inclusionAI/Ring-lite-rl-data huggingface dataset:
An inspector rejects 0.02% of the meters as defective. How many meters will he examine to reject a certain number of meters, if he examined 10,000 meters to reject 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5,000}
Given proposition p: ∃x ≥ 0, 2^x = 3, what is its negation? Express your answer as a logical proposition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∀x ≥ 0, 2^x ≠ 3}
Given non-zero real numbers \( x \) and \( y \) that satisfy
\[
(5x + y)^{2019} + x^{2019} + 30x + 5y = 0,
\]
find the value of \(\frac{x}{y}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{6}}
A person buys an article at a certain price. He sells the article at a price that gives him a profit of 35%, which is Rs. 675. What was the original price of the article?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1928.57}
A solid rectangular block is created using $N$ congruent 1-cm cubes adhered face-to-face. When observing the block to maximize visibility of its surfaces, exactly $252$ of the 1-cm cubes remain hidden from view. Determine the smallest possible value of $N.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{392}
Felicity, Adhira, and Benjamin took separate road trips in their respective vehicles. Felicity's car runs on gasoline, Adhira's on diesel, and Benjamin's on a mix of ethanol and biodiesel. The total fuel consumption for their trips are related as follows: Felicity used 2 times the amount of gasoline as Adhira used diesel, while Benjamin used a mix of 35% ethanol and 65% biodiesel, wherein the combined volume was 5 gallons less than Adhira's diesel consumption.

Their vehicles have different fuel consumption rates: Felicity's car uses 35 miles per gallon (mpg) for gasoline, Adhira's uses 25 mpg for diesel, and Benjamin's uses 30 mpg for ethanol and 20 mpg for biodiesel.

Each of them drove different distances, with Adhira covering 150 miles more than Felicity, who in turn covered 50 miles more than Benjamin. Together, they covered a total distance of 1750 miles. Calculate the number of gallons of gasoline Felicity used on her trip.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
In the Cartesian coordinate system $x O y$, consider the set $K=\{(x, y) \mid x, y=-1,0,1\}$. If three points are randomly chosen from $K$, what is the probability that there exists a pair of points among them with a distance of $\sqrt{5}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{7}}
Given that the geometric sequence $\{a\_n\}$ has a common ratio of $q$, and $S\_n$, $S\_{n-1}$, $S\_{n+1}$ form an arithmetic sequence, find the value of $\frac{a\_5 + a\_7}{a\_3 + a\_5}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Without stoppages, a train travels a certain distance with an average speed of some km/h, and with stoppages, it covers the same distance with an average speed of 125 km/h. The train stops for 30 minutes per hour. What is the average speed of the train without stoppages?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Compute $(2 \cos 30^\circ + 2i \sin 30^\circ)^{10}.$ Enter your answer in rectangular form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512 - 512i\sqrt{3}}
Three fair six-sided dice, each numbered 1 through 6, are rolled. What is the probability that the three numbers that come up can form the sides of a triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{37}{72}}
一个"位置"是指直角坐标平面上的一个点 (x,y)，其中 x,y 都是不超过 20 的正整数.最初时，所有 400 个位置都是空的. 甲乙两人轮流摆放石子，由甲先进行. 每次轮到甲时，他在一个空的位置上摆上一个新的红色石子，要求任意两个红色石子所在位置之间的距离都不等于 \sqrt{5}. 每次轮到乙时，他在任意一个空的位置上摆上一个新的蓝色石子，蓝色石子所在位置与其它石子所在位置之间的距离可以是任意值. 如此这般进行下去直至某人无法再摆放石子.求最大的整数 k，使得无论乙如何摆放蓝色石子，甲总能保证至少摆放 k 个红色石子.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
The height \( PO \) of the regular quadrilateral pyramid \( PABC D \) is 4, and the side of the base \( ABCD \) is 6. Points \( M \) and \( K \) are the midpoints of segments \( BC \) and \( CD \). Find the radius of the sphere inscribed in the pyramid \( PMKC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12}{13+\sqrt{41}}}
Evaluate the following limit: $$\lim \limits_{n \to \infty}\sum_{k=1}^{n}\frac{k}{n^2}\sin{\frac{k}{n^2}}\cos{\frac{k}{n}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
1. It is known that there is a prime number between $10^{17}$ and $10^{17}+10$. Try to find this prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^{17}+3}
On the lateral side \( CD \) of trapezoid \( ABCD \) (\( AD \parallel BC \)), a point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is drawn to segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
The radius of the base of a cone is $R$. Two mutually perpendicular generators divide the lateral surface area of the cone into parts with the ratio $1: 2$. Find the volume of the cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi R^3 \sqrt{2}}{6}}
24. Brother Bear and Little Bear went on an adventure to find treasure and obtained a treasure chest. This treasure chest requires a ten-digit password consisting only of the digits 0 and 1 to open. Brother Bear first entered “1000000000”, and the treasure chest indicated that 1 digit was incorrect; Little Bear then entered “0110000000”, and the treasure chest indicated that 2 digits were incorrect. If the treasure chest no longer gives hints for subsequent password attempts, how many more times do the clever Brother Bear and Little Bear need to try to definitely open the treasure chest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the limit of the sequence $\{a_n\}$ defined by $$a_1 = 1, a_{n+1} = \frac{3n-1}{3n} a_n + \frac{1}{n^2}, \quad \forall n\ge 1.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
2. [7] Suppose that $a, b, c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of
$$
20\left(a^{2}+b^{2}+c^{2}+d^{2}\right)-\sum_{\text {sym }} a^{3} b,
$$
where the sum is over all 12 symmetric terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
4. In how many ways can two knights, two bishops, two rooks, a queen, and a king be arranged on the first row of a chessboard so that the following conditions are met:
1) The bishops stand on squares of the same color;
2) The queen and the king stand on adjacent squares. (20 points).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
Given points O(0, 0), A(2, 0), B(1, $-2\sqrt{3}$), and P, a moving point on the curve $y = \sqrt{1 - \frac{x^2}{4}}$, determine the range of values for $\overrightarrow{OP} \cdot \overrightarrow{BA}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 4]}
Consider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j > 1$, $a_j = m > 1$, then $m-1$ appears in the sequence $a_1,a_2,\ldots,a_{j-1}$. Such sequences include $1,1,2,1,3,2$ and $1,2,3,1,4,1$ but not $1,2,2,4,3,2$. How many such sequences of six positive integers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{203}
For how many three-element sets of distinct positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$40$}
If a rectangle is formed by a wire with a length of $32cm$, then the maximum value of the area of the rectangle is ______ $cm^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Given vectors $\overset{⇀}{m}=(\cos x,-\sin x)$ and $\overset{⇀}{n}=(\cos x,\sin x-2 \sqrt{3}\cos x)$, where $x\in\mathbb{R}$, let $f(x)= \overset{⇀}{m}\cdot \overset{⇀}{n} = \frac{24}{13}$, and $x\in\left[ \frac{\pi}{4}, \frac{\pi}{2}\right]$, then the value of $\sin 2x$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12 \sqrt{3}+5}{26}}
Find the largest 12-digit number \( N \) that satisfies the following two conditions:

- The decimal representation of \( N \) contains six digits "4" and six digits "7";
- In the decimal representation of \( N \), no four consecutive digits form the number "7444".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{777744744744}
已知$f ( x ) = a \sin x + b \sqrt [ 3 ] { x } + c \ln ( x + \sqrt { x ^ { 2 } + 1 } ) + 1 0 0 3 ( a 、 b 、$c为实数），且$f ( \lg \log _ { 3 } 1 0 ) = 1$,则$f ( \lg 1 g 3 ) = \underline { }$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005}
某国有 36 座城市和 5 家航空公司，任意两座城市之间恰由一家航空公司执行双向飞行的航班. 如果一家航空公司在城市 A, B 与 B, C 之间均执行航班，则称三元组 (A,B,C) 是"恰当连通的". 求最大的正整数 k，使得至少有 k 个三元组是恰当连通的.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3780}
In convex quadrilateral  $ABCD$ ,  $\angle BAD = \angle BCD = 90^o$ , and  $BC = CD$ . Let  $E$  be the intersection of diagonals  $\overline{AC}$  and  $\overline{BD}$ . Given that  $\angle AED = 123^o$ , find the degree measure of  $\angle ABD$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78^\circ}
Let \(ABCDEF\) be a regular hexagon and let point \(O\) be the center of the hexagon. How many ways can you color these seven points either red or blue such that there doesn't exist any equilateral triangle with vertices of all the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given an arithmetic sequence $\left\{a_{n}\right\}$ with the sum of the first 12 terms being 60, find the minimum value of $\left|a_{1}\right| + \left|a_{2}\right| + \cdots + \left|a_{12}\right|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Find the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that Jo climbs a flight of 8 stairs, and Jo can take the stairs 1, 2, 3, or 4 at a time, or a combination of these steps that does not exceed 4 with a friend, determine the number of ways Jo and the friend can climb the stairs together, including instances where either Jo or the friend climbs alone or together.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
How many fractions less than 1 are there, where the numerator and the denominator are single-digit positive integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, \( \frac{4}{3} \) seconds, \( \frac{5}{3} \) seconds, and 2 seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in 1 minute? (Include first and last.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. Find the area of the region bounded by these $6$ reflected arcs. The original answer is in the form of k\sqrt{m} - n\pi, please provide the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
方程 \df{1}{x^2}+\df{1}{y}+\df{1}{z}=\df{4}{3} 有__________组整数解.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find all positive integers \( k \) such that for any positive numbers \( a, b, c \) satisfying \( k(ab + bc + ca) > 5(a^2 + b^2 + c^2) \), there exists a triangle with side lengths \( a, b, c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A sequence consists of the digits \(122333444455555 \ldots\) such that each positive integer \(n\) is repeated \(n\) times, in increasing order. Find the sum of the 4501st and 4052nd digits of this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{204}
已知正整数 $x_{1}<x_{2}<\cdots<x_{9}$ 且 $x_{1}+x_{2}+\cdots x_{9}=220$, 则在 $x_{1}+x_{2}+\cdots+x_{5}$ 取得最大值的情况下, $x_{9}-x_{1}$ 的最大值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In cyclic pentagon \(ABCDE\), \(\angle ABD = 90^\circ\), \(BC = CD\), and \(AE\) is parallel to \(BC\). If \(AB = 8\) and \(BD = 6\), find \(AE^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{338}{5}}
There are 100 people in a room with ages 1, 2, ..., 100. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
## Task 10/75

Given is a regular pyramid with a square base. Each side face forms an angle $\alpha$ with the base. Two spheres are inscribed in this pyramid such that one sphere touches all five faces of the pyramid, while the other touches the four side faces of the pyramid and the surface of the first sphere.

In what ratio do the volumes of the two spheres stand to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\tan^{6}\frac{\alpha}{2}}
In the triangle \(ABC\), medians \(AN\) and \(CM\) are drawn, and \(\angle ABC = 120^\circ\). A circle passing through points \(A\), \(M\), and \(N\) also passes through point \(C\). The radius of this circle is 7. Find the area of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7\sqrt{3}}
On a table, there are three cones standing on their bases, touching each other. The radii of their bases are 6, 24, and 24. A truncated cone is placed on the table with its smaller base downward, and it has a common generatrix with each of the other cones. Find the radius of the smaller base of the truncated cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A target consisting of five zones is hanging on the wall: a central circle (bullseye) and four colored rings. The width of each ring equals the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye scores 315 points. How many points does hitting the blue (second to last) zone score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
A circle constructed on the smaller leg of an isosceles trapezoid as its diameter touches the longer leg, which is equal to \(a\).

Find the midline of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{2}}
Real numbers $x, y$, and $z$ are chosen from the interval $[-1, 1]$ independently and uniformly at random. What is the probability that

$$\vert x\vert +\vert y\vert +\vert z\vert +\vert x+y+z\vert=\vert x+y\vert +\vert x+z\vert +\vert y+z\vert$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
已知数列 $\left\{(-1)^{n} \frac{3 n+2}{n(n+1) \cdot 2^{n+1}}\right\}$ 的前 $n$ 项和为 $S_{n}$. 计算 $S_{18}$ 的值。标准答案格式为$\frac{a}{b^{c} \times d}-\frac{e}{f}$，计算a+b+c+d+e+f的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Let $ABCD$ be a convex quadrilateral with $AB=2$, $AD=7$, and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}$. Find the square of the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Let positive integers a, b, and c have the following properties:

When a is divided by 29, the remainder is 7.
When b is divided by 47, the remainder is 22.
When (a + b) is divided by 23, the remainder is c.

Determine how many possible combinations of a, b, and c are less than 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{735}
Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 - x$. Let $$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)} $$ for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3984}
Six chairs sit in a row.  Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left.  There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right.  The probability that this will [b]not[/b] happen is given by $\frac{m}{n}$ 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{1106}
Find all pairs  $(a,b)$  of positive integers  such that  $a!+b$  and  $b!+a$  are both powers of  $5$ .

*Nikola Velov, North Macedonia*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5, 5)}
Find the residue of the function f(z) at z = 2, where f(z) is defined as: $$f(z)=(ze^{z-3}+(3-z)e^z)e^{(z-2)^{2}+(z-2)^{-2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Intersect an octahedron with a plane so that the cross-section is a regular hexagon. Calculate the area of this hexagon if the edge length of the octahedron is \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{3}}{8} a^2}
Given a non-constant sequence $\left\{a_{i}\right\}$ satisfying $a_{i+1}^{2} - a_{i} a_{i+1} + a_{i}^{2} = 0$, and $a_{i+1} \neq a_{i-1}$ for $i = 1, 2, \cdots, n$, with the conditions that for a given natural number $n$, $a_{1} = a_{n+1} = 1$, find the value of $\sum_{i=0}^{n-1} a_{i}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine all primes \( p \) such that 
\[ 5^p + 4 \cdot p^4 \]
is a perfect square, i.e., the square of an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine how many zeros precede the last digit of \(2009^{2011}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
97**. A piece is randomly broken off from each of three identical rods. What is the probability that these three pieces can be used to form an acute triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\frac{\pi}{4}}
How many positive integers divide $5n^{11} - 2n^5 - 3n$ for all positive integers $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$
for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4946}
Let $ABC$ be a triangle with area $5$ and $BC = 10$. Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G$. Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2 + AC^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
The set of five-digit numbers $\left\{ N_1, N_k \right\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, shares at least one digit in at least one position with at least one of the numbers $N_1, N_k$. Find the smallest possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
On the section of the river from $A$ to $B$, the current is so small that it can be ignored; on the section from $B$ to $C$, the current affects the movement of the boat. The boat covers the distance downstream from $A$ to $C$ in 6 hours, and upstream from $C$ to $A$ in 7 hours. If the current on the section from $A$ to $B$ were the same as on the section from $B$ to $C$, the entire journey from $A$ to $C$ would take 5.5 hours. How much time would the boat take to travel upstream from $C$ to $A$ under these conditions? The boat's own speed remains unchanged in all cases.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.7 \text{ hours}}
There are some guys who have an equal number of bullets. All of them shoot 4 bullets. The remaining bullets are equal to the total number of bullets each had after division. Each one originally had 25 bullets. How many guys are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Simplify the following expression:
\[ \frac{1-\sqrt{2 t}}{\frac{1-\sqrt[4]{8 t^{3}}}{1-\sqrt[4]{2 t}}-\sqrt{2 t}} \cdot\left(\frac{\sqrt[4]{\frac{1}{\frac{1}{2}}+\sqrt[4]{4 t^{2}}}}{1+\sqrt[4]{\frac{1}{2 t}}}-\sqrt{2 t}\right)^{-1}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Suppose that property P is satisfied by at least three of $a_1,b_1,c_1$, by at least three of $a_2,b_2,c_2,\cdots$, and by at least three of $a_5,b_5,c_5$. Determine the largest value of $m$ for which it is guaranteed that at least $m$ of the $a's$, or $m$ of the $b's$, or $m$ of the $c's$ satisfy property P.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
18. $[\mathbf{9}]$ Let $f$ be a function that takes in a triple of integers and outputs a real number. Suppose that $f$ satisfies the equations
$$
\begin{aligned}
f(a, b, c) & =\frac{f(a+1, b, c)+f(a-1, b, c)}{2} \\
f(a, b, c) & =\frac{f(a, b+1, c)+f(a, b-1, c)}{2} \\
f(a, b, c) & =\frac{f(a, b, c+1)+f(a, b, c-1)}{2}
\end{aligned}
$$
for all integers $a, b, c$. What is the minimum number of triples at which we need to evaluate $f$ in order to know its value everywhere?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If the least common multiple (L.C.M.) of 76 and \( d \) is 456 and the highest common factor (H.C.F.) of 76 and \( d \) is \( c \), determine the value of the positive integer \( d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
The domain of the function $f(x)$ is $\{x|x\in R$, and $x\neq 1\}$. It is known that $f(x+1)$ is an odd function. When $x \lt 1$, $f(x)=2x^{2}-x+1$. Then, when $x \gt 1$, the decreasing interval of $f(x)$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{7}{4}, +\infty)}