inclusionAI/Ring-lite-rl-data huggingface dataset:
Given an ellipse \( C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0) \) and a hyperbola \( C_{2}: \frac{x^{2}}{m^{2}}-\frac{y^{2}}{n^{2}}=1(m>0, n>0) \) with the same foci \( F_{1}, F_{2} \), where \( F_{1} \) is the left focus, point \( P \) is the point of intersection of the two curves in the first quadrant, and \( e_{1}, e_{2} \) are the eccentricities of curves \( C_{1} \) and \( C_{2} \) respectively. If triangle \( \triangle P F_{1} F_{2} \) is an isosceles triangle with \( P F_{1} \) as the base, then the range of values for \( e_{2} - e_{1} \) is \(\qquad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{2}{3}, +\infty \right)}
Nicky and Jill are trading baseball cards internationally. Nicky, who lives in the United States, trades two cards worth $8 each and three cards worth $5 each. Jill, who lives in Canada, trades 1 card worth CAD $21 and 2 cards worth CAD $6 each. If the exchange rate is 1 CAD = 0.8 USD and there is a 5% tax rate on the total value of the traded cards in USD, how many dollars of profit or loss does Nicky make on this trade?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$7.47}
11. (20 points) For any real numbers $a_{1}, a_{2}, \cdots, a_{5}$ $\in[0,1]$, find the maximum value of $\prod_{1 \leq i<j \leq 5}\left|a_{i}-a_{j}\right|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{21}}{38416}}
A point is randomly thrown on the segment [3, 8] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/15}
A triangle has an area of 114 square units and sides of integer length. What is the perimeter of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
Given the function $f(x)=\sin (ωx+φ)(ω > 0,|φ|\leqslant \dfrac {π}{2})$, $y=f(x- \dfrac {π}{4})$ is an odd function, $x= \dfrac {π}{4}$ is the symmetric axis of the graph of $y=f(x)$, and $f(x)$ is monotonic in $(\dfrac {π}{14}, \dfrac {13π}{84})$, determine the maximum value of $ω$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given a finite group $G$ with $100$ elements, what is the maximum probability that the equation $ab = cba$ holds for a randomly chosen pair of elements $a$ and $b$, where $c$ is a fixed element of $G$ that is not the identity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
A mahogany bookshelf has four identical-looking books which are $200$, $400$, $600$, and $800$ pages long. Velma picks a random book off the shelf, flips to a random page to read, and puts the book back on the shelf. Later, Daphne also picks a random book off the shelf and flips to a random page to read. Given that Velma read page $122$ of her book and Daphne read page $304$ of her book, the probability that they chose the same book is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6425}
Square $ABCD$ has side length $s$, and a circle centered at $E$ has radius $r$. Both $r$ and $s$ are rational numbers. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF = \sqrt{9 + 5\sqrt{2}}$. Find the ratio \(\frac{r}{s}\) in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In the tetrahedron \(A B C D\), it is known that \(DA \perp\) the base \(ABC\), the face \(ABD \perp\) the face \(BCD\), and \(BD = BC = 2\). The sum of the squares of the areas of the three faces \(DAB\), \(DBC\), and \(DCA\) is 8. Find \(\angle ADB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
The sequence $\left\{a_{n}\right\}_{n \geq 1}$ is defined by $a_{n+2}=7 a_{n+1}-a_{n}$ for positive integers $n$ with initial values $a_{1}=1$ and $a_{2}=8$. Another sequence, $\left\{b_{n}\right\}$, is defined by the rule $b_{n+2}=3 b_{n+1}-b_{n}$ for positive integers $n$ together with the values $b_{1}=1$ and $b_{2}=2$. Find \operatorname{gcd}\left(a_{5000}, b_{501}\right).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
有 $m$ 个男孩, $n$ 个女孩, ( $m, n$ 为正整数) 围坐在一个圆周上。我们在两端为男孩的弧段上插上一个数字 2 , 在相邻两端为女孩的弧段上插上一个数字 -2 , 相邻两端为一男一女的弧段上插上数字 0 , 把所插的数字总和记作 $a$, 则 $a$ 的可取值为 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2(m-n)$}
Let $w, x, y,$ and $z$ be nonzero complex numbers, and let $n$ be a positive integer. Suppose that the following conditions hold:

- $\frac{1}{w} + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3,$
- $wx + wy + wz + xy + xz + yz = 14,$
- $(w + x)^3 + (w + y)^3 + (w + z)^3 + (x + y)^3 + (x + z)^3 + (y + z)^3 = 2160,$
- $w + x + y + z + i\sqrt{n} \in \mathbb{R}.$

Compute $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties: \begin{enumerate} \item[(a)] $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\dots,t_n$; \item[(b)] $f(t_0) = 1/2$; \item[(c)] $\lim_{t \to t_k^+} f'(t) = 0$ for $0 \leq k \leq n$; \item[(d)] For $0 \leq k \leq n-1$, we have $f''(t) = k+1$ when $t_k < t< t_{k+1}$, and $f''(t) = n+1$ when $t>t_n$. \end{enumerate} Considering all choices of $n$ and $t_0,t_1,\dots,t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f(t_0+T) = 2023$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
If the function $y = x^2 - 2ax + a$ has an inverse function on the interval $x \in [1, 3]$ and $|a-1| + |a-3| \leq 4$, find the range of possible values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1] \cup [3, 4]}
Problem 2. (4 points) In a box, there are red, blue, yellow, and white balls, a total of 111. If 100 balls are drawn from the box with closed eyes, among them there will definitely be four balls, each two of which are of different colors. How many balls at least need to be drawn from the box (with closed eyes) to be able to assert with certainty that among them there are three balls, each two of which are of different colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88}
Calculate $1,000,000,000,000 - 888,777,888,777$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111,222,111,223}
How many distinct strings of length 8 can be formed using the letters $\{a, b, c, d\}$ if each string contains exactly one pair of adjacent letters that are identical?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20412}
Find the weight function for the Sturm-Liouville problem: $$\frac{d}{dx}\left[(1+x^2)\frac{dy(x)}{dx}\right] + \lambda \sigma (x)y(x)=0,\; y'(0)=y'(1)=0,\; 0 \leq x \leq 1.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{1 + x^2}}
Given the integral $I_n=\int_0^{\frac{\pi}{4}}e^{nx}(\tan^{n-1}x+\tan^nx+\tan^{n+1}x)\,dx$, determine the limit $\lim\limits_{n\to \infty} n\left(\sqrt[n^2]{nI_n}-1\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{4}\)}
Mad scientist Kyouma writes $N$ positive integers on a board. Each second, he chooses two numbers $x, y$ written on the board with $x > y$, and writes the number $x^2 - y^2$ on the board. After some time, he sends the list of all the numbers on the board to Christina. She notices that all the numbers from 1 to 1000 are present on the list. Aid Christina in finding the minimum possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Let \( x \) be a positive integer, and write \( a = \left\lfloor \log_{10} x \right\rfloor \) and \( b = \left\lfloor \log_{10} \frac{100}{x} \right\rfloor \). Here \( \lfloor c \rfloor \) denotes the greatest integer less than or equal to \( c \). Find the largest possible value of \( 2a^2 - 3b^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find the area of the region of the \( xy \)-plane defined by the inequality \( |x|+|y|+|x+y| \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/4}
Write $2017$ numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, \ldots, -\frac{1}{1008}, 0, \frac{1}{1008}, \frac{2}{1008}, \ldots, \frac{1007}{1008}, \frac{1008}{1008}$.

One processes some steps as follows: erase two arbitrary numbers $x, y$ on the blackboard and then write the number $x + 7xy + y$. After $2016$ steps, there is only one number left. Find the last number on the blackboard in the form of a simplified fraction \(-\frac{k}{m}\). Please provide the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a population of $100$ individuals randomly numbered from $0$ to $99$, and a sample of size $10$ is drawn, with the units digit of the number drawn from the $k$-th group being the same as the units digit of $m + k$, where $m = 6$, find the number drawn from the 7-th group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
There are a batch of wooden strips with lengths of \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10,\) and 11 centimeters, with an adequate quantity of each length. If you select 3 strips appropriately to form a triangle with the requirement that the base is 11 centimeters long, how many different triangles can be formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Trodgor the dragon is burning down a village consisting of $ 90$ cottages.  At time $ t \equal{} 0$ an angry peasant arises from each cottage, and every $ 8$ minutes ($ 480$ seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage.  It takes Trodgor $ 5$ seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many [b]seconds[/b] does it take Trodgor to burn down the entire village?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1920}
A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the midpoint of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ is the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A girl multiplies 987 by a certain number. If in the answer, both 9's are wrong but the other digits are correct, and the correct answer is 555681. What is the number she multiplied 987 by?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{563}
If $\frac{x-y}{x+y}=5$, what is the value of $\frac{2x+3y}{3x-2y}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the maximum area of the projection of a regular tetrahedron with edge length \(a\) onto a plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2}{2}}
On a quiz, every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions on the quiz.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
8. Find the largest real number $a$ such that for all positive integers $n$ and all real numbers $x_{0}, x_{1}, \cdots, x_{n}\left(0=x_{0}<x_{1}<\cdots\right.$ $\left.<x_{n}\right)$, we have
$$
\sum_{i=1}^{n} \frac{1}{x_{i}-x_{i-1}} \geqslant a \sum_{i=1}^{n} \frac{i+1}{x_{i}} .
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{9}}
Seven students in a class receive one failing grade every two days of school, while nine other students receive one failing grade every three days. The remaining students in the class never receive failing grades. From Monday to Friday, 30 new failing grades appeared in the class register. How many new failing grades will appear in the class register on Saturday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
设方程 xyz=900 的全部正整数解为 (x_i,y_i,z_i) (1\leq i\leq n). 则 \ds{k=1}{n}(x_k+y_k+z_k)=\underline{\qquad\qquad}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22572}
A robot vacuum cleaner "Orthogonal" decides to clean the coordinate plane. It starts from point $O$ located at the origin and travels a straight distance $X_{1}$. At the point it reaches, the vacuum cleaner turns and describes a circle centered at the origin. After completing the circle, it continues travelling in the same direction for a straight distance $X_{2}$. It then turns again to describe a circle centered at the origin, and after completing it, travels straight for a distance $X_{3}$. This process continues. The quantities $X_{1}, X_{2}, X_{3}, \ldots$ follow some distribution with expected value $a$ and variance $d$. Find the expected value of the area of the circle bounded by the $n$-th circle that the vacuum cleaner travels.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi n (d + a^2)}
Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots$, find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1026}
Given an ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 0 \), with the left focus at \( F \). A tangent to the ellipse is drawn at a point \( A \) on the ellipse and intersects the \( y \)-axis at point \( Q \). Let \( O \) be the origin of the coordinate system. If \( \angle QFO = 45^\circ \) and \( \angle QFA = 30^\circ \), find the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{3}}
A $10$-digit number is called interesting if its digits are distinct and the number is divisible by $11111$. Determine the total number of interesting numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3456}
Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{16}}
In a map with a scale of $1:50000$, if the area of a polygonal land is $100cm^{2}$, what is the actual area of the land in $m^{2}$ (represented in scientific notation)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5 \times 10^{7} m^{2}}
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by \[ q(x) = \sum_{k=1}^{p-1} a_k x^k, \] where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{p-1}{2}}
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{987^2+1597^2}
Let \( AL, BM, CN \) be the medians of triangle \( ABC \), intersecting at point \( K \). It is known that a circumcircle can be drawn around quadrilateral \( CLK M \), and \( AB = 2 \). Find the length of the median \( CN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
A car A is chasing a car B, both moving at the same speed. Car B is moving directly north, while car A is initially located 1 mile west of car B. If car A always heads directly towards car B, find the distance between the cars as time approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The ratio of irises to roses in Nikki's garden is 3:7. Initially, she has 35 roses but gives away 15 roses. After this, she plans to add 25 more roses and wants to plant enough new irises to maintain the same ratio. How many irises will she need to have in total after these adjustments?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A train of a certain length passes an oak tree in 11.999040076793857 seconds. The speed of the train is 54 km/hr. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{179.99}
Find the number of unitary ring homomorphisms from the ring \( \mathbb{Z}[X]/(X^3-1) \) to \( \mathbb{Z}_{12} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given n points on a plane, it is known that the distances between some pairs of points are 1, 2, 4, 8, 16, 32. What is the smallest possible value of n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the value of the expression \( 2a - \left( \frac{2a - 3}{a + 1} - \frac{a + 1}{2 - 2a} - \frac{a^2 + 3}{2a^{2-2}} \right) \cdot \frac{a^3 + 1}{a^2 - a} + \frac{2}{a} \) when \( a = 1580 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the value of the sum: $$\sum_{k=1}^{\infty}\frac{(k-1)(k-2)(k-3)\cdot\ldots\cdot(k-n)}{n!z^k}$$ where $n,z>1$ are fixed natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{(z-1)^{n+1}}\)}
Casey wants to decorate her toenails and fingernails. She wants to do a base coat, a coat of paint, and a coat of glitter on every nail. It takes some time to apply each coat and the same amount of time for each coat to dry before applying the next one. It takes her 120 minutes to finish decorating her fingernails and toenails. How many minutes does it take to apply and dry each coat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
On April 22, 2015, the Asia-Africa Leaders' Meeting was held in Jakarta, Indonesia. Among the leaders of five countries—A, B, C, D, and E—, all pairs except for B & E and D & E had one-on-one meetings. These meetings were arranged to take place over two days, either in the morning or the afternoon sessions, with each leader participating in at most one meeting per session. How many different ways can these one-on-one meetings be arranged?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Monroe Vitamin Shop sold equal quantities of Vitamin A, Vitamin B, and Vitamin C supplements yesterday morning. The Vitamin A supplements come in packs of 11, the Vitamin B supplements come in packs of 23, and the Vitamin C supplements come in packs of 37. What is the smallest number of each type of vitamin that the store could have possibly sold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9361}
In a regular triangular pyramid, the side of the base is equal to \(a\), and the angle between the slant height (apothem) and a lateral face is \(\frac{\pi}{4}\). Find the height of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a \sqrt{6}}{6}}
[ Transfer of side, diagonal, etc.] The Law of Sines

In trapezoid $ABCD (AB \| CD)$, $AB + CD = b$, diagonals $AC$ and $BD$ are related by $5AC = 3BD$. Find the area of the trapezoid if $\angle BAC = 2 \angle DBA$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\sqrt{11}}{64}b^{2}}
Consider the function $f(x)$ defined as follows:
- $f(x)$ has domain $\mathbb{R}$.
- $f(x+2)=4f(x)$ for all $x \in \mathbb{R}$.
- $f(x)=x^2-1$ for $x \in [0,2]$.

Find the maximum value of $k$ such that $f(x) \leq 15$ for all $x \in [k, \infty)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given two sets with common elements that are not subsets of each other, we say these sets "intersect". For the sets $M=\{x|ax^{2}-1=0,a > 0\}$ and $N=\{- \frac {1}{2}, \frac {1}{2},1\}$, if sets $M$ and $N$ "intersect", then $a=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations, the urn contains six balls. Calculate the probability that the urn contains three balls of each color. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}}
The side lengths $a, b, c$ of a triangle $ABC$ are positive integers. Let:

$$T_{n} = (a+b+c)^{2n} - (a-b+c)^{2n} - (a+b-c)^{2n} + (a-b-c)^{2n}$$

for any positive integer $n$. If $\frac{T_{2}}{2T_{1}} = 2023$ and $a > b > c$, determine all possible perimeters of the triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
A $8 \times 8 \times 8$ cube has three of its faces painted red and the other three faces painted blue (ensuring that any three faces sharing a common vertex are not painted the same color), and then it is cut into 512 $1 \times 1 \times 1$ smaller cubes. Among these 512 smaller cubes, how many have both a red face and a blue face?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Alexander goes to school at 12 pm. On Tuesday, Alexander's classes are in the following order : Maths, History, Geography, Science and Music. Each of his classes varies in duration: Maths is 45 minutes, History lasts 1 hour and 15 minutes, Geography is 30 minutes long, Science class takes 50 minutes, and there is a 10-minute break between each class. Just now, Alexander's Science class got over. Can you estimate the time now in pm?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:50}