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listlengths 1
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Given that $\sqrt{2+ \frac{2}{3}} =2\cdot\sqrt{ \frac{2}{3}}$, $\sqrt{3+ \frac{3}{8}} =3\cdot\sqrt{ \frac{3}{8}}$, $\sqrt{4+ \frac{4}{15}} =4\cdot\sqrt{ \frac{4}{15}}$, ..., if $\sqrt{8+ \frac{a}{t}} =8\cdot\sqrt{ \frac{a}{t}}$ (where $a,t$ are positive real numbers), according to the pattern above, then $a+t=$_______.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{71}
| null |
math
| null | null | null | null |
fef49cb5dd561684e71501340d30ea68
|
8,9
The altitudes of an acute-angled triangle $ABC$ intersect at point $O$. A circle with radius $R$ and center at point $O$ passes through vertex $A$, touches side $BC$, and intersects side $AC$ at point $M$ such that $AM: MC = 4: 1$. Find the length of side $AB$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{2R\sqrt{2}}
| null |
math
| null | null | null | null |
a72f1a0fe6b3232a72fb090fefd18255
|
Two students in a class are preparing to register for the independent admission tests of Zhejiang University, Fudan University, and Shanghai Jiao Tong University, with the requirement that each student can choose up to two schools. Find the total number of different registration results. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{36}
| null |
math
| null | null | null | null |
9eded87fb3acbf0f25e70a678b1a1894
|
From 22 kg of fresh mushrooms, 2.5 kg of dried mushrooms are obtained, which still contain 12% water. What percent of water is in the fresh mushrooms?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{90\%}
| null |
math
| null | null | null | null |
05757c4e76983ac754469735575cea99
|
Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{12}
| null |
math
| null | null | null | null |
51188dd7fe29da8c912dd0eace7bf901
|
In three-dimensional space, let \( S \) be the region of points \( (x, y, z) \) satisfying \( -1 \leq z \leq 1 \). Let \( S_{1}, S_{2}, \ldots, S_{2022} \) be 2022 independent random rotations of \( S \) about the origin \( (0,0,0) \). The expected volume of the region \( S_{1} \cap S_{2} \cap \cdots \cap S_{2022} \) can be expressed as \( \frac{a \pi}{b} \), for relatively prime positive integers \( a \) and \( b \). Compute \( 100a+b \).
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{271619}
| null |
math
| null | null | null | null |
4cbaa52d2141160a0cf874ac4b7c36eb
|
Find the highest degree $k$ of $1991$ for which $1991^k$ divides the number
\[ 1990^{1991^{1992}} + 1992^{1991^{1990}}. \]
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{1991}
| null |
math
| null | null | null | null |
ae1683ba968c54416b79990de7f64f3e
|
North Pond and Lake Michigan are homes to three different species of ducks: Mallard, Pintail, and Teal. Six more than twice as many Mallard ducks go to North Pond as go to Lake Michigan. In addition, the number of Pintail ducks at North Pond is four times the number of Mallard ducks at Lake Michigan. Moreover, the number of Teal ducks is the product of the number of Mallard and Pintail ducks at North Pond.
A biological constraint states that the number of Teal ducks in either pond cannot exceed the square root of the product of the number of Mallard and Pintail ducks in that pond. If Lake Michigan has 100 Mallard ducks and 75 Pintail ducks, and the constraint is satisfied for both ponds, how many ducks can you safely estimate there will be at North Pond?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{893}
| null |
math
| null | null | null | null |
b4bdf612e2f80dd1e72c3db0042572c2
|
There are 111 balls in a box, each being red, green, blue, or white. It is known that if 100 balls are drawn, it ensures getting balls of all four colors. Find the smallest integer $N$ such that if $N$ balls are drawn, it can ensure getting balls of at least three different colors.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{88}
| null |
math
| null | null | null | null |
226b9935e9aace627c17e03828fe0a3b
|
A square with side length \( a \) is given. On each side of the square, a trapezoid is constructed outside of it, such that the upper bases of these trapezoids and their lateral sides form a regular dodecagon. Calculate the area of the dodecagon.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{3a^2}{2}}
| null |
math
| null | null | null | null |
393ef315c232a54a08715030ee1fd1a7
|
Given the universal set \( U = \{1, 2, 3, 4, 5\} \) and the set \( I = \{X \mid X \subseteq U\} \), two different elements \( A \) and \( B \) are randomly selected from set \(I\). What is the probability that the intersection \( A \cap B \) has exactly 3 elements?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{5}{62}}
| null |
math
| null | null | null | null |
bedfca4e9c0681006cb802e7e63bfe4c
|
Ramesh purchased a refrigerator for Rs. 13500 after getting a certain discount on the labelled price. He spent Rs. 125 on transport and Rs. 250 on installation. To earn a 10% profit if no discount was offered, he should sell it for Rs. 18975. What was the percentage of discount Ramesh received on the labelled price?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{21.74\%}
| null |
math
| null | null | null | null |
5c538256c1e96d68ffb6d7b8f297206d
|
初始时，天使有 100 堆石头，每堆有 100 块.每天早上，天使执行如下两种操作之一：(a) 取走某堆中的所有石头；(b) 从每堆中各取走一块石头.然而每天晚上，恶魔执行如下两种操作之一：(a) 给每个非空的堆各增加一块石头；(b) 创造一个新的堆并加入一块石头.问：至少需要多少天，天使才能确保在该天早上取走所有石头？
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{199}
| null |
math
| null | null | null | null |
f59ea4bf57f14df5d184c08b19b662d9
|
On an island, there are knights who always tell the truth and liars who always lie. One day, 15 natives, among whom there were both knights and liars, formed a circle. Each of them said, "Of the two people standing opposite me, one is a knight and the other is a liar." How many knights are there among them?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{10}
| null |
math
| null | null | null | null |
42cd7c8038836c30871e374d846d326e
|
The sum of the digits in the product of $\overline{A A A A A A A A A} \times \overline{B B B B B B B B B}$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{81}
| null |
math
| null | null | null | null |
30794aa7e06fa897d7458935179f094b
|
Rohan's total marks obtained in Mathematics, Physics, and Biology is 90. His score in Chemistry is 10 marks more than that in Physics, and his Biology score is 5 marks less than his Chemistry score. If the weighted average system is applied with 40% weightage for Mathematics, 30% for Chemistry, and 30% for Biology, find the weighted average marks scored in Mathematics, Chemistry, and Biology together.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{38.5}
| null |
math
| null | null | null | null |
ad40869e0c8c7008f93cb1a89341e9c5
|
A student has 7 reference books, including 2 Chinese books, 2 English books, and 3 math books. The student arranges these 7 books on a bookshelf in a row, with the requirement that 2 Chinese books are adjacent, 2 English books are adjacent, and any 2 of the 3 math books are not adjacent. Calculate the total number of different ways the books can be arranged.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{48}
| null |
math
| null | null | null | null |
a67c9cf0b6c94c50aa41cddf1854fcb8
|
A coin is flipped multiple times until an odd number of heads followed by a tail is obtained. Given $n \in \mathbb{N}^{*}$, find the number of sequences of $n$ flips.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{F_{n-1}}
| null |
math
| null | null | null | null |
da177ed0261224f1983fdd19bdf100ee
|
Determine all real numbers $a$ such that \[4\lfloor an\rfloor =n+\lfloor a\lfloor an\rfloor \rfloor \; \text{for all}\; n \in \mathbb{N}.\]
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{ a = 2 + \sqrt{3} }
| null |
math
| null | null | null | null |
6c3b6271dcca3588af4c067411400265
|
Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$ .
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{ \{3, 5, 7, 13\} }
| null |
math
| null | null | null | null |
354af54412739a53118794553e661010
|
Given that \( f(x) \) is an odd function defined on \(\mathbf{R} \), with \( f(1) = 1 \), and for any \( x < 0 \), it holds that
$$
f\left( \frac{x}{x-1} \right) = x f(x).
$$
Find the value of \( \sum_{i=1}^{50} f\left( \frac{1}{i} \right) f\left( \frac{1}{101-i} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{2^{98}}{99!}}
| null |
math
| null | null | null | null |
5b94e6482fe140de386c94ecd80cdc76
|
Three equally spaced parallel lines intersect with a circle, resulting in three chords of lengths 38, 38, 34 units. Find the distance between two adjacent parallel chords. Express your answer in units as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{6}
| null |
math
| null | null | null | null |
9ae9e8c6d44a11bc1be4f301b14d34e8
|
6. Try to advance as far as possible in the analytical solution of the problem given below. In case $B$ is necessary, a computer may be used at the final stage.
Point $A$ is located on a meadow, point $B$ - on a sandy wasteland. The distance between the points is 24 km. The boundary between the wasteland and the meadow is a straight line. The distance from point $A$ to the boundary is 8 km, and the distance from point $B$ to the boundary is 4 km. Find the minimum time it takes for a pedestrian to get from point $A$ to point $B$, if their maximum speed on the wasteland is 3 km/hour, and on the meadow 6 km/hour.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{4.89}
| null |
math
| null | null | null | null |
31ac8c0a31929eb95fb12dcefe516469
|
Write the number \( 123456789101112 \cdots 19941995 \) on the blackboard to form an integer \( N_1 \). Erase the digits in the even positions of \( N_1 \) to form \( N_2 \). Then, erase the digits in the odd positions of \( N_2 \) to form \( N_3 \). Continue this process of erasing digits in even and odd positions alternately until only one digit remains on the blackboard. Determine this final digit. (Note: Positions are counted from left to right; for example, in \( 12345 \), \( 1 \) is in the first position, \( 2 \) is in the second position, and so on.) (Australian Mathematics Olympiad, 1995)
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{9}
| null |
math
| null | null | null | null |
c6523150a354cc3c90d5a7e01f07432d
|
Given that the dihedral angle $\alpha-l-\beta$ is $60^{\circ}$, points $P$ and $Q$ are on planes $\alpha$ and $\beta$ respectively. The distance from $P$ to plane $\beta$ is $\sqrt{3}$, and the distance from $Q$ to plane $\alpha$ is $2 \sqrt{3}$. What is the minimum distance between points $P$ and $Q$?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{2\sqrt{3}}
| null |
math
| null | null | null | null |
c36e43682593d4d692a840f8140d582d
|
A Boeing 757 flies a direct route from Dallas, TX, to Phoenix, AZ and then returns to Dallas, TX. The flight is 900 miles one way. It took some time for the flight to Phoenix and 2 hrs for the flight back to Dallas. The speed of the wind was 25 mph. How long did it take for the flight to Phoenix?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{2.25}
| null |
math
| null | null | null | null |
1a9e368230a976a83b92e5e74b323066
|
Given the functions y = x^3 and y = (1/2)^x, find the interval containing the x-coordinate of their intersection point. Express your answer as an interval (a, b) using standard interval notation.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{(0,1)}
| null |
math
| null | null | null | null |
0ba3bdd5544be8ec91883cfff54b02f3
|
In the Cartesian coordinate plane, there are four fixed points \(A(-3,0), B(1,-1), C(0,3), D(-1,3)\) and a moving point \(P\). What is the minimum value of \(|PA| + |PB| + |PC| + |PD|\)?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{3\sqrt{2} + 2\sqrt{5}}
| null |
math
| null | null | null | null |
df4dfc74c08d0dea93568de33818ab4c
|
6. Let the non-real complex number $z$, satisfy $z^{23}=1$. Then $\sum_{k=0}^{22} \frac{1}{1+z^{k}+z^{2 k}}=$ $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{46}{3}}
| null |
math
| null | null | null | null |
8b6e3a55fcb873830937c7d84b9b4aaf
|
For every integer $n \ge 1$ , the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$ , $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$ . Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$ . Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$ . (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$ .)
*Proposed by Lewis Chen*
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{26}
| null |
math
| null | null | null | null |
06d16e991619a99dfdbab315fc344c2e
|
If the equation \( \lg (x-1) + \lg (3-x) = \lg (a-x) \) with respect to \( x \) has exactly one real root, find the range of the real number \( a \).
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{1 < a \leq \frac{13}{4}}
| null |
math
| null | null | null | null |
e33ad39a6a3dad0ab91acc4f18da79aa
|
Find all positive integer pairs $(x, y)$ such that $x^{2}-y^{119}=1$. Please provide the sum of all possible $x$ and $y$ values from the solutions.If the equation has no solution, then the answer is 0.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{0}
| null |
math
| null | null | null | null |
b2fa580873ba0bf31549020825eee151
|
A regular pyramid with a side length of 1 and a base perimeter of 4 has a lateral surface area of __________.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\sqrt{3}}
| null |
math
| null | null | null | null |
2624ecfd150f19147accbafc38d1ab4d
|
Given a parabola $y^2=2px$ with focus $F$, let $A(x_1, y_1)$ and $B(x_2, y_2)$ be two moving points on the parabola. If $x_1+x_2+p=\frac{2\sqrt{3}}{3}|AB|$, find the maximum value of $\angle AFB$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{2\pi}{3}}
| null |
math
| null | null | null | null |
cc316abb9cf3b368e11bd346b129652d
|
Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{$0$}
| null |
math
| null | null | null | null |
0e8c5abbd907fa02f203674aefd54e5e
|
An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent song is 30 seconds longer than the previous song. Her favorite song is 3 minutes, 30 seconds long. The o-Pod will play all the songs in random order before repeating any song. What is the probability that she hears the first 4 minutes, 30 seconds of music - there are no pauses between songs - without hearing every second of her favorite song? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\dfrac{79}{90}}
| null |
math
| null | null | null | null |
47a969caac68abdf235125c35d8576a1
|
Given that a four-digit integer $MMMM$, with all identical digits, is multiplied by the one-digit integer $M$, the result is the five-digit integer $NPMPP$. Assuming $M$ is the largest possible single-digit integer that maintains the units digit property of $M^2$, find the greatest possible value of $NPMPP$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{89991}
| null |
math
| null | null | null | null |
b53a5632a1a3881d3ff09317fff6bb1c
|
Determine the real value of $t$ that minimizes the expression
\[
\sqrt{t^2 + (t^2 - 1)^2} + \sqrt{(t-14)^2 + (t^2 - 46)^2}.
\]
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{7/2}
| null |
math
| null | null | null | null |
2b69502b40aa1f91e8ec31c5d2829eb4
|
Given the complex number (a+2i)(1+i) with a modulus of 4, calculate the value of the real number 'a'. Express your answer as a real number, including the ± symbol if applicable.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{±2}
| null |
math
| null | null | null | null |
ebb9e5858ce38785d48470c70b665113
|
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Calculate the maximum value of $b-a$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{4\pi}{3}}
| null |
math
| null | null | null | null |
fb43725a455c13a882dfac70e4c8e2c2
|
Alex wrote all natural divisors of a natural number \( n \) on the board in ascending order. Dima erased several of the first and several of the last numbers of the resulting sequence so that 151 numbers remained. What is the maximum number of these 151 divisors that could be fifth powers of natural numbers?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{31}
| null |
math
| null | null | null | null |
14dfde900034688a59b0f9a84366639b
|
If the product $\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9$, what is the sum of $a$ and $b$? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{35}
| null |
math
| null | null | null | null |
073779b5a736fe8546c263b69d882977
|
Vova has 19 grades in his journal for math, all of which are 2s and 3s, with the first four grades being 2s. It turns out that among Vova's consecutive four grades, all 16 possible combinations of four 2s and 3s occur. What are Vova's last four grades?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{3222}
| null |
math
| null | null | null | null |
9676a2200cc1f19b5e55834f15f32d63
|
The graph of $y=g(x)$, defined on a limited domain shown, is conceptualized through the function $g(x) = \frac{(x-6)(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{945} - 2.5$. If each horizontal grid line represents a unit interval, determine the sum of all integers $d$ for which the equation $g(x) = d$ has exactly six solutions.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{-5}
| null |
math
| null | null | null | null |
6d4585cf422b35fc657266534058dcbd
|
The lateral edge of a regular triangular pyramid is twice the length of the side of the base. Find the angle between the pyramid's slant height and a height of the triangle in the base that does not intersect the slant height.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\arccos \left( \frac{\sqrt{5}}{30} \right)}
| null |
math
| null | null | null | null |
53cb164d0c0ff72eeef9138ee0365f5a
|
We define $N$ as the set of natural numbers $n<10^6$ with the following property:
There exists an integer exponent $k$ with $1\le k \le 43$ , such that $2012|n^k-1$ .
Find $|N|$ .
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{1988}
| null |
math
| null | null | null | null |
026790302b209881a05db932f2fb1e12
|
In triangle $ABC$, if $a=2$, $c=2\sqrt{3}$, and $\angle A=30^\circ$, then the area of $\triangle ABC$ is equal to __________.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\sqrt{3}}
| null |
math
| null | null | null | null |
983501fd98f960ac48b46e282cec5011
|
What is the sum (in base $10$) of all the natural numbers less than $64$ which have exactly three ones in their base $2$ representation?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{630}
| null |
math
| null | null | null | null |
fef0b06368075ce5f026c5bd222f51bd
|
A semicircle with a radius of 1 is drawn inside a semicircle with a radius of 2. A circle is drawn such that it touches both semicircles and their common diameter. What is the radius of this circle?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{8}{9}}
| null |
math
| null | null | null | null |
18a6e17ef45939651ea0998d1235a83e
|
Two circles with radii 5 and 3 touch each other internally. A chord of the larger circle touches the smaller circle and is divided by the point of tangency in the ratio \( 3:1 \). Find the length of this chord.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{8}
| null |
math
| null | null | null | null |
a9e977397a56064299d8ae41e7da12ef
|
Calculate the indefinite integral:
$$
\int \frac{x^{3}-5 x^{2}+5 x+23}{(x-1)(x+1)(x-5)} \, dx
$$
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{x - 3 \ln |x-1| + \ln |x+1| + 2 \ln |x-5| + C}
| null |
math
| null | null | null | null |
a525d512745bce13298891d7a87a1daf
|
Given rectangle ABCD with AD = 13, DE = 5, and EA = 12, calculate the area of rectangle ABCD. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{60}
| null |
math
| null | null | null | null |
d7c893d4571f3f20ea466b0fead85810
|
Let $p(x)$ be a nonzero polynomial of degree less than 1992 having no noncon-
stant factor in common with $x^3-x.$ Let
$$\frac{d^{1992}}{dx^{1992}}\left(\frac{p(x)}{x^3-x}\right)=\frac{f(x)}{g(x)}$$
for polynomials $f(x)$ and $g(x).$ Find the smallest possible degree of $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{3984}
| null |
math
| null | null | null | null |
fd4bcfcbe048c7c0bdf52d2c1e9a4110
|
Given a tetrahedron \(A B C D\) with the following edge lengths: \(A B = 2\), \(C D = 2\sqrt{5}\), \(A C = B D = 3\), and \(A D = B C = \sqrt{5}\), find the volume of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{4/3}
| null |
math
| null | null | null | null |
b39c1bd307d3bb565994c39505592001
|
Paul fills in a $7 \times 7$ grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same thing again (to obtain two different random arrangements of the numbers in the grid). What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{147 / 2}
| null |
math
| null | null | null | null |
fa822148cb689ad52d032b0c65a43c06
|
A sports club has allocated $n$ rubles to buy bicycles. Due to a price reduction, the cost of each bicycle decreased by $a$ rubles, resulting in the purchase of $b$ more bicycles than initially planned. How many bicycles were bought?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{ab + \sqrt{ab(ab + 4n)}}{2a}}
| null |
math
| null | null | null | null |
d4bb42f5f250c7f305afd6ccc55d195d
|
已知函数$f(x)=\frac{x}{{{e}^{x}}}$在$x={{x}_{0}}$处切线方程为$y=h\left( x \right)$,若$[f(x)-h(x)]\cdot (x-{{x0}})\ge 0$对$x\in R$恒成立，则${{x}_{0}}=$_________.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{$2$}
| null |
math
| null | null | null | null |
5e9bac8ea2ccf615fcae4376d3fdce9f
|
In English class, you have discovered a mysterious phenomenon: if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$, but if you spend $35$ minutes on the essay you somehow do not earn any points.
It is 4 AM, your English class starts at 8:05 AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{75}
| null |
math
| null | null | null | null |
36fc618fa1bc31f04030cf767f3f60a8
|
In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP = PQ = QB = BC.$ Find the ratio of $\angle ACB$ to $\angle APQ.$ Enter your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{4}{7}}
| null |
math
| null | null | null | null |
1b708c12f809827b43e92fa34f6a3799
|
If $x = \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 89^\circ$ and $y = \cos 2^\circ \cos 6^\circ \cos 10^\circ \ldots \cos 86^\circ$, then what is the integer nearest to $\frac{2}{7} \log_2 \frac{y}{x}$?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{19}
| null |
math
| null | null | null | null |
5a0628dc3ffa60c90c7c4cc678eb709a
|
Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle, congruent to the other five, is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by the other five circles, then it will turn through an angle of $k$ degrees before it returns to its starting position. Find $k$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{720}
| null |
math
| null | null | null | null |
f760a3ccfd3fa05fd22d408322ccf06c
|
A circle and a line touch at point \( M \). From points \( A \) and \( B \) on this circle, perpendiculars are drawn to the line, with lengths \( a \) and \( b \) respectively. Find the distance from point \( M \) to the line \( A B \).
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\sqrt{a \cdot b}}
| null |
math
| null | null | null | null |
ec071427129a1fa54e6a0836316be942
|
An equilateral triangle \( ABC \) is inscribed in the ellipse \( \frac{x^2}{p^2} + \frac{y^2}{q^2} = 1 \), such that vertex \( B \) is at \( (0, q) \), and \( \overline{AC} \) is parallel to the \( x \)-axis. The foci \( F_1 \) and \( F_2 \) of the ellipse lie on sides \( \overline{BC} \) and \( \overline{AB} \), respectively. Given \( F_1 F_2 = 2 \), find the ratio \( \frac{AB}{F_1 F_2} \).
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{8}{5}}
| null |
math
| null | null | null | null |
3a230d93430fc8fb34c436ea48c63f87
|
$AL$, $BM$, and $CN$ are the medians of $\triangle ABC$. $K$ is the intersection of the medians. If $C, K, L, M$ are concyclic and $AB = \sqrt{3}$, find the length of the median $CN$. The original answer is in the form \(\frac{k\sqrt{3}}{n}\), where k and n are integers. Please find the value of k + n.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{5}
| null |
math
| null | null | null | null |
42825a74ece1fa00228a7d7ab528460a
|
Let $a$ be the number of numbers that satisfy the condition "there is no other number whose absolute value is equal to the absolute value of $a$," $b$ be the number of numbers that satisfy the condition "there is no other number whose square is equal to the square of $b$," and $c$ be the number of numbers that satisfy the condition "there is no other number that, when multiplied by $c$, results in a product greater than 1." Then, $a+b+c=$ ?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{0}
| null |
math
| null | null | null | null |
f684218ec2f90d1696f397ffda51c9a4
|
Let \(\mathcal{V}\) be the volume enclosed by the graph
\[
x^{2016} + y^{2016} + z^{2} = 2016
\]
Find \(\mathcal{V}\) rounded to the nearest multiple of ten.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{360}
| null |
math
| null | null | null | null |
e997d26a9e8961e47f276c4a781ad9db
|
Given an odd function $y=f\left(x\right)$ defined on $R$, which is strictly decreasing on the interval $\left[0,+\infty \right)$. If for any $x\in R$, we always have $f(kx^{2}+2)+f\left(kx+k\right)\leqslant 0$ holds, then the range of real number $k$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{[0,+\infty)}
| null |
math
| null | null | null | null |
4a880ec425d3842c150a95a33bb2898a
|
Natural numbers are arranged in a spiral, turning the first bend at 2, the second bend at 3, the third bend at 5, and so on. What is the number at the twentieth bend?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{71}
| null |
math
| null | null | null | null |
f3bdf480221a54a552c3b870deed4fb0
|
In \(\triangle ABC\), \(AB = 9\), \(BC = 8\), and \(AC = 7\). The bisector of \(\angle A\) meets \(BC\) at \(D\). The circle passing through \(A\) and touching \(BC\) at \(D\) cuts \(AB\) and \(AC\) at \(M\) and \(N\) respectively. Find \(MN\).
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{6}
| null |
math
| null | null | null | null |
42fff990ba32508894c02a9d80b71ac1
|
A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, their sum will be:
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{180^{\circ}}
| null |
math
| null | null | null | null |
61fd5c3f8f5faa23e6c184147549ea8e
|
6. (10 points) Doudou's family has 4 people. This year, Doudou's brother is 3 years older than Doudou, and Doudou's mother is 2 years younger than Doudou's father. 5 years ago, the total age of the family was 59 years, and 5 years later, the total age of the family will be 97 years. Doudou's mother is $\qquad$ years old this year.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{33}
| null |
math
| null | null | null | null |
800a232dc5e764d3df0ea736c75adf97
|
Given that $a_n$ is the coefficient of the term containing $x^n$ in the expansion of $(1+x)^{n+2}$, find the sum of the first $n$ terms of the sequence $\{\frac{1}{a_n}\}$. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{n}{n+2}}
| null |
math
| null | null | null | null |
3214fb4383327b09b0b34d01cfa5b5d3
|
Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. Find the value of $c_1$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{11}
| null |
math
| null | null | null | null |
46357116f63114b5a79f534ffa89db95
|
Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{$431$}
| null |
math
| null | null | null | null |
1721dd758877f85ee7c6c5de26c8f172
|
All three vertices of an equilateral triangle are on the parabola $y = x^2$, and one of its sides has a slope of $2$. The $x$-coordinates of the three vertices have a sum of $m/n$, where $m$ and $n$ are relatively prime positive integers. What is the value of $m + n$?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{14}
| null |
math
| null | null | null | null |
9977ff99241933e5b9e06e3b1c4f247e
|
A circle is circumscribed around a regular $2n$-gon. A set of three distinct vertices of the polygon is called one-sided if there exists a semicircle on which these vertices lie (the endpoints of the semicircle are included). What is the probability that a randomly chosen set of three vertices is one-sided?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{3n}{2(2n-1)}}
| null |
math
| null | null | null | null |
7e868c9164934ef1c146e2063f8795a5
|
Because of the migration of salmon, the total number of salmon in river Trout has increased by ten times the number initially in the river. Now, there are 5500 salmons in the river. How many salmons were initially in the river?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{500}
| null |
math
| null | null | null | null |
a0e1ae8cebeebee6b55c286ea390c2cb
|
4. Let $n \in \mathbf{Z}_{+}$. A volleyball team has $n$ male players and $n$ female players. Initially, each player is assigned to one of the positions numbered $1,2, \cdots, 2 n$, with only positions 1 and $n+1$ being outside the court. During the game, position swaps occur, and each swap involves moving the player in position $i(i=1,2, \cdots, 2 n)$ to position $i+1$, where position $2 n+1$ is the same as position 1. How many different initial allocation methods are there such that, regardless of how many rounds of swapping occur, at least $n-1$ female players are on the court?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{2^{n}(n!)^{2}}
| null |
math
| null | null | null | null |
6f7d16c6093092bfb05a16a2627d62b7
|
The shape shown is made up of three similar right-angled triangles. The smallest triangle has two sides of side-length 2, as shown. What is the area of the shape?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{14}
| null |
math
| null | null | null | null |
910df80fed696297e7c966d1d46a0764
|
A cube has an edge length of 3. A square hole with a side length of 1, which goes through to the opposite side, is drilled in the exact center of each face, and the edges of the hole are parallel to the edges of the cube. Find the volume of the resulting solid.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{18}
| null |
math
| null | null | null | null |
b042e8a9bf00a4ceba75be4ca10841ef
|
There are 10 children, 5 boys and 5 girls, imagining playing the following game: First, let 4 children (not all boys) stand at 4 equidistant positions on a circle. If two adjacent children are of the same gender, then a boy is placed between them; otherwise, a girl is placed between them. Then, the original 4 children temporarily exit, which counts as one activity. This activity continues according to the above rules until the 4 children standing on the circle are all boys. The maximum number of times this activity can be performed is ___.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{4}
| null |
math
| null | null | null | null |
c55fcea94dd32b76dc61552c18fa7603
|
Given real numbers $a$ and $b$ satisfy $\ln (b+1)+a-3b=0$, and real numbers $c$ and $d$ satisfy $2d-c+ \sqrt {5}=0$, then the minimum value of $(a-c)^{2}+(b-d)^{2}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{1}
| null |
math
| null | null | null | null |
e6a01148941989847203f4ca6a32a044
|
In how many ways can a lottery ticket be filled out such that the sum of the 5 marked numbers is at least 228?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{21974634}
| null |
math
| null | null | null | null |
a620871d289753bf9ce80c98c79d7996
|
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.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{{998}.}
| null |
math
| null | null | null | null |
ade66974d7e66c3c1af4624ccf20d590
|
求最小的正整数 k，使得对集合 S=\{1,2,\cdots,2012\} 的任意一个 k 元子集 A，都存在 S 的三个不同的元素 a,b,c，使得 a+b, b+c, c+a 均在 A 中.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{1008}
| null |
math
| null | null | null | null |
d4a72f2097dbf959375c425f064783e2
|
From point \(A\), lying on a circle, chords \(AB\), \(AC\), and tangent \(AD\) are drawn. The angle between the chords is \(\frac{\pi}{6}\), and the angle between the chord \(AC\) and the tangent \(AD\), which does not contain the chord \(AB\), is \(\frac{5\pi}{12}\). Compute the integer part of the radius of the circle if the area of the triangle \(ABC\) is 32.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{4}
| null |
math
| null | null | null | null |
206bd640ffbcf362184bb37525dc8834
|
In triangle ABC, the side lengths opposite to angles A, B, and C are a, b, and c, respectively. Given that the height h from vertex A to side BC equals c - a, calculate the value of sin((C - A)/2) + cos((C + A)/2).
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{1}
| null |
math
| null | null | null | null |
3eb7d450c86e28079a7deb7fbb0b771c
|
On each non-boundary unit segment of an $8 \times 8$ chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{0}
| null |
math
| null | null | null | null |
643ccb93ed2f2dc6050e405e7544d340
|
A set of five-digit numbers $\{N_1, ..., N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1, ..., N_k$. Find the smallest possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{9}
| null |
math
| null | null | null | null |
e32c7f75672048cc4e3b91900dcc924c
|
Find the greatest number which is such that when 794, 858 and a certain number are divided by it, the remainders are all the same. The greatest number is 1. What is the third number?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{922}
| null |
math
| null | null | null | null |
0efb68cc69c4951582e41b28cd2de455
|
In bridge, a standard 52-card deck is dealt to 4 players. Each hand is assigned a number of "points" based on the formula
\[ 4 \times(\# \text{A's}) + 3 \times(\# \text{K's}) + 2 \times(\# \text{Q's}) + 1 \times(\# \text{J's}). \]
Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{197}{1820}}
| null |
math
| null | null | null | null |
da9baf4bd8cccfbf6e0e3ab4e380a854
|
Of the land owned by a farmer, 85 percent was cleared for planting. Of the cleared land, 10 percent was planted with grapes, 70 percent was planted with potatoes, and 5 percent was planted with corn. If the remaining 500 acres of cleared land was planted with tomatoes, how many acres did the farmer own?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{3921.57}
| null |
math
| null | null | null | null |
6867ae59258c029216f53316562aa026
|
Find the maximum value of the function $f(x_1, x_2, ..., x_T) = \sum_{t=1}^T 2^{-t}\sqrt{x_t}$ subject to the constraints $\sum_{t=1}^T x_t \leq 1$ and $x_t \geq 0$ for all $t = 1, 2, ..., T$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\sqrt{\frac{1-4^{-T}}{3}}}
| null |
math
| null | null | null | null |
f669c6b511a1f6f548b98a09009c20f6
|
3. In the Cartesian coordinate system, the set of points $(x, y)$ that satisfy $x<y,|x|<3,|y|<3$, and make the equation $\left(x^{3}-y^{3}\right) t^{4}+$ $(3 x+y) t^{2}+\frac{1}{x-y}=0$ have no real roots is denoted as $N$. Then the area of the region formed by the point set $N$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{81}{5}}
| null |
math
| null | null | null | null |
6750c955fd4cd04b7698b6aa750e6a16
|
In a certain kingdom, there were 32 knights. Some of them were vassals of others (a vassal can have only one liege, and a liege is always wealthier than their vassal). A knight with at least four vassals held the title of baron. What is the maximum number of barons that could exist under these conditions?
(In the kingdom, the law stated: "a vassal of my vassal is not my vassal.")
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{7}
| null |
math
| null | null | null | null |
07dde21b062717ea5a52ce7b4cf12a80
|
There are integers $x$ that satisfy the inequality $|x-2000|+|x| \leq 9999$. Find the number of such integers $x$.
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{9999}
| null |
math
| null | null | null | null |
21834c1f4678f28c087e6624c6a2c7a7
|
Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{\frac{33}{105}}
| null |
math
| null | null | null | null |
5809c1313ace4fa664a48301ceeaf3de
|
There are 111 StarCraft progamers. The StarCraft team SKT starts with a given set of eleven progamers on it, and at the end of each season, it drops a progamer and adds a progamer (possibly the same one). At the start of the second season, SKT has to field a team of five progamers to play the opening match. How many different lineups of five players could be fielded if the order of players on the lineup matters?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{4015440}
| null |
math
| null | null | null | null |
81385f697f2940efa02ec77fab7e39e2
|
Lizzy's mother gave her 80 cents, and her father gave her 40 cents for keeping to her promise. She went to her uncle's and spent 50 cents buying candy for her little cousin. Her uncle gave her another 70 cents. Then, she went to a toy store, saw a toy she wanted to buy. The price tag on the toy was 90 cents but there was a discount of 20% on toys that day. With the discount applied, she paid for the toy and the cashier gave her the change in dollars and cents: 1 dollar and 10 cents. How many cents does she have now?
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{178}
| null |
math
| null | null | null | null |
d693e6998017b5f332f8ef70b25793ae
|
Given the hyperbola $\frac {x^{2}}{16}- \frac {y^{2}}{9}=1$, and a chord AB with a length of 6 connected to the left focus F₁, calculate the perimeter of △ABF₂ (F₂ being the right focus).
Please reason step by step, and put your final answer within \boxed{}.
|
\boxed{28}
| null |
math
| null | null | null | null |
28ad8810c6a514aadfc54fe294efa88e
|
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