}{3}
| 12.5 |
11,233 |
In triangle \( \triangle ABC \), given \( AB = 4 \), \( AC = 3 \), and \( P \) is a point on the perpendicular bisector of \( BC \), find \( \overrightarrow{BC} \cdot \overrightarrow{AP} \).
|
-\frac{7}{2}
| 41.40625 |
11,234 |
On a table, there are 210 different cards each with a number from the sequence \(2, 4, 6, \ldots, 418, 420\) (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the sum of the numbers on the chosen cards is divisible by \(7\)?
|
3135
| 71.09375 |
11,235 |
Find the smallest natural number with the following properties:
a) It ends in digit 6 in decimal notation;
b) If you remove the last digit 6 and place this digit 6 at the beginning of the remaining number, the result is four times the original number.
|
153846
| 96.875 |
11,236 |
On a table, there are 2004 boxes, each containing one ball. It is known that some of the balls are white, and their number is even. You are allowed to point to any two boxes and ask if there is at least one white ball in them. What is the minimum number of questions needed to guarantee the identification of a box that contains a white ball?
|
2003
| 5.46875 |
11,237 |
Find the smallest natural number that starts with the digit 5, which, when this 5 is removed from the beginning of its decimal representation and appended to its end, becomes four times smaller.
|
512820
| 88.28125 |
11,238 |
A chessboard of size $8 \times 8$ is considered.
How many ways are there to place 6 rooks such that no two rooks are ever on the same row or column?
|
564480
| 43.75 |
11,239 |
Determine the average daily high temperature in Addington from September 15th, 2008, through September 21st, 2008, inclusive. The high temperatures for the days are as follows: 51, 64, 60, 59, 48, 55, 57 degrees Fahrenheit. Express your answer as a decimal to the nearest tenth.
|
56.3
| 7.03125 |
11,240 |
Using the six digits 0, 1, 2, 3, 4, 5, we can form _ _ _ _ _ different five-digit numbers that do not have repeating digits and are divisible by 5.
|
216
| 93.75 |
11,241 |
Given \(\sin \alpha + \sin (\alpha + \beta) + \cos (\alpha + \beta) = \sqrt{3}\), where \(\beta \in \left[\frac{\pi}{4}, \pi\right]\), find the value of \(\beta\).
|
\frac{\pi}{4}
| 10.9375 |
11,242 |
Ten football teams played each other exactly once. As a result, each team ended up with exactly $x$ points.
What is the largest possible value of $x$? (A win earns 3 points, a draw earns 1 point, and a loss earns 0 points.)
|
13
| 82.03125 |
11,243 |
A game board is constructed by shading three of the regions formed by the diagonals of a regular pentagon. What is the probability that the tip of the spinner will come to rest in a shaded region? Assume the spinner can land in any region with equal likelihood.
|
\frac{3}{10}
| 21.875 |
11,244 |
Given the sequence ${a_n}$ is an arithmetic sequence, with $a_1 \geq 1$, $a_2 \leq 5$, $a_5 \geq 8$, let the sum of the first n terms of the sequence be $S_n$. The maximum value of $S_{15}$ is $M$, and the minimum value is $m$. Determine $M+m$.
|
600
| 10.15625 |
11,245 |
Point \( M \) is the midpoint of the hypotenuse \( AC \) of right triangle \( ABC \). Points \( P \) and \( Q \) on lines \( AB \) and \( BC \) respectively are such that \( AP = PM \) and \( CQ = QM \). Find the measure of angle \( \angle PQM \) if \( \angle BAC = 17^{\circ} \).
|
17
| 10.9375 |
11,246 |
A regular hexagon has a side length of 8 cm. Calculate the area of the shaded region formed by connecting two non-adjacent vertices to the center of the hexagon, creating a kite-shaped region.
[asy]
size(100);
pair A,B,C,D,E,F,O;
A = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240); F = dir(300); O = (0,0);
fill(A--C--O--cycle,heavycyan);
draw(A--B--C--D--E--F--A);
draw(A--C--O);
[/asy]
|
16\sqrt{3}
| 6.25 |
11,247 |
The difference of the logarithms of the hundreds digit and the tens digit of a three-digit number is equal to the logarithm of the difference of the same digits, and the sum of the logarithms of the hundreds digit and the tens digit is equal to the logarithm of the sum of the same digits, increased by 4/3. If you subtract the number, having the reverse order of digits, from this three-digit number, their difference will be a positive number, in which the hundreds digit coincides with the tens digit of the given number. Find this number.
|
421
| 0.78125 |
11,248 |
Two identical cylindrical sheets are cut open along the dotted lines and glued together to form one bigger cylindrical sheet. The smaller sheets each enclose a volume of 100. What volume is enclosed by the larger sheet?
|
400
| 6.25 |
11,249 |
In triangle \( \triangle ABC \), \(\angle ABC = 50^\circ\), \(\angle ACB = 30^\circ\), \(M\) is a point inside the triangle such that \(\angle MCB = 20^\circ\), \(\angle MAC = 40^\circ\). Find the measure of \(\angle MBC\).
(Problem 1208 from Mathematical Bulletin)
|
30
| 5.46875 |
11,250 |
Juan is measuring the diameter of a large rather ornamental plate to cover it with a decorative film. Its actual diameter is 30cm, but his measurement tool has an error of up to $30\%$. Compute the largest possible percent error, in percent, in Juan's calculated area of the ornamental plate.
|
69
| 57.8125 |
11,251 |
In the plane rectangular coordinate system \(x O y\), the equation of the ellipse \(C\) is \(\frac{x^{2}}{9}+\frac{y^{2}}{10}=1\). Let \(F\) be the upper focus of \(C\), \(A\) be the right vertex of \(C\), and \(P\) be a moving point on \(C\) located in the first quadrant. Find the maximum area of the quadrilateral \(O A P F\).
|
\frac{3}{2} \sqrt{11}
| 0 |
11,252 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{\sin A}{\sqrt{3} \sin C}$.
(1) Find the value of $b$.
(2) If $\cos B + \sqrt{3} \sin B = 2$, find the maximum area of triangle $ABC$.
|
\frac{3\sqrt{3}}{4}
| 84.375 |
11,253 |
A cheetah takes strides of 2 meters each, while a fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. If the distance between the cheetah and the fox is 30 meters, how many meters must the cheetah run to catch up with the fox?
|
120
| 60.15625 |
11,254 |
31 people attended a class club afternoon event, and after the program, they danced. Ági danced with 7 boys, Anikó with 8, Zsuzsa with 9, and so on, with each subsequent girl dancing with one more boy than the previously mentioned one. Finally, Márta danced with all but 3 boys. How many boys were at the club afternoon?
|
20
| 7.03125 |
11,255 |
What is the value of $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \cdots + [\sqrt{1989 \cdot 1990}] + [-\sqrt{1}] + [-\sqrt{2}] + [-\sqrt{3}] + \cdots + [-\sqrt{1989 \cdot 1990}]$?
(The 1st "Hope Cup" Mathematics Contest, 1990)
|
-3956121
| 6.25 |
11,256 |
You have 68 coins of different weights. Using 100 weighings on a balance scale without weights, find the heaviest and the lightest coin.
|
100
| 2.34375 |
11,257 |
The greatest prime number that is a divisor of $16,385$ can be deduced similarly, find the sum of the digits of this greatest prime number.
|
19
| 1.5625 |
11,258 |
A club has 99 members. Find the smallest positive integer $n$ such that if the number of acquaintances of each person is greater than $n$, there must exist 4 people who all know each other mutually (here it is assumed that if $A$ knows $B$, then $B$ also knows $A$).
|
66
| 18.75 |
11,259 |
If a sequence $\{a_n\}$ satisfies $$\frac {1}{a_{n+1}}- \frac {1}{a_{n}}=d$$ (where $n\in\mathbb{N}^*$, $d$ is a constant), then the sequence $\{a_n\}$ is called a "harmonic sequence". Given that the sequence $\{\frac {1}{x_{n}}\}$ is a "harmonic sequence", and $x_1+x_2+\ldots+x_{20}=200$, the maximum value of $x_3x_{18}$ is \_\_\_\_\_\_.
|
100
| 35.9375 |
11,260 |
Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many students are in their class. The responses were "Twenty-five" five times, "Twenty-six" six times, and "Twenty-seven" seven times. Can we determine the actual number of students in their class based on their answers?
|
27
| 7.8125 |
11,261 |
Given that the equation \(2x^3 - 7x^2 + 7x + p = 0\) has three distinct roots, and these roots form a geometric progression. Find \(p\) and solve this equation.
|
-2
| 26.5625 |
11,262 |
In the triangle $\triangle ABC$, $\angle A = 60^{\circ}$ and $\angle B = 45^{\circ}$. A line $DE$, with $D$ on $AB$ and $E$ on $BC$, such that $\angle ADE =75^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. Determine the ratio $\frac{AD}{AB}$.
A) $\frac{1}{2}$
B) $\frac{1}{\sqrt{3}}$
C) $\frac{1}{\sqrt{6}}$
D) $\frac{1}{3}$
E) $\frac{1}{4}$
|
\frac{1}{\sqrt{6}}
| 26.5625 |
11,263 |
The value of $999 + 999$ is
|
1998
| 69.53125 |
11,264 |
A father and son were walking one after the other along a snow-covered road. The father's step length is $80 \mathrm{~cm}$, and the son's step length is $60 \mathrm{~cm}$. Their steps coincided 601 times, including at the very beginning and at the end of the journey. What distance did they travel?
|
1440
| 50.78125 |
11,265 |
Given $0 \leq a_k \leq 1$ for $k=1,2,\ldots,2020$, and defining $a_{2021}=a_1, a_{2022}=a_2$, find the maximum value of $\sum_{k=1}^{2020}\left(a_{k}-a_{k+1} a_{k+2}\right)$.
|
1010
| 60.15625 |
11,266 |
On side \(AD\) of rectangle \(ABCD\), a point \(E\) is marked. On segment \(EC\) there is a point \(M\) such that \(AB = BM\) and \(AE = EM\). Find the length of side \(BC\), given that \(ED = 16\) and \(CD = 12\).
|
20
| 33.59375 |
11,267 |
Given complex numbers \( x \) and \( y \), find the maximum value of \(\frac{|3x+4y|}{\sqrt{|x|^{2} + |y|^{2} + \left|x^{2}+y^{2}\right|}}\).
|
\frac{5\sqrt{2}}{2}
| 37.5 |
11,268 |
On the complex plane, consider the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ where the area of the parallelogram is $\frac{24}{25}.$ If the real part of $z$ is positive, determine the smallest possible value of $\left| z + \frac{1}{z} \right|.$ Compute the square of this value.
|
\frac{36}{25}
| 7.03125 |
11,269 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. If $c\cos B + b\cos C = 2a\cos A$, $M$ is the midpoint of $BC$, and $AM=1$, find the maximum value of $b+c$.
|
\frac{4\sqrt{3}}{3}
| 50 |
11,270 |
$48n$ is the smallest positive integer that satisfies the following conditions:
1. $n$ is a multiple of 75;
2. $n$ has exactly 75 positive divisors (including 1 and itself).
Find $\frac{n}{75}$.
|
432
| 66.40625 |
11,271 |
Find the natural number that is divisible by 9 and 5 and has 14 distinct divisors.
|
3645
| 47.65625 |
11,272 |
If the graph of the function $f(x) = \sin \omega x \cos \omega x + \sqrt{3} \sin^2 \omega x - \frac{\sqrt{3}}{2}$ ($\omega > 0$) is tangent to the line $y = m$ ($m$ is a constant), and the abscissas of the tangent points form an arithmetic sequence with a common difference of $\pi$.
(Ⅰ) Find the values of $\omega$ and $m$;
(Ⅱ) Find the sum of all zeros of the function $y = f(x)$ in the interval $x \in [0, 2\pi]$.
|
\frac{11\pi}{3}
| 51.5625 |
11,273 |
Convert the binary number $101111011_{(2)}$ to its decimal equivalent.
|
379
| 92.96875 |
11,274 |
Calculate the limit of the function:
\[ \lim _{x \rightarrow \frac{1}{2}} \frac{\sqrt[3]{\frac{x}{4}}-\frac{1}{2}}{\sqrt{\frac{1}{2}+x}-\sqrt{2x}} \]
|
-\frac{2}{3}
| 16.40625 |
11,275 |
A sphere with radius $r$ is inside a cone, whose axial section is an equilateral triangle with the sphere inscribed in it. The ratio of the total surface area of the cone to the surface area of the sphere is \_\_\_\_\_\_.
|
9:4
| 0 |
11,276 |
A group of 8 boys and 8 girls was paired up randomly. Find the probability that there is at least one pair with two girls. Round your answer to the nearest hundredth.
|
0.98
| 22.65625 |
11,277 |
The calculation result of the expression \(143 \times 21 \times 4 \times 37 \times 2\) is $\qquad$.
|
888888
| 71.09375 |
11,278 |
Compute the least possible value of $ABCD - AB \times CD$ , where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$ , $B$ , $C$ , and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.)
|
109
| 0 |
11,279 |
In a football tournament, each team is supposed to play one match against each of the other teams. However, during the tournament, half of the teams were disqualified and did not participate further. As a result, a total of 77 matches were played, and the disqualified teams managed to play all their matches against each other, with each disqualified team having played the same number of matches. How many teams were there at the beginning of the tournament?
|
14
| 28.90625 |
11,280 |
In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with a side length of 1, points $E$ and $F$ are located on $A A_{1}$ and $C C_{1}$ respectively, such that $A E = C_{1} F$. Determine the minimum area of the quadrilateral $E B F D_{1}$.
|
\frac{\sqrt{6}}{2}
| 60.9375 |
11,281 |
Given a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. The sequence satisfies the conditions $a\_1=23$, $a\_2=-9$, and $a_{n+2}=a\_n+6\times(-1)^{n+1}-2$ for all $n \in \mathbb{N}^*$.
(1) Find the general formula for the terms of the sequence $\{a\_n\}$;
(2) Find the value of $n$ when $S\_n$ reaches its maximum.
|
11
| 27.34375 |
11,282 |
If $α$ is an acute angle, and $\sin (α - \frac{π}{4})= \frac{3}{5}$, then $\cos 2α$ equals ______.
|
-\frac{24}{25}
| 65.625 |
11,283 |
Find all natural numbers having exactly six divisors, the sum of which equals 3500.
|
1996
| 60.15625 |
11,284 |
Calculate the sum of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}$ with an accuracy of $10^{-3}$.
|
0.973
| 21.875 |
11,285 |
Trapezoid \(ABCD\) is inscribed in the parabola \(y = x^2\) such that \(A = (a, a^2)\), \(B = (b, b^2)\), \(C = (-b, b^2)\), and \(D = (-a, a^2)\) for some positive reals \(a, b\) with \(a > b\). If \(AD + BC = AB + CD\), and \(AB = \frac{3}{4}\), what is \(a\)?
|
\frac{27}{40}
| 42.1875 |
11,286 |
Three flower beds, X, Y, and Z, contain X = 600, Y = 480, and Z = 420 plants, respectively. The plants are shared as follows: Beds X and Y share 60 plants, Beds Y and Z share 70 plants, and Beds X and Z share 80 plants. Also, there are 30 plants common to all three beds. Find the total number of unique plants.
|
1320
| 83.59375 |
11,287 |
Given the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$$ ($a>0$, $b>0$) with its right focus at $F(c, 0)$.
(1) If one of the asymptotes of the hyperbola is $y=x$ and $c=2$, find the equation of the hyperbola;
(2) With the origin $O$ as the center and $c$ as the radius, draw a circle. Let the intersection of the circle and the hyperbola in the first quadrant be $A$. Draw the tangent line to the circle at $A$, with a slope of $-\sqrt{3}$. Find the eccentricity of the hyperbola.
|
\sqrt{2}
| 65.625 |
11,288 |
The reciprocal of $\frac{2}{3}$ is ______, the opposite of $-2.5$ is ______.
|
2.5
| 16.40625 |
11,289 |
Given the function $f(x)=e^{x}$, for real numbers $m$, $n$, $p$, it is known that $f(m+n)=f(m)+f(n)$ and $f(m+n+p)=f(m)+f(n)+f(p)$. Determine the maximum value of $p$.
|
2\ln2-\ln3
| 0 |
11,290 |
Given that m > 0, p: 0 < x < m, q: x(x - 1) < 0, if p is a sufficient but not necessary condition for q, then the value of m can be _______. (Only one value of m that satisfies the condition is needed)
|
\frac{1}{2}
| 78.90625 |
11,291 |
Solve the equation \( 2 \sqrt{2} \sin ^{3}\left(\frac{\pi x}{4}\right) = \sin \left(\frac{\pi}{4}(1+x)\right) \).
How many solutions of this equation satisfy the condition: \( 2000 \leq x \leq 3000 \)?
|
250
| 53.125 |
11,292 |
On the International Space Station, there was an electronic clock displaying time in the HH:MM format. Due to an electromagnetic storm, the device started malfunctioning, causing each digit on the display to either increase by 1 or decrease by 1. What was the actual time when the storm occurred, if immediately after the storm the clock showed 09:09?
|
18:18
| 5.46875 |
11,293 |
Given an ellipse $C: \frac{x^{2}}{4} + y^{2} = 1$, with $O$ being the origin of coordinates, and a line $l$ intersects the ellipse $C$ at points $A$ and $B$, and $\angle AOB = 90^{\circ}$.
(Ⅰ) If the line $l$ is parallel to the x-axis, find the area of $\triangle AOB$;
(Ⅱ) If the line $l$ is always tangent to the circle $x^{2} + y^{2} = r^{2} (r > 0)$, find the value of $r$.
|
\frac{2\sqrt{5}}{5}
| 14.0625 |
11,294 |
Calculate the value of $\dfrac{\sqrt[3]{81}}{\sqrt[4]{81}}$ in terms of 81 raised to what power?
|
\frac{1}{12}
| 6.25 |
11,295 |
Given the function $y=\cos (x+\frac{π}{3})$, determine the horizontal shift of the graph of the function $y=\sin x$.
|
\frac{5\pi}{6}
| 34.375 |
11,296 |
Given that connecting all the vertices of a polygon from a point on one of the edges results in 2022 triangles, determine the number of sides of this polygon.
|
2023
| 21.875 |
11,297 |
A high school basketball team has 16 players, including a set of twins named Bob and Bill, and another set of twins named Tom and Tim. How many ways can we choose 5 starters if no more than one player from each set of twins can be chosen?
|
3652
| 0 |
11,298 |
In square ABCD, an isosceles triangle AEF is inscribed; point E lies on side BC, point F lies on side CD, and AE = AF. The tangent of angle AEF is 3. Find the cosine of angle FAD.
|
\frac{2\sqrt{5}}{5}
| 8.59375 |
11,299 |
A circle made of wire and a rectangle are arranged in such a way that the circle passes through two vertices $A$ and $B$ and touches the side $CD$. The length of side $CD$ is 32.1. Find the ratio of the sides of the rectangle, given that its perimeter is 4 times the radius of the circle.
|
4:1
| 0 |
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