inclusionAI/Ring-lite-rl-data huggingface dataset:
Given 5 differently colored balls and 3 different boxes, with the requirement that no box is empty, calculate the total number of different ways to place 4 balls into the boxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
The first three numbers of a sequence are \(1, 7, 8\). Every subsequent number is the remainder obtained when the sum of the previous three numbers is divided by 4. Find the sum of the first 2011 numbers in this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3028}
Given the function $f(x)=\sqrt{2}\sin(2\omega x-\frac{\pi}{12})+1$ ($\omega > 0$) has exactly $3$ zeros in the interval $\left[0,\pi \right]$, determine the minimum value of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{3}}
If n is a positive integer and the product of all integers from 1 to n, inclusive, is a multiple of a certain number, the least possible value of n is 7. What is the multiple?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5040}
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is 11.879049676025918 seconds. If the length of the second train is 160 m, what is the length of the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{170}
Let \( f(x) = ax^3 + bx + c \) (where \( a, b, c \) are real numbers). When \( 0 \leq x \leq 1 \), \( 0 \leq f(x) \leq 1 \). Determine the maximum possible value of \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 \sqrt{3}}{2}}
Find all prime numbers \( p, q, \) and \( r \) such that \( pqr = 5(p + q + r) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 5, 7)}
19. Let the number $a$ have the following property: For any four real numbers $x_{1}, x_{2}, x_{3}, x_{4}$, there always exist integers $k_{1}, k_{2}, k_{3}, k_{4}$, such that $\sum_{1 \leqslant i<j \leqslant 4}\left(\left(x_{i}-k_{i}\right)-\left(x_{j}-k_{j}\right)\right)^{2} \leqslant a$, find the minimum value of such $a$. (1990
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{4}}
What is the ratio between perimeters of two squares, one having 2 times the diagonal than the other, and the other square having doubled side lengths compared to the original squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:1}
Find the smallest natural decimal number \(n\) whose square starts with the digits 19 and ends with the digits 89.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1383}
Given that a normal vector of line $l$ is $\overrightarrow{n}=(\sqrt{3}, -1)$, find the size of the slope angle of line $l$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
For any $x \in \mathbb{R}$, the function $f(x)$ satisfies $f(x+1)= \sqrt {f(x)-[f(x)]^{2}}+ \frac {1}{2}$. Let $a_{n}=[f(n)]^{2}-f(n)$, and the sum of the first $15$ terms of the sequence $\{a_{n}\}$ is $- \frac {31}{16}$. Find $f(15)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {3}{4}}
Arthur decided to treat himself to dinner at a nice restaurant to celebrate his recent promotion. He ordered a nice appetizer for $8, a delicious ribeye steak for his entrée at $30, had two glasses of nice red wine with dinner for $4 each, and a slice of caramel cheesecake for dessert for $7. He had a voucher that offered a 40% discount on his entrée, and another voucher for a 10% discount on his total bill before tax. The local tax rate is 8%. 

Arthur thoughtfully tipped his waiter a full 20% of what the cost of his meal would have been without any discounts, but including tax. How much, including the tip, did Arthur spend on dinner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$51.30}
Maria must work 8 hours each day, excluding her 45-minute lunch break. If she starts working at 7:25 A.M. and takes her lunch break at noon, what time will her workday end? Express your answer in the format of time (e.g., 4:10 P.M.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:10 P.M.}
Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. What is $a + b?$ (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$151$}
Compute \(\left\lfloor \frac{1000!}{1!+2!+\cdots+999!} \right\rfloor.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
Evaluate the triple integral \( \iiint_D x^2yz \,dx\,dy\,dz \) over the region \( D = \{(x,y,z):0 \leq x \leq y+z \leq z \leq 1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{420}}
In each cell of a $5 \times 5$ table, an invisible ink natural number is written. It is known that the sum of all the numbers is 200, and the sum of the three numbers located inside any $1 \times 3$ rectangle is 23. What is the central number in the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In triangle \(ABC\), a median \(AM\) is drawn (point \(M\) lies on side \(BC\)). It is known that the angle \(\angle CAM\) is \(30^\circ\) and side \(AC\) is 2. Find the distance from point \(B\) to line \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Cyclists Petya, Vlad, and Timur simultaneously started a warm-up race on a circular bike path. Their speeds are 27 km/h, 30 km/h, and 32 km/h, respectively. After what minimum time will they find themselves at the same point on the track again? (The length of the bike path is 400 meters.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24 \text{ minutes}}
Find the following limit $$\lim_{n \rightarrow \infty} (\frac{2^{1/n}}{n+1} +\frac{2^{2/n}}{n+1/2}+\dots+\frac{2^{n/n}}{n+{1/n}})$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{\ln 2}\)}
A natural number, when raised to the sixth power, has digits which, when arranged in ascending order, are:

$$
0,2,3,4,4,7,8,8,9
$$

What is this number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Find the limit of the sequence \( x_n = \left(\frac{e\sqrt[3]{e}\cdots\sqrt[n]{e}}{n}\right) \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\gamma - \frac{1}{2}}}
John has three part-time jobs. From job A, he initially makes $60 a week, and after a raise, he now makes $80 a week. Job B gives biweekly raises, so he initially makes $100 a week, after the first raise, he makes $110, and after the second raise, he makes $130 a week. Job C pays hourly, and John works 20 hours per week initially at a rate of $8 per hour, then his hourly rate is increased to $10. What is the overall average percentage increase in his weekly income after all the raises?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29.44\%}
Use a parameterisation to find the flux of $F = (3xy, 0, -z)$ outward (normal away from the z-axis) through the cone $z = 9\sqrt{x^2 + y^2}$, $0 \le z \le 9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi}
Factorize: $a+(a+b) x+(a+2 b) x^{2}+(a+3 b) x^{3}+3 b x^{4}+2 b x^{5}+b x^{6}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 + x)(1 + x^2)(a + bx + bx^2 + bx^3)}
Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. For integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Words cannot cross between lines but may be repeated. Compute the number of distinct haikus Ted can make, and find the remainder when this number is divided by $100$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Let $M=\{1,2, \cdots, 2005\}$. Subset $A$ of $M$ satisfies the condition: if $x \in A$, then $15x \notin A$. What is the maximum number of elements in $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1880}
Spencer is making burritos, each consisting of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos. In how many ways can he make exactly five burritos?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
On the game show $\text{\emph{Wheel of Fortunes}}$, you encounter a spinner divided equally into 6 regions labeled as follows: "Bankrupt," "$600", "$100", "$2000", "$150", "$700". What is the probability that you will accumulate exactly $1450$ in your first three spins without landing on "Bankrupt"? Assume each spin is independent and all outcomes have equal likelihood.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{125}}
Calculate the following expression (accurate to 8 decimal places):
$$
16\left(\frac{1}{5}-\frac{1}{3} \times \frac{1}{5^{3}}+\frac{1}{5} \times \frac{1}{5^{5}}-\frac{1}{7} \times \frac{1}{5^{7}}+\frac{1}{9} \times \frac{1}{5^{9}}-\frac{1}{11} \times \frac{1}{5^{11}}\right)-4\left(\frac{1}{239}-\frac{1}{3} \times \frac{1}{239^{3}}\right)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.14159265}
Simplify first, then evaluate $\left(\frac{{a^2-4}}{{a^2-4a+4}}-\frac{1}{{2-a}}\right) \div \frac{2}{{a^2-2a}}$, where $a$ satisfies $a^{2}+3a-2=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find $B^2$, where $B$ is the sum of the absolute values of all roots of the following equation:
\[ x = \sqrt{29} + \frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{x}}}}}}}}}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{513}
Two trains 140 m and another train of certain length run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is 10.799136069114471 seconds. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Express the following as a common fraction: $\sqrt[3]{5\div 15.75}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt[3]{20}}{\sqrt[3]{63}}}
In the diagram, P is on RS so that QP bisects ∠SQR. Also, PQ = PR, ∠RSQ = 2y°, and ∠RPQ = 3y°. Find the measure of ∠RPQ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Long ago, nine identical books cost 11 rubles and some kopecks, and thirteen such books cost 15 rubles and some kopecks. How much did one book cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123 \text{ kopecks}}
Four runners start at the same position on a circular track that is 1 km long, running in the same direction. Their speeds are \( \sqrt{5} \), \( e \), 3, and \( \pi \) meters per second. Determine the first time, in seconds, when the runner with speed \( e \) is at least 250 meters away from each of the other runners.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{250}{3 - e}}
Determine the interval in which the solution to the equation \(e^x = \frac{1}{x}\) lies for \(x > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0.5, 0.6)}
Given a trapezoid \(ABCD\) where angle \(BAD\) is a right angle. A circle is constructed with diameter \(AB\) and it intersects the diagonal \(BD\) at point \(M\). It is known that \(AB = 3\), \(AD = 4\), and \(BC = 1\). Find the angle \(CAM\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arcsin \frac{1}{\sqrt{10}}}
Calculate the definite integral:

$$
\int_{0}^{\frac{2\pi}{3}} \frac{\cos^2 x \, dx}{(1 + \cos x + \sin x)^2}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2} - \ln 2}
Given that the point $\left(\frac{1}{2}, 2\right)$ is on the graph of the power function $y=f(x)$, and the point $(-2, \frac{1}{4})$ is on the graph of the power function $y=g(x)$, find the value of $f(2)+g(-1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
We define \( a @ b = a \times (a + 1) \times \ldots \times (a + b - 1) \). Given \( x @ y @ 2 = 420 \), then \( y @ x = \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let  $\theta=\frac{2\pi}{2015}$ , and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form  $\frac{b}{2^a}$ , where  $a$  is a non-negative integer and  $b$  is an odd integer (not necessarily positive). Find  $a+b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1441}
Let $P$ and $A$ be $3 × 3$ real matrices such that $PAP^t = -A^t$, where $P^t$ denotes the transpose of $P$. Find the possible values of $\det(P)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the sequence \( \{ a_n \} \) of real numbers for \( n \ge 1 \) defined by the recurrence relation:
\[ a_{n+2} = a_{n+1} - a_n \]
and the conditions:
\[ \sum_{n=1}^{999} a_n = 1003 \]
\[ \sum_{n=1}^{1003} a_n = -999 \]
find the value of \( \sum_{n=1}^{2002} a_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2002}
Given an equilateral triangle \( \triangle ABC \) with side length 1 and \( PA \perp \) plane \( ABC \), and \( PA = \frac{\sqrt{6}}{4} \). Let \( A' \) be the reflection of point \( A \) over plane \( PBC \). Determine the angle formed between line \( A'C \) and \( AB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
I have found that the curve given by $x^3+x+1=y^2$ over $\mathbb{F_5}$ has 9 points. Now I am supposed to find the number of points of the same curve on $\mathbb{F}_{125}$. Using Hasse and the fact that 9 has to divide this order I have 5 possibilities: $108,117,126,135,144$ but here I don't know how to proceed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Find the norm of the linear operator \( T: C([0,1],\mathbb{R}) \rightarrow C([0,1],\mathbb{R}) \) defined by \( Tf = gf \), where \( g \in C([0,1],\mathbb{R}) \), with respect to the \( L^2 \) norm \( \|f\|_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|g\|_\infty}
Compute the limit: \[ \lim_{n\to\infty}\sum_{i=1}^{2n^4+n^2}\frac{5n^2+1}{n^4+i} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the integral \( I = \int_{0}^{1} f(x) \, dx \) given that the function \( f : [0,1] \rightarrow \mathbb{R} \) satisfies \( f'(x) = \sqrt{1+f^2(x)} \) for all \( x \in [0,1] \) and \( f(0) + f(1) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Juan is measuring the diameter of a large rather ornamental plate to cover it with a decorative film. Its actual diameter is 30cm, but his measurement tool has an error of up to $30\%$. Compute the largest possible percent error, in percent, in Juan's calculated area of the ornamental plate.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69}
Given a permutation group \( G \) acting on a finite set \( D \), and a finite set \( R \) of colors, a permutation group \( \hat{G} \) is induced to act on \( R^D \) (the set of functions from \( D \) to \( R \)). For any \( \sigma \in G \), the function \( \hat{\sigma} \) is defined by \( \hat{\sigma}(F)(d) = F(\sigma(d)) \) for all \( F \in R^D \). Determine the relationship between \( |G| \) and \( |\hat{G}| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|\hat{G}| = |G|}
In the quadrilateral PQRS, the length of PQ is 11 cm, the length of QR is 7 cm, the length of RS is 9 cm, and the length of SP is 3 cm. Both ∠QRS and ∠SPQ are 90°. What is the area of the quadrilateral PQRS?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine the number of real roots of the polynomial \( P(x) = x^7 + x^6 - 7x^5 - 7x^4 + 8x^3 + 8x^2 + 16x + 16 \), given that its Sturm's sequence ends with \( P_5(x) = -x^2 + 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a}|=2$, $| \overrightarrow {b}|=1$, and $\overrightarrow {b} \perp ( \overrightarrow {a}+ \overrightarrow {b})$, determine the projection of vector $\overrightarrow {a}$ onto vector $\overrightarrow {b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Starting from the natural numbers beginning with 1, remove multiples of 4, and then group the remaining numbers into sets of three. Determine the group number and position within the group where the number 2022 falls. Express your answer in the format "group number, position within group" (e.g., 506th group, 2nd number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{506th group, 2nd number}
求最小的正整数$n$，满足：

(1)$n$恰有144个不同的正因数；
(2)在$n$的正因数中有10个连续整数．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110880}
From vertex \( B \) of the triangle \( ABC \), a line segment antiparallel to \( BC \) is drawn and intersects side \( b \) at point \( B_1 \). From \( B_1 \), a similar segment parallel to \( BC \) is drawn, intersecting side \( c \) at point \( C_1 \). This procedure can be repeated infinitely.

What is the sum of \( \overline{B C}+\overline{B B_{1}}+B_{1} C_{1}+\ldots \) to infinity, if \( b > c \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{ab}{b - c}}
Three uncoordinated aliens launch a three-day attack on four galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galaxy is destroyed by the end of the attack can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \), what is \( m+n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1151}
Given two sets $A=\{1,x^{- \frac {1}{2}}\}$ and $B=\{0,1,2\}$, if $A \subseteq B$, then $x =$ __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Marcy has three pets: a cat, a dog, and a parrot. She spends 12 minutes petting her cat, 1/3 of that time combing it, 1/4 of the combing time brushing its teeth, and 1/2 of the petting time playing with it. In addition to that, she feeds her cat for 5 minutes and spends 2/5 of the feeding time cleaning its food bowl. 

She spends 18 minutes walking her dog, 2/3 of that time playing fetch, and 9 minutes grooming it, with 1/3 of the time trimming its nails. She spends 1/4 of the walking time training the dog for new tricks. 

Her parrot needs 15 minutes of daily speech training, where she divides the time equally between repeating words and playing short tunes on a xylophone. Marcy also spends 8 minutes per day cleaning the parrot's cage, with half of that time spent setting up its toys in the cage. 

How many minutes does Marcy spend with her pets in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103.5}
The integers whose absolute values are not greater than $3$ are ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, \pm 1, \pm 2, \pm 3}
On a circle with a radius of 1, mark a point \( O \) and use a compass to make a mark to the right with a radius of \( l \). From the obtained point \( O_{1} \), make a second mark in the same direction with the same radius, and continue this process 1968 times. After this, the circle is cut at all 1968 marks, resulting in 1968 arcs. How many different lengths of arcs can be obtained in this way?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Some camels cost as much as 24 horses, 16 horses cost as much as 4 oxen, and 6 oxen cost as much as 4 elephants. The cost of 10 elephants is Rs. 110000. If the cost of a camel is Rs. 4400, how many camels are we talking about?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
From point $A$, two rays emerge at an angle $\alpha$. Two points $B$ and $B_{1}$ are taken on one ray, and two points $C$ and $C_{1}$ are taken on the other ray. Find the length of the common chord of the circles circumscribed around triangles $ABC$ and $AB_{1}C_{1}$, if $|AB|-|AC|=\left|AB_{1}\right|-\left|AC_{1}\right|=a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{2 \sin \left( \frac{\alpha}{2} \right)}}
Calculate the spectral radius of the Volterra operator \( A \) defined on \( L^2[0,1] \) by \( A(f)(x) = \int_0^x f(t) \, dt \). Use the formula \( \rho(A) = \lim_{n\rightarrow \infty} \|A^n\|^{1/n} \) and the fact that \( \|(x-s)^n\|_\infty = 1 \) for \( s \in [0,x] \), \( x \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The height of a right triangle, dropped to the hypotenuse, is $h$.

What is the minimum length that the median, drawn from the vertex of the larger acute angle, can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3h}{2}}
Determine whether the map \( \phi : L^3([0,1], \mathbb{C}) \to \mathbb{C} \) defined by \( \phi(f) = \int_{0}^{1} f(x^2) \, dx \) is a bounded linear functional on the Banach space \( L^3([0,1], \mathbb{C}) \). If it is bounded, find its norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{1/3}}
If ${e}^{x+1}-\frac{lnx+2k}{x}-k≥0$ always holds for any $x \gt 0$, then the range of real number $k$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1]}
Given two independent standard normal random variables, $X$ and $Z$, calculate the probability $P(X+Z<0, Z>0)$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
From a pile of 20 100-yuan bills that includes 5 counterfeit bills, two bills are randomly selected. If one of the two bills is found to be counterfeit when checked with a counterfeit detector, the probability that both bills are counterfeit is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{17}}