Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
25,700 | Given in a cube ABCD-A1B1C1D1 with edge length 1, P is a moving point inside the cube (including the surface),
if $x + y + z = s$, and $0 \leq x \leq y \leq z \leq 1$, then the volume of the geometric body formed by all possible positions of point P is $\_\_\_\_\_\_\_\_\_\_$. | \frac{1}{6} | 79.6875 |
25,701 | Let ellipse M be defined by the equation $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ where $a>b>0$. The eccentricity of ellipse M and the eccentricity of the hyperbola defined by $x^{2}-y^{2}=1$ are reciprocals of each other, and ellipse M is inscribed in the circle defined by $x^{2}+y^{2}=4$.
(1) Find the equation of ellipse M;
(2) If the line $y= \sqrt {2}x+m$ intersects ellipse M at points A and B, and there is a point $P(1, \sqrt {2})$ on ellipse M, find the maximum area of triangle PAB. | \sqrt {2} | 0 |
25,702 | The digits from 1 to 9 are each used exactly once to write three one-digit integers and three two-digit integers. The one-digit integers are equal to the length, width and height of a rectangular prism. The two-digit integers are equal to the areas of the faces of the same prism. What is the surface area of the rectangular prism? | 198 | 0.78125 |
25,703 | A U-shaped number is a special type of three-digit number where the units digit and the hundreds digit are equal and greater than the tens digit. For example, 818 is a U-shaped number. How many U-shaped numbers are there? | 36 | 1.5625 |
25,704 | Given the function $f(x) = \begin{cases} \log_{10} x, & x > 0 \\ x^{-2}, & x < 0 \end{cases}$, if $f(x\_0) = 1$, find the value of $x\_0$. | 10 | 43.75 |
25,705 | In a factor tree, each value is the product of the two values below it, unless a value is a prime number or a preset integer product of primes. Using this structure, calculate the value of $X$ in the factor tree provided:
[asy]
draw((-2,-.3)--(0,0)--(2,-.3),linewidth(1));
draw((-3,-1.3)--(-2,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(2,-.8)--(3,-1.3),linewidth(1));
label("X",(0,0),N);
label("F",(-2,-.8),N);
label("7",(-3,-1.3),S);
label("G",(2,-.8),N);
label("4",(-1,-1.3),S);
label("11",(1,-1.3),S);
label("H",(3,-1.3),S);
draw((-2,-2.3)--(-1,-1.8)--(0,-2.3),linewidth(1));
draw((2,-2.3)--(3,-1.8)--(4,-2.3),linewidth(1));
label("7",(-2,-2.3),S);
label("2",(0,-2.3),S);
label("11",(2,-2.3),S);
label("2",(4,-2.3),S);
[/asy] | 6776 | 2.34375 |
25,706 | The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$ , that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$ . | 9/14 | 0.78125 |
25,707 | The number of elderly employees in a sample of 32 young employees from a workplace with a total of 430 employees, 160 of whom are young and the number of middle-aged employees is twice the number of elderly employees, can be found by determining the ratio of young employees in the population and the sample. | 18 | 9.375 |
25,708 | In the rhombus \(ABCD\), point \(Q\) divides side \(BC\) in the ratio \(1:3\) starting from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CF\) of triangle \(CEQ\) is equal to \(2\sqrt{2}\), and \(EQ = \sqrt{2}\). Find the radius of the circle inscribed in rhombus \(ABCD\). | \frac{\sqrt{7}}{2} | 0 |
25,709 | In $\triangle ABC$, the sides opposite to the three internal angles are $a$, $b$, and $c$, respectively. Given that $\cos A= \frac{ \sqrt {10}}{10}$ and $a\sin A+b\sin B-c\sin C= \frac{ 2 \sqrt {5}}{5}a\sin B$.
1. Find the value of $B$;
2. If $b=10$, find the area $S$ of $\triangle ABC$. | 60 | 37.5 |
25,710 | In the diagram, $ABCD$ and $EFGD$ are squares each with side lengths of 5 and 3 respectively, and $H$ is the midpoint of both $BC$ and $EF$. Calculate the total area of the polygon $ABHFGD$. | 25.5 | 9.375 |
25,711 | Define a new operation: \( x \odot y = 18 + x - a \times y \), where \( a \) is a constant. For example:
\[ 1 \odot 2 = 18 + 1 - a \times 2. \]
If \( 2 \odot 3 = 8 \), then what is \( 3 \odot 5 \) and \( 5 \odot 3 \)? | 11 | 95.3125 |
25,712 | Given the system of equations for the positive numbers \(x, y, z\):
$$
\left\{\begin{array}{l}
x^{2}+xy+y^{2}=108 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=124
\end{array}\right.
$$
Find the value of the expression \(xy + yz + xz\). | 48 | 87.5 |
25,713 | In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$, respectively. If the function $f(x)=\frac{1}{3} x^{3}+bx^{2}+(a^{2}+c^{2}-ac)x+1$ has no extreme points, then the maximum value of angle $B$ is \_\_\_\_\_ | \frac{\pi}{3} | 67.1875 |
25,714 | Two particles move along the edges of a square $ABCD$ with \[A \Rightarrow B \Rightarrow C \Rightarrow D \Rightarrow A,\] starting simultaneously and moving at the same speed. One starts at vertex $A$, and the other starts at the midpoint of side $CD$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of square $ABCD$?
A) $\frac{1}{16}$
B) $\frac{1}{12}$
C) $\frac{1}{9}$
D) $\frac{1}{6}$
E) $\frac{1}{4}$ | \frac{1}{4} | 79.6875 |
25,715 | In triangle \(ABC\), angle \(C\) equals \(30^\circ\), and angle \(A\) is acute. A line perpendicular to side \(BC\) is drawn, cutting off triangle \(CNM\) from triangle \(ABC\) (point \(N\) lies between vertices \(B\) and \(C\)). The areas of triangles \(CNM\) and \(ABC\) are in the ratio \(3:16\). Segment \(MN\) is half the height \(BH\) of triangle \(ABC\). Find the ratio \(AH:HC\). | 1/3 | 0 |
25,716 |
Vasya wrote consecutive natural numbers \( N \), \( N+1 \), \( N+2 \), and \( N+3 \) in rectangles. Under each rectangle, he wrote the sum of the digits of the corresponding number in a circle.
The sum of the numbers in the first two circles turned out to be 200, and the sum of the numbers in the third and fourth circles turned out to be 105. What is the sum of the numbers in the second and third circles? | 103 | 1.5625 |
25,717 | Given $f(x) = -4x^2 + 4ax - 4a - a^2$ has a maximum value of $-5$ in the interval $[0, 1]$, find the value of $a$. | -5 | 3.90625 |
25,718 | In a regular tetrahedron \( P-ABCD \), where each face is an equilateral triangle with side length 1, points \( M \) and \( N \) are the midpoints of edges \( AB \) and \( BC \), respectively. Find the distance between the skew lines \( MN \) and \( PC \). | \frac{\sqrt{2}}{4} | 15.625 |
25,719 | Find the inclination angle of the line $\sqrt {2}x+ \sqrt {6}y+1=0$. | \frac{5\pi}{6} | 61.71875 |
25,720 | Let $a$ and $b$ be positive real numbers such that $a + 3b = 2.$ Find the minimum value of
\[\frac{2}{a} + \frac{4}{b}.\] | 14 | 0.78125 |
25,721 | Given that $AC$ and $CE$ are two diagonals of a regular hexagon $ABCDEF$, and points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. If points $B$, $M$, and $N$ are collinear, find the value of $r$. | \frac{1}{\sqrt{3}} | 0 |
25,722 | How many lattice points lie on the hyperbola \( x^2 - y^2 = 1800^2 \)? | 150 | 1.5625 |
25,723 | Given a biased coin with probabilities of $\frac{3}{4}$ for heads and $\frac{1}{4}$ for tails, and outcomes of tosses being independent, calculate the probabilities of winning Game A and Game B. | \frac{1}{4} | 1.5625 |
25,724 | Given \(\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt{3}\), representing a cube root of unity, specifically \(\omega = e^{2\pi i / 3}\). Let \(T\) denote all points in the complex plane of the form \(a + b\omega + c\omega^2\), where \(0 \leq a \leq 2\), \(0 \leq b \leq 1\), and \(0 \leq c \leq 1\). Determine the area of \(T\). | 2\sqrt{3} | 17.1875 |
25,725 | 29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once with each partner). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned? | 29 | 5.46875 |
25,726 | A cube has six faces, and each face has two diagonals. From these diagonals, choose two to form a pair. Among these pairs, how many form an angle of $60^\circ$? | 48 | 1.5625 |
25,727 | Given that there is a geometric sequence $\{a_n\}$ with a common ratio $q > 1$ and the sum of the first $n$ terms is $S_n$, $S_3 = 7$, the sequence $a_1+3$, $3a_2$, $a_3+4$ forms an arithmetic sequence. The sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n$, and $6T_n = (3n+1)b_n + 2$ for $n \in \mathbb{N}^*$.
(1) Find the general term formula for the sequence $\{a_n\}$.
(2) Find the general term formula for the sequence $\{b_n\}$.
(3) Let $A = \{a_1, a_2, \ldots, a_{10}\}$, $B = \{b_1, b_2, \ldots, b_{40}\}$, and $C = A \cup B$. Calculate the sum of all elements in the set $C$. | 3318 | 2.34375 |
25,728 | The equations $x^3 + Cx + 20 = 0$ and $x^3 + Dx^2 + 100 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$ | 15 | 19.53125 |
25,729 | In an infinite increasing sequence of natural numbers, each number is divisible by at least one of the numbers 1005 and 1006, but none is divisible by 97. Additionally, any two consecutive numbers differ by no more than $k$. What is the smallest possible $k$ for this scenario? | 2011 | 0.78125 |
25,730 | Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity $e = \frac{\sqrt{3}}{3}$. The left and right foci are $F_1$ and $F_2$, respectively, with $F_2$ coinciding with the focus of the parabola $y^2 = 4x$.
(I) Find the standard equation of the ellipse;
(II) If a line passing through $F_1$ intersects the ellipse at points $B$ and $D$, and another line passing through $F_2$ intersects the ellipse at points $A$ and $C$, with $AC \perp BD$, find the minimum value of $|AC| + |BD|$. | \frac{16\sqrt{3}}{5} | 1.5625 |
25,731 | Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$ | 227.052227052227 | 0 |
25,732 | In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are located on sides $YZ$, $XZ$, and $XY$, respectively. The cevians $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$. Given that $\frac{XP}{PX'}+\frac{YP}{PY'}+\frac{ZP}{PZ'}=100$, find the value of $\frac{XP}{PX'} \cdot \frac{YP}{PY'} \cdot \frac{ZP}{PZ'}$. | 98 | 3.90625 |
25,733 | In the isosceles triangle \(ABC\) with the sides \(AB = BC\), the angle \(\angle ABC\) is \(80^\circ\). Inside the triangle, a point \(O\) is taken such that \(\angle OAC = 10^\circ\) and \(\angle OCA = 30^\circ\). Find the angle \(\angle AOB\). | 70 | 38.28125 |
25,734 | An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$ , $Q\in\overline{AC}$ , and $N,P\in\overline{BC}$ .
Suppose that $ABC$ is an equilateral triangle of side length $2$ , and that $AMNPQ$ has a line of symmetry perpendicular to $BC$ . Then the area of $AMNPQ$ is $n-p\sqrt{q}$ , where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n+10p+q$ .
*Proposed by Michael Ren* | 5073 | 0 |
25,735 | The volume of the solid generated by rotating the circle $x^2 + (y + 1)^2 = 3$ around the line $y = kx - 1$ for one complete revolution is what? | 4\sqrt{3}\pi | 24.21875 |
25,736 | Let the set \( I = \{1, 2, \cdots, n\} (n \geqslant 3) \). If two non-empty proper subsets \( A \) and \( B \) of \( I \) satisfy \( A \cap B = \varnothing \) and \( A \cup B = I \), then \( A \) and \( B \) are called a partition of \( I \). If for any partition \( A \) and \( B \) of the set \( I \), there exist two numbers in \( A \) or \( B \) such that their sum is a perfect square, then \( n \) must be at least \(\qquad\). | 15 | 69.53125 |
25,737 | Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $EFGH$, has its sides parallel to the corresponding sides of the larger square, and each side of $EFGH$ is tangent to one of the semicircles. Provide the area of square $EFGH$.
A) $36$
B) $64$
C) $100$
D) $144$
E) $256$ | 144 | 1.5625 |
25,738 | $10 \cdot 52 \quad 1990-1980+1970-1960+\cdots-20+10$ equals: | 1000 | 27.34375 |
25,739 | A grid consists of multiple squares, as shown below. Count the different squares possible using the lines in the grid.
[asy]
unitsize(0.5 cm);
int i, j;
for(i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
}
for(i = 1; i <= 4; ++i) {
for (j = 1; j <= 4; ++j) {
if ((i > 1 && i < 4) || (j > 1 && j < 4)) continue;
draw((i,j)--(i,j+1)--(i+1,j+1)--(i+1,j)--cycle);
}
}
[/asy] | 54 | 0.78125 |
25,740 | A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | 2 + \sqrt{2} | 3.125 |
25,741 | $ABCD$ is a regular tetrahedron. Let $N$ be the midpoint of $\overline{AB}$, and let $G$ be the centroid of triangle $ACD$. What is $\cos \angle BNG$? | \frac{1}{3} | 21.09375 |
25,742 | An artist arranges 1000 dots evenly around a circle, with each dot being either red or blue. A critic counts faults: each pair of adjacent red dots counts as one fault, and each pair of blue dots exactly two apart (separated by one dot) counts as another fault. What is the smallest number of faults the critic could find? | 250 | 3.125 |
25,743 | Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sqrt{3}a\cos C=c\sin A$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $a > 2$ and $b-c=1$, find the minimum perimeter of triangle $\triangle ABC$. | 9 + 6\sqrt{2} | 0 |
25,744 | For a natural number $b$ , let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$ ? | 240 | 74.21875 |
25,745 | Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is to the left, right, up, or down, all four equally likely. Let $q$ be the probability that the object reaches $(3,3)$ in eight or fewer steps. Write $q$ in the form $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b.$ | 4151 | 1.5625 |
25,746 | Given the product \( S = \left(1+2^{-\frac{1}{32}}\right)\left(1+2^{-\frac{1}{16}}\right)\left(1+2^{-\frac{1}{8}}\right)\left(1+2^{-\frac{1}{4}}\right)\left(1+2^{-\frac{1}{2}}\right) \), calculate the value of \( S \). | \frac{1}{2}\left(1 - 2^{-\frac{1}{32}}\right)^{-1} | 0 |
25,747 | A capacitor with a capacitance of $C_{1} = 20 \mu$F is charged to a voltage $U_{1} = 20$ V. A second capacitor with a capacitance of $C_{2} = 5 \mu$F is charged to a voltage $U_{2} = 5$ V. The capacitors are connected with opposite-charged plates. Determine the voltage that will be established across the plates. | 15 | 48.4375 |
25,748 | Let \( x \) and \( y \) be non-zero real numbers such that
\[ \frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9 \pi}{20}. \]
(1) Find the value of \(\frac{y}{x}\).
(2) In triangle \( \triangle ABC \), if \( \tan C = \frac{y}{x} \), find the maximum value of \( \sin 2A + 2 \cos B \). | \frac{3}{2} | 0.78125 |
25,749 | Given the function $f(x) = 2\sin^2\left(\frac{\pi}{4} + x\right) - \sqrt{3}\cos{2x} - 1$, where $x \in \mathbb{R}$:
1. If the graph of function $h(x) = f(x + t)$ is symmetric about the point $\left(-\frac{\pi}{6}, 0\right)$, and $t \in \left(0, \frac{\pi}{2}\right)$, find the value of $t$.
2. In an acute triangle $ABC$, if angle $A$ satisfies $h(A) = 1$, find the range of $(\sqrt{3} - 1)\sin{B} + \sqrt{2}\sin{C}$. | \frac{\pi}{3} | 0.78125 |
25,750 | Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009. | 42 | 48.4375 |
25,751 | In a debate competition with four students participating, the rules are as follows: Each student must choose one question to answer from two given topics, Topic A and Topic B. For Topic A, answering correctly yields 100 points and answering incorrectly results in a loss of 100 points. For Topic B, answering correctly yields 90 points and answering incorrectly results in a loss of 90 points. If the total score of the four students is 0 points, how many different scoring situations are there? | 36 | 34.375 |
25,752 | The cells of a $50 \times 50$ table are colored in $n$ colors such that for any cell, the union of its row and column contains cells of all $n$ colors. Find the maximum possible number of blue cells if
(a) $n=2$
(b) $n=25$. | 1300 | 0.78125 |
25,753 | How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? | 240 | 2.34375 |
25,754 | Let \[P(x) = (3x^5 - 45x^4 + gx^3 + hx^2 + ix + j)(4x^3 - 60x^2 + kx + l),\] where $g, h, i, j, k, l$ are real numbers. Suppose that the set of all complex roots of $P(x)$ includes $\{1, 2, 3, 4, 5, 6\}$. Find $P(7)$. | 51840 | 1.5625 |
25,755 | A cube is inscribed in a regular octahedron in such a way that its vertices lie on the edges of the octahedron. By what factor is the surface area of the octahedron greater than the surface area of the inscribed cube? | \frac{2\sqrt{3}}{3} | 38.28125 |
25,756 | Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a>0$ and $b>0$. If the point $F_2$ is symmetric with respect to the asymptote line and lies on the hyperbola, calculate the eccentricity of the hyperbola. | \sqrt{5} | 43.75 |
25,757 | For all positive integers $m>10^{2022}$ , determine the maximum number of real solutions $x>0$ of the equation $mx=\lfloor x^{11/10}\rfloor$ . | 10 | 6.25 |
25,758 | Multiply the first eight positive composite integers, add the first prime number to this product, then divide by the product of the next eight positive composite integers after adding the second prime number to it. Express your answer as a common fraction. | \frac{4 \cdot 6 \cdot 8 \cdot 9 \cdot 10 \cdot 12 \cdot 14 \cdot 15 + 2}{16 \cdot 18 \cdot 20 \cdot 21 \cdot 22 \cdot 24 \cdot 25 \cdot 26 + 3} | 0 |
25,759 | Given \(\triangle DEF\), where \(DE=28\), \(EF=30\), and \(FD=16\), calculate the area of \(\triangle DEF\). | 221.25 | 0 |
25,760 | From the numbers 0, 1, 2, 3, 4, select three different digits to form a three-digit number. What is the sum of the units digit of all these three-digit numbers? | 90 | 85.15625 |
25,761 | The numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\) have their last three digits discarded. How many of the first terms in the resulting sequence form an arithmetic progression? | 32 | 3.125 |
25,762 | Compute $\tan\left(\frac{\pi}{9}\right)\tan\left(\frac{2\pi}{9}\right)\tan\left(\frac{4\pi}{9}\right)$. | \frac{1}{3} | 0 |
25,763 | How many values of $x$, $-10 < x < 50$, satisfy $\cos^2 x + 3\sin^2 x = 1.5?$ (Note: $x$ is measured in radians.) | 18 | 0.78125 |
25,764 | Remove five out of twelve digits so that the remaining numbers sum up to 1111.
$$
\begin{array}{r}
111 \\
333 \\
+\quad 777 \\
999 \\
\hline 1111
\end{array}
$$ | 1111 | 7.03125 |
25,765 | Add together the numbers $7.56$ and $4.29$. Write the result as a decimal. | 11.85 | 94.53125 |
25,766 | Among the following propositions, the correct ones are __________.
(1) The regression line $\hat{y}=\hat{b}x+\hat{a}$ always passes through the center of the sample points $(\bar{x}, \bar{y})$, and at least through one sample point;
(2) After adding the same constant to each data point in a set of data, the variance remains unchanged;
(3) The correlation index $R^{2}$ is used to describe the regression effect; it represents the contribution rate of the forecast variable to the change in the explanatory variable, the closer to $1$, the better the model fits;
(4) If the observed value $K$ of the random variable $K^{2}$ for categorical variables $X$ and $Y$ is larger, then the credibility of "$X$ is related to $Y$" is smaller;
(5) For the independent variable $x$ and the dependent variable $y$, when the value of $x$ is certain, the value of $y$ has certain randomness, the non-deterministic relationship between $x$ and $y$ is called a function relationship;
(6) In the residual plot, if the residual points are relatively evenly distributed in a horizontal band area, it indicates that the chosen model is relatively appropriate;
(7) Among two models, the one with the smaller sum of squared residuals has a better fitting effect. | (2)(6)(7) | 0 |
25,767 | Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}-a_{n}=\left(-1\right)^{n+1}\frac{1}{n(n+2)}$, calculate the sum of the first 40 terms of the sequence $\{\left(-1\right)^{n}a_{n}\}$. | \frac{20}{41} | 1.5625 |
25,768 | What is the least integer whose square is 75 more than its double? | -8 | 17.96875 |
25,769 | Given that $| \vec{e} | = 1$ and it satisfies $| \vec{a} + \vec{e} | = | \vec{a} - 2\vec{e} |$, find the projection of vector $\vec{a}$ in the direction of $\vec{e}$. | \frac{1}{2} | 22.65625 |
25,770 | What is $\frac{1}{(-5^2)^3} \cdot (-5)^8 \cdot \sqrt{5}$? | 5^{5/2} | 0 |
25,771 | Two students, A and B, are playing table tennis. They have agreed on the following rules: ① Each point won earns 1 point; ② They use a three-point serve system, meaning they switch serving every three points. Assuming that when A serves, the probability of A winning a point is $\frac{3}{5}$, and when B serves, the probability of A winning a point is $\frac{1}{2}$, and the outcomes of each point are independent. According to the draw result, A serves first.
$(1)$ Let $X$ represent the score of A after three points. Find the distribution table and mean of $X$;
$(2)$ Find the probability that A has more points than B after six points. | \frac{441}{1000} | 0.78125 |
25,772 | Given that spinner A contains the numbers 4, 5, 6, spinner B contains the numbers 1, 2, 3, 4, 5, and spinner C can take numbers from the set 7, 8, 9 if spinner B lands on an odd number and the set {6, 8} if spinner B lands on an even number, find the probability that the sum of the numbers resulting from the rotation of spinners A, B, and C is an odd number. | \frac{4}{15} | 4.6875 |
25,773 | Find the number of integers $n$ with $1\le n\le 100$ for which $n-\phi(n)$ is prime. Here $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$ .
*Proposed by Mehtaab Sawhney* | 13 | 0.78125 |
25,774 | Given that the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\cdot b^{t}$, where $a$ and $b$ are positive constants, and the decomposition rate is $5\%$ after $6$ months and $10\%$ after $12$ months, calculate the time it takes for this type of garbage to completely decompose. | 32 | 1.5625 |
25,775 | Each of 100 students sends messages to 50 different students. What is the least number of pairs of students who send messages to each other? | 50 | 10.15625 |
25,776 | 8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder. If the recorder is seated between the leader and vice leader, how many different seating arrangements are possible (considering that arrangements that can be obtained by rotation are identical)? | 240 | 55.46875 |
25,777 | Rectangle PQRS and right triangle SRT share side SR and have the same area. Rectangle PQRS has dimensions PQ = 4 and PS = 8. Find the length of side RT. | 16 | 10.9375 |
25,778 | 10 times 0.1 equals to ____, 10 times 0.01 equals to ____, 10 times 0.001 equals to ____. | 0.01 | 11.71875 |
25,779 | In the sequence $\{a_n\}$, $a_1 = 1$, $a_2 = 2$, $a_{n+2}$ is equal to the remainder of $a_n + a_{n+1}$ divided by 3. Find the sum of the first 89 terms of $\{a_n\}$. | 100 | 19.53125 |
25,780 | Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \ldots + k^2$ is a multiple of $360$. | 175 | 0 |
25,781 | From the natural numbers 1 to 2008, the maximum number of numbers that can be selected such that the sum of any two selected numbers is not divisible by 3 is ____. | 671 | 36.71875 |
25,782 | A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points? | 1979 | 0.78125 |
25,783 | Find the positive integer $N$, such that numbers $N$ and $N^2$ end in the same sequence of four digits $abcd$ where $a$ is not zero, under the modulus $8000$. | 625 | 0 |
25,784 | Given the function $f\left(x\right)=x^{2}-2bx+3$, where $b\in R$.
$(1)$ Find the solution set of the inequality $f\left(x\right) \lt 4-b^{2}$.
$(2)$ When $x\in \left[-1,2\right]$, the function $y=f\left(x\right)$ has a minimum value of $1$. Find the maximum value of the function $y=f\left(x\right)$ when $x\in \left[-1,2\right]$. | 4 + 2\sqrt{2} | 15.625 |
25,785 | Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and no two adjacent vertices can have the same color, nor can the vertices at the ends of each diagonal. Calculate the total number of different colorings possible. | 5040 | 0 |
25,786 | Find the value of $k$ for the ellipse $\frac{x^2}{k+8} + \frac{y^2}{9} = 1$ with an eccentricity of $\frac{1}{2}$. | -\frac{5}{4} | 13.28125 |
25,787 | Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:
- The sum of the fractions is equal to $2$ .
- The sum of the numerators of the fractions is equal to $1000$ .
In how many ways can Pedro do this?
| 200 | 8.59375 |
25,788 | Let $U$ be a positive integer whose only digits are 0s and 1s. If $Y = U \div 18$ and $Y$ is an integer, what is the smallest possible value of $Y$? | 61728395 | 50.78125 |
25,789 | A right circular cone is sliced into three pieces by planes parallel to its base, each piece having equal height. The pieces are labeled from top to bottom; hence the smallest piece is at the top and the largest at the bottom. Calculate the ratio of the volume of the smallest piece to the volume of the largest piece. | \frac{1}{27} | 3.125 |
25,790 | Given a sequence $\{a\_n\}$ with its first $n$ terms sum $S\_n$, where $a\_1=1$ and $3S\_n = a_{n+1} - 1$.
1. Find the general formula for the sequence $\{a\_n\}$.
2. Consider an arithmetic sequence $\{b\_n\}$ with its first $n$ terms sum $T\_n$, where $a\_2 = b\_2$ and $T\_4 = 1 + S\_3$. Find the value of $\frac{1}{b\_1 \cdot b\_2} + \frac{1}{b\_2 \cdot b\_3} + \dots + \frac{1}{b_{10}b_{11}}$. | \frac{10}{31} | 91.40625 |
25,791 | The maximum and minimum values of the function $y=2x^{3}-3x^{2}-12x+5$ on the interval $[0,3]$ need to be determined. | -15 | 87.5 |
25,792 | In triangle \(ABC\), point \(O\) is the center of the circumcircle, and point \(L\) is the midpoint of side \(AB\). The circumcircle of triangle \(ALO\) intersects the line \(AC\) at point \(K\). Find the area of triangle \(ABC\) if \(\angle LOA = 45^\circ\), \(LK = 8\), and \(AK = 7\). | 112 | 0 |
25,793 | The sides of triangle $DEF$ are in the ratio of $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. What is the length, in inches, of the longer subsegment of side $DE$ if the length of side $DE$ is $12$ inches? Express your answer as a common fraction. | \frac{48}{7} | 3.125 |
25,794 | On a table, there are 20 cards numbered from 1 to 20. Each time, Xiao Ming picks out 2 cards such that the number on one card is 2 more than twice the number on the other card. What is the maximum number of cards Xiao Ming can pick? | 12 | 13.28125 |
25,795 | Let $(a_1,a_2,\ldots, a_{13})$ be a permutation of $(1, 2, \ldots, 13)$ . Ayvak takes this permutation and makes a series of *moves*, each of which consists of choosing an integer $i$ from $1$ to $12$ , inclusive, and swapping the positions of $a_i$ and $a_{i+1}$ . Define the *weight* of a permutation to be the minimum number of moves Ayvak needs to turn it into $(1, 2, \ldots, 13)$ .
The arithmetic mean of the weights of all permutations $(a_1, \ldots, a_{13})$ of $(1, 2, \ldots, 13)$ for which $a_5 = 9$ is $\frac{m}{n}$ , for coprime positive integers $m$ and $n$ . Find $100m+n$ .
*Proposed by Alex Gu* | 13703 | 0 |
25,796 | Points $A=(8,15)$ and $B=(14,9)$ lie on circle $\omega$ in the plane. Suppose the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the x-axis. Find the area of $\omega$. | 306\pi | 3.125 |
25,797 | A box contains 4 labels marked with the numbers $1$, $2$, $3$, and $4$. Two labels are randomly selected according to the following conditions. Find the probability that the numbers on the two labels are consecutive integers:
1. The selection is made without replacement;
2. The selection is made with replacement. | \frac{3}{16} | 0.78125 |
25,798 | Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | 15 | 16.40625 |
25,799 | Calculate $1010101_2 + 110011_2$ and express your answer in base $10$. | 136 | 57.8125 |
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