inclusionAI/Ring-lite-rl-data huggingface dataset:
A line passing through the focus \( F \) of the parabola \( y^2 = 2px \) intersects the parabola at points \( A \) and \( B \). The projections of points \( A \) and \( B \) onto the directrix of the parabola are \( A_1 \) and \( B_1 \), respectively. Find the angle \( \angle A_1 F B_1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Calculate the flux of the vector field \(\mathbf{a} = y^2 \mathbf{j} + z \mathbf{k}\) through the surface defined by \(z = x^2 + y^2\), bounded by the plane \(z=2\). Assume the normal vector is outward with respect to the region enclosed by the paraboloid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
Evaluate the limit $\lim_{x\to 0} \dfrac{\ln(1+ax+x^2)}{\sqrt{x+b}-1}$ given that $a,b\in\mathbb{R}$ and $a+b=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the area of the region defined by the inequalities: \(0 \leq x \leq \sqrt{y}\), \(0 \leq y \leq \sqrt{x}\), and \(x+y \leq \frac{3}{4}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{96}}
Consider the graph in 3-space of $0=xyz(x+y)(y+z)(z+x)(x-y)(y-z)(z-x)$. This graph divides 3-space into $N$ connected regions. What is $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
7. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Find the number of elements in the set $\left\{n \left\lvert\, n=\left[\frac{k^{2}}{2005}\right]\right., 1 \leqslant k \leqslant 2004, k \in \mathbf{N}\right\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1503}
Given $A=\left\{ x\mid 2x^{2}-3x+1\leqslant 0 \right\}$ and $B=\left\{ x^{2}-(2a+1)x+a(a+1)\leqslant 0 \right\}$, if $(C_{R}B\cap A=\varnothing)$, find the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0\leqslant a\leqslant \frac{1}{2}}
A semicircle has diameter $\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\overline{AD}$, and points $E$ and $F$ lie on the arc of the semicircle. The two right triangles $\vartriangle BCF$ and $\vartriangle CDE$ are congruent. The area of $\vartriangle BCF$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Find the minimum of $\sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2$ where $x$ is a real numbers
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1435
}
Arrange the numbers 1, 2, 3, 4, 5, 6 in a row, denoting the $i$-th number as $a_i$ ($i=1,2,\ldots,6$). If $a_1 \neq 1$, $a_3 \neq 3$, $a_5 \neq 5$, and $a_1 < a_3 < a_5$, then the number of different arrangements is ___ (answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
There were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Determine the volume, in cubic light years, of the set of all possible locations for a base such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27 \sqrt{6} \pi}{8}}
What is the largest number of integers that can be selected from the natural numbers from 1 to 3000 such that the difference between any two of them is different from 1, 4, and 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
At a certain university, 70% of the professors are women, and 70% of the professors are tenured. If a certain percentage of the professors are women, tenured, or both, and 50% of the men are tenured, what percent of the professors are women, tenured, or both?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85\%}
Determine the cardinality of the set of all infinite binary sequences that do not contain the subsequence 101.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\aleph_0$}
John uses the bathroom every certain minutes. During a 2.5-hour movie, he uses the bathroom 3 times. How many minutes are there between each bathroom visit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37.5}
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that $1$ belongs to $\mathcal{S}$ and that $2002$ is the largest element of $\mathcal{S},$ what is the greatest number of elements that $\mathcal{S}$ can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the number of integer solutions \((x, y)\) to the equation \(23107\ | \ 5x^2 + 35y^2 + 2023\) such that \(0 \leq x, y \leq 23106.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23100}
A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles than the area of the small semicircles? Express your answer to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{178}
The new dormitory manager received 10 keys to open 10 rooms. He knows that each key can only open one of the rooms, but he does not know which key opens which room. Now, to open all the locked rooms, he needs to try at most     times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Given a triangular prism \( P-ABC \) where the three face angles at the top vertex are all \( 60^\circ \), i.e., \( \angle APB = \angle BPC = \angle CPA = 60^\circ \), and the areas of the three lateral faces are \( \frac{\sqrt{3}}{2}, 2, \text{and}\, 1 \), find the volume of the triangular prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{2}}{3 \sqrt{3}}}
Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14th jump lands at the origin?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{613470}
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is the value of $a+b+c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given a graph with N vertices (3 ≤ N ≤ 1000) and M edges, determine the minimum number of additional vertices that need to be added to ensure that every vertex is either included in the new set of vertices or is adjacent to a vertex in the new set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are 4 pieces of part $A$ weighing 5 tons each, 6 pieces of part $B$ weighing 4 tons each, 11 pieces of part $C$ weighing 3 tons each, and 7 pieces of part $D$ weighing 1 ton each. If all the parts are to be transported at once, what is the minimum number of trucks, each with a capacity of 6 tons, required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Professor Rackbrain recently asked his young friends to find all five-digit perfect squares for which the sum of the numbers formed by the first two digits and the last two digits equals a perfect cube. For example, if we take the square of the number 141, which is equal to 19881, and add 81 to 19, we get 100 – a number that unfortunately is not a perfect cube.

How many such solutions exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Suppose that $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all possible values of $x^2 + y^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-90}
2. 103 Use 94 bricks, each of size $4 \times 10 \times 19$, to stack into a tower 94 bricks high, where each brick can be oriented to provide a height of 4, 10, or 19. If all 94 bricks are used, how many different heights can the tower be (in inches)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{465}
Let $\Gamma$ be the maximum possible value of $a+3b+9c$ among all triples $(a,b,c)$ of positive real numbers such that
\[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\]
If $\Gamma = \frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, then find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the closed form solution for the sequence $$S_n = \frac{x}{1-x^2}+\frac{x^2}{1-x^4}+ \ldots + \frac{x^{2^{n-1}}}{1-x^{2^{n}}}$$ given that $S_n$ follows the recurrence relation $S_n = \frac{S_{n-1}}{1-x^{2^n}} + \frac{x^{2^n}}{1-x^{2^{n+1}}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x}{1-x}-\frac{x^{2^{n}}}{1-x^{2^{n}}}}
An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs (they cannot split again). At a certain moment, there are 21 heads and 73 legs on the battlefield. How many monsters are there at this moment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Three points are placed independently and at random in a unit square. What is the expected value of the area of the triangle formed by these three points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{144}}
The whole arc of a circle with radius $R$ is divided into four large and four small parts, which alternate one after another. The large part is twice as long as the small part. Determine the area of the octagon whose vertices are the division points of the arc of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R^2 (1 + \sqrt{3})}
The sum of two natural numbers and their greatest common divisor is equal to their least common multiple. Determine the ratio of the two numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 : 2}
Rebecca drinks soda daily, and her consumption varies between 1/4 and 1/2 of a bottle each day. She recently bought a combination of two different sizes of soda bottles: five 6-packs of 500mL sodas and two 12-packs of 1L sodas from a store.

If Rebecca consumes an equal amount of the two different sized sodas and her average daily consumption fluctuates within the given range, approximately how many bottles of soda will she have left after five weeks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
What is the maximum number of diagonals of a regular  $12$ -gon which can be selected such that no two of the chosen diagonals are perpendicular?

Note: sides are not diagonals and diagonals which intersect outside the  $12$ -gon at right angles are still considered perpendicular.

*2018 CCA Math Bonanza Tiebreaker Round #1*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Determine the maximum number of fleas that can be placed on a $10 \times 10$ chessboard such that each flea jumps to an adjacent square every minute in one of four directions (up, down, left, right) and reverses direction when it cannot continue. During one hour, no two fleas should ever occupy the same square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Eleven girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Ami, starts with the ball, skips the next three girls and throws to the fifth girl, who then skips the next three girls and throws the ball to the ninth girl. If the throwing pattern continues, including Ami's initial throw, how many total throws are necessary for the ball to return to Ami?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the total number of five-digit numbers \(abcde\) that satisfy the following conditions: \( a < b, \, b > c > d, \, d < e, \, a > d, \, b > e. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2892}
A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper measuring 5 cm \( \times 8 \) cm. The first sheet is placed at the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet touches the top-right corner. What is the length \( x \) in centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
There are 2n people seated around a circular table, and m cookies are distributed among them. The cookies may be passed around under the following rules:
- Each person may only pass cookies to his or her neighbours.
- Each time someone passes a cookie, he or she must also eat a cookie.

Let A be one of these people. Find the least m such that no matter how m cookies are distributed to begin with, there is a strategy to pass cookies so that A receives at least one cookie.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
George went to a movie theater to see a movie. He paid $16 for the ticket and decided to buy some snacks for himself and a friend. The nachos were half the price of the ticket, and he decided to buy 2 servings. The small popcorn was 25% cheaper than the nachos, and he bought 3 bags. The soda was 75% of the popcorn price, and they got 4 sodas in total. Additionally, they purchased a combo deal that included 2 boxes of candy at a fixed price of $7. There was also a 10% discount on the total food bill, except for the combo deal, and a 5% sales tax applied to the entire purchase. How much did George pay in total for the visit to the movie theater?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$73.29}
Yasmin's bank account had an initial balance of X dollars. On Monday, she withdrew half of her balance, and on Tuesday, she deposited $30 into her account. On Wednesday, she deposited a $50 birthday check from her grandmother. This check was worth a quarter of her new balance after the check was added. Finally, on Thursday, Yasmin withdrew $20 from her account. After all these transactions, her final balance was $160. Find the initial balance (X) of Yasmin's account in dollars before any transactions took place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$200}
A function $f\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ is linear in each of the $x_{i}$ and $f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\frac{1}{x_{1} x_{2} \cdots x_{n}}$ when $x_{i} \in\{3,4\}$ for all $i$. In terms of $n$, what is $f(5,5, \ldots, 5)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6^{n}}}
3. For any four-element subset $\mathrm{P}$ of the set $\{1,2,3, \ldots, 12\}$, denote

$$
\mathrm{Q}=\{3 x: x \in \mathrm{P}\} \quad \text { and } \quad \mathrm{R}=\{4 x: x \in \mathrm{P}\} \text {. }
$$

Determine the number of such sets $P$ for which the numbers from $P, Q, R$ give all possible non-zero remainders when divided by 13.

(Jaromír Šimša)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Teams A and B play a basketball match. The rule is that if one team wins 4 games, that team wins the match and the game ends (best of seven). The probability of teams A and B winning each game is 1/2. Let ξ be the number of games required. Calculate Eξ.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{93}{16}}
How many sequences \( a_{1}, a_{2}, \ldots, a_{8} \) of zeroes and ones have \( a_{1}a_{2} + a_{2}a_{3} + \cdots + a_{7}a_{8} = 5 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given the parabola equation y^2 = 8x and a line passing through the focus F(2,0) with a slope angle of 60 degrees, calculate the length of the chord intercepted by the parabola. Express your answer as a single fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{32}{3}}
Mac needs to fill 2012 barrels with apple cider. He starts with 0 energy. Each minute, he can either rest to gain 1 energy or use his current energy. If he has \( n \) energy, he can expend \( k \) energy \((0 \leq k \leq n)\) to fill up to \( n(k+1) \) barrels. What is the minimal number of minutes required for Mac to fill all the barrels?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
George Arfken's book: Mathematical Methods for Physicists has the following problem in a chapter on contour integration: $\displaystyle \int_0^{\infty} \dfrac{(\log x)^2}{x^2 + 1} dx$. Their suggestion is to make the substitution $x \rightarrow z=e^t$. What is the value of the integral?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi^3}{8}\)}
Let $C$ be a circumference of radius $a$ in the plane $2x+2y+z=4$, centered at the point (1,2,-2). Given the vector field $F(x,y,z)=(y-x,z-x,x-y)$, determine the value of $a$ such that the line integral $\oint_C F \cdot dr = -\frac{8\pi}{3}$. Use Stokes' Theorem to solve the problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the recursive function defined by \( C_n = C_{n-1} + iC_{n-2} \), with initial conditions \( C_1 = 1 \) and \( C_2 = 1 \). If \( C_{10} \) is expressed in the form \( a + bi \), find the value of \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12}
In three 90-minute training matches, seven players \( \mathrm{A}_{1}, \mathrm{A}_{2}, \ldots, \mathrm{A}_{7} \) from country \( F \) are observed. At any moment, exactly one player is on the field. Players \( \mathrm{A}_{1}, \mathrm{A}_{2}, \mathrm{A}_{3}, \mathrm{A}_{4} \) have total playing times divisible by 7, and players \( \mathrm{A}_{5}, \mathrm{A}_{6}, \mathrm{A}_{7} \) have total playing times divisible by 13. With unlimited substitutions, how many different total playing time combinations are possible for each player?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208806}
Let  $P$  be a  $10$ -degree monic polynomial with roots  $r_1, r_2, . . . , r_{10} \ne $  and let  $Q$  be a  $45$ -degree monic polynomial with roots  $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$  where  $i < j$  and  $i, j \in \{1, ... , 10\}$ . If  $P(0) = Q(1) = 2$ , then  $\log_2 (|P(1)|)$  can be written as  $a/b$  for relatively prime integers  $a, b$ . Find  $a + b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let \( n \) be an integer and 
$$
m = (n-1001)(n-2001)(n-2002)(n-3001)(n-3002)(n-3003).
$$

Given that \( m \) is positive, find the minimum number of digits of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
In a trapezoid, the diagonals intersect at a right angle, and one of them is equal to the midsegment. Determine the angle that this diagonal forms with the bases of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Given that $||\vec{a}+\vec{b}||=||\vec{a}-\vec{b}||=2||\vec{a}||$, find the angle between the vectors $(\vec{a}-\vec{b})$ and $\vec{b}$. Express your answer in radians, as a fraction of $\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{6}}
The volume of the solid generated by rotating the circle $x^2 + (y + 1)^2 = 3$ around the line $y = kx - 1$ for one complete revolution is what?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{3}\pi}
A checkered paper square of size \( 8 \times 8 \) was folded several times along the lines of the grid to form a small \( 1 \times 1 \) square. This small square was then cut along a segment connecting the midpoints of two opposite sides of the square. Into how many parts could the original square break apart by this cut?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In the ring $R=\mathbb{Z}_5[x]$, consider the element $b=(x-4)(x-3)^2$. Let $A$ be the set of elements $a$ in $R$ such that the equation $at=b$ has a solution $t\in R$. Determine the number of elements in the set $A$. \(\text{Express your answer as an integer.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A trader starts with 500 bags of rice in stock. On the first day, she sells 25% of her stock and restocks twice the number of sold bags. On the second day, she sells 30% of her new stock and restocks 50% more rice bags than she sold on that day. By the end of the third day, the total number of bags she has left is 20% less than her stock after the second day restocking. How many bags of rice did she sell on the third day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
What is the units digit when $5^4$ is multiplied out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a vector \( \vec{x} \in \mathbb{R}^n \) (or \( \mathbb{C}^n \)), determine the maximum possible rank of a matrix \( A \in \mathbb{R}^{m \times n} \) (or \( \mathbb{C}^{m \times n} \)) such that \( A\vec{x} = \vec{0} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min(m, n - 1)}
A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the flux of the vector field \( \vec{F} = -9\hat{j} - 3\hat{k} \) across the surface \( z = y \) bounded by the sphere \( x^2 + y^2 + z^2 = 16 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48\sqrt{2}\pi}
A train that is 456 meters long is running at a speed of 43 km/hour. In what time will it pass a bridge that is 254 meters long?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59.44}
If $g(x)$ is a cubic polynomial with roots $p, q, r$, and its derivative $g'(x)$ divides $g(2x)$, find the value of $C$ such that $Cpq = 16r^2$. Given $g(x) = A(x-p)(x-q)(x-r)$ and $g(2x) = A(2x-p)(2x-q)(2x-r)$, determine the value of $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that a vowel (v) is followed by a consonant (c) 75% of the time and a consonant is followed by a vowel 65% of the time, determine the most likely sequence of vowels and consonants when three successive letters are chosen.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{cvc}