inclusionAI/Ring-lite-rl-data huggingface dataset:
When Vitya was a first grader, he had a set of 12 number cards: two cards with the number 1, two cards with the number 2, and so on up to the number 6. Vitya placed them on the table in a random order from left to right, and then removed the first 1, the first 2, the first 3, and so on. For example, if Vitya initially had the sequence 434653625112, the resulting sequence would be 436512. What is the probability that the sequence left on the table is 123456?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{720}}
Johnny buys some packs of colored pencils for his class. Each pack has a red, yellow, and green pencil inside. When he gets home he notices that 3 of the packs have two extra red pencils inside. Johnny bought 21 red colored pencils. How many packs of colored pencils did Johnny buy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The function $y=|\log_{2}x|$ is meaningful and not monotonic in the interval $(k-1, k+1)$. Find the range of values for $k$. Express your answer as an interval in standard notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
A train leaves its departure station with 6 coal cars, some iron cars, and 2 wood cars. Between each station, there are 6 miles, and it takes 25 minutes to travel between them. At each station, the train can deposit up to 2 coal cars, 3 iron cars, and 1 wood car. It takes 100 minutes to deliver all the cars. How many iron cars does the train have initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A goes to place X and B goes to place Y. After A travelled 180 km more than B, they together meet at a point between X and Y. They together had their lunch for half an hour. After that, A took some time to reach X and B took 4.5 hrs to reach Y. The distance between X and Y is 900 km. How long did A take to reach X after lunch?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.5}
From a 12 × 12 grid, a 4 × 4 square has been cut out, located at the intersection of horizontals from the fourth to the seventh and the same verticals. What is the maximum number of rooks that can be placed on this board such that no two rooks attack each other, given that rooks do not attack across the cut-out cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1983}
Let $A_{12}$ denote the answer to problem $12$.  There exists a unique triple of digits $(B,C,D)$ such that $10>A_{12}>B>C>D>0$ and \[\overline{A_{12}BCD}-\overline{DCBA_{12}}=\overline{BDA_{12}C},\] where $\overline{A_{12}BCD}$ denotes the four digit base $10$ integer.  Compute $B+C+D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
What is the limit of the sequence $S_n$ defined as follows? $$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If the acute angle \(\alpha\) satisfies \(\frac{1}{\sqrt{\tan \frac{\alpha}{2}}}=\sqrt{2 \sqrt{3}} \sqrt{\tan 10^{\circ}}+\sqrt{\tan \frac{\alpha}{2}}\), then the measure of the angle \(\alpha\) in degrees is \(\qquad\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Find the largest integer  $n$  for which  $2^n$  divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \]*Proposed by Evan Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{193}
In triangle \( \triangle ABC \), given that \( \sin A = 10 \sin B \sin C \) and \( \cos A = 10 \cos B \cos C \), find the value of \( \tan A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A circle with a radius of 2 is inscribed in triangle \(ABC\) and touches side \(BC\) at point \(D\). Another circle with a radius of 4 touches the extensions of sides \(AB\) and \(AC\), as well as side \(BC\) at point \(E\). Find the area of triangle \(ABC\) if the measure of angle \(\angle ACB\) is \(120^{\circ}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{56}{\sqrt{3}}}
Let point P $(x, 2)$ be a point on the terminal side of angle $\alpha$, and it satisfies $\sin\alpha= \frac{2}{3}$. Then, the value of $x$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{5}}
Consider the system of equations:

\[
8x - 6y = a,
\]
\[
12y - 18x = b.
\]
If there's a solution $(x, y)$ where both $x$ and $y$ are nonzero, determine $\frac{a}{b}$, assuming $b$ is nonzero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{4}{9}}
对于任何集合 $S$, 以 $|S|$ 表示集合 $S$ 的元素个数, 以 $n(S)$ 表示集合 $S$ 的子集的个数（包括空集与 $S$ 本身）。如果 $A, B$ 和 $C$ 是三个集合，满足条件：

$$
n(A)+n(B)+n(C)=n(A \cup B \cup C), \quad|A|=|B|=100,
$$


求 $|A \cap B \cap C|$ 的最小可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
What is the correct relationship between the number 2 and the set {0, 1, 2}? Express your answer using the appropriate symbol (e.g., ∈, ⊆, =).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 ∈ {0, 1, 2}}
Sacha wants to cut a \(6 \times 7\) rectangle into squares that all have integer length sides. What is the smallest number of squares he could obtain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Rotating an isosceles trapezoid around its axis of symmetry forms a \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{frustum}
8. Given the set $A=\{1,2, \cdots, 104\}, S$ is a subset of $A$. If $x \in S$, and at the same time $x-1 \notin S$ and $x+1 \notin S$, then $x$ is called an "isolated point" of $S$. The number of all 5-element subsets of $A$ with no "isolated points" is . $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
The ancient Chinese mathematical text, "The Nine Chapters on the Mathematical Art," presents the following problem: "There is a gold bar that is 5 feet long. When 1 foot is cut from the thicker end, it weighs 4 pounds; when 1 foot is cut from the thinner end, it weighs 2 pounds. What is the weight of each successive foot?" Assume that the gold bar's thickness and weight decrease uniformly from the thicker end to the thinner end. Let M be the total weight of the gold bar. If the bar is divided into 10 equal segments, let a<sub>i</sub> denote the weight of the i-th segment, where a<sub>1</sub> < a<sub>2</sub> <... < a<sub>10</sub>. If 48a<sub>i</sub> = 5M, then find the value of i. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the residue of the function $\frac{z^9}{z^{10} -1}$ at infinity using the formula for the residue at infinity involving the function evaluated at the reciprocal of the variable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let  $ABC$  be a triangle.  The incircle  $\omega$  of  $\triangle ABC$ , which has radius  $3$ , is tangent to  $\overline{BC}$  at  $D$ .  Suppose the length of the altitude from  $A$  to  $\overline{BC}$  is  $15$  and  $BD^2 + CD^2 = 33$ .  What is  $BC$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{7}}
Alberto, Bernardo, and Carlos participated in a race, each running at a constant speed throughout the entire distance. When Alberto crossed the finish line, Bernardo and Carlos were 36 meters and 46 meters behind him, respectively. When Bernardo crossed the finish line, Carlos was 16 meters behind him. Calculate the length of the track in meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The trains will be clear of each other in 21.998240140788738 seconds from the moment they meet. If the first train is 120 m long, how long is the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{319.96}
A mathematician left point A for point B. After some time, a physicist left point A as well. The physicist caught up with the mathematician after 20 km, and without stopping, continued to point B and then turned back. They met for the second time 20 km away from point B. Each then continued in their respective directions. Upon reaching points A and B respectively, they turned back and headed towards each other again. At how many kilometers from point A will they meet for the third time, if the distance between points A and B is 100 km?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Haylee has 3 dozen guppies in her aquarium. Jose has half as many guppies as the square root of Haylee's guppies in his tank. Charliz has one-third of what Jose has in his tank, but Charliz sells 1.5 guppies to a neighbor. Nicolai has 4 times as many guppies as Charliz in his pond, minus 2 guppies. Alice has 5 more guppies than Nicolai in her fishbowl, while Bob has half the guppies of ((Jose's guppies)^2)/3 + (Charliz's guppies after selling) in his aquarium. Sara has 3i guppies (where i is an imaginary unit) in her experimental tank. Cameron has 2^(Haylee's dozens of guppies) guppies in his fish tank. How many guppies do these eight friends have altogether, excluding imaginary guppies?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
If the function $y=f(x)$ defined on $R$ satisfies that there exist a finite number of non-zero independent variable values $x_{0}$ such that $f(-x_{0})=-f(x_{0})$, then $y=f(x)$ is called a class odd function. If the function $y=x^{4}+(a^{2}-1)x^{2}+a\sin x$ is a class odd function, then the range of real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,1)}
During Bill's counting process, 6 toddlers are taken to the playground by another teacher, and 4 new toddlers join the daycare midway through his counting. Additionally, Bill double-counts 9 toddlers and misses 5 who are hiding. If Bill thinks he counted 28 toddlers, considering the added difficulties, how many are there really?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Bill and Ted head to the river to throw stuff into the water. Bill throws 6 more sticks into the river than Ted does, but Ted tosses twice as many rocks into the river as Bill. If Bill tossed 21 objects into the river, how many sticks did Ted toss into the river?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
When the expression \(\left[(1+x)\left(1+2 x^{3}\right)\left(1+4 x^{9}\right)\left(1+8 x^{27}\right)\left(1+16 x^{81}\right)\left(1+32 x^{243}\right)\left(1+64 x^{729}\right)\right]^{2}\) is expanded and simplified, what is the coefficient of \(x^{2003}\)? Express your answer as a power of 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{30}}
The median $A D$ and the angle bisector $C E$ of the right triangle $A B C \left(\angle B=90^{\circ}\right)$ intersect at point $M$. Find the area of triangle $A B C$ if $C M =8$, and $M E =5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1352}{15}}
If $yz:zx:xy = 1:2:3$, find the ratio $\frac{x}{yz}:\frac{y}{zx}$. The original answer is in the format m:n, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $ABC$ be an isosceles triangle with the apex at $A$. Let $M$ be the midpoint of segment $[BC]$. Let $D$ be the symmetric point of $M$ with respect to the segment $[AC]$. Let $x$ be the angle $\widehat{BAC}$. Determine, as a function of $x$, the value of the angle $\widehat{MDC}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{x}{2}}
Arrange the digits from 0 to 9 into a ten-digit number where the first digit is not zero and all digits are unique. How many such ten-digit numbers are divisible by 99?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{285120}
Find the number of perfect squares of five digits whose last two digits are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
If angle $\alpha = 2$, then $\alpha$ is an angle in the \_\_\_\_\_\_ quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{second}}
3 people run for president. John manages to capture 150 votes. James captures a certain percentage of the remaining vote. There were 1150 people voting, and the third guy got 150 more votes than John. What percentage of the remaining vote did James capture?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70\%}
Two circles touch each other externally at point $K$. On their common internal tangent, a point $P$ is marked such that $K P = 14$. Through point $P$, two secants are drawn to the circles such that one of them cuts a chord $AB = 45$ on the first circle, and the other cuts a chord $CD = 21$ on the second circle, with point $A$ lying between points $B$ and $P$, and point $C$ lying between points $D$ and $P$. Find the ratio $BC: AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.75}
In how many ways can 6 people be seated at a round table, considering the ways different if at least some of the people get new neighbors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Problem 6. Calculate the maximum number of distinct real roots that a polynomial $P$ can have, which satisfies the following property: the product of two distinct roots of $P$ is still a root of $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given $f\left(x\right)=-\ln x+\frac{a}{x}-ex+4$, $g\left(x\right)=\frac{1}{3}{x^3}-{x^2}+2$. If for all $x_{1}\in \left(0,1\right]$, there exists $x_{2}\in \left[-1,1\right]$, such that $g(x_{2})\geqslant f(x_{1})$, then the range of $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -\frac{2}{e}]}
In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Holly needs to take 2 insulin pills per day, 3 blood pressure pills per day, and twice as many anticonvulsants as blood pressure pills each day. She takes calcium supplements every other day, consuming three times as many calcium pills as insulin pills. Also, she needs to take one multivitamin once every three days, and anxiety pills, which are taken once every four days. How many pills does Holly take in a week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A natural number $n$ is called individual if $n$ has no primitive roots and at least $4$ squares are needed so that it is expressed as a sum of squares. How many natural numbers $n$ between $2$ and $60$ are there that are individual?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the number of solutions for the Diophantine equation $x^4 - 1785y^2 = 1$ using Cohn's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \(ABCD\) be a rectangle with \(AB = 8\) and \(AD = 20\). Two circles of radius 5 are drawn with centers in the interior of the rectangle - one tangent to \(AB\) and \(AD\), and the other passing through both \(C\) and \(D\). What is the area inside the rectangle and outside of both circles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112 - 25\pi}
Given the function $f(x)=\sin 3x$, where $\frac{f(x_{i})}{x_{i}}=m$, $x_{i}\in[-2π,2π]$, $i=1$, $2$, $...n$, $n∈\mathbb{N}^{*}$, find the maximum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $f$ be a continuously differentiable function on $[0, 1]$ and $m \in \mathbb{N}$. Let $A = f(1)$ and let $B=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx$. Calculate $$\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{n^m}\right)\right)$$in terms of $A$ and $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{m}{2}A - \frac{m-1}{2}B}
For a bijective function $g : \mathbb{R} \to \mathbb{R}$, we say that a function $f : \mathbb{R} \to \mathbb{R}$ is its superinverse if it satisfies the following identity: 
\[(f \circ g)(x) = g^{-1}(x)\]
where $g^{-1}$ is the inverse of $g$. Given:
\[g(x) = x^3 + 9x^2 + 27x + 81\]
and $f$ is its superinverse, find:
\[|f(-289)|\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
For an upcoming holiday, the weather forecast indicates a probability of $30\%$ chance of rain on Monday and a $60\%$ chance of rain on Tuesday. Moreover, once it starts raining, there is an additional $80\%$ chance that the rain will continue into the next day without interruption. Calculate the probability that it rains on at least one day during the holiday period. Express your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72\%}
60. In what ratio does a plane, parallel to two skew edges of a triangular pyramid and dividing one of the other edges in the ratio $2: 1$, divide the volume of the pyramid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{20}{7}}
Find all real values of $x$ that satisfy $\frac{1}{x+1} + \frac{3}{x+7} \ge \frac23.$ (Give your answer in interval notation.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-7, -4] \cup (-1, 2]}
How many ways are there to insert plus signs (+) between the digits of 1111111111111111 (fifteen 1's) so that the result will be a multiple of 30?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2002}
there are 5 boxes in a cargo . the weight of the 1 st box is 200 kg , the weight of the 2 nd box is 20 % higher than the third box , whose weight is 25 % higher than the 1 st box weight . the 4 th box which weighs 350 kg is 30 % lighter than the 5 th box . find the difference in average weight of the 4 heaviest boxes and the 4 lightest boxes .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Two circles with radii $15$ and $95$ have their centers $d$ units apart. The external tangents to the circles intersect at an angle of $60$ degrees. What is the distance $d$ between the centers of the circles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Find the number of positive integers less than 1,000,000 that are divisible by some perfect cube greater than 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168089}
Let $a_1, a_2, \ldots, a_{2005}$ be real numbers such that $$\begin{array}{ccccccccccc}
a_1\cdot 1 &+ &a_2 \cdot 2 &+& a_3 \cdot 3 &+ &\cdots& + &a_{2005} \cdot 2005 &=& 0 \\
a_1\cdot 1^2 &+& a_2\cdot 2^2 &+& a_3 \cdot 3^2 &+ &\cdots& + & a_{2005} \cdot 2005^2 &=& 0 \\
a_1 \cdot 1^3 &+& a_2 \cdot 2^3 &+& a_3 \cdot 3^3 &+ &\cdots& + & a_{2005} \cdot 2005^3 &=& 0 \\ \vdots&&\vdots&&\vdots&&&&\vdots&&\vdots \\
a_1\cdot 1^{2004} &+& a_2\cdot 2^{2004} &+& a_3\cdot 3^{2004} &+ &\cdots& + &a_{2005} \cdot 2005^{2004} &=& 0
\end{array}$$and $$\begin{array}{ccccccccccc}
a_1 \cdot 1^{2005}& +& a_2\cdot 2^{2005} &+& a_3\cdot 3^{2005} &+ &\cdots& + &a_{2005} \cdot 2005^{2005} &=& 1.
\end{array}$$What is the value of $a_1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2004!}}
Let \(ABCD\) be a quadrilateral with an inscribed circle \(\omega\). Let \(I\) be the center of \(\omega\), and suppose \(IA = 12\), \(IB = 16\), \(IC = 14\), and \(ID = 11\). Let \(M\) be the midpoint of segment \(AC\). Compute \(\frac{IM}{IN}\), where \(N\) is the midpoint of segment \(BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{21}{22}}
Let $x$ be a real number between 0 and $\tfrac{\pi}{2}$ for which the function  $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$ obtains its maximum value, $M$. Find the value of $M + 100\cos^2x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Let \(x, y, z\) be positive real numbers such that \(xyz = 1\). Find the maximum value of 
\[
\frac{x^2y}{x+y} + \frac{y^2z}{y+z} + \frac{z^2x}{z+x}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
12.172. A trapezoid is inscribed in a circle. The larger base of the trapezoid forms an angle $\alpha$ with a lateral side, and an angle $\beta$ with a diagonal. Find the ratio of the area of the circle to the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2\sin^2\alpha\sin2\beta}}
For each integer $k\ge2$, the decimal expansions of the numbers $1024, 1024^2, \dots, 1024^k$ are concatenated, in that order, to obtain a number $X_k$. (For example, $X_2 = 10241048576$.) If $ \frac{X_n}{1024^n}$ is an odd integer, find the smallest possible value of $n$, where $n\ge2$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find all the common points of the graphs

$$
y=8 \cos \pi x \cdot \cos 2 \pi x \cdot \cos ^{2} 4 \pi x \quad \text { and } \quad y=\cos 11 \pi x
$$

with abscissas belonging to the interval $[0, 1]$. In your answer, provide the sum of the abscissas of the found points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.5}
In the Cartesian coordinate system, the graph of the quadratic function $y=ax^{2}+bx+c\left(a \lt 0\right)$ passes through the points $\left(m,b\right)$ and $\left(m+1,a\right)$. When $b\geqslant a$ and $m \lt 0$, the graph of the quadratic function $y=ax^{2}+bx+c$ has a maximum value of $-2$. The maximum value of $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{8}{3}}
All faces of a triangular pyramid are congruent isosceles triangles, and the height of the pyramid coincides with the height of one of its lateral faces. Find the volume of the pyramid if the distance between the longest opposite edges is 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}