inclusionAI/Ring-lite-rl-data huggingface dataset:
The trunk of the sky-high tree branches into two at a height of one foot. In the following, we consider the part between two branches as a branch, on which there is no further branching. Every branch of the sky-high tree is straight and ends one foot higher than its end closer to the ground. We consider the children of a branch to be the branches starting from the higher end of the branch, which are also considered siblings to each other. Every branch of the sky-high tree has at least two children, and if it does not have exactly two children, then it has a sibling who has exactly two children. Siblings always have a different number of children. If a branch has more than two children, then it has a sibling who has exactly one child less than it. How many branches start from the branches at a height of $n$ feet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{2n+2}{n+1}-\binom{2n+2}{n}}
The base of a hill located beside a river is 300m above the seabed. This depth is a certain fraction of the vertical distance between the riverbed and the peak of the hill. The hill is 900m high. What is the ratio of the depth from the base of the hill to the seabed to the vertical distance between the riverbed and the peak of the hill?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:3}
In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Calculate the area of trapezoid $EFCD$ if the height from $E$ to line $CD$ is 10 units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,\!000$th number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$996,506$}
Given the sequence ${a_n}=C_{200}^n⋅{({\sqrt[3]{6}})^{200-n}}⋅{({\frac{1}{{\sqrt{2}}}})^n}$ for $n=1,2,\ldots,95$, calculate the number of integer terms in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A tempo is insured with fixed coverage percentages and premium rates for all three years. In the first year, it is insured to an extent of 5/7 of its original value at a premium rate of 3%. In the second year, it is insured to the same extent of its original value at a premium rate of 4%. In the third year, it is insured to an extent of 5/7 of its original value at a premium rate of 5%. If the total premium paid for all three years amounts to $950, calculate the original value of the tempo.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$11,083.33$}
A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) If $d$ to the nearest tenth is $x$, what is the value of $10x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A sector representing a quarter of a circle has been cut out. From this sector and the remaining part of the circle, the lateral surfaces of two cones have been made. Find the ratio of the heights of these cones.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{15}}{\sqrt{7}}}
Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its column ("addition table"). What is the maximum number of sums in this table that could end up being rational numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
Steve's new book sells 1,000,000 copies.  He got an advance to pay for 100,000 copies. He gets $2 for each copy of the book sold.  His agent takes 10% of that.  How much money did he keep not counting the money from the advance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1620000}
Given \(2n\) positive real numbers \(a_{1} \geq a_{2} \geq \cdots \geq a_{n}\), \(b_{1} \geq b_{2} \geq \cdots \geq b_{n}\) and \(0 \leq \lambda \leq 2\), define
$$
M = \sum_{i=1}^{n} \sqrt{a_{i}^2 + b_{i}^2 - \lambda a_{i} b_{i}}, \quad N = \sum_{i=1}^{n} \sqrt{a_{i}^2 + b_{i}^2 - \lambda a_{i+1} b_{i}},
$$
where \(a_{n+1} = a_{1}\). Determine the relationship between \(M\) and \(N\), specifying whether \(M \leq N\), \(M \geq N\), or if the relationship is indeterminate.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M \leq N}
Find the value of the expression \(\cos ^{4} \frac{5 \pi}{24}+\cos ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{19 \pi}{24}+\sin ^{4} \frac{13 \pi}{24}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Andrey, Maxim, Igor, and Kolya competed in a bicycle race. To the question of who took which place, they answered:

Andrey: - I was neither first nor last.

Maxim: - I was not last.

Igor: - I was first.

Kolya: - I was last.

It is known that three boys answered honestly and only one lied. Who among the boys lied?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Igor}
Find the distance between the foci of the ellipse
\[\frac{x^2}{20} + \frac{y^2}{4} = 7.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 \sqrt{7}}
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
A triangle has sides of length  $48$ ,  $55$ , and  $73$ .  A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle.  If  $c$  and  $d$  are relatively prime positive integers such that  $c/d$  is the length of a side of the square, find the value of  $c+d$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200689}
The first digit of a string of 2002 digits is a 1. Any two-digit number formed by consecutive digits within this string is divisible by 19 or 31. What is the largest possible last digit in this string?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Calculate the integral $$\int_{0}^{\infty}\frac{\mathrm{d}x}{\left(x + 2\right)\left(x^{2} + 1\right)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi - \ln(2)}{5}\)}
Select numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be the results when $a, b$ and $c$, respectively, are rounded to the nearest integer. Find the probability that $A + B = C$. The original answer is in \frac{k}{m} format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Suppose a group $G$ has irreducible complex representations of dimensions $1,1,2,3$, and $d$. Find $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
设 $S=\{1,2, \cdots, 98\}$, 求最小正整数 $n$, 使得 $S$ 的任意一个 $n$ 元子集中都可以选出 10 个数, 无论怎样将 10 个数均分成两组, 总有一组中存在一个数与另外 4 个数互质, 而另一组中总有一个数与另外 4 个数不互质。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
A bug starts at a vertex of a grid made of equilateral triangles of side length \(1\). At each step, the bug moves in one of the \(6\) possible directions along the grid lines randomly and independently with equal probability. What is the probability that after \(5\) moves the bug never will have been more than \(1\) unit away from the starting position? Provide your answer as a fraction in simplest form \(\frac{k}{m}\), and calculate the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Find the last three digits of the sum of all the real values of $m$ such that the ellipse $x^2 + xy + y^2 = m$ intersects the hyperbola $xy = n$ only at its two vertices, as $n$ ranges over all non-zero integers $-81 \le n \le 81$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{284}
Given a triangle with side lengths a, b, and c, an area equal to 1/4, and the radius of its circumcircle is 1, let s = √a + √b + √c and t = 1/a + 1/b + 1/c. Determine the relationship between s and t. Express your answer as an inequality (e.g., s < t, s > t, s = t).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s < t}
We call a number $10^3 < n < 10^6$ a balanced number if the sum of its last three digits is equal to the sum of its other digits. Find the sum of all balanced numbers in $\bmod{13}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The distinct positive integers $a$ and $b$ have the property that $\frac{a+b}{2}$, $\sqrt{ab}$, and $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ are all positive integers. Find the smallest possible value of $|a-b|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In a certain wire service, there are reporters covering local politics in three different countries A, B, and C. The percentage of reporters covering local politics in each country are as follows: 20% cover local politics in country A, 25% cover local politics in country B, and 15% cover local politics in country C. Additionally, 20% of the reporters who cover politics for the wire service do not cover local politics in any of these three countries. What percent of the reporters for the wire service do not cover politics?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
All edges of a regular quadrilateral pyramid \(S A B C D\) with base \(A B C D\) are of equal length. Plane \(\alpha\) is perpendicular to line \(S A\), and plane \(\beta\) is parallel to line \(C D\). Determine the smallest possible angle between planes \(\alpha\) and \(\beta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
In the vertices of a cube, the numbers \(\pm 1\) are assigned, and on its faces, the numbers are equal to the product of the numbers at the vertices of that face. Find all possible values that the sum of these 14 numbers can take. Provide their product in the answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-20160}
Determine the number of homomorphisms from $S_{3} \rightarrow \mathbb{Z}_{2} \times \mathbb{Z}_{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
John wants to buy a $100 trouser at the store, but he thinks it's too expensive. Finally, it goes on sale for $10. What is the percent decrease in the price of the trouser?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90\%}
In the game of Dubblefud, yellow chips are worth a certain number of points, blue chips are worth 4 points, and green chips are worth 5 points. In a certain selection of chips, the product of the point values of the chips is 16000. The number of blue chips in this selection equals the number of green chips, and there are 4 yellow chips. How many points is a yellow chip worth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the hyperbola $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1 (m < 0 < n)$ with asymptote equations $y = \pm \sqrt{2}x$, calculate the hyperbola's eccentricity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Let $S = \left\{A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i} = 0 \text{ or } 1, i = 1, 2, \cdots, 8\right\}$. For two elements $A = \left(a_{1}, a_{2}, \cdots, a_{8}\right)$ and $B = \left(b_{1}, b_{2}, \cdots, b_{8}\right)$ in $S$, denote $d(A, B) = \sum_{i=1}^{8}\left|a_{i}-b_{i}\right|$, and refer to it as the distance between $A$ and $B$. How many elements can be selected from $S$ such that the distance between any two of them is at least 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
8.7. Let's consider the equilateral triangle $A B C$. Points $D, E$, and $F$ are located on the sides $(B C),(C A)$, and, respectively, $(A B)$ such that the lines $D F$ and $A B$ are perpendicular, and the line $D E$ is the perpendicular bisector of the segment $C F$. Find the measure in degrees of the angle $D E F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Evaluate $$\iint_{S} (x^3+y^3)\,dy\,dz + (z^3+y^3)\,dz\,dx + (x^3+z^3)\,dx\,dy$$ where $S$ is the surface of the sphere $x^2+y^2+z^2= a^2$. Spherical coordinates help me in this question, $r^2=a^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{12}{5}\pi a^5\)}
Let $BCB'C'$ be a rectangle, let $M$ be the midpoint of $B'C'$, and let $A$ be a point on the circumcircle of the rectangle. Let triangle $ABC$ have orthocenter $H$, and let $T$ be the foot of the perpendicular from $H$ to line $AM$. Suppose that $AM=2$, $[ABC]=2020$, and $BC=10$. Then $AT=\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.

[i]Proposed by Ankit Bisain[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2102}
Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{141}
II. (50 points)
Let $a, b, c \in \mathbf{R}^{+}$, and $a b c + a + c = b$. Determine the maximum value of $P = \frac{2}{a^{2}+1} - \frac{2}{b^{2}+1} + \frac{3}{c^{2}+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3}}
The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111$, $R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=\frac{R_{24}}{R_4}$ is an integer whose base-ten representation is a sequence containing only ones and zeros. Find the number of zeros in $Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In a sealed transparent cubic container with a volume of 6, there is a liquid inside. If the shape of the liquid surface is not a triangle regardless of how the cube is rotated, then the range of the liquid volume is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,5)}
Let set $A=\{x|x^2-2ax+a=0, x\in \mathbb{R}\}$, and set $B=\{x|x^2-4x+a+5=0, x\in \mathbb{R}\}$. If exactly one of the sets $A$ and $B$ is empty, then the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 0] \cup [1, +\infty)}
Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons, and regular octagons. Tim makes a random sequence \( F_{0}, F_{1}, F_{2}, F_{3}, \ldots \) of figures as follows:
- \( F_{0} \) is a square.
- For every positive integer \( i \), \( F_{i} \) is randomly chosen to be one of the 2 figures distinct from \( F_{i-1} \) (each chosen with equal probability \(\frac{1}{2}\)).
- Tim takes 4 seconds to make squares, 6 to make hexagons, and 8 to make octagons. He makes one figure after another, with no breaks in between.

Suppose that exactly 17 seconds after he starts making \( F_{0} \), Tim is making a figure with \( n \) sides. What is the expected value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the smallest positive integer $n$ such that there exist rational numbers $a_i$ and $b_i$ ($i=1, 2, \ldots, n$) satisfying
\[
x^{2} + x + 4 = \sum_{i=1}^{n}\left(a_{i} x + b_{i}\right)^{2}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
At a race track, a maximum of 6 horses can race together at a time. There is a total of 30 horses. Races not only depend on speed, but also rely on track conditions, unpredictable horse behavior during each race, and asymmetric track layouts, where each horse encounters unique challenges with varying difficulty levels. Track conditions and layouts change for each race. What is the minimum number w of races we need to conduct to confidently determine the top 3 fastest horses under different conditions and asymmetric layouts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A technician makes a round-trip to and from a certain service center by the same route. On the way to the center, the technician encounters a traffic jam that causes a 15% increase in travel time and a construction detour that causes an additional 10% increase in travel time. If the technician completes the drive to the center and then completes 20% of the drive back from the center without any delays, determine the completion percentage of the round-trip, accounting for the delays experienced on the way to the center.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72.5\%}
Given the parabola y = 3 - x^2 and the line of symmetry x + y = 0, find the distance |AB| between two distinct points A and B on the parabola that are symmetric with respect to the line.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
Dima is driving on a straight highway from point A to point B. From point B, a traffic jam is extending towards Dima at a speed of \( v \) km/h. The speed of the car in the traffic jam is 10 km/h, and outside the traffic jam it is 60 km/h. The navigator in the car continuously shows the remaining time for Dima to reach point B based on the current length of the traffic jam. At a certain moment (before reaching the traffic jam), Dima noticed that the navigator showed the same time as it did a few minutes ago. Find \( v \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 \text{ km/h}}
Determine the number of angles $\theta$ between 0 and $2 \pi$, other than integer multiples of $\pi / 2$, such that the quantities $\sin \theta, \cos \theta$, and $\tan \theta$ form a geometric sequence in some order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
# Task 7. (14 points)

In a certain company, no two employees have jobs of the same difficulty, and no two employees receive the same salary. On April 1, each employee made two statements:

(a) There will not be 12 employees with more difficult jobs.

(b) At least 30 employees have a higher salary. How many employees are in the company if some employees told the truth twice, and the rest lied twice.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Example 5 Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9, \cdots, n\}$. Take 3 numbers from $A$ and 2 numbers from $B$ to form a set $A_{i}(i=1,2, \cdots, 20)$ with 5 elements, such that $\left|A_{i} \cap A_{j}\right| \leqslant 2,1 \leqslant i<j \leqslant 20$, find the minimum value of $n$. (2002 China National Training Team for IMO)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Gary has $2$ children. We know one is a boy born on a Friday. Assume birthing boys and girls are equally likely, being born on any day of the week is equally likely, and that these properties are independent of each other, as well as independent from child to child. The probability that both of Gary's children are boys is $\frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Find $a + b.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
给正方体的 8 个顶点染上 $k$ 个红点, $8-k$ 个蓝点 $(1 \leqslant k$ $<8)$. 凡两端为红色的棱记上数字 $\frac{-1+\sqrt{3} i}{2}$, 凡两端为蓝色的棱记上数字 $\frac{-1-\sqrt{3} i}{2}$, 凡两端异色的棱记上数字 1 , 这 12 个数字之积的所有可取值为 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
在坐标平面上有两个区域 $M$ 和 $N$。$M$ 是由 $y \geqslant 0, y \leqslant x$ 和 $y \leqslant 2-x$ 这三个不等式确定的，$N$ 是随 $t$ 变化的区域，它由不等式 $t \leqslant x \leqslant t+1$ 所确定，$t$ 的取值范围是 $0 \leqslant t \leqslant 1$。设 $M$ 和 $N$ 的公共面积是函数 $f(t)$。原始答案的形式是 $-at^{2}+bt+\frac{c}{d}$，请给出 $a+b+c+d$ 的值，其中 $\frac{c}{d}$ 是最简分数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Eastbound traffic flows at 80 miles per hour and westbound traffic flows at 60 miles per hour. An eastbound driver observes 30 westbound vehicles in a 10-minute period. Calculate the number of westbound vehicles in a 150-mile section of the highway.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{193}
Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\frac{12}5\sqrt2$. What is the volume of the tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{24}5$}
Barney's grocery store sold out all of its items at the beginning of the pandemic, so they ordered 4458 items to restock the shelves. They sold some items that day, and have 575 items in the storeroom. They have 3,472 items left in the whole store. How many items did they sell that day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1561}
A square of side length $1$ and a circle of radius $\dfrac{\sqrt{3}}{3}$ share the same center. Find the area inside the circle, but outside the square. The original answer is in the form \frac{k\pi}{m} - \frac{\sqrt{n}}{p}, please give the value of k + m + n + p.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
如果 $$ax^{2} + bx + c > 0 (a ≠ 0)$$ 的解集为 {$x | x < -2 或 x > 4$}, 设 $$f(x) = ax^{2} + bx + c$$, 试比较 $f(-1), f(2), f(5)$ 的大小.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(5) > f(-1) > f(2)}
A circle with a radius of 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect the diagonal $AC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{375}{442}}
设正整数 $m, n \leqslant 2021$ ，并且 $\frac{m}{n+1}<\sqrt{2}<$ $\frac{m+1}{n}$, 则这样的数组 $(m, n)$ 的个数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3449}
如果在区间 $[1,2]$ 上函数 $f(x)=x^{2}+p x+q$ 与 $g(x)=x+\frac{1}{x^{2}}$ 在同一点取相同的最小值，那么 $f(x)$ 在该区间上的最大值是多少？答案的格式为 \(a - \frac{b}{c} \sqrt[3]{d} + \sqrt[3]{e}\)，请计算a + b + c + d + e的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
A savings card's password consists of 6 digits, each of which can be chosen from 0 to 9. When someone tries to withdraw money from an ATM, they forget the last digit of the password. If they randomly press the last digit, calculate the probability of pressing the correct digit in no more than 2 attempts. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{5}}