theprint/PyRe · Datasets at Hugging Face

source stringclasses 3 values	instruction stringlengths 23 3.97k	output stringlengths 1 3.75k
agentica-org/DeepScaleR-Preview-Dataset	A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee with 3 Republicans and 2 Democrats?	840
agentica-org/DeepScaleR-Preview-Dataset	Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$	7
agentica-org/DeepScaleR-Preview-Dataset	Given that $f(x)= \frac {4}{4^{x}+2}$, $S\_n$ is the sum of the first $n$ terms of the sequence $\{a\_n\}$, and $\{a\_n\}$ satisfies $a\_1=0$, and when $n \geqslant 2$, $a\_n=f( \frac {1}{n})+f( \frac {2}{n})+f( \frac {3}{n})+…+f( \frac {n-1}{n})$, find the maximum value of $\frac {a_{n+1}}{2S\_n+a\_6}$.	\frac {2}{7}
agentica-org/DeepScaleR-Preview-Dataset	Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\},$ the second from $\{A,I,O\},$ and the third from $\{D,M,N,T\}.$ When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of ADDITIONAL license plates that can be made by adding two letters?	40
agentica-org/DeepScaleR-Preview-Dataset	A club has 30 members, which includes 12 females and 18 males. In how many ways can a 5-person executive committee be formed such that at least one member must be female?	133,938
agentica-org/DeepScaleR-Preview-Dataset	Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?	\frac{1}{5}
agentica-org/DeepScaleR-Preview-Dataset	Given the function $f(x) = \sin x + \cos x$, where $x \in \mathbb{R}$, - (I) Find the value of $f\left(\frac{\pi}{2}\right)$. - (II) Determine the smallest positive period of the function $f(x)$. - (III) Calculate the minimum value of the function $g(x) = f\left(x + \frac{\pi}{4}\right) + f\left(x + \frac{3\pi}{4}\right)$.	-2
agentica-org/DeepScaleR-Preview-Dataset	A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction.	\frac{9}{2}
agentica-org/DeepScaleR-Preview-Dataset	Given the function $f(x)=\ln x+\frac{1}{2}x^2-ax$, where $a\in \mathbb{R}$, it has two extreme points at $x=x_1$ and $x=x_2$, with $x_1 < x_2$. (I) When $a=3$, find the extreme values of the function $f(x)$. (II) If $x_2\geqslant ex_1$ ($e$ is the base of the natural logarithm), find the maximum value of $f(x_2)-f(x_1)$.	1-\frac{e}{2}+\frac{1}{2e}
agentica-org/DeepScaleR-Preview-Dataset	Define a power cycle to be a set $S$ consisting of the nonnegative integer powers of an integer $a$, i.e. $S=\left\{1, a, a^{2}, \ldots\right\}$ for some integer $a$. What is the minimum number of power cycles required such that given any odd integer $n$, there exists some integer $k$ in one of the power cycles such that $n \equiv k$ $(\bmod 1024) ?$	10
agentica-org/DeepScaleR-Preview-Dataset	Given that point $P$ is a moving point on the parabola $y^2=2x$, find the minimum value of the sum of the distance from point $P$ to point $A(0,2)$ and the distance from $P$ to the directrix of the parabola.	\frac{\sqrt{17}}{2}
agentica-org/DeepScaleR-Preview-Dataset	In how many ways can the number 1500 be represented as a product of three natural numbers? Variations where the factors are the same but differ in order are considered identical.	30
agentica-org/DeepScaleR-Preview-Dataset	For real numbers $x$ and $y$, define $x \spadesuit y = (x+y)(x-y)$. What is $3 \spadesuit (4 \spadesuit 5)$?	-72
agentica-org/DeepScaleR-Preview-Dataset	Positive real numbers \( x, y, z \) satisfy: \( x^{4} + y^{4} + z^{4} = 1 \). Find the minimum value of the algebraic expression \( \frac{x^{3}}{1-x^{8}} + \frac{y^{3}}{1-y^{8}} + \frac{z^{3}}{1-z^{8}} \).	\frac{9 \sqrt[4]{3}}{8}
agentica-org/DeepScaleR-Preview-Dataset	Alex and Bob have 30 matches. Alex picks up somewhere between one and six matches (inclusive), then Bob picks up somewhere between one and six matches, and so on. The player who picks up the last match wins. How many matches should Alex pick up at the beginning to guarantee that he will be able to win?	2
agentica-org/DeepScaleR-Preview-Dataset	A standard six-sided die is rolled 3 times. If the sum of the numbers rolled on the first two rolls equals the number rolled on the third roll, what is the probability that at least one of the numbers rolled is a 2?	$\frac{8}{15}$
agentica-org/DeepScaleR-Preview-Dataset	In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when $N$ is divided by $1000$.	16
agentica-org/DeepScaleR-Preview-Dataset	At 11:00 a.m. how many degrees are in the smaller angle formed by the minute hand and the hour hand of the clock?	30
agentica-org/DeepScaleR-Preview-Dataset	In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 15 and 30 units respectively, and the altitude is 18 units. Points $E$ and $F$ divide legs $AD$ and $BC$ into thirds respectively, with $E$ one third from $A$ to $D$ and $F$ one third from $B$ to $C$. Calculate the area of quadrilateral $EFCD$.	360
agentica-org/DeepScaleR-Preview-Dataset	How many different ways can the five vertices S, A, B, C, and D of a square pyramid S-ABCD be colored using four distinct colors so that each vertex is assigned one color and no two vertices sharing an edge have the same color?	72
agentica-org/DeepScaleR-Preview-Dataset	Determine the number of digits in the value of $2^{15} \times 5^{10} \times 3^2$.	13
agentica-org/DeepScaleR-Preview-Dataset	The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers?	19
agentica-org/DeepScaleR-Preview-Dataset	Xiao Ming forgot the last two digits of his WeChat login password. He only remembers that the last digit is one of the letters \\(A\\), \\(a\\), \\(B\\), or \\(b\\), and the other digit is one of the numbers \\(4\\), \\(5\\), or \\(6\\). The probability that Xiao Ming can successfully log in with one attempt is \_\_\_\_\_\_.	\dfrac{1}{12}
agentica-org/DeepScaleR-Preview-Dataset	Let $T$ be a right triangle with sides having lengths $3$ , $4$ , and $5$ . A point $P$ is called awesome if P is the center of a parallelogram whose vertices all lie on the boundary of $T$ . What is the area of the set of awesome points?	1.5
agentica-org/DeepScaleR-Preview-Dataset	The value $b^n$ has both $b$ and $n$ as positive integers less than or equal to 15. What is the greatest number of positive factors $b^n$ can have?	496
agentica-org/DeepScaleR-Preview-Dataset	In triangle $ABC$, $AB = AC = 15$ and $BC = 14$. Points $D, E, F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE} \parallel \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. What is the perimeter of parallelogram $ADEF$?	30
agentica-org/DeepScaleR-Preview-Dataset	Sequence $A$ is a geometric sequence. Sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $300.$ What is the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B?$ $\bullet$ Sequence $A:$ $2,$ $4,$ $8,$ $16,$ $32,$ $\ldots$ $\bullet$ Sequence $B:$ $20,$ $40,$ $60,$ $80,$ $100,$ $\ldots$	4
agentica-org/DeepScaleR-Preview-Dataset	Given real numbers $x$, $y$ satisfying $x > y > 0$, and $x + y \leqslant 2$, the minimum value of $\dfrac{2}{x+3y}+\dfrac{1}{x-y}$ is	\dfrac {3+2 \sqrt {2}}{4}
agentica-org/DeepScaleR-Preview-Dataset	Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram which is the same as the previous one. If $AB = 36$ units and $CD = 72$ units, what is the area of square $BCFE$?	2592
agentica-org/DeepScaleR-Preview-Dataset	For functions $f(x)$ and $g(x)$, let $m\in \{x\|f(x)=0\}$, $n\in \{x\|g(x)=0\}$. If there exist $m$ and $n$ such that $\|m-n\|\leqslant 1$, then $f(x)$ and $g(x)$ are called "zero-point related functions". If the functions $f(x)=e^{x-2}+\ln(x-1)-1$ and $g(x)=x(\ln x-ax)-2$ are "zero-point related functions", then the minimum value of the real number $a$ is ____.	-2
agentica-org/DeepScaleR-Preview-Dataset	Let \( M \) and \( m \) be the maximum and minimum elements, respectively, of the set \( \left\{\left.\frac{3}{a}+b \right\rvert\, 1 \leq a \leq b \leq 2\right\} \). Find the value of \( M - m \).	5 - 2\sqrt{3}
agentica-org/DeepScaleR-Preview-Dataset	We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?	606
agentica-org/DeepScaleR-Preview-Dataset	The prime factorization of 2160 is $2^4 \times 3^3 \times 5$. How many of its positive integer factors are perfect squares?	6
agentica-org/DeepScaleR-Preview-Dataset	From the digits 0, 1, 2, 3, 4, 5, 6, select 2 even numbers and 1 odd number to form a three-digit number without repeating digits. The number of such three-digit numbers that are divisible by 5 is ____. (Answer with a number)	27
agentica-org/DeepScaleR-Preview-Dataset	Each of the letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "RHYME"? Express your answer as a common fraction.	\frac{1}{2}
agentica-org/DeepScaleR-Preview-Dataset	If \begin{align} a + b + c &= 1, \\ a^2 + b^2 + c^2 &= 2, \\ a^3 + b^3 + c^3 &= 3, \end{align}find $a^4 + b^4 + c^4.$	\frac{25}{6}
agentica-org/DeepScaleR-Preview-Dataset	Given the inequality $(\|x\|-1)^2+(\|y\|-1)^2<2$, determine the number of lattice points $(x, y)$ that satisfy it.	16
agentica-org/DeepScaleR-Preview-Dataset	Given real numbers $a$ and $b$ satisfying $a^{2}b^{2}+2ab+2a+1=0$, calculate the minimum value of $ab\left(ab+2\right)+\left(b+1\right)^{2}+2a$.	-\frac{3}{4}
agentica-org/DeepScaleR-Preview-Dataset	A conference center is setting up chairs in rows for a seminar. Each row can seat $13$ chairs, and currently, there are $169$ chairs set up. They want as few empty seats as possible but need to maintain complete rows. If $95$ attendees are expected, how many chairs should be removed?	65
agentica-org/DeepScaleR-Preview-Dataset	When two numbers are sequentially and randomly picked from the set {1, 2, 3, 4}, what is the probability that the product of the two picked numbers is even?	\frac{5}{6}
agentica-org/DeepScaleR-Preview-Dataset	Solve the equation \[-x^2 = \frac{3x+1}{x+3}.\]Enter all solutions, separated by commas.	-1
agentica-org/DeepScaleR-Preview-Dataset	A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is:	y+3x-4=0
agentica-org/DeepScaleR-Preview-Dataset	In response to the national medical and health system reform and the "Three Visits to the Countryside" cultural and scientific activities nationwide in 2023, to truly implement the concept of "putting the people first" and promote the transfer and decentralization of high-quality medical resources, continuously enhance the "depth" and "warmth" of medical services. The People's Hospital of our city plans to select 3 doctors from the 6 doctors recommended by each department to participate in the activity of "Healthy Countryside Visit, Free Clinic Warming Hearts." Among these 6 doctors, there are 2 surgeons, 2 internists, and 2 ophthalmologists. - $(1)$ Find the probability that the number of selected surgeons is greater than the number of selected internists. - $(2)$ Let $X$ represent the number of surgeons selected out of the 3 people. Find the mean and variance of $X$.	\frac{2}{5}
agentica-org/DeepScaleR-Preview-Dataset	How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?	65
agentica-org/DeepScaleR-Preview-Dataset	Carl decided to fence his rectangular flowerbed using 24 fence posts, including one on each corner. He placed the remaining posts spaced exactly 3 yards apart along the perimeter of the bed. The bed’s longer side has three times as many posts compared to the shorter side, including the corner posts. Calculate the area of Carl’s flowerbed, in square yards.	144
agentica-org/DeepScaleR-Preview-Dataset	Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence?	192
agentica-org/DeepScaleR-Preview-Dataset	A three-digit natural number with digits in the hundreds, tens, and units places denoted as $a$, $b$, $c$ is called a "concave number" if and only if $a > b$, $b < c$, such as $213$. If $a$, $b$, $c \in \{1,2,3,4\}$, and $a$, $b$, $c$ are all different, then the probability of this three-digit number being a "concave number" is ____.	\frac{1}{3}
agentica-org/DeepScaleR-Preview-Dataset	Let \(\triangle ABC\) be inscribed in the unit circle \(\odot O\), with the center \(O\) located within \(\triangle ABC\). If the projections of point \(O\) onto the sides \(BC\), \(CA\), and \(AB\) are points \(D\), \(E\), and \(F\) respectively, find the maximum value of \(OD + OE + OF\).	\frac{3}{2}
agentica-org/DeepScaleR-Preview-Dataset	Find $(4^4 \div 4^3) \cdot 2^8$.	1024
agentica-org/DeepScaleR-Preview-Dataset	$1.$ A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high. Find the height of the bottle.	10
agentica-org/DeepScaleR-Preview-Dataset	On a clock, there are two instants between $12$ noon and $1 \,\mathrm{PM}$ , when the hour hand and the minute hannd are at right angles. The difference in minutes between these two instants is written as $a + \dfrac{b}{c}$ , where $a, b, c$ are positive integers, with $b < c$ and $b/c$ in the reduced form. What is the value of $a+b+c$ ?	51
agentica-org/DeepScaleR-Preview-Dataset	Let \( d = \overline{xyz} \) be a three-digit number that cannot be divisible by 10. If the sum of \( \overline{xyz} \) and \( \overline{zyx} \) is divisible by \( c \), find the largest possible value of this integer \( d \).	979
agentica-org/DeepScaleR-Preview-Dataset	If a school bus leaves school with 48 students on board, and one-half of the students get off the bus at each of the first three stops, how many students remain on the bus after the third stop?	6
agentica-org/DeepScaleR-Preview-Dataset	Cut a 12cm long thin iron wire into three segments with lengths a, b, and c, (1) Find the maximum volume of the rectangular solid with lengths a, b, and c as its dimensions; (2) If these three segments each form an equilateral triangle, find the minimum sum of the areas of these three equilateral triangles.	\frac {4 \sqrt {3}}{3}
agentica-org/DeepScaleR-Preview-Dataset	If $x=3$, $y=2x$, and $z=3y$, what is the value of $z$?	18
agentica-org/DeepScaleR-Preview-Dataset	A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$ . Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular.	1 + \frac{3\sqrt{3}}{4}
agentica-org/DeepScaleR-Preview-Dataset	Suppose the edge length of a regular tetrahedron $ABC D$ is 1 meter. A bug starts at point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to this vertex with equal probability and crawls along this edge to the next vertex. What is the probability that the bug will be back at point $A$ after crawling for 4 meters?	7/27
agentica-org/DeepScaleR-Preview-Dataset	Sasa wants to make a pair of butterfly wings for her Barbie doll. As shown in the picture, she first drew a trapezoid and then drew two diagonals, which divided the trapezoid into four triangles. She cut off the top and bottom triangles, and the remaining two triangles are exactly a pair of beautiful wings. If the areas of the two triangles that she cut off are 4 square centimeters and 9 square centimeters respectively, then the area of the wings that Sasa made is $\qquad$ square centimeters.	12
agentica-org/DeepScaleR-Preview-Dataset	The minimum positive period of the function $y=\sin x \cdot \|\cos x\|$ is __________.	2\pi
agentica-org/DeepScaleR-Preview-Dataset	Given that $\alpha, \beta, \gamma$ are all acute angles and $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$, find the minimum value of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$.	2\sqrt{2}
agentica-org/DeepScaleR-Preview-Dataset	Find the root $x$ of the equation $\log x = 4 - x$ where $x \in (k, k+1)$, and $k \in \mathbb{Z}$. What is the value of $k$?	k = 3
agentica-org/DeepScaleR-Preview-Dataset	We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids?	\frac{2}{9}
agentica-org/DeepScaleR-Preview-Dataset	There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$.	80
agentica-org/DeepScaleR-Preview-Dataset	Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$ . Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.	23
agentica-org/DeepScaleR-Preview-Dataset	Nine balls numbered $1, 2, \cdots, 9$ are placed in a bag. These balls differ only in their numbers. Person A draws a ball from the bag, the number on the ball is $a$, and after returning it to the bag, person B draws another ball, the number on this ball is $b$. The probability that the inequality $a - 2b + 10 > 0$ holds is $\qquad$ .	61/81
agentica-org/DeepScaleR-Preview-Dataset	In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} x=4+ \frac { \sqrt {2}}{2}t \\ y=3+ \frac { \sqrt {2}}{2}t\end{cases}$$ (t is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ²(3+sin²θ)=12. 1. Find the general equation of line l and the rectangular coordinate equation of curve C. 2. If line l intersects curve C at points A and B, and point P is defined as (2,1), find the value of $$\frac {\|PB\|}{\|PA\|}+ \frac {\|PA\|}{\|PB\|}$$.	\frac{86}{7}
agentica-org/DeepScaleR-Preview-Dataset	Given vectors $\vec{a}$ and $\vec{b}$ with magnitudes $\|\vec{a}\|=2$ and $\|\vec{b}\|=\sqrt{3}$, respectively, and the equation $( \vec{a}+2\vec{b}) \cdot ( \vec{b}-3\vec{a})=9$: (1) Find the dot product $\vec{a} \cdot \vec{b}$. (2) In triangle $ABC$, with $\vec{AB}=\vec{a}$ and $\vec{AC}=\vec{b}$, find the length of side $BC$ and the projection of $\vec{AB}$ onto $\vec{AC}$.	-\sqrt{3}
agentica-org/DeepScaleR-Preview-Dataset	Suppose that $a^2$ varies inversely with $b^3$. If $a=7$ when $b=3$, find the value of $a^2$ when $b=6$.	6.125
agentica-org/DeepScaleR-Preview-Dataset	In three independent repeated trials, the probability of event $A$ occurring in each trial is the same. If the probability of event $A$ occurring at least once is $\frac{63}{64}$, then the probability of event $A$ occurring exactly once is $\_\_\_\_\_\_$.	\frac{9}{64}
agentica-org/DeepScaleR-Preview-Dataset	If one vertex and the two foci of an ellipse form an equilateral triangle, determine the eccentricity of this ellipse.	\dfrac{1}{2}
agentica-org/DeepScaleR-Preview-Dataset	Let \( p, q, r, s, t, u \) be positive real numbers such that \( p + q + r + s + t + u = 10 \). Find the minimum value of \[ \frac{1}{p} + \frac{9}{q} + \frac{4}{r} + \frac{16}{s} + \frac{25}{t} + \frac{36}{u}. \]	44.1
agentica-org/DeepScaleR-Preview-Dataset	Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$ , $OG = 1$ and $OG \parallel BC$ . (As usual $O$ is the circumcenter and $G$ is the centroid.)	12
agentica-org/DeepScaleR-Preview-Dataset	In the diagram, \( S \) lies on \( R T \), \( \angle Q T S = 40^{\circ} \), \( Q S = Q T \), and \( \triangle P R S \) is equilateral. The value of \( x \) is	80
agentica-org/DeepScaleR-Preview-Dataset	Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition: For any three elements in $M$ , there exist two of them $a$ and $b$ such that $a\|b$ or $b\|a$ . Determine the maximum value of $\|M\|$ where $\|M\|$ denotes the number of elements in $M$	18
agentica-org/DeepScaleR-Preview-Dataset	The symbol \( R_{k} \) represents an integer whose decimal representation consists of \( k \) consecutive 1s. For example, \( R_{3} = 111 \), \( R_{5} = 11111 \), and so on. If \( R_{4} \) divides \( R_{24} \), the quotient \( Q = \frac{R_{24}}{R_{4}} \) is an integer, and its decimal representation contains only the digits 1 and 0. How many zeros are in \( Q \)?	15
agentica-org/DeepScaleR-Preview-Dataset	Find all real values of $x$ that satisfy $\frac{x(x+1)}{(x-4)^2} \ge 12.$ (Give your answer in interval notation.)	[3, 4) \cup \left(4, \frac{64}{11}\right]
agentica-org/DeepScaleR-Preview-Dataset	Simplify $(5^7+3^6)(1^5-(-1)^4)^{10}$.	0
agentica-org/DeepScaleR-Preview-Dataset	In the diagram, the largest circle has a radius of 10 meters. Seven congruent smaller circles are symmetrically aligned in such a way that in an east-to-west and north-to-south orientation, the diameter of four smaller circles equals the diameter of the largest circle. What is the radius in meters of one of the seven smaller circles?	2.5
agentica-org/DeepScaleR-Preview-Dataset	The sequence ${a_n}$ satisfies $a_1=1$, $a_{n+1} \sqrt { \frac{1}{a_{n}^{2}}+4}=1$. Let $S_{n}=a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}$. If $S_{2n+1}-S_{n}\leqslant \frac{m}{30}$ holds for any $n\in\mathbb{N}^{*}$, find the minimum value of the positive integer $m$.	10
agentica-org/DeepScaleR-Preview-Dataset	The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity \[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \] Determine all possible values of $Q \left( - \frac{1}{2} \right)$.	2^n
agentica-org/DeepScaleR-Preview-Dataset	Given the line $y=a (0 < a < 1)$ and the function $f(x)=\sin \omega x$ intersect at 12 points on the right side of the $y$-axis. These points are denoted as $(x\_1)$, $(x\_2)$, $(x\_3)$, ..., $(x\_{12})$ in order. It is known that $x\_1= \dfrac {\pi}{4}$, $x\_2= \dfrac {3\pi}{4}$, and $x\_3= \dfrac {9\pi}{4}$. Calculate the sum $x\_1+x\_2+x\_3+...+x\_{12}$.	66\pi
agentica-org/DeepScaleR-Preview-Dataset	What is the remainder when $2^{2005}$ is divided by 7?	2
agentica-org/DeepScaleR-Preview-Dataset	Compute $$\sum_{k=1}^{2000} k(\lceil \log_{2}{k}\rceil- \lfloor\log_{2}{k} \rfloor).$$	1998953
agentica-org/DeepScaleR-Preview-Dataset	Observe the following equations: \\(① \dfrac {1}{ \sqrt {2}+1}= \dfrac { \sqrt {2}-1}{( \sqrt {2}+1)( \sqrt {2}-1)}= \sqrt {2}-1\\); \\(② \dfrac {1}{ \sqrt {3}+ \sqrt {2}}= \dfrac { \sqrt {3}- \sqrt {2}}{( \sqrt {3}+ \sqrt {2})( \sqrt {3}- \sqrt {2})}= \sqrt {3}- \sqrt {2}\\); \\(③ \dfrac {1}{ \sqrt {4}+ \sqrt {3}}= \dfrac { \sqrt {4}- \sqrt {3}}{( \sqrt {4}+ \sqrt {3})( \sqrt {4}- \sqrt {3})}= \sqrt {4}- \sqrt {3}\\);\\(…\\) Answer the following questions: \\((1)\\) Following the pattern of the equations above, write the \\(n\\)th equation: \_\_\_\_\_\_ ; \\((2)\\) Using the pattern you observed, simplify: \\( \dfrac {1}{ \sqrt {8}+ \sqrt {7}}\\); \\((3)\\) Calculate: \\( \dfrac {1}{1+ \sqrt {2}}+ \dfrac {1}{ \sqrt {2}+ \sqrt {3}}+ \dfrac {1}{ \sqrt {3}+2}+…+ \dfrac {1}{3+ \sqrt {10}}\\).	\sqrt {10}-1
agentica-org/DeepScaleR-Preview-Dataset	The arithmetic mean of an odd number of consecutive odd integers is $y$. Find the sum of the smallest and largest of the integers in terms of $y$.	2y
agentica-org/DeepScaleR-Preview-Dataset	Nine people sit down at random seats around a round table. Four of them are math majors, three others are physics majors, and the two remaining people are biology majors. What is the probability that all four math majors sit in consecutive seats?	\frac{1}{14}
agentica-org/DeepScaleR-Preview-Dataset	How many such pairs of numbers \((n, k)\) are there, for which \(n > k\) and the difference between the internal angles of regular polygons with \(n\) and \(k\) sides is \(1^{\circ}\)?	52
agentica-org/DeepScaleR-Preview-Dataset	Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer $n$ for which $n+9,16n+9,27n+9$ are also perfect squares.	280
agentica-org/DeepScaleR-Preview-Dataset	A hexagon inscribed in a circle has three consecutive sides, each of length 3, and three consecutive sides, each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides, each of length 3, and the other with three sides, each of length 5, has length equal to $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.	409
agentica-org/DeepScaleR-Preview-Dataset	Calculate: $(243)^{\frac35}$	27
agentica-org/DeepScaleR-Preview-Dataset	Find the smallest positive integer $n$ such that $1^{2}+2^{2}+3^{2}+4^{2}+\cdots+n^{2}$ is divisible by 100.	24
agentica-org/DeepScaleR-Preview-Dataset	A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid in cubic units?	288\sqrt{3}
agentica-org/DeepScaleR-Preview-Dataset	Each square of an $n \times n$ grid is coloured either blue or red, where $n$ is a positive integer. There are $k$ blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form $S_B$ . He then performs the same calculation on the red cells to compute $S_R$ . If $S_B- S_R = 50$ , determine (with proof) all possible values of $k$ .	313
agentica-org/DeepScaleR-Preview-Dataset	When $0.\overline{36}$ is expressed as a common fraction in lowest terms, what is the sum of the numerator and denominator?	15
agentica-org/DeepScaleR-Preview-Dataset	Simplify first, then evaluate: $[\left(2x-y\right)^{2}-\left(y+2x\right)\left(y-2x\right)]\div ({-\frac{1}{2}x})$, where $x=\left(\pi -3\right)^{0}$ and $y={({-\frac{1}{3}})^{-1}}$.	-40
agentica-org/DeepScaleR-Preview-Dataset	A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015},$ how far from $P_0$ is she, in inches?	1008 \sqrt{6} + 1008 \sqrt{2}
agentica-org/DeepScaleR-Preview-Dataset	If the graph of the line $y = ax + b$ passes through the points $(4,5)$ and $(8,17)$, what is $a - b$?	10
agentica-org/DeepScaleR-Preview-Dataset	Two adjacent faces of a tetrahedron, each of which is a regular triangle with a side length of 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. (12 points)	\frac{\sqrt{3}}{4}
agentica-org/DeepScaleR-Preview-Dataset	Given the polar equation of a circle is $\rho=2\cos \theta$, the distance from the center of the circle to the line $\rho\sin \theta+2\rho\cos \theta=1$ is ______.	\dfrac { \sqrt {5}}{5}
agentica-org/DeepScaleR-Preview-Dataset	On a one-way single-lane highway, cars travel at the same speed and maintain a safety distance such that for every 20 kilometers per hour or part thereof in speed, there is a distance of one car length between the back of one car and the front of the next. Each car is 5 meters long. A sensor on the side of the road counts the number of cars that pass in one hour. Let $N$ be the maximum whole number of cars that can pass the sensor in one hour. Determine the quotient when $N$ is divided by 10.	400

agentica-org/DeepScaleR-Preview-Dataset

A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee with 3 Republicans and 2 Democrats?

840

agentica-org/DeepScaleR-Preview-Dataset

Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$

7

agentica-org/DeepScaleR-Preview-Dataset

Given that $f(x)= \frac {4}{4^{x}+2}$, $S\_n$ is the sum of the first $n$ terms of the sequence $\{a\_n\}$, and $\{a\_n\}$ satisfies $a\_1=0$, and when $n \geqslant 2$, $a\_n=f( \frac {1}{n})+f( \frac {2}{n})+f( \frac {3}{n})+…+f( \frac {n-1}{n})$, find the maximum value of $\frac {a_{n+1}}{2S\_n+a\_6}$.

\frac {2}{7}

agentica-org/DeepScaleR-Preview-Dataset

Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\},$ the second from $\{A,I,O\},$ and the third from $\{D,M,N,T\}.$ When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of ADDITIONAL license plates that can be made by adding two letters?

40

agentica-org/DeepScaleR-Preview-Dataset

A club has 30 members, which includes 12 females and 18 males. In how many ways can a 5-person executive committee be formed such that at least one member must be female?

133,938

agentica-org/DeepScaleR-Preview-Dataset

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?

\frac{1}{5}

agentica-org/DeepScaleR-Preview-Dataset

Given the function $f(x) = \sin x + \cos x$, where $x \in \mathbb{R}$, - (I) Find the value of $f\left(\frac{\pi}{2}\right)$. - (II) Determine the smallest positive period of the function $f(x)$. - (III) Calculate the minimum value of the function $g(x) = f\left(x + \frac{\pi}{4}\right) + f\left(x + \frac{3\pi}{4}\right)$.

-2

agentica-org/DeepScaleR-Preview-Dataset

A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction.

\frac{9}{2}

agentica-org/DeepScaleR-Preview-Dataset

Given the function $f(x)=\ln x+\frac{1}{2}x^2-ax$, where $a\in \mathbb{R}$, it has two extreme points at $x=x_1$ and $x=x_2$, with $x_1 < x_2$. (I) When $a=3$, find the extreme values of the function $f(x)$. (II) If $x_2\geqslant ex_1$ ($e$ is the base of the natural logarithm), find the maximum value of $f(x_2)-f(x_1)$.

1-\frac{e}{2}+\frac{1}{2e}

agentica-org/DeepScaleR-Preview-Dataset

Define a power cycle to be a set $S$ consisting of the nonnegative integer powers of an integer $a$, i.e. $S=\left\{1, a, a^{2}, \ldots\right\}$ for some integer $a$. What is the minimum number of power cycles required such that given any odd integer $n$, there exists some integer $k$ in one of the power cycles such that $n \equiv k$ $(\bmod 1024) ?$

10

agentica-org/DeepScaleR-Preview-Dataset

Given that point $P$ is a moving point on the parabola $y^2=2x$, find the minimum value of the sum of the distance from point $P$ to point $A(0,2)$ and the distance from $P$ to the directrix of the parabola.

\frac{\sqrt{17}}{2}

agentica-org/DeepScaleR-Preview-Dataset

In how many ways can the number 1500 be represented as a product of three natural numbers? Variations where the factors are the same but differ in order are considered identical.

30

agentica-org/DeepScaleR-Preview-Dataset

For real numbers $x$ and $y$, define $x \spadesuit y = (x+y)(x-y)$. What is $3 \spadesuit (4 \spadesuit 5)$?

-72

agentica-org/DeepScaleR-Preview-Dataset

Positive real numbers $ x, y, z $ satisfy: $ x^{4} + y^{4} + z^{4} = 1 $. Find the minimum value of the algebraic expression $ \frac{x^{3}}{1-x^{8}} + \frac{y^{3}}{1-y^{8}} + \frac{z^{3}}{1-z^{8}} $.

\frac{9 \sqrt[4]{3}}{8}

agentica-org/DeepScaleR-Preview-Dataset

Alex and Bob have 30 matches. Alex picks up somewhere between one and six matches (inclusive), then Bob picks up somewhere between one and six matches, and so on. The player who picks up the last match wins. How many matches should Alex pick up at the beginning to guarantee that he will be able to win?

2

agentica-org/DeepScaleR-Preview-Dataset

A standard six-sided die is rolled 3 times. If the sum of the numbers rolled on the first two rolls equals the number rolled on the third roll, what is the probability that at least one of the numbers rolled is a 2?

$\frac{8}{15}$

agentica-org/DeepScaleR-Preview-Dataset

In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when $N$ is divided by $1000$.

16

agentica-org/DeepScaleR-Preview-Dataset

At 11:00 a.m. how many degrees are in the smaller angle formed by the minute hand and the hour hand of the clock?

30

agentica-org/DeepScaleR-Preview-Dataset

In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 15 and 30 units respectively, and the altitude is 18 units. Points $E$ and $F$ divide legs $AD$ and $BC$ into thirds respectively, with $E$ one third from $A$ to $D$ and $F$ one third from $B$ to $C$. Calculate the area of quadrilateral $EFCD$.

360

agentica-org/DeepScaleR-Preview-Dataset

How many different ways can the five vertices S, A, B, C, and D of a square pyramid S-ABCD be colored using four distinct colors so that each vertex is assigned one color and no two vertices sharing an edge have the same color?

72

agentica-org/DeepScaleR-Preview-Dataset

Determine the number of digits in the value of $2^{15} \times 5^{10} \times 3^2$.

13

agentica-org/DeepScaleR-Preview-Dataset

The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers?

19

agentica-org/DeepScaleR-Preview-Dataset

Xiao Ming forgot the last two digits of his WeChat login password. He only remembers that the last digit is one of the letters \$A\$, \$a\$, \$B\$, or \$b\$, and the other digit is one of the numbers \$4\$, \$5\$, or \$6\$. The probability that Xiao Ming can successfully log in with one attempt is \_\_\_\_\_\_.

\dfrac{1}{12}

agentica-org/DeepScaleR-Preview-Dataset

Let $T$ be a right triangle with sides having lengths $3$ , $4$ , and $5$ . A point $P$ is called *awesome* if P is the center of a parallelogram whose vertices all lie on the boundary of $T$ . What is the area of the set of awesome points?

1.5

agentica-org/DeepScaleR-Preview-Dataset

The value $b^n$ has both $b$ and $n$ as positive integers less than or equal to 15. What is the greatest number of positive factors $b^n$ can have?

496

agentica-org/DeepScaleR-Preview-Dataset

In triangle $ABC$, $AB = AC = 15$ and $BC = 14$. Points $D, E, F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE} \parallel \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. What is the perimeter of parallelogram $ADEF$?

30

agentica-org/DeepScaleR-Preview-Dataset

Sequence $A$ is a geometric sequence. Sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $300.$ What is the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B?$ $\bullet$ Sequence $A:$ $2,$ $4,$ $8,$ $16,$ $32,$ $\ldots$ $\bullet$ Sequence $B:$ $20,$ $40,$ $60,$ $80,$ $100,$ $\ldots$

4

agentica-org/DeepScaleR-Preview-Dataset

Given real numbers $x$, $y$ satisfying $x > y > 0$, and $x + y \leqslant 2$, the minimum value of $\dfrac{2}{x+3y}+\dfrac{1}{x-y}$ is

\dfrac {3+2 \sqrt {2}}{4}

agentica-org/DeepScaleR-Preview-Dataset

Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram which is the same as the previous one. If $AB = 36$ units and $CD = 72$ units, what is the area of square $BCFE$?

2592

agentica-org/DeepScaleR-Preview-Dataset

For functions $f(x)$ and $g(x)$, let $m\in \{x|f(x)=0\}$, $n\in \{x|g(x)=0\}$. If there exist $m$ and $n$ such that $|m-n|\leqslant 1$, then $f(x)$ and $g(x)$ are called "zero-point related functions". If the functions $f(x)=e^{x-2}+\ln(x-1)-1$ and $g(x)=x(\ln x-ax)-2$ are "zero-point related functions", then the minimum value of the real number $a$ is ____.

-2

agentica-org/DeepScaleR-Preview-Dataset

Let $ M $ and $ m $ be the maximum and minimum elements, respectively, of the set $ \left\{\left.\frac{3}{a}+b \right\rvert\, 1 \leq a \leq b \leq 2\right\} $. Find the value of $ M - m $.

5 - 2\sqrt{3}

agentica-org/DeepScaleR-Preview-Dataset

We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?

606

agentica-org/DeepScaleR-Preview-Dataset

The prime factorization of 2160 is $2^4 \times 3^3 \times 5$. How many of its positive integer factors are perfect squares?

6

agentica-org/DeepScaleR-Preview-Dataset

From the digits 0, 1, 2, 3, 4, 5, 6, select 2 even numbers and 1 odd number to form a three-digit number without repeating digits. The number of such three-digit numbers that are divisible by 5 is ____. (Answer with a number)

27

agentica-org/DeepScaleR-Preview-Dataset

Each of the letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "RHYME"? Express your answer as a common fraction.

\frac{1}{2}

agentica-org/DeepScaleR-Preview-Dataset

If \begin{align*} a + b + c &= 1, \\ a^2 + b^2 + c^2 &= 2, \\ a^3 + b^3 + c^3 &= 3, \end{align*}find $a^4 + b^4 + c^4.$

\frac{25}{6}

agentica-org/DeepScaleR-Preview-Dataset

Given the inequality $(|x|-1)^2+(|y|-1)^2<2$, determine the number of lattice points $(x, y)$ that satisfy it.

16

agentica-org/DeepScaleR-Preview-Dataset

Given real numbers $a$ and $b$ satisfying $a^{2}b^{2}+2ab+2a+1=0$, calculate the minimum value of $ab\left(ab+2\right)+\left(b+1\right)^{2}+2a$.

-\frac{3}{4}

agentica-org/DeepScaleR-Preview-Dataset

A conference center is setting up chairs in rows for a seminar. Each row can seat $13$ chairs, and currently, there are $169$ chairs set up. They want as few empty seats as possible but need to maintain complete rows. If $95$ attendees are expected, how many chairs should be removed?

65

agentica-org/DeepScaleR-Preview-Dataset

When two numbers are sequentially and randomly picked from the set {1, 2, 3, 4}, what is the probability that the product of the two picked numbers is even?

\frac{5}{6}

agentica-org/DeepScaleR-Preview-Dataset

Solve the equation \[-x^2 = \frac{3x+1}{x+3}.\]Enter all solutions, separated by commas.

-1

agentica-org/DeepScaleR-Preview-Dataset

A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is:

y+3x-4=0

agentica-org/DeepScaleR-Preview-Dataset

In response to the national medical and health system reform and the "Three Visits to the Countryside" cultural and scientific activities nationwide in 2023, to truly implement the concept of "putting the people first" and promote the transfer and decentralization of high-quality medical resources, continuously enhance the "depth" and "warmth" of medical services. The People's Hospital of our city plans to select 3 doctors from the 6 doctors recommended by each department to participate in the activity of "Healthy Countryside Visit, Free Clinic Warming Hearts." Among these 6 doctors, there are 2 surgeons, 2 internists, and 2 ophthalmologists. - $(1)$ Find the probability that the number of selected surgeons is greater than the number of selected internists. - $(2)$ Let $X$ represent the number of surgeons selected out of the 3 people. Find the mean and variance of $X$.

\frac{2}{5}

agentica-org/DeepScaleR-Preview-Dataset

How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?

65

agentica-org/DeepScaleR-Preview-Dataset

Carl decided to fence his rectangular flowerbed using 24 fence posts, including one on each corner. He placed the remaining posts spaced exactly 3 yards apart along the perimeter of the bed. The bed’s longer side has three times as many posts compared to the shorter side, including the corner posts. Calculate the area of Carl’s flowerbed, in square yards.

144

agentica-org/DeepScaleR-Preview-Dataset

Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence?

192

agentica-org/DeepScaleR-Preview-Dataset

A three-digit natural number with digits in the hundreds, tens, and units places denoted as $a$, $b$, $c$ is called a "concave number" if and only if $a > b$, $b < c$, such as $213$. If $a$, $b$, $c \in \{1,2,3,4\}$, and $a$, $b$, $c$ are all different, then the probability of this three-digit number being a "concave number" is ____.

\frac{1}{3}

agentica-org/DeepScaleR-Preview-Dataset

Let $\triangle ABC$ be inscribed in the unit circle $\odot O$, with the center $O$ located within $\triangle ABC$. If the projections of point $O$ onto the sides $BC$, $CA$, and $AB$ are points $D$, $E$, and $F$ respectively, find the maximum value of $OD + OE + OF$.

\frac{3}{2}

agentica-org/DeepScaleR-Preview-Dataset

Find $(4^4 \div 4^3) \cdot 2^8$.

1024

agentica-org/DeepScaleR-Preview-Dataset

$1.$ A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high. Find the height of the bottle.

10

agentica-org/DeepScaleR-Preview-Dataset

On a clock, there are two instants between $12$ noon and $1 \,\mathrm{PM}$ , when the hour hand and the minute hannd are at right angles. The difference *in minutes* between these two instants is written as $a + \dfrac{b}{c}$ , where $a, b, c$ are positive integers, with $b < c$ and $b/c$ in the reduced form. What is the value of $a+b+c$ ?

51

agentica-org/DeepScaleR-Preview-Dataset

Let $ d = \overline{xyz} $ be a three-digit number that cannot be divisible by 10. If the sum of $ \overline{xyz} $ and $ \overline{zyx} $ is divisible by $ c $, find the largest possible value of this integer $ d $.

979

agentica-org/DeepScaleR-Preview-Dataset

If a school bus leaves school with 48 students on board, and one-half of the students get off the bus at each of the first three stops, how many students remain on the bus after the third stop?

6

agentica-org/DeepScaleR-Preview-Dataset

Cut a 12cm long thin iron wire into three segments with lengths a, b, and c, (1) Find the maximum volume of the rectangular solid with lengths a, b, and c as its dimensions; (2) If these three segments each form an equilateral triangle, find the minimum sum of the areas of these three equilateral triangles.

\frac {4 \sqrt {3}}{3}

agentica-org/DeepScaleR-Preview-Dataset

If $x=3$, $y=2x$, and $z=3y$, what is the value of $z$?

18

agentica-org/DeepScaleR-Preview-Dataset

A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$ . Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular.

1 + \frac{3\sqrt{3}}{4}

agentica-org/DeepScaleR-Preview-Dataset

Suppose the edge length of a regular tetrahedron $ABC D$ is 1 meter. A bug starts at point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to this vertex with equal probability and crawls along this edge to the next vertex. What is the probability that the bug will be back at point $A$ after crawling for 4 meters?

7/27

agentica-org/DeepScaleR-Preview-Dataset

Sasa wants to make a pair of butterfly wings for her Barbie doll. As shown in the picture, she first drew a trapezoid and then drew two diagonals, which divided the trapezoid into four triangles. She cut off the top and bottom triangles, and the remaining two triangles are exactly a pair of beautiful wings. If the areas of the two triangles that she cut off are 4 square centimeters and 9 square centimeters respectively, then the area of the wings that Sasa made is $\qquad$ square centimeters.

12

agentica-org/DeepScaleR-Preview-Dataset

The minimum positive period of the function $y=\sin x \cdot |\cos x|$ is __________.

2\pi

agentica-org/DeepScaleR-Preview-Dataset

Given that $\alpha, \beta, \gamma$ are all acute angles and $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$, find the minimum value of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$.

2\sqrt{2}

agentica-org/DeepScaleR-Preview-Dataset

Find the root $x$ of the equation $\log x = 4 - x$ where $x \in (k, k+1)$, and $k \in \mathbb{Z}$. What is the value of $k$?

k = 3

agentica-org/DeepScaleR-Preview-Dataset

We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids?

\frac{2}{9}

agentica-org/DeepScaleR-Preview-Dataset

There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$.

80

agentica-org/DeepScaleR-Preview-Dataset

Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$ . Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.

23

agentica-org/DeepScaleR-Preview-Dataset

Nine balls numbered $1, 2, \cdots, 9$ are placed in a bag. These balls differ only in their numbers. Person A draws a ball from the bag, the number on the ball is $a$, and after returning it to the bag, person B draws another ball, the number on this ball is $b$. The probability that the inequality $a - 2b + 10 > 0$ holds is $\qquad$ .

61/81

agentica-org/DeepScaleR-Preview-Dataset

In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} x=4+ \frac { \sqrt {2}}{2}t \\ y=3+ \frac { \sqrt {2}}{2}t\end{cases}$$ (t is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ²(3+sin²θ)=12. 1. Find the general equation of line l and the rectangular coordinate equation of curve C. 2. If line l intersects curve C at points A and B, and point P is defined as (2,1), find the value of $$\frac {|PB|}{|PA|}+ \frac {|PA|}{|PB|}$$.

\frac{86}{7}

agentica-org/DeepScaleR-Preview-Dataset

Given vectors $\vec{a}$ and $\vec{b}$ with magnitudes $|\vec{a}|=2$ and $|\vec{b}|=\sqrt{3}$, respectively, and the equation $( \vec{a}+2\vec{b}) \cdot ( \vec{b}-3\vec{a})=9$: (1) Find the dot product $\vec{a} \cdot \vec{b}$. (2) In triangle $ABC$, with $\vec{AB}=\vec{a}$ and $\vec{AC}=\vec{b}$, find the length of side $BC$ and the projection of $\vec{AB}$ onto $\vec{AC}$.

-\sqrt{3}

agentica-org/DeepScaleR-Preview-Dataset

Suppose that $a^2$ varies inversely with $b^3$. If $a=7$ when $b=3$, find the value of $a^2$ when $b=6$.

6.125

agentica-org/DeepScaleR-Preview-Dataset

In three independent repeated trials, the probability of event $A$ occurring in each trial is the same. If the probability of event $A$ occurring at least once is $\frac{63}{64}$, then the probability of event $A$ occurring exactly once is $\_\_\_\_\_\_$.

\frac{9}{64}

agentica-org/DeepScaleR-Preview-Dataset

If one vertex and the two foci of an ellipse form an equilateral triangle, determine the eccentricity of this ellipse.

\dfrac{1}{2}

agentica-org/DeepScaleR-Preview-Dataset

Let $ p, q, r, s, t, u $ be positive real numbers such that $ p + q + r + s + t + u = 10 $. Find the minimum value of \[ \frac{1}{p} + \frac{9}{q} + \frac{4}{r} + \frac{16}{s} + \frac{25}{t} + \frac{36}{u}. \]

44.1

agentica-org/DeepScaleR-Preview-Dataset

Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$ , $OG = 1$ and $OG \parallel BC$ . (As usual $O$ is the circumcenter and $G$ is the centroid.)

12

agentica-org/DeepScaleR-Preview-Dataset

In the diagram, $ S $ lies on $ R T $, $ \angle Q T S = 40^{\circ} $, $ Q S = Q T $, and $ \triangle P R S $ is equilateral. The value of $ x $ is

80

agentica-org/DeepScaleR-Preview-Dataset

Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition: For any three elements in $M$ , there exist two of them $a$ and $b$ such that $a|b$ or $b|a$ . Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$

18

agentica-org/DeepScaleR-Preview-Dataset

The symbol $ R_{k} $ represents an integer whose decimal representation consists of $ k $ consecutive 1s. For example, $ R_{3} = 111 $, $ R_{5} = 11111 $, and so on. If $ R_{4} $ divides $ R_{24} $, the quotient $ Q = \frac{R_{24}}{R_{4}} $ is an integer, and its decimal representation contains only the digits 1 and 0. How many zeros are in $ Q $?

15

agentica-org/DeepScaleR-Preview-Dataset

Find all real values of $x$ that satisfy $\frac{x(x+1)}{(x-4)^2} \ge 12.$ (Give your answer in interval notation.)

[3, 4) \cup \left(4, \frac{64}{11}\right]

agentica-org/DeepScaleR-Preview-Dataset

Simplify $(5^7+3^6)(1^5-(-1)^4)^{10}$.

0

agentica-org/DeepScaleR-Preview-Dataset

In the diagram, the largest circle has a radius of 10 meters. Seven congruent smaller circles are symmetrically aligned in such a way that in an east-to-west and north-to-south orientation, the diameter of four smaller circles equals the diameter of the largest circle. What is the radius in meters of one of the seven smaller circles?

2.5

agentica-org/DeepScaleR-Preview-Dataset

The sequence ${a_n}$ satisfies $a_1=1$, $a_{n+1} \sqrt { \frac{1}{a_{n}^{2}}+4}=1$. Let $S_{n}=a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}$. If $S_{2n+1}-S_{n}\leqslant \frac{m}{30}$ holds for any $n\in\mathbb{N}^{*}$, find the minimum value of the positive integer $m$.

10

agentica-org/DeepScaleR-Preview-Dataset

The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity \[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \] Determine all possible values of $Q \left( - \frac{1}{2} \right)$.

2^n

agentica-org/DeepScaleR-Preview-Dataset

Given the line $y=a (0 < a < 1)$ and the function $f(x)=\sin \omega x$ intersect at 12 points on the right side of the $y$-axis. These points are denoted as $(x\_1)$, $(x\_2)$, $(x\_3)$, ..., $(x\_{12})$ in order. It is known that $x\_1= \dfrac {\pi}{4}$, $x\_2= \dfrac {3\pi}{4}$, and $x\_3= \dfrac {9\pi}{4}$. Calculate the sum $x\_1+x\_2+x\_3+...+x\_{12}$.

66\pi

agentica-org/DeepScaleR-Preview-Dataset

What is the remainder when $2^{2005}$ is divided by 7?

2

agentica-org/DeepScaleR-Preview-Dataset

Compute $$\sum_{k=1}^{2000} k(\lceil \log_{2}{k}\rceil- \lfloor\log_{2}{k} \rfloor).$$

1998953

agentica-org/DeepScaleR-Preview-Dataset

Observe the following equations: \$① \dfrac {1}{ \sqrt {2}+1}= \dfrac { \sqrt {2}-1}{( \sqrt {2}+1)( \sqrt {2}-1)}= \sqrt {2}-1\$; \$② \dfrac {1}{ \sqrt {3}+ \sqrt {2}}= \dfrac { \sqrt {3}- \sqrt {2}}{( \sqrt {3}+ \sqrt {2})( \sqrt {3}- \sqrt {2})}= \sqrt {3}- \sqrt {2}\$; \$③ \dfrac {1}{ \sqrt {4}+ \sqrt {3}}= \dfrac { \sqrt {4}- \sqrt {3}}{( \sqrt {4}+ \sqrt {3})( \sqrt {4}- \sqrt {3})}= \sqrt {4}- \sqrt {3}\$;\$…\$ Answer the following questions: \$(1)\$ Following the pattern of the equations above, write the \$n\$th equation: \_\_\_\_\_\_ ; \$(2)\$ Using the pattern you observed, simplify: \$ \dfrac {1}{ \sqrt {8}+ \sqrt {7}}\$; \$(3)\$ Calculate: \$ \dfrac {1}{1+ \sqrt {2}}+ \dfrac {1}{ \sqrt {2}+ \sqrt {3}}+ \dfrac {1}{ \sqrt {3}+2}+…+ \dfrac {1}{3+ \sqrt {10}}\$.

\sqrt {10}-1

agentica-org/DeepScaleR-Preview-Dataset

The arithmetic mean of an odd number of consecutive odd integers is $y$. Find the sum of the smallest and largest of the integers in terms of $y$.

2y

agentica-org/DeepScaleR-Preview-Dataset

Nine people sit down at random seats around a round table. Four of them are math majors, three others are physics majors, and the two remaining people are biology majors. What is the probability that all four math majors sit in consecutive seats?

\frac{1}{14}

agentica-org/DeepScaleR-Preview-Dataset

How many such pairs of numbers $(n, k)$ are there, for which $n > k$ and the difference between the internal angles of regular polygons with $n$ and $k$ sides is $1^{\circ}$?

52

agentica-org/DeepScaleR-Preview-Dataset

Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer $n$ for which $n+9,16n+9,27n+9$ are also perfect squares.

280

agentica-org/DeepScaleR-Preview-Dataset

A hexagon inscribed in a circle has three consecutive sides, each of length 3, and three consecutive sides, each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides, each of length 3, and the other with three sides, each of length 5, has length equal to $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

409

agentica-org/DeepScaleR-Preview-Dataset

Calculate: $(243)^{\frac35}$

27

agentica-org/DeepScaleR-Preview-Dataset

Find the smallest positive integer $n$ such that $1^{2}+2^{2}+3^{2}+4^{2}+\cdots+n^{2}$ is divisible by 100.

24

agentica-org/DeepScaleR-Preview-Dataset

A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid in cubic units?

288\sqrt{3}

agentica-org/DeepScaleR-Preview-Dataset

Each square of an $n \times n$ grid is coloured either blue or red, where $n$ is a positive integer. There are $k$ blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form $S_B$ . He then performs the same calculation on the red cells to compute $S_R$ . If $S_B- S_R = 50$ , determine (with proof) all possible values of $k$ .

313

agentica-org/DeepScaleR-Preview-Dataset

When $0.\overline{36}$ is expressed as a common fraction in lowest terms, what is the sum of the numerator and denominator?

15

agentica-org/DeepScaleR-Preview-Dataset

Simplify first, then evaluate: $[\left(2x-y\right)^{2}-\left(y+2x\right)\left(y-2x\right)]\div ({-\frac{1}{2}x})$, where $x=\left(\pi -3\right)^{0}$ and $y={({-\frac{1}{3}})^{-1}}$.

-40

agentica-org/DeepScaleR-Preview-Dataset

A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015},$ how far from $P_0$ is she, in inches?

1008 \sqrt{6} + 1008 \sqrt{2}

agentica-org/DeepScaleR-Preview-Dataset

If the graph of the line $y = ax + b$ passes through the points $(4,5)$ and $(8,17)$, what is $a - b$?

10

agentica-org/DeepScaleR-Preview-Dataset

Two adjacent faces of a tetrahedron, each of which is a regular triangle with a side length of 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. (12 points)

\frac{\sqrt{3}}{4}

agentica-org/DeepScaleR-Preview-Dataset

Given the polar equation of a circle is $\rho=2\cos \theta$, the distance from the center of the circle to the line $\rho\sin \theta+2\rho\cos \theta=1$ is ______.

\dfrac { \sqrt {5}}{5}

agentica-org/DeepScaleR-Preview-Dataset

On a one-way single-lane highway, cars travel at the same speed and maintain a safety distance such that for every 20 kilometers per hour or part thereof in speed, there is a distance of one car length between the back of one car and the front of the next. Each car is 5 meters long. A sensor on the side of the road counts the number of cars that pass in one hour. Let $N$ be the maximum whole number of cars that can pass the sensor in one hour. Determine the quotient when $N$ is divided by 10.

400