inclusionAI/Ring-lite-rl-data huggingface dataset:
An artist who needs to buy only paint and canvas finds that the cost of canvas has decreased by 40 percent and the cost of paint has decreased by 60 percent. If paint previously accounted for some times as much of the expense of painting as did canvas, the total cost for paint and canvas decreased by 55.99999999999999 percent. What is the ratio of the expense of paint to the expense of canvas before the decrease in cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:1}
现有 11 位同学报名博物馆的志愿讲解活动，活动从上午 9 点开始到下午 5 点结束，每小时安排一场公益小讲堂，每场需要 1 位同学为参观的游客提供讲解服务. 为避免同学们劳累，馆方在排班时不会让同一人连续讲解 2 场，并且第一场与最后一场需要两位不同的同学负责. 则馆方共有__________种排班方式.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100000010}
A round cake is cut into \( n \) pieces with 3 cuts. Find the product of all possible values of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{840}
On the island of Unfortune, there live knights who always tell the truth and liars who always lie. One day, 2022 natives gathered at a round table, and each of them made the statement:

"Next to me sit a knight and a liar!"

It is known that three knights made a mistake (i.e., unintentionally lied). What is the maximum number of knights that could have been at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1349}
In a soap company, a soap is manufactured with 11 parts. For making one soap, you will get some parts as scrap. At the end of the day, they have 251 such scraps. From that, they can manufacture 25 more soaps. How many parts are obtained as scrap for making one soap?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A corner of a chessboard of size $(2n+1) \times (2n+1)$ is cut off. For which values of $n$ can the remaining squares be covered by $2 \times 1$ dominos in such a way that half of the dominos are horizontal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \text{ is even}}
A cylindrical can has a circumference of 24 inches and a height of 7 inches. A spiral strip is painted on the can such that it winds around the can precisely once, reaching from the bottom to the top. However, instead of reaching directly above where it started, it ends 3 inches horizontally to the right. What is the length of the spiral strip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{778}}
Consider a box with dimensions $10\, \text{cm} \times 16\, \text{cm} \times 1\, \text{cm}$. Determine the maximum number of balls with a diameter of $1\, \text{cm}$ that the box can contain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Let $ABCD$ be a square with side length $6$. Circles $X$, $Y$, and $Z$ are congruent circles with centers inside the square such that:

- Circle $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$.
- Circle $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$.
- Circle $Z$ is tangent to side $\overline{CD}$ and both circles $X$ and $Y$.

The radius of circle $X$ can be expressed as $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{195}
A circle of radius \( t \) is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius \( r = 1 + \sin \frac{\pi}{8} \). Find \( rt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 + \sqrt{2}}{4}}
In what time will a train 250 m long cross an electric pole, if its speed is 85 km/h?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.59}
A finite increasing sequence \(a_{1}, a_{2}, \ldots, a_{n}\) of natural numbers is given, where \(n \geq 3\), and for all \(k \leq n-2\) the following equality holds: \(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\). The sequence must include \(a_{k} = 2022\). Determine the maximum number of three-digit numbers, divisible by 4, that this sequence can contain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
Let $M$ be the least common multiple of all the integers $10$ through $30$, inclusive. Let $N$ be the least common multiple of $M$, $32$, $33$, $34$, $35$, $36$, $37$, $38$, $39$, and $40$. What is the value of $\frac{N}{M}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
Solve the equation:

$$(x+1)^{63}+(x+1)^{62}(x-1)+(x+1)^{61}(x-1)^{2}+\ldots+(x-1)^{63}=0$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
将函数 \(y=\sqrt{4+6 x-x^{2}}-2(x \in[0, 6])\) 的图象绕坐标原点逆时针方向旋转角 \(\theta(0 \leqslant \theta \leqslant \alpha)\)，得到曲线 \(C\)。若对于每一个旋转角 \(\theta\)，曲线 \(C\) 都是一个函数的图象，则 \(\alpha\) 的最大值为多少？答案的形式为\(\arctan \frac{k}{m}\)，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a function $f(x) = \frac {1}{2}\sin x - \frac {\sqrt {3}}{2}\cos x$ defined on the interval $[a, b]$, the range of $f(x)$ is $[-\frac {1}{2}, 1]$. Find the maximum value of $b-a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\pi}{3}}
In the mathematical text "Zhang Qiujian Suanjing" from the Northern and Southern Dynasties of China, there is a problem: "There are ten ranks of people, and the palace gives gold to each rank in descending order of difference. The first three people enter and receive 4 pounds of gold, and the last three people enter and receive 3 pounds of gold. The middle four people have not yet received their share, but they will also receive gold according to their rank." Calculate the total amount of gold received by the fifth and sixth ranked people. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{3}}
In how many ways can 8 identical rooks be placed on an $8 \times 8$ chessboard symmetrically with respect to the diagonal that passes through the lower-left corner square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{139448}
For the sequence $\{c_n\}$, if there exists an arithmetic sequence $\{a_n\}$ with all terms being positive integers and a geometric sequence $\{b_n\}$ with all terms being positive integers such that $c_n = a_n + b_n$, then the sequence $\{c_n\}$ is called a "DQ sequence". It is known that the sequence $\{e_n\}$ is a "DQ sequence", and its first 5 terms are: 3, 6, 11, 20, 37. Then $e_n = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 2^n}
7. Given the equation $8 t^{3}-4 t^{2}-4 t+1=0$ has a root $x$ in $\left(0, \frac{\pi}{13}\right)$, then $x=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin\frac{\pi}{14}}
Jack plays a game in which he first rolls a fair six-sided die and gets some number $n$, then, he flips a coin until he flips $n$ heads in a row and wins, or he flips $n$ tails in a row in which case he rerolls the die and tries again. What is the expected number of times Jack must flip the coin before he wins the game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given $60\%$ of students like dancing and the rest dislike it, $80\%$ of those who like dancing say they like it and the rest say they dislike it, also $90\%$ of those who dislike dancing say they dislike it and the rest say they like it. Calculate the fraction of students who say they dislike dancing but actually like it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
In the convex quadrilateral $ABCD$, points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ respectively. Segments $AE$, $AF$, and $EF$ divide the quadrilateral into 4 triangles whose areas are consecutive natural numbers. What is the greatest possible area of triangle $ABD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
2. 103 Use 94 large and small bricks, each with dimensions $4 \times 10 \times 19$, to stack into a tower 94 bricks high, where each brick can be placed to provide a height of 4, 10, or 19. If all 94 bricks are used, how many different heights can the tower be (height unit in inches)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{465}
10. For positive integer $n$ and real number $x (0 \leqslant x < n)$, define
$$
f(n, x)=(1-\{x\}) \cdot C_{n}^{[x]}+\{x\} \cdot C_{n}^{[x]+1},
$$

where $[x]$ denotes the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. If integers $m, n \geqslant 2$ satisfy
$$
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,
$$

find the value of $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
Given a sequence with the general term formula $a_n=n^2-(6+2\lambda)n+2014$, if $a_6$ or $a_7$ is the minimum term of the sequence $\{a_n\}$, then the range of values for the real number $\lambda$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{5}{2}, \frac{9}{2} \right)}
A line \( MN \) parallel to the base \( AB \) is drawn through the center of the circle inscribed in triangle \( ABC \) (\(M\) lies on \( BC \), \(N\) on \( AC \)). Find the perimeter of the quadrilateral \( ABMN \), given that \( AB = 5 \) and \( MN = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A square $EFGH$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 16$ and the $x$-axis. Find the area of square $EFGH$.

[asy]
unitsize(0.8 cm);

real parab2 (real x) {
  return(x^2 - 8*x + 16);
}

pair E, F, G, H;
real x = -2 + 2*sqrt(3);

E = (4 - x,0);
F = (4 + x,0);
G = (4 + x,-2*x);
H = (4 - x,-2*x);

draw(graph(parab2,0.5,7.5));
draw(E--H--G--F);
draw((0,0)--(8,0));

label("$E$", E, N);
label("$F$", F, N);
label("$G$", G, SE);
label("$H$", H, SW);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given an ellipse \(\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) with \(a > b > 0\). The left and right endpoints of the major axis are \(A\) and \(B\), respectively, and the left and right foci are \(F_{1}\) and \(F_{2}\), respectively. If there exists a point \(P\) on the ellipse \(\Gamma\) such that \(P F_{1}\) and \(P F_{2}\) trisect \(\angle APB\), find the number of distinct eccentricities \(e\) of the ellipse \(\Gamma\) that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If \( x \) and \( y \) are two coprime positive integers such that
\[ x + \frac{2016}{x} = 32y + \frac{63}{y}, \]
then how many pairs \((x, y)\) satisfy the condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
## Task B-2.7.

On the graph of the function $f(x)=-x^{2}+2022 x$, all integer points $(x, y)$ for which $y \geqslant-2023$ are marked in blue. On the x-axis, all integer points $(x, 0)$ are marked in red such that $a<x<b, f(a)=f(b)=0$. How many different triangles are there with vertices at two blue points and one red point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4141633279}
For a positive integer $n$, define a function $f_n(x)$ on the interval $[0, n+1]$ as follows:
\[ f_n(x) = \left( \sum_{i=1}^{n} |x-i| \right)^2 - \sum_{i=1}^{n} (x-i)^2. \]
Let $a_n$ be the minimum value of $f_n(x)$. Find the value of:
\[ \sum_{n=1}^{11} (-1)^{n+1} a_n. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{450}
Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given $30$ students such that each student has at most $5$ friends and for every $5$ students there is a pair of students that are not friends, determine the maximum $k$ such that for all such possible configurations, there exists $k$ students who are all not friends.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate $\int_{0}^{2\pi} \frac{\sin{3\theta}}{5+4\sin{\theta}}\,d{\theta}$ using complex integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The diameter of a cylindrical tin is some value and the height is 5 cm. The volume of the cylinder is 125.00000000000001 cm³. What is the diameter of the cylindrical tin?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.64}
The radius of the circumcircle of the acute-angled triangle \(ABC\) is 1. It is known that on this circumcircle lies the center of another circle passing through the vertices \(A\), \(C\), and the orthocenter of triangle \(ABC\). Find \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
A fashion designer sold a pair of jeans to a wholesale distributor for 35 percent more than it cost to manufacture the pair of jeans. The wholesale distributor then sold the pair of jeans to a retail store for 25 percent more than the distributor's purchase price. Finally, a customer bought the pair of jeans for 45 percent more than the retailer paid for them. The price the customer paid was what percent greater than the cost of manufacturing the jeans?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144.69\%}
Sophia started saving a certain amount of money every day. In 20 days, she would have saved 2 dimes in total. How much money does she save each day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Each of two angles of a triangle is $60^{\circ}$ and the included side is $4$ inches. The area of the triangle, in square inches, is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4\sqrt{3}$}
Given 1 coin of 0.1 yuan, 1 coin of 0.2 yuan, 1 coin of 0.5 yuan, 4 coins of 1 yuan, and 2 coins of 5 yuan, how many different amounts of money can be paid using any combination of these coins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Adnan is trying to remember his four-digit PIN. He is sure it contains the digits 5, 3, 7, and 0 but can't recall the order in which they appear. How many different arrangements are possible for his PIN?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A certain store sells a kind of product. Due to a decrease in the purchase price by 8% and an increase in the profit margin by 10%, what was the original profit margin of this product? (Answer in percentage) (Purchase Price × Profit Margin = Profit)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15\%}
Given a hyperbola $C_{1}$ defined by $2x^{2}-y^{2}=1$, find the area of the triangle formed by a line parallel to one of the asymptotes of $C_{1}$, the other asymptote, and the x-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{8}}
4. A positive integer n is given.Find the smallest  $k$  such that we can fill a  $3*k$  gird with non-negative integers 
such that: $\newline$   $i$ ) Sum of the numbers in  each column is  $n$ .
  $ii$ ) Each of the numbers  $0,1,\dots,n$  appears at least once in each row.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lceil \frac{3(n+1)}{2} \right\rceil}
The \emph{Stooge sort} is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order. We then (if $n \ge 3$) Stooge sort the first $\lceil \frac{2n}{3} \rceil$ elements, then the last $\lceil \frac{2n}{3} \rceil$, then the first $\lceil \frac{2n}{3} \rceil$ elements again. Given that this runs in $O(n^\alpha)$, where $\alpha$ is minimal, find the value of $(243/32)^\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
If the point $\left(m,n\right)$ in the first quadrant is symmetric with respect to the line $x+y-2=0$ and lies on the line $2x+y+3=0$, calculate the minimum value of $\frac{1}{m}+\frac{8}{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{25}{9}}
A regular triangular prism \(ABC A_{1}B_{1}C_{1}\) is inscribed in a sphere with the base \(ABC\) and lateral edges \(AA_{1}\), \(BB_{1}\), \(CC_{1}\). The segment \(C_{1}D\) is the diameter of this sphere, and point \(K\) is the midpoint of edge \(CC_{1}\). Find the volume of the prism if \(DK=2\) and \(DA=\sqrt{6}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For positive integers $x$, let $g(x)$ be the number of blocks of consecutive 1's in the binary expansion of $x$. For example, $g(19)=2$ because $19=10011_{2}$ has a block of one 1 at the beginning and a block of two 1's at the end, and $g(7)=1$ because $7=111_{2}$ only has a single block of three 1's. Compute $g(1)+g(2)+g(3)+\cdots+g(256)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{577}
We have a five-digit positive integer  $N$ . We select every pair of digits of  $N$  (and keep them in order) to obtain the  $\tbinom52 = 10$  numbers  $33$ ,  $37$ ,  $37$ ,  $37$ ,  $38$ ,  $73$ ,  $77$ ,  $78$ ,  $83$ ,  $87$ .  Find  $N$ .

*Proposed by Lewis Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37837}
Given that a high school senior year has 12 classes, with exactly 8 classes to be proctored by their own homeroom teachers, find the number of different proctoring arrangements for the math exam.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4455}
What is the value of the sum
\[
  \sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,
\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{49}{4}}
In rectangle \(ABCD\), point \(E\) is on side \(BC\) and point \(F\) is on side \(CD\), satisfying \(AE = AF\). Given that the area of \(\triangle ADF\) is 2, the area of \(\triangle ABE\) is 4, and the area of \(\triangle CEF\) is 8, find the value of \(7 \cos \angle EAF\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given an arithmetic-geometric sequence $\{a_{n}\}$ where $S_{n}$ represents the sum of the first $n$ terms, if $27a_{3}-a_{6}=0$, then find the value of $\frac{S_{6}}{S_{3}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
5.66 A deck of cards has a total of 54 cards. The magician divides them into several piles, and the audience writes a natural number on each card, which is equal to the number of cards in the pile where the card is located. Then the magician mixes the cards in a special way and divides them into several piles again. The audience writes a natural number on each card again, which is also equal to the number of cards in the pile where the card is now located. This process continues. How many times does this process of dividing piles and writing numbers need to be repeated at least, so that the (unordered) number arrays written on each card are all different?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
\(ABCD\) is a convex quadrilateral in which \(AC\) and \(BD\) meet at \(P\). Given \(PA = 1\), \(PB = 2\), \(PC = 6\), and \(PD = 3\). Let \(O\) be the circumcenter of \(\triangle PBC\). If \(OA\) is perpendicular to \(AD\), find the circumradius of \(\triangle PBC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
若复数 \(z\) 满足 \(\frac{z}{2}\) 与 \(\frac{2}{z}\) 的实部、虚部均在区间 \([-1,1]\) 上取值，则 \(z\) 在复平面上形成轨迹的面积是多少？答案的形式为 \(a-b\pi\)，请给出\(a+b\)的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In triangle  $ABC$ , let  $P$  and  $R$  be the feet of the perpendiculars from  $A$  onto the external and internal bisectors of  $\angle ABC$ , respectively; and let  $Q$  and  $S$  be the feet of the perpendiculars from  $A$  onto the internal and external bisectors of  $\angle ACB$ , respectively. If  $PQ = 7, QR = 6$  and  $RS = 8$ , what is the area of triangle  $ABC$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Bobby initially had 250 pieces of candy and 175 pieces of chocolate. He ate 38 pieces of candy and then ate 36 more. After that, he shared 12 pieces of candy with his friends. Meanwhile, he ate 16 pieces of chocolate and bought 28 more. How many more pieces of candy than chocolate did Bobby have left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
某足球邀请赛有十六个城市参加, 每市派出甲、乙两个队, 根据比赛规则, 比赛若干天后进行统计, 发现除 $A$ 市甲队外, 其它各队已比赛过的场数各不相同. 问 $A$ 市乙队已赛过多少场？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given $S=(x-a)^{2}+(\ln x-a)^{2}$ $(a\in\mathbb{R})$, then the minimum value of $S$ is ______________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
A rectangular wall is covered entirely with three kinds of decorative tiles: small, regular, and jumbo. The small tiles have a length-to-width ratio of 2:1, while the regular and jumbo tiles have the same length-to-width ratio of 3:1. The wall consists of 2/5 small tiles, 3/10 regular tiles, and 1/4 jumbo tiles. Jumbo tiles have a length three times that of regular tiles. If regular tiles cover 90 square feet of the wall, and no tiles overlap, what is the area of the entire wall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
In an acute triangle $\triangle ABC$, altitudes $\overline{AD}$ and $\overline{BE}$ intersect at point $H$. Given that $HD=6$ and $HE=3$, calculate $(BD)(DC)-(AE)(EC)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
A trader bought a car at a 20% discount on its original price. He then made a few modifications to the car and increased the price by 15%. Due to market conditions, he had to offer a 10% discount before finally selling the car at a 45% increase on the last discounted price. What percent of profit did he make on the original price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20.06\%}
Kevin colors three distinct squares in a  $3\times 3$  grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let $\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $\mathcal{P}_{i}$. In other words, find the maximum number of points that can lie on two or more of the parabolas $\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}