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40.3k
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stringlengths 10
5.15k
| ground_truth
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float64 0
100
|
|---|---|---|---|
9,900 |
In the unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, points $E, F, G$ are the midpoints of edges $A A_{1}, C_{1} D_{1}$, and $D_{1} A_{1}$, respectively. Find the distance from point $B_{1}$ to the plane $E F G$.
|
\frac{\sqrt{3}}{2}
| 71.09375 |
9,901 |
Determine the number of three-element subsets of the set \(\{1, 2, 3, 4, \ldots, 120\}\) for which the sum of the three elements is a multiple of 3.
|
93640
| 52.34375 |
9,902 |
Given the function $f(x)=x^{3}+ax^{2}+bx+a^{2}-1$ has an extremum of $9$ at $x=1$, find the value of $f(2)$.
|
17
| 39.84375 |
9,903 |
Given the sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) that satisfies \( a_{1}=a_{2}=1, a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \). Furthermore, it is given that \( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( a_{1} + a_{2} + \cdots + a_{100} \).
|
200
| 54.6875 |
9,904 |
It is known that there are four different venues $A$, $B$, $C$, $D$ at the Flower Expo. Person A and person B each choose 2 venues to visit. The probability that exactly one venue is the same in their choices is ____.
|
\frac{2}{3}
| 91.40625 |
9,905 |
From the numbers \(1,2, \cdots, 14\), select \(a_{1}, a_{2}, a_{3}\) in ascending order such that \(a_{2} - a_{1} \geq 3\) and \(a_{3} - a_{2} \geq 3\). How many different ways are there to select the numbers satisfying these conditions?
|
120
| 56.25 |
9,906 |
The quadratic polynomial \( f(x) = a x^{2} + b x + c \) has exactly one root, and the quadratic polynomial \( f(3x + 2) - 2f(2x - 1) \) also has exactly one root. Find the root of the polynomial \( f(x) \).
|
-7
| 50.78125 |
9,907 |
\[
\frac{\sin ^{2}\left(135^{\circ}-\alpha\right)-\sin ^{2}\left(210^{\circ}-\alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-2 \alpha\right)}{\cos ^{2}\left(225^{\circ}+\alpha\right)-\cos ^{2}\left(210^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)}=-1
\]
|
-1
| 67.96875 |
9,908 |
Given that $n\in N^{*}$, select $k(k\in N, k\geqslant 2)$ numbers $j\_1$, $j\_2$, $...$, $j\_k$ from the set ${1,2,3,...,n}$ such that they simultaneously satisfy the following two conditions: $①1\leqslant j\_1 < j\_2 < ...j\_k\leqslant n$; $②j_{i+1}-j_{i}\geqslant m(i=1,2,…,k-1)$. Then the array $(j\_1, j\_2, ..., j\_k)$ is called a combination of selecting $k$ elements with a minimum distance of $m$ from $n$ elements, denoted as $C_{ n }^{ (k,m) }$. For example, from the set ${1,2,3}$, we have $C_{ 3 }^{ (2,1) }=3$. For the given set ${1,2,3,4,5,6,7}$, find $C_{ 7 }^{ (3,2) }=$ \_\_\_\_\_\_.
|
10
| 94.53125 |
9,909 |
In a school cafeteria line, there are 16 students alternating between boys and girls (starting with a boy, followed by a girl, then a boy, and so on). Any boy, followed immediately by a girl, can swap places with her. After some time, all the girls end up at the beginning of the line and all the boys are at the end. How many swaps were made?
|
36
| 16.40625 |
9,910 |
Given that $f(x)$ is an even function defined on $\mathbb{R}$, and $g(x)$ is an odd function defined on $\mathbb{R}$ that passes through the point $(-1, 3)$ and $g(x) = f(x-1)$, find the value of $f(2007) + f(2008)$.
|
-3
| 24.21875 |
9,911 |
Points $A_{1}$ and $C_{1}$ are located on the sides $BC$ and $AB$ of triangle $ABC$. Segments $AA_{1}$ and $CC_{1}$ intersect at point $M$.
In what ratio does line $BM$ divide side $AC$, if $AC_{1}: C_{1}B = 2: 3$ and $BA_{1}: A_{1}C = 1: 2$?
|
1:3
| 32.8125 |
9,912 |
A 20-quart container is fully filled with water. Five quarts are removed and replaced with pure antifreeze liquid. Then, five quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated three more times (for a total of five times). Determine the fractional part of the final mixture that is water.
|
\frac{243}{1024}
| 78.90625 |
9,913 |
A company needs to transport two types of products, $A$ and $B$, with the following volumes and masses per unit as shown in the table:
| | Volume $(m^{3}/$unit) | Mass (tons$/$unit) |
|----------|-----------------------|--------------------|
| $A$ type | $0.8$ | $0.5$ |
| $B$ type | $2$ | $1$ |
1. Given a batch of products containing both $A$ and $B$ types with a total volume of $20m^{3}$ and a total mass of $10.5$ tons, find the number of units for each type of product.
2. A logistics company has trucks with a rated load of $3.5$ tons and a capacity of $6m^{3}$. The company offers two payment options:
- Charging per truck: $600$ yuan per truck to transport goods to the destination.
- Charging per ton: $200$ yuan per ton to transport goods to the destination.
Determine how the company should choose to transport the products from part (1) in one or multiple shipments to minimize the shipping cost, and calculate the cost under the chosen method.
|
2100
| 19.53125 |
9,914 |
If the product of 6 consecutive odd numbers is 135135, what is the sum of these 6 numbers? $\qquad$
|
48
| 13.28125 |
9,915 |
Find the maximum value of the function
$$
f(x) = \sqrt{3} \sin 2x + 2 \sin x + 4 \sqrt{3} \cos x.
$$
|
\frac{17}{2}
| 0 |
9,916 |
Given a $24$-inch by $30$-inch pan of brownies, cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of pieces of brownie the pan contains.
|
60
| 40.625 |
9,917 |
Find the smallest six-digit number that is divisible by 3, 7, and 13 without a remainder.
|
100191
| 45.3125 |
9,918 |
There are 1000 rooms in a row along a long corridor. Initially, the first room contains 1000 people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?
|
61
| 60.9375 |
9,919 |
A digit is inserted between the digits of a two-digit number to form a three-digit number. Some two-digit numbers, when a certain digit is inserted in between, become three-digit numbers that are $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$?
|
19
| 32.8125 |
9,920 |
In an isosceles triangle \(ABC\), the angle at the vertex \(B\) is \(20^\circ\). Points \(D\) and \(K\) are taken on the sides \(AB\) and \(BC\) respectively such that \(\angle KAC = 50^\circ\) and \(\angle DCA = 60^\circ\). Calculate \(\angle CDK\).
|
30
| 61.71875 |
9,921 |
In the expansion of $((x^2+1)^2(x-1)^6)$, find the coefficient of the $x^3$ term.
|
-32
| 32.8125 |
9,922 |
Kolya was supposed to square a certain natural number for his homework. Instead, he mistakenly doubled the number and got a two-digit number, written with the same digits as the square of the number but in reverse order. What should be the correct answer?
|
81
| 0 |
9,923 |
Evaluate \( \frac{18}{4.9 \times 106} \).
|
\frac{18}{519.4}
| 0 |
9,924 |
A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01?
|
11475
| 10.9375 |
9,925 |
Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are eliminated, and the other team wins. Determine the total number of possible sequences of matches.
|
3432
| 51.5625 |
9,926 |
Given \( z \in \mathbf{C} \) and \( z^{7} = 1 \) (where \( z \neq 1 \)), find the value of \( \cos \alpha + \cos 2 \alpha + \cos 4 \alpha \), where \(\alpha\) is the argument of \(z\).
|
-\frac{1}{2}
| 93.75 |
9,927 |
Determine the value of the following sum:
$$
\log _{3}\left(1-\frac{1}{15}\right)+\log _{3}\left(1-\frac{1}{14}\right)+\log _{3}\left(1-\frac{1}{13}\right)+\cdots+\log _{3}\left(1-\frac{1}{8}\right)+\log _{3}\left(1-\frac{1}{7}\right)+\log _{3}\left(1-\frac{1}{6}\right)
$$
(Note that the sum includes a total of 10 terms.)
|
-1
| 92.96875 |
9,928 |
Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$, tangent to circle $C$ at point $R$, and a different secant from $Q$ intersects $C$ at points $D$ and $E$ such that $QD < QE$. If $QD = 5$ and the length of the tangent from $Q$ to $R$ ($QR$) is equal to $DE - QD$, calculate $QE$.
|
\frac{15 + 5\sqrt{5}}{2}
| 1.5625 |
9,929 |
In bag A, there are 3 white balls and 2 red balls, while in bag B, there are 2 white balls and 4 red balls. If a bag is randomly chosen first, and then 2 balls are randomly drawn from that bag, the probability that the second ball drawn is white given that the first ball drawn is red is ______.
|
\frac{17}{32}
| 62.5 |
9,930 |
Given a finite sequence \(P = \left(p_{1}, p_{2}, \cdots, p_{n}\right)\), the Caesar sum (named after a mathematician Caesar) is defined as \(\frac{s_{1}+s_{2}+\cdots+s_{n}}{n}\), where \(s_{k} = p_{1} + p_{2} + \cdots + p_{k}\) for \(1 \leq k \leq n\). If a sequence of 99 terms \(\left(p_{1}, p_{2}, \cdots, p_{99}\right)\) has a Caesar sum of 1000, determine the Caesar sum of the 100-term sequence \(\left(1, p_{1}, p_{2}, \cdots, p_{99}\right)\).
|
991
| 85.9375 |
9,931 |
First, a number \( a \) is randomly selected from the set \(\{1,2,3, \cdots, 99,100\}\), then a number \( b \) is randomly selected from the same set. Calculate the probability that the last digit of \(3^{a} + 7^{b}\) is 8.
|
\frac{3}{16}
| 60.15625 |
9,932 |
Given the imaginary number \( z \) satisfies \( z^3 + 1 = 0 \), \( z \neq -1 \). Then \( \left( \frac{z}{z-1} \right)^{2018} + \left( \frac{1}{z-1} \right)^{2018} = \) .
|
-1
| 60.9375 |
9,933 |
There are several soldiers forming a rectangular formation with exactly eight columns. If adding 120 people or removing 120 people from the formation can both form a square formation, how many soldiers are there in the original rectangular formation?
|
136
| 3.125 |
9,934 |
A sphere is inscribed in a cone whose axial cross-section is an equilateral triangle. Find the volume of the cone if the volume of the sphere is \( \frac{32\pi}{3} \ \text{cm}^3 \).
|
24 \pi
| 18.75 |
9,935 |
Let $n = 2^{35}3^{17}$. How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$?
|
594
| 61.71875 |
9,936 |
A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle?
|
\frac{3\sqrt{7}}{4}
| 75.78125 |
9,937 |
Given $f(x)= \frac{1}{2^{x}+ \sqrt {2}}$, use the method for deriving the sum of the first $n$ terms of an arithmetic sequence to find the value of $f(-5)+f(-4)+…+f(0)+…+f(5)+f(6)$.
|
3 \sqrt {2}
| 0 |
9,938 |
In an isosceles trapezoid, the bases are 40 and 24, and its diagonals are mutually perpendicular. Find the area of the trapezoid.
|
1024
| 5.46875 |
9,939 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $(2c-a)\cos B=b\cos A$.
(1) Find angle $B$;
(2) If $b=6$ and $c=2a$, find the area of $\triangle ABC$.
|
6 \sqrt{3}
| 91.40625 |
9,940 |
The vertical drops of six roller coasters at Fractal Fun Park are listed in the table below:
\begin{tabular}{|l|c|}
\hline
The Spiral Slide & 173 feet \\
\hline
The Velocity Vortex & 125 feet \\
\hline
The Cyclone & 150 feet \\
\hline
The Apex Jump & 310 feet \\
\hline
The Quantum Leap & 205 feet \\
\hline
The Zero Gravity & 180 feet \\
\hline
\end{tabular}
What is the positive difference between the mean and the median of these values?
|
14
| 5.46875 |
9,941 |
A prime number of the form \( 2^{p} - 1 \) is called a Mersenne prime, where \( p \) is also a prime number. To date, the largest known Mersenne prime is \( 2^{82589933} - 1 \). Find the last two digits of this number.
|
91
| 58.59375 |
9,942 |
In a right triangle where the ratio of the legs is 1:3, a perpendicular is dropped from the vertex of the right angle to the hypotenuse. Find the ratio of the segments created on the hypotenuse by this perpendicular.
|
1:9
| 42.96875 |
9,943 |
Find a six-digit number $\overline{xy243z}$ that is divisible by 396.
|
432432
| 4.6875 |
9,944 |
The sum of an infinite geometric series is \( 16 \) times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio?
|
-\frac{1}{4}
| 30.46875 |
9,945 |
In the diagram, \( P Q = 19 \), \( Q R = 18 \), and \( P R = 17 \). Point \( S \) is on \( P Q \), point \( T \) is on \( P R \), and point \( U \) is on \( S T \) such that \( Q S = S U \) and \( U T = T R \). The perimeter of \(\triangle P S T\) is equal to:
|
36
| 49.21875 |
9,946 |
Given that $α$ and $β$ are acute angles, and $\cos(α+β)=\frac{3}{5}$, $\sin α=\frac{5}{13}$, find the value of $\cos β$.
|
\frac{56}{65}
| 64.0625 |
9,947 |
Determine the degree of the polynomial resulting from $(5x^6 - 4x^5 + x^2 - 18)(2x^{12} + 6x^9 - 11x^6 + 10) - (x^3 + 4)^6$ when this expression is expanded and simplified.
|
18
| 85.9375 |
9,948 |
Calculate the areas of the regions bounded by the curves given in polar coordinates.
$$
r=\cos 2 \phi
$$
|
\frac{\pi}{2}
| 47.65625 |
9,949 |
In a division problem, the dividend is 12, and the divisor is a natural number less than 12. What is the sum of all possible different remainders?
|
15
| 26.5625 |
9,950 |
Calculate the value of the expression \(\sin \frac{b \pi}{36}\), where \(b\) is the sum of all distinct numbers obtained from the number \(a = 987654321\) by cyclic permutations of its digits (in a cyclic permutation, all the digits of the number, except the last one, are shifted one place to the right, and the last digit moves to the first place).
|
\frac{\sqrt{2}}{2}
| 16.40625 |
9,951 |
How many unordered pairs of edges of a given octahedron determine a plane?
|
66
| 0 |
9,952 |
Bethany has 11 pound coins and some 20 pence coins and some 50 pence coins in her purse. The mean value of the coins is 52 pence. Which could not be the number of coins in the purse?
A) 35
B) 40
C) 50
D) 65
E) 95
|
40
| 61.71875 |
9,953 |
If \( a, b, c \) are the three real roots of the equation
\[ x^{3} - x^{2} - x + m = 0, \]
then the minimum value of \( m \) is _____.
|
-\frac{5}{27}
| 77.34375 |
9,954 |
Find the volume of the solid $T$ consisting of all points $(x, y, z)$ such that $|x| + |y| \leq 2$, $|x| + |z| \leq 2$, and $|y| + |z| \leq 2$.
|
\frac{32}{3}
| 60.15625 |
9,955 |
In the Cartesian coordinate system $xoy$, the parametric equation of line $l$ is $\begin{cases} x=1- \frac { \sqrt {3}}{2}t \\ y= \frac {1}{2}t\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the equation of circle $C$ is $\rho=2 \sqrt {3}\sin \theta$.
$(1)$ Write the standard equation of line $l$ and the Cartesian coordinate equation of circle $C$;
$(2)$ If the Cartesian coordinates of point $P$ are $(1,0)$, and circle $C$ intersects line $l$ at points $A$ and $B$, find the value of $|PA|+|PB|$.
|
2 \sqrt {3}
| 0 |
9,956 |
Given \( x, y, z > 0 \) and \( x + y + z = 1 \), find the maximum value of
$$
f(x, y, z) = \sum \frac{x(2y - z)}{1 + x + 3y}.
$$
|
1/7
| 60.15625 |
9,957 |
Shift the graph of the function $y=3\sin (2x+ \frac {\pi}{6})$ to the graph of the function $y=3\cos 2x$ and determine the horizontal shift units.
|
\frac {\pi}{6}
| 55.46875 |
9,958 |
Snow White entered a room where 30 chairs were arranged around a circular table. Some of the chairs were occupied by dwarfs. It turned out that Snow White could not sit in such a way that there was no one next to her. What is the minimum number of dwarfs that could have been at the table? (Explain how the dwarfs must have been seated and why there would be a chair with no one next to it if there were fewer dwarfs.)
|
10
| 31.25 |
9,959 |
Two students, A and B, are preparing to have a table tennis match during their physical education class. Assuming that the probability of A winning against B in each game is $\frac{1}{3}$, the match follows a best-of-three format (the first player to win two games wins the match). What is the probability of A winning the match?
|
\frac{7}{27}
| 97.65625 |
9,960 |
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
|
\frac{7}{2}
| 94.53125 |
9,961 |
Given a $3 \times 3$ grid (like a Tic-Tac-Toe board), four randomly selected cells have been randomly placed with four tokens.
Find the probability that among these four tokens, three are aligned in a row either vertically, horizontally, or diagonally.
|
8/21
| 59.375 |
9,962 |
The numeral $65$ in base $c$ represents the same number as $56$ in base $d$. Assuming that both $c$ and $d$ are positive integers, find the least possible value of $c+d$.
|
13
| 36.71875 |
9,963 |
Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files.
|
12
| 7.03125 |
9,964 |
In $\triangle ABC$, the sides opposite to angles $A, B, C$ are denoted as $a, b, c$, respectively, and $a=1, A=\frac{\pi}{6}$.
(Ⅰ) When $b=\sqrt{3}$, find the magnitude of angle $C$;
(Ⅱ) Find the maximum area of $\triangle ABC$.
|
\frac{2+ \sqrt{3}}{4}
| 39.0625 |
9,965 |
How many consecutive "0"s are there at the end of the product \(5 \times 10 \times 15 \times 20 \times \cdots \times 2010 \times 2015\)?
|
398
| 5.46875 |
9,966 |
Given $a= \int_{ 0 }^{ \pi }(\sin x-1+2\cos ^{2} \frac {x}{2})dx$, find the constant term in the expansion of $(a \sqrt {x}- \frac {1}{ \sqrt {x}})^{6}\cdot(x^{2}+2)$.
|
-332
| 69.53125 |
9,967 |
Given the set \( M = \{1, 2, \cdots, 2017\} \) which consists of the first 2017 positive integers, if one element is removed from \( M \) such that the sum of the remaining elements is a perfect square, what is the removed element?
|
1677
| 6.25 |
9,968 |
Convert the binary number $110101_{(2)}$ to decimal.
|
53
| 96.09375 |
9,969 |
Given: Circle $C$ passes through point $D(0,1)$, $E(-2,1)$, $F(-1,\sqrt{2})$, $P$ is any point on the line $l_{1}: y=x-2$, and the line $l_{2}: y=x+1$ intersects circle $C$ at points $A$ and $B$. <br/>$(Ⅰ)$ Find the equation of circle $C$;<br/>$(Ⅱ)$ Find the minimum value of $|PA|^{2}+|PB|^{2}$.
|
13
| 30.46875 |
9,970 |
A paper equilateral triangle of side length 2 on a table has vertices labeled \(A\), \(B\), and \(C\). Let \(M\) be the point on the sheet of paper halfway between \(A\) and \(C\). Over time, point \(M\) is lifted upwards, folding the triangle along segment \(BM\), while \(A\), \(B\), and \(C\) remain on the table. This continues until \(A\) and \(C\) touch. Find the maximum volume of tetrahedron \(ABCM\) at any time during this process.
|
\frac{\sqrt{3}}{6}
| 2.34375 |
9,971 |
If \(\frac{1}{4} + 4\left(\frac{1}{2013} + \frac{1}{x}\right) = \frac{7}{4}\), find the value of \(1872 + 48 \times \left(\frac{2013 x}{x + 2013}\right)\).
|
2000
| 58.59375 |
9,972 |
How many non-similar quadrilaterals have angles whose degree measures are distinct positive integers in an arithmetic progression?
|
29
| 88.28125 |
9,973 |
Find $k$ where $2^k$ is the largest power of $2$ that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\]
|
2007
| 100 |
9,974 |
Given the function $f(x)=\sin x\cos x- \sqrt {3}\cos ^{2}x.$
(I) Find the smallest positive period of $f(x)$;
(II) When $x\in[0, \frac {π}{2}]$, find the maximum and minimum values of $f(x)$.
|
- \sqrt {3}
| 0 |
9,975 |
Given that $b = 8$ and $n = 15$, calculate the number of positive factors of $b^n$ where both $b$ and $n$ are positive integers, with $n$ being 15. Determine if this choice of $b$ and $n$ maximizes the number of factors compared to similar calculations with other bases less than or equal to 15.
|
46
| 89.0625 |
9,976 |
Given the function $f(x)= \frac{1}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))(n∈N^{})$ where $N^{}$ represents the set of positive integers, vector $ \overrightarrow{i}=(0,1)$, and $θ_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{i}$, determine the value of $\frac{cosθ_{1}}{sinθ_{1}}+ \frac{cosθ_{2}}{sinθ_{2}}+…+\frac{cosθ_{2017}}{sinθ_{2017}}$.
|
\frac{2017}{2018}
| 86.71875 |
9,977 |
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the integer part of $x$, which is the greatest integer not exceeding $x$. This function, $\lfloor x \rfloor$, is called the "floor function". Calculate the sum $\lfloor \log_3 1 \rfloor + \lfloor \log_3 2 \rfloor + \lfloor \log_3 3 \rfloor + \lfloor \log_3 4 \rfloor + \ldots + \lfloor \log_3 243 \rfloor$.
|
857
| 38.28125 |
9,978 |
Compute $\sin 870^\circ$ and $\cos 870^\circ$.
|
-\frac{\sqrt{3}}{2}
| 35.15625 |
9,979 |
Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 planks so that any two adjacent planks are different colors and he uses all three colors? Provide a justification for your answer.
|
1530
| 27.34375 |
9,980 |
In the arithmetic sequence $\{a_n\}$, the common difference is $\frac{1}{2}$, and $a_1+a_3+a_5+\ldots+a_{99}=60$. Find the value of $a_2+a_4+a_6+\ldots+a_{100}$.
|
85
| 81.25 |
9,981 |
The following diagram shows a square where each side has four dots that divide the side into three equal segments. The shaded region has area 105. Find the area of the original square.
[center][/center]
|
135
| 3.125 |
9,982 |
Given that $x, y > 0$ and $\frac{1}{x} + \frac{1}{y} = 2$, find the minimum value of $x + 2y$.
|
\frac{3 + 2\sqrt{2}}{2}
| 26.5625 |
9,983 |
Given the circle \(\Gamma: x^{2} + y^{2} = 1\) with two intersection points with the \(x\)-axis as \(A\) and \(B\) (from left to right), \(P\) is a moving point on circle \(\Gamma\). Line \(l\) passes through point \(P\) and is tangent to circle \(\Gamma\). A line perpendicular to \(l\) passes through point \(A\) and intersects line \(BP\) at point \(M\). Find the maximum distance from point \(M\) to the line \(x + 2y - 9 = 0\).
|
2\sqrt{5} + 2
| 50.78125 |
9,984 |
Last year, 10% of the net income from our school's ball was allocated to clubs for purchases, and the remaining part covered the rental cost of the sports field. This year, we cannot sell more tickets, and the rental cost remains the same, so increasing the share for the clubs can only be achieved by raising the ticket price. By what percentage should the ticket price be increased to make the clubs' share 20%?
|
12.5
| 43.75 |
9,985 |
Given a random variable $ξ$ follows a normal distribution $N(2,σ^{2})$, and $P(ξ \leqslant 4-a) = P(ξ \geqslant 2+3a)$, solve for the value of $a$.
|
-1
| 57.8125 |
9,986 |
Determine the volume of the region in space defined by
\[|x + y + z| + |x + y - z| \le 12\]
and \(x, y, z \ge 0.\)
|
108
| 58.59375 |
9,987 |
A rock is dropped off a cliff of height $ h $ As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be $ kh $ . If the number of photographs taken is huge, find $ k $ . That is: what is the time-average of the distance traveled divided by $ h $ , dividing by $h$ ?
*Problem proposed by Ahaan Rungta*
|
$\dfrac{1}{3}$
| 0 |
9,988 |
For a natural number \( x \), five statements are made:
$$
3x > 91
$$
$$
\begin{aligned}
& x < 120 \\
& 4x > 37 \\
& 2x \geq 21 \\
& x > 7
\end{aligned}
$$
It is known that only three of these statements are true, and two are false. Determine \( x \).
|
10
| 28.125 |
9,989 |
Six soccer teams play at most one match between any two teams. If each team plays exactly 2 matches, how many possible arrangements of these matches are there?
|
70
| 1.5625 |
9,990 |
Given the point \( A(0,1) \) and the curve \( C: y = \log_a x \) which always passes through point \( B \), if \( P \) is a moving point on the curve \( C \) and the minimum value of \( \overrightarrow{AB} \cdot \overrightarrow{AP} \) is 2, then the real number \( a = \) _______.
|
e
| 51.5625 |
9,991 |
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, and a point $P(1,\frac{{\sqrt{2}}}{2})$ on the ellipse, satisfying $|PF_{1}|+|PF_{2}|=2\sqrt{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ A line $l$ passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. Find the maximum area of $\triangle AOB$.
|
\frac{\sqrt{2}}{2}
| 22.65625 |
9,992 |
Assume integers \( u \) and \( v \) satisfy \( 0 < v < u \), and let \( A \) be \((u, v)\). Points are defined as follows: \( B \) is the reflection of \( A \) over the line \( y = x \), \( C \) is the reflection of \( B \) over the \( y \)-axis, \( D \) is the reflection of \( C \) over the \( x \)-axis, and \( E \) is the reflection of \( D \) over the \( y \)-axis. The area of pentagon \( ABCDE \) is 451. Find \( u+v \).
|
21
| 19.53125 |
9,993 |
Given the parabola $C: y^2 = 16x$ with the focus $F$, and the line $l: x = -1$, if a point $A$ lies on $l$ and the line segment $AF$ intersects the parabola $C$ at point $B$ such that $\overrightarrow{FA} = 5\overrightarrow{FB}$, then find the length of $|AB|$.
|
28
| 69.53125 |
9,994 |
A frog sits at the point $(2, 3)$ on a grid within a larger square bounded by points $(0,0), (0,6), (6,6)$, and $(6,0)$. Each jump the frog makes is parallel to one of the coordinate axes and has a length $1$. The direction of each jump (up, down, left, right) is not necessarily chosen with equal probability. Instead, the probability of jumping up or down is $0.3$, and left or right is $0.2$ each. The sequence of jumps ends when the frog reaches any side of the square. What is the probability that the sequence of jumps ends on a vertical side of the square?
**A**) $\frac{1}{2}$
**B**) $\frac{5}{8}$
**C**) $\frac{2}{3}$
**D**) $\frac{3}{4}$
**E**) $\frac{7}{8}$
|
\frac{5}{8}
| 19.53125 |
9,995 |
Let $ABC$ be a triangle with $AC = 28$ , $BC = 33$ , and $\angle ABC = 2\angle ACB$ . Compute the length of side $AB$ .
*2016 CCA Math Bonanza #10*
|
16
| 24.21875 |
9,996 |
Consider two sets of consecutive integers. Let $A$ be the least common multiple (LCM) of the integers from $15$ to $25$ inclusive. Let $B$ be the least common multiple of $A$ and the integers $26$ to $45$. Compute the value of $\frac{B}{A}$.
A) 4536
B) 18426
C) 3 * 37 * 41 * 43
D) 1711
E) 56110
|
3 \cdot 37 \cdot 41 \cdot 43
| 1.5625 |
9,997 |
Find the number of natural numbers not exceeding 2022 and not belonging to either the arithmetic progression \(1, 3, 5, \ldots\) or the arithmetic progression \(1, 4, 7, \ldots\).
|
674
| 71.875 |
9,998 |
Given vectors $\overrightarrow{a}=(\cos \alpha,\sin \alpha)$, $\overrightarrow{b}=(\cos x,\sin x)$, $\overrightarrow{c}=(\sin x+2\sin \alpha,\cos x+2\cos \alpha)$, where $(0 < \alpha < x < \pi)$.
$(1)$ If $\alpha= \frac {\pi}{4}$, find the minimum value of the function $f(x)= \overrightarrow{b} \cdot \overrightarrow{c}$ and the corresponding value of $x$;
$(2)$ If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac {\pi}{3}$, and $\overrightarrow{a} \perp \overrightarrow{c}$, find the value of $\tan 2\alpha$.
|
- \frac { \sqrt {3}}{5}
| 0 |
9,999 |
A two-digit number is divided by the sum of its digits. The result is a number between 2.6 and 2.7. Find all of the possible values of the original two-digit number.
|
29
| 78.125 |
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