Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
9,300 |
There is a unique triplet of positive integers \((a, b, c)\) such that \(a \leq b \leq c\) and
$$
\frac{25}{84}=\frac{1}{a}+\frac{1}{a b}+\frac{1}{a b c}.
$$
Determine \(a + b + c\).
|
17
| 71.875 |
9,301 |
Let $A,B,C,D$ , be four different points on a line $\ell$ , so that $AB=BC=CD$ . In one of the semiplanes determined by the line $\ell$ , the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the angle $\angle MBN$ .
|
60
| 32.03125 |
9,302 |
Evaluate the expression $\left(b^b - b(b-1)^b\right)^b$ when $b=4$.
|
21381376
| 2.34375 |
9,303 |
Find the midsegment (median) of an isosceles trapezoid, if its diagonal is 25 and its height is 15.
|
20
| 55.46875 |
9,304 |
Given $tan({θ+\frac{π}{{12}}})=2$, find $sin({\frac{π}{3}-2θ})$.
|
-\frac{3}{5}
| 23.4375 |
9,305 |
Given \( x_{1}, x_{2}, \cdots, x_{1993} \) satisfy:
\[
\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993,
\]
and
\[
y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k} \quad (k=1,2,\cdots,1993),
\]
what is the maximum possible value of \( \left|y_{1}-y_{2}\right|+\left|y_{2}-y_{3}\right|+\cdots+\left|y_{1992}-y_{1993}\right| \)?
|
1992
| 62.5 |
9,306 |
Given that the sequence of positive integers $a_{1}, a_{2}, a_{3}, a_{4}$ forms a geometric progression with a common ratio $r$ which is not an integer and $r > 1$, find the minimum possible value of $a_{4}$.
|
27
| 69.53125 |
9,307 |
If $a$, $b$, $c$, $d$, $e$, and $f$ are integers such that $8x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$?
|
356
| 42.1875 |
9,308 |
Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$.
|
680
| 35.9375 |
9,309 |
An infinite arithmetic progression of positive integers contains the terms 7, 11, 15, 71, 75, and 79. The first term in the progression is 7. Kim writes down all the possible values of the one-hundredth term in the progression. What is the sum of the numbers Kim writes down?
|
714
| 25 |
9,310 |
Given that $\alpha, \beta, \gamma$ are all acute angles and $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$, find the minimum value of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$.
|
2\sqrt{2}
| 93.75 |
9,311 |
Given that $D$ is the midpoint of side $AB$ of $\triangle ABC$ with an area of $1$, $E$ is any point on side $AC$, and $DE$ is connected. Point $F$ is on segment $DE$ and $BF$ is connected. Let $\frac{DF}{DE} = \lambda_{1}$ and $\frac{AE}{AC} = \lambda_{2}$, with $\lambda_{1} + \lambda_{2} = \frac{1}{2}$. Find the maximum value of $S$, where $S$ denotes the area of $\triangle BDF$.
|
\frac{1}{32}
| 44.53125 |
9,312 |
(1) Find the constant term in the expansion of ${\left( \frac{1}{x}- \sqrt{\frac{x}{2}}\right)}^{9}$;
(2) Given that ${x}^{10}={a}_{0}+{a}_{1}\left( x+2 \right)+{a}_{2}{\left( x+2 \right)}^{2}+... +{a}_{10}{\left( x+2 \right)}^{10}$, find the value of ${{a}_{1}+{a}_{2}+{a}_{3}+... +{a}_{10}}$.
|
-1023
| 91.40625 |
9,313 |
Use the Horner's method to calculate the value of the polynomial $f(x) = 5x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$ when $x=1$ and find the value of $v_3$.
|
7.9
| 50 |
9,314 |
From the set $\{10, 11, 12, \ldots, 19\}$, 5 different numbers were chosen, and from the set $\{90, 91, 92, \ldots, 99\}$, 5 different numbers were also chosen. It turned out that the difference of any two numbers from the ten chosen numbers is not divisible by 10. Find the sum of all 10 chosen numbers.
|
545
| 16.40625 |
9,315 |
Given vectors $\overrightarrow{a}=(\sin \theta, 2)$ and $\overrightarrow{b}=(\cos \theta, 1)$, which are collinear, where $\theta \in (0, \frac{\pi}{2})$.
1. Find the value of $\tan (\theta + \frac{\pi}{4})$.
2. If $5\cos (\theta - \phi)=3 \sqrt{5}\cos \phi, 0 < \phi < \frac{\pi}{2}$, find the value of $\phi$.
|
\frac{\pi}{4}
| 99.21875 |
9,316 |
Given a convex quadrilateral \( ABCD \) with \( X \) being the midpoint of the diagonal \( AC \). It is found that \( CD \parallel BX \). Find \( AD \) given that \( BX = 3 \), \( BC = 7 \), and \( CD = 6 \).
|
14
| 14.0625 |
9,317 |
Find the least positive integer $n$ such that $$\frac 1{\sin 30^\circ\sin 31^\circ}+\frac 1{\sin 32^\circ\sin 33^\circ}+\cdots+\frac 1{\sin 88^\circ\sin 89^\circ}+\cos 89^\circ=\frac 1{\sin n^\circ}.$$
|
n = 1
| 72.65625 |
9,318 |
Let $a_0=29$ , $b_0=1$ and $$ a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1} $$ for $n\geq 1$ . Determine the smallest positive integer $k$ for which $29$ divides $\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does not divide $b_1$ .
|
28
| 43.75 |
9,319 |
Find the smallest natural number that ends with the digit 6 such that moving this digit to the front increases the number exactly fourfold.
|
153846
| 100 |
9,320 |
Given the conditions $a+acosC=\sqrt{3}csinA$, $\left(a+b+c\right)\left(a+b-c\right)=3ab$, $\left(a-b\right)\sin \left(B+C\right)+b\sin B=c\sin C$. Choose any one of these three conditions and complete the following question, then solve it. In triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, _____. Find the value of angle $C$; If the angle bisector of angle $C$ intersects $AB$ at point $D$ and $CD=2\sqrt{3}$, find the minimum value of $2a+b$.
|
6 + 4\sqrt{2}
| 17.1875 |
9,321 |
Let \( x \) and \( y \) be real numbers with \( x > y \) such that \( x^{2} y^{2} + x^{2} + y^{2} + 2xy = 40 \) and \( xy + x + y = 8 \). Find the value of \( x \).
|
3 + \sqrt{7}
| 64.0625 |
9,322 |
For a row of six students, calculate:
(1) How many different arrangements are there if student A cannot be in the first or the last position?
(2) How many different arrangements are there if students A, B, and C cannot stand next to each other? (Formulate your answer with expressions before computing the numerical results.)
|
144
| 0 |
9,323 |
In the expansion of $({x-1}){({\frac{1}{{{x^{2022}}}}+\sqrt{x}+1})^8}$, the coefficient of $x^{2}$ is _____. (Provide your answer as a number).
|
-42
| 10.9375 |
9,324 |
There are 5 people standing in a row. The number of ways to arrange them so that there is exactly one person between A and B.
|
36
| 35.9375 |
9,325 |
In the quadrilateral pyramid $S-ABCD$ with a right trapezoid as its base, where $\angle ABC = 90^\circ$, $SA \perp$ plane $ABCD$, $SA = AB = BC = 1$, and $AD = \frac{1}{2}$, find the tangent of the angle between plane $SCD$ and plane $SBA$.
|
\frac{\sqrt{2}}{2}
| 38.28125 |
9,326 |
The average of \( p, q, r \) is 12. The average of \( p, q, r, t, 2t \) is 15. Find \( t \).
\( k \) is a real number such that \( k^{4} + \frac{1}{k^{4}} = t + 1 \), and \( s = k^{2} + \frac{1}{k^{2}} \). Find \( s \).
\( M \) and \( N \) are the points \( (1, 2) \) and \( (11, 7) \) respectively. \( P(a, b) \) is a point on \( MN \) such that \( MP:PN = 1:s \). Find \( a \).
If the curve \( y = ax^2 + 12x + c \) touches the \( x \)-axis, find \( c \).
|
12
| 10.15625 |
9,327 |
The numbers 407 and 370 equal the sum of the cubes of their digits. For example, \( 4^3 = 64 \), \( 0^3 = 0 \), and \( 7^3 = 343 \). Adding 64, 0, and 343 gives you 407. Similarly, the cube of 3 (27), added to the cube of 7 (343), gives 370.
Could you find a number, not containing zero and having the same property? Of course, we exclude the trivial case of the number 1.
|
153
| 30.46875 |
9,328 |
Among 51 consecutive odd numbers $1, 3, 5, \cdots, 101$, select $\mathrm{k}$ numbers such that their sum is 1949. What is the maximum value of $\mathrm{k}$?
|
44
| 57.03125 |
9,329 |
A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers that were initially written on the board.
|
998
| 0.78125 |
9,330 |
A conference center is setting up chairs in rows for a seminar. Each row can seat $13$ chairs, and currently, there are $169$ chairs set up. They want as few empty seats as possible but need to maintain complete rows. If $95$ attendees are expected, how many chairs should be removed?
|
65
| 27.34375 |
9,331 |
Given that Lucas's odometer showed 27372 miles, which is a palindrome, and 3 hours later it showed another palindrome, calculate Lucas's average speed, in miles per hour, during this 3-hour period.
|
33.33
| 68.75 |
9,332 |
The "Tiao Ri Method", invented by mathematician He Chengtian during the Southern and Northern Dynasties of China, is an algorithm for finding a more accurate fraction to represent a numerical value. Its theoretical basis is as follows: If the deficient approximate value and the excess approximate value of a real number $x$ are $\frac{b}{a}$ and $\frac{d}{c}$ ($a, b, c, d \in \mathbb{N}^*$) respectively, then $\frac{b+d}{a+c}$ is a more accurate deficient approximate value or excess approximate value of $x$. We know that $\pi = 3.14159...$, and if we let $\frac{31}{10} < \pi < \frac{49}{15}$, then after using the "Tiao Ri Method" once, we get $\frac{16}{5}$ as a more accurate excess approximate value of $\pi$, i.e., $\frac{31}{10} < \pi < \frac{16}{5}$. If we always choose the simplest fraction, then what approximate fraction can we get for $\pi$ after using the "Tiao Ri Method" four times?
|
\frac{22}{7}
| 33.59375 |
9,333 |
Given that the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=1+\frac{1}{2}t}\\{y=\sqrt{3}+\frac{{\sqrt{3}}}{2}t}\end{array}}\right.$ (where $t$ is a parameter), establish a polar coordinate system with the origin as the pole and the non-negative x-axis as the polar axis. The polar coordinate equation of curve $C$ is $\rho =4\sin \theta$.
$(1)$ Find the rectangular coordinate equation of curve $C$ and the polar coordinate equation of line $l$.
$(2)$ If $M(1,\sqrt{3})$, and the line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $\frac{{|MB|}}{{|MA|}}+\frac{{|MA|}}{{|MB|}}$.
|
\frac{3\sqrt{3} - 1}{2}
| 6.25 |
9,334 |
A certain number is written in the base-12 numeral system. For which divisor \( m \) is the following divisibility rule valid: if the sum of the digits of the number is divisible by \( m \), then the number itself is divisible by \( m \)?
|
11
| 53.90625 |
9,335 |
Let \([x]\) represent the integral part of the decimal number \(x\). Given that \([3.126] + \left[3.126 + \frac{1}{8}\right] + \left[3.126 + \frac{2}{8}\right] + \ldots + \left[3.126 + \frac{7}{8}\right] = P\), find the value of \(P\).
|
25
| 75 |
9,336 |
The digits of a three-digit number form a geometric progression with distinct terms. If this number is decreased by 200, the resulting three-digit number has digits that form an arithmetic progression. Find the original three-digit number.
|
842
| 46.875 |
9,337 |
Four cars $A, B, C,$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ travel clockwise, while $C$ and $D$ travel counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at that same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. After how much time from the start of the race will $C$ and $D$ meet for the first time?
|
53
| 7.03125 |
9,338 |
Given real numbers \( a, b, c \) satisfy the system of inequalities
\[
a^{2}+b^{2}-4a \leqslant 1, \quad b^{2}+c^{2}-8b \leqslant -3, \quad c^{2}+a^{2}-12c \leqslant -26,
\]
calculate the value of \( (a+b)^{c} \).
|
27
| 21.875 |
9,339 |
Given that $\{a_{n}\}$ is an arithmetic sequence, with the sum of its first $n$ terms denoted as $S_{n}$, and $a_{4}=-3$, choose one of the following conditions as known: <br/>$(Ⅰ)$ The arithmetic sequence $\{a_{n}\}$'s general formula; <br/>$(Ⅱ)$ The minimum value of $S_{n}$ and the value of $n$ when $S_{n}$ reaches its minimum value. <br/>Condition 1: $S_{4}=-24$; <br/>Condition 2: $a_{1}=2a_{3}$.
|
-30
| 7.03125 |
9,340 |
The distance between points A and B is 1200 meters. Dachen starts from point A, and 6 minutes later, Xiaogong starts from point B. After another 12 minutes, they meet. Dachen walks 20 meters more per minute than Xiaogong. How many meters does Xiaogong walk per minute?
|
28
| 72.65625 |
9,341 |
Yangyang leaves home for school. If she walks 60 meters per minute, she arrives at school at 6:53. If she walks 75 meters per minute, she arrives at school at 6:45. What time does Yangyang leave home?
|
6:13
| 17.96875 |
9,342 |
Person A, Person B, and Person C start from point $A$ to point $B$. Person A starts at 8:00, Person B starts at 8:20, and Person C starts at 8:30. They all travel at the same speed. After Person C has been traveling for 10 minutes, Person A is exactly halfway between Person B and point $B$, and at that moment, Person C is 2015 meters away from point $B$. Find the distance between points $A$ and $B$.
|
2418
| 14.84375 |
9,343 |
A bag contains $4$ identical small balls, of which there is $1$ red ball, $2$ white balls, and $1$ black ball. Balls are drawn from the bag with replacement, randomly taking one each time.
(1) Find the probability of drawing a white ball two consecutive times;
(2) If drawing a red ball scores $2$ points, drawing a white ball scores $1$ point, and drawing a black ball scores $0$ points, find the probability that the sum of the scores from three consecutive draws is $4$ points.
|
\frac{15}{64}
| 18.75 |
9,344 |
Given a moving circle that passes through the fixed point $F(1,0)$ and is tangent to the fixed line $l$: $x=-1$.
(1) Find the equation of the trajectory $C$ of the circle's center;
(2) The midpoint of the chord $AB$ formed by the intersection of line $l$ and $C$ is $(2,1}$. $O$ is the coordinate origin. Find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$ and $| \overrightarrow{AB}|$.
|
\sqrt{35}
| 39.0625 |
9,345 |
Point \( M \) lies on the parabola \( y = 2x^2 - 3x + 4 \), and point \( F \) lies on the line \( y = 3x - 4 \).
Find the minimum value of \( MF \).
|
\frac{7 \sqrt{10}}{20}
| 3.125 |
9,346 |
Find the minimum value of the function \( f(x) = \operatorname{tg}^{2} x - 4 \operatorname{tg} x - 12 \operatorname{ctg} x + 9 \operatorname{ctg}^{2} x - 3 \) on the interval \(\left( -\frac{\pi}{2}, 0 \right)\).
|
3 + 8\sqrt{3}
| 0 |
9,347 |
Square \( ABCD \) has center \( O \). Points \( P \) and \( Q \) are on \( AB \), \( R \) and \( S \) are on \( BC \), \( T \) and \( U \) are on \( CD \), and \( V \) and \( W \) are on \( AD \), so that \( \triangle APW \), \( \triangle BRQ \), \( \triangle CTS \), and \( \triangle DVU \) are isosceles and \( \triangle POW \), \( \triangle ROQ \), \( \triangle TOS \), and \( \triangle VOU \) are equilateral. What is the ratio of the area of \( \triangle PQO \) to that of \( \triangle BRQ \)?
|
1:1
| 0 |
9,348 |
Let \(\triangle ABC\) be inscribed in the unit circle \(\odot O\), with the center \(O\) located within \(\triangle ABC\). If the projections of point \(O\) onto the sides \(BC\), \(CA\), and \(AB\) are points \(D\), \(E\), and \(F\) respectively, find the maximum value of \(OD + OE + OF\).
|
\frac{3}{2}
| 57.8125 |
9,349 |
Let $S_{n}$ and $T_{n}$ represent the sum of the first $n$ terms of the arithmetic sequences ${a_{n}}$ and ${b_{n}}$, respectively. Given that $\frac{S_{n}}{T_{n}} = \frac{n+1}{2n-1}$, where $n \in \mathbb{N}^*$, find the value of $\frac{a_{3} + a_{7}}{b_{1} + b_{9}}$.
|
\frac{10}{17}
| 30.46875 |
9,350 |
There are 2021 balls in a crate, numbered from 1 to 2021. Erica calculates the digit sum for each ball. For example, the digit sum of 2021 is 5, since \(2+0+2+1=5\). Balls with equal digit sums have the same color and balls with different digit sums have different colors. How many different colors of balls are there in the crate?
|
28
| 92.1875 |
9,351 |
Two sides of a triangle are $8 \mathrm{dm}$ and $5 \mathrm{dm}$; the angle opposite to the first side is twice as large as the angle opposite to the second side. What is the length of the third side of the triangle?
|
7.8
| 34.375 |
9,352 |
Given that line $l$ passes through point $P(-1,2)$ with a slope angle of $\frac{2\pi}{3}$, and the circle's equation is $\rho=2\cos (\theta+\frac{\pi}{3})$:
(1) Find the parametric equation of line $l$;
(2) Let line $l$ intersect the circle at points $M$ and $N$, find the value of $|PM|\cdot|PN|$.
|
6+2\sqrt{3}
| 30.46875 |
9,353 |
Let \( x \in \left(-\frac{3\pi}{4}, \frac{\pi}{4}\right) \), and \( \cos \left(\frac{\pi}{4} - x\right) = -\frac{3}{5} \). Find the value of \( \cos 2x \).
|
-\frac{24}{25}
| 74.21875 |
9,354 |
A point is randomly thrown onto the segment [6, 11], and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2k-24\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$.
|
2/3
| 27.34375 |
9,355 |
Calculate the sum of the distances from one vertex of a rectangle with length 3 and width 4 to the midpoints of each of its sides.
A) $6 + \sqrt{5}$
B) $7 + \sqrt{12}$
C) $7.77 + \sqrt{13}$
D) $9 + \sqrt{15}$
|
7.77 + \sqrt{13}
| 27.34375 |
9,356 |
Find $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$
|
\sqrt[4]{27}
| 8.59375 |
9,357 |
Please choose one of the following two sub-questions to answer. If multiple choices are made, the score will be based on the first chosen question.
$(①)$ The sum of the internal angles of a regular hexagon is $ $ degrees.
$(②)$ Xiaohua saw a building with a height of $(137)$ meters at its signboard. From the same horizontal plane at point $B$, he measured the angle of elevation to the top of the building $A$ to be $30^{\circ}$. The distance from point $B$ to the building is $ $ meters (rounded to the nearest whole number, and ignore the measuring instrument error, $\sqrt{3} \approx 1.732$).
|
237
| 61.71875 |
9,358 |
A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?
|
67
| 84.375 |
9,359 |
Let \( y = (17 - x)(19 - x)(19 + x)(17 + x) \), where \( x \) is a real number. Find the smallest possible value of \( y \).
|
-1296
| 64.0625 |
9,360 |
In triangle \(ABC\), angle \(A\) is the largest angle. Points \(M\) and \(N\) are symmetric to vertex \(A\) with respect to the angle bisectors of angles \(B\) and \(C\) respectively. Find \(\angle A\) if \(\angle MAN = 50^\circ\).
|
80
| 35.9375 |
9,361 |
Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$ . Find the square of the area of triangle $ADC$ .
|
192
| 34.375 |
9,362 |
Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion?
|
999000000000
| 84.375 |
9,363 |
Given an arithmetic sequence $\{a_n\}$, it is known that $\frac{a_{11}}{a_{10}} + 1 < 0$. Determine the maximum value of $n$ for which $S_n > 0$ holds.
|
19
| 64.0625 |
9,364 |
Find the ratio of AD:DC in triangle ABC, where AB=6, BC=8, AC=10, and D is a point on AC such that BD=6.
|
\frac{18}{7}
| 53.125 |
9,365 |
If parallelogram ABCD has area 48 square meters, and E and F are the midpoints of sides AB and CD respectively, and G and H are the midpoints of sides BC and DA respectively, calculate the area of the quadrilateral EFGH in square meters.
|
24
| 84.375 |
9,366 |
Each of the three cutlets needs to be fried on a pan on both sides for five minutes each side. The pan can hold only two cutlets at a time. Is it possible to fry all three cutlets in less than 20 minutes (ignoring the time for flipping and transferring the cutlets)?
|
15
| 19.53125 |
9,367 |
Given that $\sin \alpha$ is a root of the equation $5x^2 - 7x - 6 = 0$, find the value of $\frac{\sin \left(\alpha - \frac{\pi}{2}\right)\sin^2\left(-\alpha + \pi\right)\sin \left( \frac{3\pi}{2}-\alpha\right)\tan \left(\alpha - \pi\right)}{\sin \left( \frac{\pi}{2}-\alpha\right)\cos \left(\alpha + \frac{\pi}{2}\right)\cos \left(\pi + \alpha\right)\tan \left(-\alpha + 5\pi\right)}$.
|
\frac{3}{5}
| 34.375 |
9,368 |
Inside a unit cube, eight equal spheres are placed. Each sphere is inscribed in one of the trihedral angles of the cube and touches the three spheres corresponding to the adjacent vertices of the cube. Find the radii of the spheres.
|
1/4
| 24.21875 |
9,369 |
Given the event "Randomly select a point P on the side CD of rectangle ABCD, such that the longest side of ΔAPB is AB", with a probability of 1/3, determine the ratio of AD to AB.
|
\frac{\sqrt{5}}{3}
| 19.53125 |
9,370 |
The length of the hypotenuse of a right-angled triangle is 2cm longer than one of its legs, and the other leg is 6cm long. Find the length of its hypotenuse.
|
10
| 97.65625 |
9,371 |
A couch cost 62,500 rubles. Its price was adjusted by 20% each month, either increasing or decreasing. Over six months, the price increased three times and decreased three times (the order of these changes is unknown). Can you uniquely determine the price of the couch after six months? If so, what was its price?
|
55296
| 69.53125 |
9,372 |
The ratio of the land area to the ocean area on the Earth's surface is 29:71. If three-quarters of the land is in the northern hemisphere, then calculate the ratio of the ocean area in the southern hemisphere to the ocean area in the northern hemisphere.
|
171:113
| 0 |
9,373 |
Calculate: \(\frac{2 \times 4.6 \times 9 + 4 \times 9.2 \times 18}{1 \times 2.3 \times 4.5 + 3 \times 6.9 \times 13.5} =\)
|
\frac{18}{7}
| 22.65625 |
9,374 |
Lucas wants to buy a book that costs $28.50. He has two $10 bills, five $1 bills, and six quarters in his wallet. What is the minimum number of nickels that must be in his wallet so he can afford the book?
|
40
| 85.9375 |
9,375 |
The mass of the first cast iron ball is $1462.5\%$ greater than the mass of the second ball. By what percentage less paint is needed to paint the second ball compared to the first ball? The volume of a ball with radius $R$ is $\frac{4}{3} \pi R^{3}$, and the surface area of a ball is $4 \pi R^{2}$.
|
84
| 38.28125 |
9,376 |
Let $S=1+ \frac{1}{ \sqrt {2}}+ \frac{1}{ \sqrt {3}}+...+ \frac{1}{ \sqrt {1000000}}$. Determine the integer part of $S$.
|
1998
| 92.96875 |
9,377 |
Consider a regular tetrahedron $ABCD$. Find $\sin \angle BAC$.
|
\frac{2\sqrt{2}}{3}
| 11.71875 |
9,378 |
Given the ellipse $\Gamma$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its right focus at $F(3,0)$, and its upper and lower vertices at $A$ and $B$ respectively. The line $AF$ intersects $\Gamma$ at another point $M$. If the line $BM$ intersects the $x$-axis at the point $N(12,0)$, find the eccentricity of $\Gamma$.
|
\frac{1}{2}
| 50 |
9,379 |
There are 35 groups of students, each group containing 3 students. Among these groups:
- 10 groups have only 1 boy.
- 19 groups have at least 2 boys.
- The number of groups with 3 boys is twice the number of groups with 3 girls.
How many boys are there?
|
60
| 70.3125 |
9,380 |
Find the integer that is closest to $2000\sum_{n=2}^{5000}\frac{1}{n^2-1}$.
|
1500
| 62.5 |
9,381 |
Twelve points are spaced around a $3 \times 3$ square at intervals of one unit. Two of the 12 points are chosen at random. Find the probability that the two points are one unit apart.
|
\frac{2}{11}
| 23.4375 |
9,382 |
Let $m$ be the product of all positive integers less than $5!$ which are invertible modulo $5!$. Find the remainder when $m$ is divided by $5!$.
|
119
| 2.34375 |
9,383 |
Determine all six-digit numbers \( p \) that satisfy the following properties:
(1) \( p, 2p, 3p, 4p, 5p, 6p \) are all six-digit numbers;
(2) Each of the six-digit numbers' digits is a permutation of \( p \)'s six digits.
|
142857
| 98.4375 |
9,384 |
Given an increasing sequence $\{a_n\}$ that satisfies $a_{n+1}a_{n-1} = a_n^2$ (for $n \geq 2$, $n \in \mathbb{N}$), the sum of the first 10 terms equals 50, and the sum of the first 15 terms is 210, calculate the sum of the first 5 terms ($S_5$).
|
10
| 41.40625 |
9,385 |
Given the sequence \\(\{a_n\}\) satisfies \\(a_{n+1}= \dfrac {2016a_n}{2014a_n+2016}(n\in N_+)\), and \\(a_1=1\), find \\(a_{2017}= \) ______.
|
\dfrac {1008}{1007\times 2017+1}
| 0 |
9,386 |
Evaluate the following expression: $$ 0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000 $$ The terms with minus signs are exactly the powers of two.
|
1996906
| 4.6875 |
9,387 |
Jarris the triangle is playing in the \((x, y)\) plane. Let his maximum \(y\) coordinate be \(k\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \(x\)-axis, find the minimum possible value of \(k\).
|
24/5
| 1.5625 |
9,388 |
A reconnaissance team has 12 soldiers, including 3 radio operators. The 12 soldiers are randomly divided into three groups, with group sizes of 3, 4, and 5 soldiers, respectively. What is the probability that each group has exactly 1 radio operator?
|
3/11
| 47.65625 |
9,389 |
Write $-\left(-3\right)-4+\left(-5\right)$ in the form of algebraic sum without parentheses.
|
3-4-5
| 27.34375 |
9,390 |
Given $l_{1}$: $ρ \sin (θ- \frac{π}{3})= \sqrt {3}$, $l_{2}$: $ \begin{cases} x=-t \\ y= \sqrt {3}t \end{cases}(t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, points $A$, $B$, and $C$ are on the ellipse $\frac{x^{2}}{4}+y^{2}=1$. $O$ is the coordinate origin, and $∠AOB=∠BOC=∠COA=120^{\circ}$, find the value of $\frac{1}{|OA|^{2}}+ \frac{1}{|OB|^{2}}+ \frac{1}{|OC|^{2}}$.
|
\frac{15}{8}
| 14.0625 |
9,391 |
Calculate the lengths of arcs of curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=2(t-\sin t) \\
y=2(1-\cos t)
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{2}
\end{aligned}
$$
|
8 - 4\sqrt{2}
| 90.625 |
9,392 |
The average density of pathogenic microbes in one cubic meter of air is 100. A sample of 2 cubic decimeters of air is taken. Find the probability that at least one microbe will be found in the sample.
|
0.181
| 0 |
9,393 |
What is the area of a square inscribed in a semicircle of radius 1, with one of its sides flush with the diameter of the semicircle?
|
\frac{4}{5}
| 82.03125 |
9,394 |
A positive integer divisor of $10!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
|
10
| 56.25 |
9,395 |
Add 78.621 to 34.0568 and round to the nearest thousandth.
|
112.678
| 88.28125 |
9,396 |
Given $f(x) = kx + \frac {2}{x^{3}} - 3$ $(k \in \mathbb{R})$, and it is known that $f(\ln 6) = 1$. Find $f\left(\ln \frac {1}{6}\right)$.
|
-7
| 89.84375 |
9,397 |
The decimal number corresponding to the binary number $111011001001_2$ is to be found.
|
3785
| 0.78125 |
9,398 |
How many paths are there from point $A$ to point $B$, if every step must be up or to the right in a grid where $A$ is at the bottom left corner and $B$ is at the top right corner of a 7x7 grid?
|
3432
| 100 |
9,399 |
Among the numbers $85_{(9)}$, $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is __________.
|
111111_{(2)}
| 94.53125 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.