inclusionAI/Ring-lite-rl-data huggingface dataset:
A rope can make 70 rounds of the circumference of a cylinder whose radius of the base is 14 cm. How many times can the rope go around a cylinder having a different radius if it can go around the second cylinder 49 times?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
9 - 3 ÷ 1 / 3 + what number equals 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the smallest value of $(a+5)^2 + (b-2)^2 + (c-9)^2$ for all real numbers $a, b, c$ satisfying $a^2 + b^2 + c^2 - ab - bc - ca = 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
A regular tetrahedron has two vertices on the body diagonal of a cube with side length 12. The other two vertices lie on one of the face diagonals not intersecting that body diagonal. Find the side length of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \sqrt{3}}
In how many ways every unit square of a $2018$ x $2018$ board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \cdot (2018!)^2}
Find the least prime number greater than $1000$ that divides $2^{1010} \cdot 23^{2020} + 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1013}
Each pan of a balance has k weights, numbered from 1 to k. The left pan is heavier. It turns out that if we interchange the places of any two weights with the same number, then either the right pan becomes heavier or the two pans reach an equilibrium. For what k this is possible? Please provide the sum of all possible values of k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Kartik wants to shade three of the squares in a grid blue, and Lucy wants to shade the remaining two squares red. How many possible finished grids are there where Lucy's red squares are next to each other? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the least positive integer with the property that the product of its digits is $9! ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2578899}
The data of sample A are as follows: 52, 54, 54, 56, 56, 56, 55, 55, 55, 55. If the data of sample B are exactly the data of sample A each increased by 6, then what statistical measure remains the same for both samples? Express your answer as a single word or phrase representing a statistical measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Standard Deviation}
A regular octahedron $A B C D E F$ is given such that $A D, B E$, and $C F$ are perpendicular. Let $G, H$, and $I$ lie on edges $A B, B C$, and $C A$ respectively such that \frac{A G}{G B}=\frac{B H}{H C}=\frac{C I}{I A}=\rho. For some choice of $\rho>1, G H, H I$, and $I G$ are three edges of a regular icosahedron, eight of whose faces are inscribed in the faces of $A B C D E F$. Find $\rho$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1+\sqrt{5}) / 2}
Let  $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$ ,  $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$ ,  $C(x)=x^3-6x^2+5x+15$ ,  $H(x)=x^4+2x^3+3x^2+4x+5$ ,  $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$ ,  $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$ ,  $O(x)=|x|\log |x|\log\log |x|$ ,  $T(x)=\sum_{n=1}^{\infty}\frac{n^x}{\left(n!\right)^3}$ , and  $Z(x)=\frac{x^{21}}{2016+20x^{16}+16x^{20}}$  for any real number  $x$  such that the functions are defined. Determine   $$ C(C(A(M(A(T(H(B(O(N(A(N(Z(A(2016)))))))))))))). $$  *2016 CCA Math Bonanza Lightning #5.3*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that the volume of the tetrahedron \(ABCD\) is 40, \(AB = 7\), \(BC = 8\), \(AC = 9\), and the orthogonal projection of vertex \(D\) onto the plane \(ABC\) is precisely the incenter \(H\) of triangle \(ABC\), what is the surface area of the tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60 + 12 \sqrt{5}}
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. What time does she finish the third task? Express your answer in time format (HH:MM PM).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:30 \text{ PM}}
Suppose a regular tetrahedron \( P-ABCD \) has all edges equal in length. Using \(ABCD\) as one face, construct a cube \(ABCD-EFGH\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \( PA \) and \( CF \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 + \sqrt{2}}{4}}
Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions:

i) $|A| = n$, $|B| = m$ and $A$ is a subset of $B$
ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$

Determine the minimum value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 3n - 3 }
How many of the numbers $1\cdot 2\cdot 3$, $2\cdot 3\cdot 4$,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
27
}
A number of women proposed once each, of whom one-eighth were widows. Consequently, a number of men were to be married, of whom one-eleventh were widowers. Of the proposals made to widowers, one-fifth were declined. All the widows were accepted. Thirty-five forty-fourths of the widows married bachelors. One thousand two hundred and twenty-one spinsters were declined by bachelors. The number of spinsters accepted by bachelors was seven times the number of widows accepted by bachelors. How many women proposed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11616}
In $\triangle ABC$, $AB > AC$, $\angle BAC = 45^\circ$. Point $E$ is the intersection of the external angle bisector of $\angle BAC$ with the circumcircle of $\triangle ABC$. Point $F$ is on $AB$ such that $EF \perp AB$. Given $AF = 1$ and $BF = 5$, find the area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \sqrt{2}}
Each of two teams has seven players numbered 1 to 7. In the first game, the two players numbered 1 play each other. The loser of each game is eliminated and replaced by the next player of the same team, until all players from one team have been eliminated. Let $N$ be the number of possible sequences of games. Find the remainder when $N$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432}
Given that there are \(2^n\) coins distributed among a group of kids, an interchange occurs when one of the kids has at least half of all the coins. During an interchange, the kid with at least half of the coins gives each of the other kids the same number of coins as they currently have. If all the coins are with one kid, no interchange is possible. What is the greatest possible number of consecutive interchanges? (\(n\) is a natural number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Define a sequence of sets \( \{S_i\}_{i=0}^{\infty} \) such that \( S_0 = \{0\} \) and \( S_i = \bigcup\limits_{k=0}^{i-1} \{S_k\} \). Find the number of brackets in the power set of \( S_4 \) if the empty set is written as \( \{\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{272}
Find a real number \(a\) such that the function \(g_a(x) = \begin{cases} x^a \sin \left(\frac{1}{x}\right) & x \neq 0 \\ 0 & x = 0 \end{cases}\) is differentiable on \(\mathbb{R}\) and its derivative \(g_a'(x)\) is unbounded on the interval \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Suppose we have a nonnegative sequence $b_m \in \ell^1$. Define a new sequence $d_n$ as:

$$d_n = \alpha^n \sum\limits_{m=n}^{\infty} b_m \binom{m}{n}\beta^{m-n}.$$

where $0 < \alpha < 1$ and $0 \leq \beta < 1$ satisfy $\alpha + \beta \leq 1$. Find a constant $L$ such that $\|d_n\|_{\ell^1} \leq L \|b_n\|_{\ell^1}$. Determine the value of $L$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the random variable $\xi$ follows the normal distribution $N(1, \sigma^2)$, if $P(\xi > 3) = 0.023$, then $P(-1 \leq \xi \leq 3)$ equals to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.954}
记 I=\ds{m=1}{\infty}\ds{n=1}{\infty}\df{1}{n!n!+m!n!}，则 [10I]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given an equilateral triangle $\triangle ABC$. Points $D$ and $E$ are taken on side $BC$ such that $BC = 3DE$. Construct an equilateral triangle $\triangle DEF$ and connect point $A$ to point $F$. Draw $DG$ parallel to $AF$ intersecting side $AB$ at point $G$, and draw $EH$ parallel to $AF$ intersecting side $AC$ at point $H$. Draw perpendiculars $GI \perp AF$ and $HJ \perp AF$. Given that the area of $\triangle BDF$ is $45$ and the area of $\triangle DEF$ is $30$, find the ratio $GI \div HJ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are 8 different books and 10 different movies in the 'crazy silly school' series. If you read a certain number of the books and watched 61 of the movies, there are 2 more movies than books in the 'crazy silly school' series. How many books did you read?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \( D \) be a regular ten-sided polygon with edges of length 1. A triangle \( T \) is defined by choosing three vertices of \( D \) and connecting them with edges. How many different (non-congruent) triangles \( T \) can be formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A digit is inserted between the digits of a two-digit number to form a three-digit number. Some two-digit numbers, when a certain digit is inserted in between, become three-digit numbers that are $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
The Wolf and Ivan Tsarevich are 20 versts away from a source of living water, and the Wolf is taking Ivan Tsarevich there at a speed of 3 versts per hour. To revive Ivan Tsarevich, one liter of water is needed, which flows from the source at a rate of half a liter per hour. At the source, there is a Raven with unlimited carrying capacity; it must gather the water, after which it will fly towards the Wolf and Ivan Tsarevich at a speed of 6 versts per hour, spilling a quarter liter of water every hour. After how many hours will it be possible to revive Ivan Tsarevich?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 	ext{ hours}}
Calculate the variance of the random variable \( X = \int_{0}^{1} W^2(t) \, dt \), where \( W(t) \) is a Wiener process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Let $[x]$ denote the greatest integer less than or equal to $x$. When $0 \leqslant x \leqslant 10$, find the number of distinct integers represented by the function $f(x) = [x] + [2x] + [3x] + [4x]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is  $\frac{m}{n}$  , where m and n are relatively prime positive integers. Find  $m + n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{843}
Find the maximum possible value of the sum \(\sum_{1 \leq i < j \leq n} |x_i-x_j|\), where \(x_1,...,x_n\) are integers in the interval \([1,35]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10404}
A school has 9 English teachers, some history, and 6 geography teachers. Each teacher can teach 2 subjects max. The minimum number of teachers required is 11. How many history teachers are there in the school?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the range of real number k such that the inequality kx^2 + 2kx - (k+2) < 0 always holds true. Express your answer in interval notation, including any relevant mathematical notation (e.g., parentheses, brackets).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 0)}
We wish to color the integers $1,2,3, \ldots, 10$ in red, green, and blue, so that no two numbers $a$ and $b$, with $a-b$ odd, have the same color. (We do not require that all three colors be used.) In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{186}
Tim and Allen are playing a match of tennis. In a match of tennis, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability \( \frac{3}{4} \), and in the even-numbered games, Allen wins with probability \( \frac{3}{4} \). What is the expected number of games in a match?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{3}}
Example 6 Given
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1 \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1 \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1 \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1
\end{array}\right.
$$

Find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine the maximum distance from the origin to a point on the curve given by the equation \(|z+e^{i\theta}w|=r_z+r_w\), where \(z\) and \(w\) are fixed complex numbers and \(\theta\) varies over the interval \([0,2\pi]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r_z + r_w}
How many total strikes do the clocks make in a day if they strike once every half an hour and strike $1,2,3 \ldots 12$ times each hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Determine the number of integral solutions for the equation \(x^4 - ax^3 - bx^2 - cx - d = 0\), given that \(a \geq b \geq c \geq d\) and \(a, b, c, d \in \mathbb{N}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find \( F^{(n+2)}(x) \) for \( F(x) = \int_0^x (x-t)^n u(t) \, dt \), where \( u(t) \) is a function of \( t \), using the Leibniz rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n! \, u'(x)}
How many circles of radius \(\frac{\pi}{4}\) can be drawn within a circle of radius \(\pi\) such that they do not intersect one another?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Evaluate \( \lim_{x\to\infty }x\int_1^{x}\frac{e^t}{t}dt-e^x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
点集 $S=\left\{(x, y) \mid x \leq 5, y \leq 4, x, y \in \mathrm{~N}^{*}\right\}$, 则由 S 中的点可组成 $\qquad$个不同的三角形。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1056}
What is the minimum order of a differential equation whose solution can be expressed as a linear combination of three trigonometric functions with different arguments?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that the number of Master's degrees awarded by faculty F of university U follows a Poisson distribution, and the funding for faculty F in a given year is proportional to the sum of the number of degrees awarded in the three preceding years, find the variance of the funding for year \((2000 + i)\), where \(i \ge 3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\lambda}
Let $x, y$ be complex numbers such that \frac{x^{2}+y^{2}}{x+y}=4$ and \frac{x^{4}+y^{4}}{x^{3}+y^{3}}=2$. Find all possible values of \frac{x^{6}+y^{6}}{x^{5}+y^{5}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 \pm 2 \sqrt{17}}
In which dimension does a Riemannian manifold with constant scalar curvature also have constant sectional curvature?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the perimeter of a rectangle if the sum of the lengths of its three different sides can be equal to 6 or 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $Q$ be a smooth hyperquadric in $\mathbb{P}^5_{\mathbb{C}}$. Determine the dimension of the variety $F_2(Q)$, which parametrizes all 2-planes contained in $Q$. Assume that each hyperquadric contains at least one plane and that the dimension of the Grassmannian $G(k,n)$ is given by $(n-k)(k+1)$. Find $\dim(F_2(Q))$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the limit: \[ \lambda = \lim_{z \to \infty} \left[ 2^z - \left(\frac{4}{3}\right)^z - \zeta(\zeta(z)) \right] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Determine the values of $n$ for which it is possible to collect all the water into a single glass in a finite number of steps, given that each of the $n$ glasses initially contains an equal amount of water and you can pour as much water from any glass into any other as the latter contains.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{5 \operatorname{tg} x+2}{2 \sin 2 x+5} d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} \ln \left(\frac{14}{5}\right)}
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1-\cos(2x)}{x^2}dx \) using the Residue Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Determine the cardinality of the set of subrings of \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
What is the molecular weight of 4 moles of K2Cr2O7?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1176.80}
Tom Brady is shy of the passing yards record in a season, which is 5999 yards. He has some passing yards with 6 games left in the season and needs to average 300 passing yards per game to beat the record. How many passing yards does he currently have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4200}
The product \( 29 \cdot 11 \), and the numbers 1059, 1417, and 2312, are each divided by \( d \). If the remainder is always \( r \), where \( d \) is an integer greater than 1, what is \( d - r \) equal to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Consider the following infinite geometric series: $$\frac{7}{8}-\frac{14}{27}+\frac{28}{81}-\dots$$ Find the common ratio of this series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{2}{3}}
Let \(\{a_{n}\}\) be an integer sequence such that for any \(n \in \mathbf{N}^{*}\), the condition \((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\) holds. Additionally, \(2008 \mid a_{2007}\). Find the smallest positive integer \(n \geqslant 2\) such that \(2008 \mid a_{n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{501}
Keesha wants to get her hair, nails, and makeup done for prom, and also receive a pedicure and facial. She decides to go to three different boutiques for these services. For hair, updos cost $50 and come with a 10% student discount. Manicures cost $30 and have a buy-one-get-one 50% off deal on pedicures, which cost $35 each. Makeup applications cost $40, with a 7% tax applied. Pedicure services are regularly $45 but are included in the buy-one-get-one deal on manicures. Facial treatments cost $60 and have a special 15% off promo. Keesha decides to tip each beautician 20%. Calculate the total cost Keesha will pay for these services, taking into account service costs, discounts, taxes, and tips.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$223.56}
After watching a movie, viewers rated the movie sequentially with an integer score from 0 to 10. At any given moment, the movie rating was calculated as the sum of all given scores divided by their number. At a certain moment $T$, the rating turned out to be an integer, and then with each new voting viewer, it decreased by one unit. What is the maximum number of viewers who could have voted after moment $T ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the limit: \( \lim_{n\to\infty}n\cdot \sin(2\pi e(n!)) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Use the divergence theorem to evaluate the surface integral \( I = \iint_S \mathbf{G} \cdot \mathbf{n} \, dS \), where \( \mathbf{G} = (x+y) \mathbf{i} +(y+z) \mathbf{j} +(z+x) \mathbf{k} \) and \( S \) is the surface of the sphere \( x^2+y^2+z^2 =9 \) between \( -2<z<2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72\pi}
A train 400 m long can cross an electric pole in 40 sec. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let $\Omega = D(0,2) \setminus \{\frac{1}{2}\}$, where $D(0,2)$ is a disc, and let $f$ be holomorphic in $\Omega$. Given that $\frac{1}{2}$ is a simple pole for $f$ with residue $1$, calculate the limit:
\[ \lim_{n \to \infty} \frac{f^{(n)}(0)}{2^{n}n!} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Calculate the variance \( \mathrm{Var}(e^{-X}) \) given that the random variable \( X \) has the probability density function \( f_\theta(x) = \exp(-(x-\theta) - \exp(-(x-\theta))) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-2\theta}}
设集合 $T=\{\left(a,b,c\right)|a,b,c\in\mathbf{Z}_{+}$,且 $a、b、c$
可构成某个三角形的三边长}.则$\sum_{(a,b,c)\in T}\frac{2^a}{3^b\times5^c}$的值为$\_$原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O.$ Let the intersection points of the perpendicular bisector of $CH$ with $AC$ and $BC$ be $X$ and $Y$ respectively. Lines $XO$ and $YO$ cut $AB$ at $P$ and $Q$ respectively. If $XP+YQ=AB+XY,$ determine $\measuredangle OHC.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$90^{\circ} $
}
Determine the image of the imaginary axis under the function \( f(z) = \frac{z+1}{z-1} \), where \( f: \mathbb{C} \to \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|w| = 1}