inclusionAI/Ring-lite-rl-data huggingface dataset:
Allen, Shiela's brother, likes to play with blocks. Shiela repainted Allen's old blocks in different colors and shapes. If Allen has 100 identical blocks, and there are 4 shapes (cubes, cylinders, spheres, and pyramids) with different sizes and 5 blocks for each color of paint used, determine the number of color and shape combinations Shiela created.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{810}
Compute the smallest positive integer $M$ such that there exists a positive integer $n$ for which:

- $M$ is the sum of the squares of some $n$ consecutive positive integers.
- $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4250}
4. Consider the game "Battleship" on a $5 \times 5$ square grid. What is the minimum number of shots needed to guarantee hitting a ship of size $1 \times 4$ cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that $\alpha$ is an angle in the third quadrant, simplify $\sqrt{ \frac{1+\cos\left(\frac{9\pi}{2}-\alpha\right)}{1+\sin(\alpha-5\pi)}} - \sqrt{ \frac{1-\cos\left(-\frac{3\pi}{2}-\alpha\right)}{1-\sin(\alpha-9\pi)}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\tan(\alpha)}
after getting 2 successive discounts , a shirt with a list price of rs 150 is available at rs 105 . if the second discount is 12.55 , find the first discount .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Given the set of 10 integers {1, 2, 3, ..., 9, 10}, choose any 3 distinct numbers to be the coefficients of the quadratic function f(x) = ax^2 + bx + c. Determine the number of ways to choose the coefficients such that f(1)/3 is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Suppose $f_n(x)=x^n\ln{x}$, $ n \in \mathbb{N}$. Find $$ \lim_{n\to\infty} \frac{f_n^{(n)}(\frac{1}{n})}{n!} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
某银行一年期定期储蓄年利率为2.25%，如果存款到期不取出继续留存于银行，银行自
动将本金及80%的利息（利息须交纳20%利息税，由银行代交）自动转存一年期定期储蓄，
某人以一年期定期储蓄存入银行20万元，则5年后，这笔钱款交纳利息税后的本利和为
________元.（精确到1元）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{218660}
16. On the plane there are two triangles, each with side lengths 18,24 and 30 . If the two triangles do not completely overlap, but share the same circumcircle as well as the same inscribed circle, find the area of the region common to both triangles.
(2 marks)
平面上有兩個三角形，邊長都是 18、24、30。若兩個三角形並非完全重疊，
但它們的外接圓相同, 內切圓也相同, 求兩個三角形重疊部分的面積。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Along the path between Neznayka's and Sineglazka's houses, there were flowers planted in a row: 15 peonies and 15 tulips mixed together. While visiting Neznayka, Sineglazka watered all the flowers she encountered. After watering the 10th tulip, she ran out of water, leaving 10 flowers unwatered. The next day, while visiting Sineglazka, Neznayka picked all the flowers he encountered. After picking the 6th tulip, he decided he had enough for a bouquet. How many flowers were left growing along the path?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
At the baseball stadium, the snack stand offers popcorn and soda. The price before tax for popcorn is $14.70 for 5 bags and the price for soda is $2 per can. If you wanted to buy 4 bags of popcorn and 3 cans of soda, a 10% discount is given if you buy more than 3 bags of popcorn and a 5% discount applies if you buy more than 2 cans of soda. The sales tax rate is 7% for soda and 6% for popcorn. Calculate the total cost for your purchases, including the respective discounts and sales tax for each item.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$17.32}
Given the function $f(x) = 2 + \log_{3}x$, where $x \in [1,9]$, find the range of the function $y = [f(x)]^{2} + f(x^{2})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[6,13]}
Among 30 people with different ages, select two groups: the first group with 12 people and the second group with 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. Determine the number of ways to make this selection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4060}
The set of real numbers $x$ for which  

$$\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1$$  

is the union of intervals of the form $a<x\le b$. What is the sum of the lengths of these intervals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given complex numbers \( z \) and \( \omega \) satisfying the following two conditions:
1. \( z + \omega + 3 = 0 \);
2. \( |z|, 2, |\omega| \) form an arithmetic sequence.

Is there a maximum value for \( \cos(\arg z - \arg \omega) \)? If so, find it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
Determine the number of $8$-tuples of nonnegative integers $(a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $0 \le a_k \le k$, for each $k = 1,2,3,4$, and $a_1 + a_2 + a_3 + a_4 + 2b_1 + 3b_2 + 4b_3 + 5b_4 = 19$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1540}
Find the constant term in the expansion of (1+x)(e^(-2x)-e^x)^9. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Consider a trapezoidal field where it's planted uniformly with wheat. The trapezoid has the following measurements: side $AB$ is 150 m, base $AD$ (the longest side) is 300 m, and the other base $BC$ is 150 m. The angle at $A$ is $75^\circ$, and the angle at $B$ is $105^\circ$. At harvest time, all the wheat is collected at the point nearest to the trapezoid's perimeter. What fraction of the crop is brought to the longest side $AD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
In triangle \( ABC \), the angle bisector \( AL \) is drawn. Points \( E \) and \( D \) are marked on segments \( AB \) and \( BL \), respectively, such that \( DL = LC \) and \( ED \parallel AC \). Find the length of segment \( ED \), given \( AE = 15 \) and \( AC = 12 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A square flag has a cross of equal width. The cross is symmetric about the diagonals of the square, with a small blue square in the center and the rest being white. If the entire cross (including the red and blue portions) occupies 36% of the flag's area, then what percentage of the flag's area is occupied by the blue square? Express your answer as a whole number percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Three friends met at a café: sculptor Belov, violinist Chernov, and artist Ryzhov. "It's remarkable that each of us has white, black, and red hair, but none of us has hair color matching our last name," said the black-haired person. "You're right," said Belov. What is the artist's hair color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{black}
3. Find the largest integer $n$ with $\lambda(n)=12$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65520}
Péter took one stamp each of values 1, 2, 3, ..., 37 forints from his stamp collection. He wants to group them so that each group has the same total value of stamps. In how many ways can he do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many numbers \( (x_1 x_2 \ldots x_{1998}) \) are there such that:

\[
(x_1 x_2 \ldots x_{1998}) = 7 \cdot 10^{1996} (x_1 + x_2 + \ldots + x_{1998})
\]

where \( (x_1 x_2 \ldots x_{1998}) \) shows a number with 1998 digits in the decimal system? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Maria must work 8 hours each day, excluding her 45-minute lunch break. If she begins working at 7:25 A.M. and takes her lunch break at noon, what time will her workday end? Express your answer in 12-hour clock format (e.g., 4:10 P.M.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:10 P.M.}
5.2. A boy presses a vertical rod against a rough horizontal surface with his thumb. Then he gradually tilts the rod, keeping the component of the force directed along the rod applied to its end unchanged. At an angle of inclination of the rod to the horizontal $\alpha=70^{\circ}$, the rod starts to slide along the surface. Determine the coefficient of friction between the surface and the rod, if in the vertical position the normal reaction force of the plane was 21 times the weight of the rod. Round your answer to the hundredths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.35}
In a single-elimination tournament consisting of \(2^9 = 512\) teams, there is a strict ordering on the skill levels of the teams, but Joy does not know that ordering. The teams are randomly put into a bracket and they play out the tournament, with the better team always beating the worse team. Joy is then given the results of all 511 matches and must create a list of teams such that she can guarantee that the third-best team is on the list. What is the minimum possible length of Joy's list?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
The derivative of the function $f(x)$ is given by $f'(x) = x(1 - x)$. Find the interval(s) over which $f(x)$ is monotonically increasing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1]}
Given the function $f(x)=x^{2}+ax+1$, if there exists $x\_0$ such that $|f(x\_0)| \leqslant \frac{1}{4}$ and $|f(x\_0+1)| \leqslant \frac{1}{4}$ hold simultaneously, then the range of the real number $a$ is $\_\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\sqrt{6}, -2] \cup [2, \sqrt{6}]}
Let $k$ be a positive integer such that $1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{13} = \frac{k}{13!}$. Find the remainder when $k$ is divided by $7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Jim starts with $80 in his investment portfolio. Over the next two years, he experiences the following growth rates and transactions:

Year 1:
- Q1: 15% growth
- Q2: 5% growth, Jim adds $28
- Q3: 6% growth, Jim withdraws $10
- Q4: 3% decline

Year 2:
- Q1: 10% growth, Jim adds $40
- Q2: 4% decline, Jim withdraws $20
- Q3: 2% growth, Jim adds $12
- Q4: 7% decline, Jim withdraws $15

Calculate Jim's final portfolio worth after two years from when he started, considering all transactions and growth/decline rates throughout the quarters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$132.23}
\( \mathrm{n} \) is a positive integer not greater than 100 and not less than 10, and \( \mathrm{n} \) is a multiple of the sum of its digits. How many such \( \mathrm{n} \) are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given a Pascal distribution with parameters r = 3 and p, where 0 < p < 1, find the maximum value of p that satisfies the condition P(X=6) >= P(X=5). Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/5}
In the independence test, suppose H0: variable X is unrelated to variable Y. Given the estimated probability p(k^2 >= 10.83) ≈ 0.001, what is the probability that variable X is related to variable Y? Express your answer as a percentage (%).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99.9%}
For each positive integer $n$ , find the number of $n$ -digit positive integers that satisfy both of the following conditions:
$\bullet$ no two consecutive digits are equal, and
$\bullet$ the last digit is a prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[
\frac{2}{5}\left(9^n + (-1)^{n+1}\right)
\]}
All the altitudes of the pyramid $ABCD$, whose faces are acute-angled triangles, are equal to each other. It is known that $AB=9$, $BC=13$, and the angle $ADC$ is $60^\circ$. Find the edge $BD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{133}}
In a certain land only one or two - letter words are used. The parliament decided to forbid the use of the seventh letter. The people of this land have lost 129 words because of the prohibition. How many different letters does the local language have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
In the sport of diving from a high platform, there is a functional relationship between the athlete's height above the water surface $h$ (m) and the time $t$ (s) after the jump: $h(t)=-4.9t^2+6.5t+10$. Determine the moment when the instantaneous velocity is $0 \text{ m/s}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{65}{98}}
Mark initially has 13 trees in his backyard. He decides to remove 3 trees due to their old age. Mark then buys 18 new trees from a nursery, but only plants 12 of them immediately, saving the rest for later. If Mark plants an additional 25% more trees than he initially planned, how many trees will he have in his backyard in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
4. (28th Russian Mathematical Olympiad) Find the smallest positive integer that can be represented as the sum of 2002 positive integers with equal sums of digits, and also as the sum of 2003 positive integers with equal sums of digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10010}
59. Let $a, b$ be positive constants. $x_{1}, x_{2}, \cdots, x_{n}$ are positive real numbers. $n \geqslant 2$ is a positive integer. Find the maximum value of $y=$ $\frac{x_{1} x_{2} \cdots x_{n}}{\left(a+x_{1}\right)\left(x_{1}+x_{2}\right) \cdots\left(x_{n-1}+x_{n}\right)\left(x_{n}+b\right)}$. (1999 Polish Mathematical Olympiad Problem)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{(\sqrt[n+1]{a}+\sqrt[n+1]{b})^{n+1}}}
There are 2 boys and 3 girls, making up a total of 5 students standing in a line. If boy A does not stand at either end, and exactly two of the three girls stand together, find the number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Dima wrote a sequence of 0's and 1's in his notebook. He then noticed that 1 follows 0 sixteen times, 0 follows 1 fifteen times, and 0 follows 01 eight times. How many times does 0 follow 11?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{172822}
Let $A X B Y$ be a cyclic quadrilateral, and let line $A B$ and line $X Y$ intersect at $C$. Suppose $A X \cdot A Y=6, B X \cdot B Y=5$, and $C X \cdot C Y=4$. Compute $A B^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{242}{15}}
Suppose that $x$ and $y$ are real numbers that satisfy the two equations: $x^{2} + 3xy + y^{2} = 909$ and $3x^{2} + xy + 3y^{2} = 1287$. What is a possible value for $x+y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Given the sets $P=\{x|\log_{2}x < -1\}$ and $Q=\{x||x| < 1\}$, determine the set $P\cap Q$. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \frac {1}{2})}
A sphere is inscribed in a cone. The ratio of the radius of the circle of tangency of the spherical and conical surfaces to the radius of the base of the cone is $k$. Find the cosine of the angle between the slant height of the cone and the plane of the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - k}
The charge for a single room at hotel P is some percent less than the charge for a single room at hotel R and 10 percent less than the charge for a single room at hotel G. The charge for a single room at hotel R is 12.500000000000004 percent greater than the charge for a single room at hotel G. What is the percentage difference between the charges for a single room at hotel P and hotel R?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
As shown in the diagram, \(E, F, G, H\) are the midpoints of the sides \(AB, BC, CD, DA\) of the quadrilateral \(ABCD\). The intersection of \(BH\) and \(DE\) is \(M\), and the intersection of \(BG\) and \(DF\) is \(N\). What is \(\frac{S_{\mathrm{BMND}}}{S_{\mathrm{ABCD}}}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
A car begins its journey traveling at a constant speed of 80 km/h for the first 120 km. Then, the car stops at a gas station for 15 minutes before resuming its trip at a constant speed of 100 km/h for the next 180 km. Finally, the car makes another stop at a checkpoint for 10 minutes before covering the last 75 km at a constant speed of 120 km/h. Given this information, what is the car's average speed for the entire journey, including stops?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86.37}
Let $\alpha$ and $\beta$ be two interior angles of a triangle, and let $\tan \alpha$ and $\tan \beta$ be the two real roots of the equation $x^{2}+p(x+1)+1=0$. Find the measure of the obtuse angle of this triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135^\circ}
From the set {0,1,2,3,4,5,6}, two distinct numbers a and b are chosen to form the complex number a+bi. How many imaginary numbers can be formed? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
In an acute triangle $ABC$ with an angle $\angle ACB = 75^\circ$, altitudes $AA_3$ and $BB_3$ intersect the circumscribed circle at points $A_1$ and $B_1$, respectively. On the lines $BC$ and $CA$, select points $A_2$ and $B_2$, respectively, such that the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$. Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
A train travels from Albany to Syracuse, a distance of 120 miles, at an average rate of 50 miles per hour. The train then continues to Rochester, which is 90 miles from Syracuse, before returning to Albany. On its way to Rochester, the train's average speed is 60 miles per hour. Finally, the train travels back to Albany from Rochester, with the total travel time of the train, including all three legs of the journey, being 9 hours and 15 minutes. What was the average rate of speed of the train on the return trip from Rochester to Albany?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39.25}
James needs to get a new pair of glasses. His frames cost $200, the lenses cost $500, and he wants to add anti-glare and scratch-resistant coatings. Each coating costs $75, but they're currently running a promotion where if you get both, they will only charge for the more expensive coating. James's insurance covers 70% of the cost of lenses and 50% of the cost of the anti-glare coating, but it doesn't cover the scratch-resistant coating. He has a $50 off coupon that is only valid on frames, a 10% off loyalty discount for being a long-time customer, but this discount is only applicable to the total cost of the lenses and coatings. He also received a $20 cash reward from his last purchase, which he can use on any item. Lastly, there's a local sales tax of 7% that's added after all the discounts have been subtracted. How much will he have to pay in total, after all the discounts have been applied and including the sales tax?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$319.66}
If 3 eighths of a number is 141, what will be the approximate value of 32.08% of this number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Evaluate the integral $\int_{\lvert z \rvert = 2} \frac{dz}{z \sin\left( \frac{1}{z} \right)}$ using residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A, B, and C are lattice points in the plane. The triangle \( \triangle ABC \) contains exactly one lattice point, \( X \), in its interior. The line \( AX \) meets \( BC \) at \( E \). What is the largest possible value of \( \frac{AX}{XE} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
For the curve  $ C: y = \frac {1}{1 + x^2}$ , Let  $ A(\alpha ,\ f(\alpha)),\ B\left( - \frac {1}{\alpha},\ f\left( - \frac {1}{\alpha} \right)\right)\ (\alpha > 0).$  Find the minimum area bounded by the line segments  $ OA,\ OB$  and  $ C,$  where  $ O$  is the origin.

Note that you are not allowed to use the integral formula of  $ \frac {1}{1 + x^2}$  for the problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi - 1}{2}}
A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
A type of bulrush grows 3 feet on the first day and subsequently decreases its growth by half each day; while a type of reed grows 1 foot on the first day and subsequently doubles its height each day. Using the reference data: lg 2 = 0.3010, lg 3 = 0.4771, calculate the number of days until the height of the bulrush and the reed are equal. Round your answer to 0.1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.6}
Let $C$ be a circle centered at point $O$, and let $P$ be a point in the interior of $C$. Let $Q$ be a point on the circumference of $C$ such that $PQ \perp OP$, and let $D$ be the circle with diameter $PQ$. Consider a circle tangent to $C$ whose circumference passes through point $P$. Let the curve $\Gamma$ be the locus of the centers of all such circles. If the area enclosed by $\Gamma$ is $1/100$ the area of $C$, then what is the ratio of the area of $C$ to the area of $D$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
Mary tried to improve her health by changing her diet. At first, she dropped a dozen pounds. Then, she added back some weight, then dropped three times more weight than she initially lost, and finally gained back half of a dozen pounds. She weighed 99 pounds at the start of her change in diet, and her final weight was 81 pounds. What is the ratio of the weight she added back to the weight she initially lost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}