inclusionAI/Ring-lite-rl-data huggingface dataset:
Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). Rectangle \( MNLK \) is inscribed in square \( ABCD \) in such a way that one of its vertices is at point \( M \), and the other three vertices are located on different sides of the base square. Rectangle \( M_1N_1L_1K_1 \) is the orthogonal projection of rectangle \( MNLK \) onto the plane of the upper face \( A_1B_1C_1D_1 \). The ratio of the side lengths \( MK_1 \) and \( MN \) of quadrilateral \( MK_1L_1N \) is \( \sqrt{54}:8 \). Find the ratio \( AM:MB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:4}
Stella works at a pharmaceutical company and gets paid on a varying monthly income due to performance-based bonuses. Her base salary is 3500 dollars a month. Last year, she had the following bonuses in different months:

January: 1200 dollars
March: 600 dollars
May: 1500 dollars
August: 900 dollars
October: 1200 dollars

In addition, Stella took an unpaid leave for 2 months and had a 5% annual tax deduction on her total income. Calculate Stella's annual income after the tax deduction last year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38380}
A vat of orange juice contains 502.75 pints. If you wanted to pour the vat into 21 glasses with the same amount in each glass, how many pints would be in each glass?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.94}
Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where the upper vertex of $C$ is $A$, and the two foci are $F_{1}$ and $F_{2}$, with an eccentricity of $\frac{1}{2}$. A line passing through $F_{1}$ and perpendicular to $AF_{2}$ intersects $C$ at points $D$ and $E$, where $|DE| = 6$. Find the perimeter of $\triangle ADE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
13. How many polynomials $P$ with integer coefficients and degree at most 5 satisfy $0 \leq P(x)<120$ for all $x \in\{0,1,2,3,4,5\} ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86400000}
Using single variable calculus, find the volume of the tetrahedron formed by the tangent plane to the hyperboloid $$\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ at a point $(x_0, y_0, z_0)$ and the coordinate axis planes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{a^2b^2c^2}{6|x_0y_0z_0|}$}
A school is getting ready to open for the year and the sports class is organizing the equipment they have been donated. In total, they have 300 donations to organize. 60 of these were basketball hoops, half of which also had basketballs included as part of the donation. 120 pool floats were donated, but a quarter of these were damaged and were discarded before the sports class arrived. There were 50 footballs, 40 tennis balls, and the remaining donations were basketballs. In total, how many basketballs have been donated?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Given the parabola $y^{2}=2px\left(x \gt 0\right)$, point $P\left(2,1\right)$ lies inside the parabola. A line $l:y=2x+m$ that does not pass through point $P$ intersects the parabola at points $A$ and $B$. Lines $AP$ and $BP$ intersect the parabola at points $C$ and $D$, respectively. If for any line $l$, there exists a $\lambda$ such that $\overrightarrow{AP}=λ\overrightarrow{PC}$ and $\overrightarrow{BP}=λ\overrightarrow{PD}(\lambda\ \ \gt 0,\lambda \neq 1)$ hold, then $p=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Using the digits $0$, $1$, $2$, $3$, $4$ to form a five-digit number without repeating any digit, the probability that the number is even and the digits $1$, $2$ are adjacent is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.25}
On the side of a triangle, a point is taken such that an angle equals another angle. What is the smallest possible distance between the centers of the circles circumscribed around triangles, if \( BC = 1 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Consider the sequence of real numbers $S_n$ defined as $\sum_{i=0}^n \frac{(\log(n+i)-\log n)^2}{n+i}$. Determine if the limit $\lim_{n\to \infty}S_n$ exists, and if so, calculate its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\log^3 2}{3}}
Given a cyclic quadrilateral \(ABCD\). Rays \(AB\) and \(DC\) intersect at point \(E\), and rays \(DA\) and \(CB\) intersect at point \(F\). Ray \(BA\) intersects the circumcircle of triangle \(DEF\) at point \(L\), and ray \(BC\) intersects the same circle at point \(K\). The length of segment \(LK\) is 5, and \(\angle EBC = 15^\circ\). Find the radius of the circumcircle of triangle \(EFK\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
What is the square root of 16? Express your answer as a numerical value, including a ± symbol if both positive and negative roots are applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±4}
There are 13.5 multiples of 8 between 100 and a certain number (both are inclusive). What is the ending number of the range?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{204}
An alarm clock runs 4 minutes slow every hour. It was set right $3 \frac{1}{2}$ hours ago. Now another clock, which is correct, shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Add $36_7 + 274_7.$ Express your answer in base 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{343_7}
Determine the minimum of the expression

$$
\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{ab+bc+ca}}
$$

under the conditions that \(ab + bc + ca > 0\), \(a + b + c = 1\), and \(a, b, c\) are distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\sqrt{6}}
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 96 }
Call a permutation $a_1, a_2, \cdots, a_n$ of the integers $1, 2, \cdots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, $53421$ and $14253$ are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1, 2, \cdots, 7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{486}
Gina's snake is an interesting creature. In the first year of its life, it eats one mouse every 4 weeks. As it grows in the second year, it starts to eat one mouse every 3 weeks. For the rest of its life, it eats a mouse every 2 weeks. Given that a non-leap year has 52 weeks and a leap year has 52 weeks and 1 day, how many mice will Gina's snake eat over a decade assuming the decade starts in a non-leap year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{238}
In the regular triangular prism ABC-A1B1C1, there are six diagonals on the sides: AB1, A1B, BC1, B1C, C1A, CA1. Given that AB1 ⊥ BC1, how many other pairs of these diagonals are also perpendicular to each other? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the function $f(x)=x^{3}$, find the derivative of $f(-2)$, denoted as $[f(-2)]'$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider triangle $A B C$ with $\angle A=2 \angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{A B}$ at $E$. If $\frac{D E}{D C}=\frac{1}{3}$, compute $\frac{A B}{A C}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{9}}
Consider a modified sequence rule:
1) If a number is 30 or less, triple the number.
2) If a number is more than 30, subtract 15 from it.

Let $G$ be the first number in a sequence generated by the new rule. $G$ is a "magic number" if 18 is not a term in the sequence that starts with $G$. Determine how many of the whole numbers 1 through 60 are "magic numbers".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Five girls ran a race. Fiona started first, followed by Gertrude, then Hannah, then India, and lastly Janice. Whenever a girl overtook another girl, she was awarded a point. India was first to finish, followed by Gertrude, Fiona, Janice, and lastly Hannah. What is the lowest total number of points that could have been awarded?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Tim has a bag of 36 orange-flavoured sweets and Peter has a bag of some grape-flavoured sweets. They have to divide up the sweets into small trays with equal number of sweets; each tray containing either orange-flavoured or grape-flavoured sweets only. If the largest possible number of sweets in each tray without any remainder is 4, how many grape-flavoured sweets does Peter have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Example 5 (2005 National High School Mathematics League Test Question) Define the function $f(k)=\left\{\begin{array}{l}0, k \text { is a perfect square } \\ {\left[\frac{1}{\{\sqrt{k}\}}\right], k \text { is not a perfect square, }}\end{array}\right.$ Find $\sum_{k=1}^{240} f(k)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768}
Arrange 8 spheres, each with a radius of 2, in two layers within a cylindrical container such that each sphere is tangent to its four neighboring spheres, as well as to one base and the side surface of the cylinder. Find the height of the cylinder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 + 2\sqrt[4]{8}}
在 $\triangle A B C$ 中, $A=90^{\circ}, A B=1, A C=\sqrt{3}$, 点 P 满足 $\frac{\overrightarrow{P A}}{|\overrightarrow{P A}|}+\frac{\overrightarrow{P B}}{|\overrightarrow{P B}|}+\frac{\overrightarrow{P C}}{|\overrightarrow{P C}|}=\mathbf{0}$。如果答案是 $x^\circ$, 求 $x$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
The angle bisector \( A L \) and the median \( B M \) of triangle \( A B C \) intersect at point \( X \). Line \( C X \) intersects side \( A B \) at point \( Y \). Find the area of triangle \( C Y L \) given that \( \angle B A C = 60^{\circ} \) and \( A L = x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x^2}{4 \sqrt{3}}}
Three planes intersect pairwise, and their three lines of intersection are parallel to each other. Determine the number of distinct parts the space is divided into as a result of these intersections. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Proctors Andy and Kristin have a PUMaC team of eight students labelled $s_1, s_2, ... , s_8$ (the PUMaC staff being awful with names). The following occurs:
$1$. Andy tells the students to arrange themselves in a line in arbitrary order.
$2$. Kristin tells each student $s_i$ to move to the current spot of student $s_j$ , where $j \equiv 3i + 1$ (mod $8$).
$3$. Andy tells each student $s_i$ to move to the current spot of the student who was in the $i$th position of the line after step $1$.
How many possible orders can the students be in now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1260}
For a natural number \( N \), if at least five of the natural numbers from 1 to 9 can divide \( N \), then \( N \) is called a "five-rule number." What is the smallest "five-rule number" greater than 2000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2004}
Let \(\mathcal{P}\) be a parabola with focus \(F\) and directrix \(\ell\). A line through \(F\) intersects \(\mathcal{P}\) at two points \(A\) and \(B\). Let \(D\) and \(C\) be the feet of the altitudes from \(A\) and \(B\) onto \(\ell\), respectively. Given that \(AB = 20\) and \(CD = 14\), compute the area of \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
A real number \(x\) is chosen uniformly at random from the interval \([0,1000]\). Find the probability that \(\left\lfloor\frac{\left\lfloor\frac{x}{2.5}\right\rfloor}{2.5}\right\rfloor=\left\lfloor\frac{x}{6.25}\right\rfloor\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{10}}
in a group of cheerleaders , the group made of 10 males and 16 females . every cheerleader gets to choose between a coke or malt . the number of those that choose malt is double the number that chose coke , if all the 6 males choose malt , how many females chose malt ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In Fifi's closet, she hangs all her clothes on colored plastic hangers. She has clothes hanging on:

- 7 pink hangers
- 4 green hangers
- One less blue hanger than there are green hangers
- One less yellow hanger than there are blue hangers
- Twice the number of orange hangers as the sum of pink and green hangers
- Three more purple hangers than the difference between blue and yellow hangers
- The arithmetic mean of pink, green, and blue hangers as the number of red hangers
- One more brown hanger than triple the number of red hangers
- 3 gray hangers for every 5 purple hangers

What is the total number of colored hangers in Fifi's closet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Given any real numbers \(a > b > c > d > 0\), to ensure \(\log_{\frac{b}{a}} 2004 + \log_{\frac{c}{b}} 2004 + \log_{\frac{d}{c}} 2004 \geq m \log_{\frac{d}{a}} 2004\) always holds, find the minimum value of \(m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Suppose we have an (infinite) cone $\mathcal{C}$ with apex $A$ and a plane $\pi$. The intersection of $\pi$ and $\mathcal{C}$ is an ellipse $\mathcal{E}$ with major axis $BC$, such that $B$ is closer to $A$ than $C$, and $BC=4, AC=5, AB=3$. Suppose we inscribe a sphere in each part of $\mathcal{C}$ cut up by $\mathcal{E}$ with both spheres tangent to $\mathcal{E}$. What is the ratio of the radii of the spheres (smaller to larger)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
A trapezoid \(K L M N\) is inscribed in a circle with center \(O\), where \(K L\) is parallel to \(M N\), \(K L = 8\), \(M N = 2\), and the angle \(\angle N K L\) is \(45^{\circ}\). The chord \(M A\) of the circle intersects the segment \(K L\) at point \(B\) such that \(K B = 3\). Find the distance from point \(O\) to the line \(A K\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{\sqrt{26}}}
Seven schools organized a science fair competition and invited students from all the schools to develop ideas and present them before a panel of judges selected by all the schools competing in the science fair. There were 90 students from Quantum High School at the competition, 2/3 times as many students from Schrödinger High as Quantum High School, 4/9 times as many students from Einstein High as Schrödinger High, 5/12 times as many students from Newton High as Einstein High, 11/20 times as many students from Galileo High as Newton High, and 13/50 times as many students from Pascal High as Galileo High. From the first six schools, the combined total number of students was x. Faraday High School sent four times the number of students from the combined total of the other six schools to the competition. Calculate the total number of students at the competition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{980}
An error of 20% in excess is made while measuring the side of a square. Additionally, the equipment used for calculating the area has an error of 4%. What is the combined percentage of error in the calculated area of the square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49.76\%}
Let $X_1, X_2, X_3, \ldots$ be a sequence of i.i.d. uniform $(0,1)$ random variables. Calculate the value of \[ \lim_{n \to \infty} P\left(-\ln(1-X_1) - \ln(1-X_2) - \cdots - \ln(1-X_n) \geq n\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
the h.c.f. of two numbers is 20 and the other three factors of their l.c.m. are 3^2, 17 and 23. The larger of the two numbers is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70380}
For positive integer $k$, define $x_k=3k+\sqrt{k^2-1}-2(\sqrt{k^2-k}+\sqrt{k^2+k})$. Then $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_{1681}}=\sqrt{m}-n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{911}
Let \( x = -\sqrt{2} + \sqrt{3} + \sqrt{5} \), \( y = \sqrt{2} - \sqrt{3} + \sqrt{5} \), and \( z = \sqrt{2} + \sqrt{3} - \sqrt{5} \). Evaluate the expression:
$$
\frac{x^{4}}{(x-y)(x-z)} + \frac{y^{4}}{(y-z)(y-x)} + \frac{z^{4}}{(z-x)(z-y)}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Solve the equation: \( 3 \operatorname{tg} 3x - 4 \operatorname{tg} 2x = \operatorname{tg}^{2} 2x \operatorname{tg} 3x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = \pi n}
Find a function $h(x)$ such that the power series expansion of $h(x)$ around $x=0$ has coefficients $b_k$ satisfying \( \sum_{k\geq 0} (k+1) b_k = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-x}}
A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors, starting and ending with the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{126}}
Given that the domain of the function y=f(2^x) is [1,2], determine the domain of the function y=f(x+1)/(x-1). Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,3]}
Consider the integer，
$$N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_{321 ~digits}.$$
Find the sum of the digits of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{342}
The base of the pyramid \( PABCD \) is the parallelogram \( ABCD \). Point \( N \) is the midpoint of edge \( AP \), point \( K \) is the midpoint of the median \( PL \) of triangle \( BPC \), and point \( M \) lies on edge \( PB \) such that \( PM = 5MB \). In what ratio does the plane passing through points \( M, N, K \) divide the volume of the pyramid \( PABCD \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25:227}
Haley's old washing machine could only wash a combination of three shirts, two sweaters, and four pairs of socks at a time. Additionally, she cannot mix whites with any colored clothes in a single load. If she had to wash nine white shirts, twelve colored shirts, eighteen white sweaters, twenty colored sweaters, sixteen white pairs of socks, and twenty-four colored pairs of socks, how many loads would she have to do, considering both capacity and color restrictions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
以 $\sqrt{2}$ 和 $1-\sqrt[3]{2}$ 为两个根的有理系数多项式的次数最低是____。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the condition that must be satisfied by the positive real numbers \(a, b, c\) such that the iterative process of constructing a new tuple \((S - 2a, S - 2b, S - 2c)\), where \(S = a + b + c\), results in a sequence of tuples with all positive elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = b = c}
Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, find the value of $x$. The original answer is in the form of $k \cdot m^n$, where $k$, $m$, and $n$ are constants. Please find the value of $k + m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Evaluate the limit \( \lim_{n \to \infty} \dfrac{1}{n^2} \sum_{k=1}^{n-1} k \cdot \left \lfloor x + \dfrac{n-k-1}{n} \right \rfloor \) to three decimal places, where \( x = \sqrt{2015} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.395}
The circle is divided by points \(A, B, C, D\) such that \(\sim AB: BC: CD: DA = 2: 3: 5: 6\). Chords \(AC\) and \(BD\) are drawn, intersecting at point \(M\).

Find the angle \(AMB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78.75}
Let $m \ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
How many liters of pure hydrochloric acid must be added to a certain amount of solution that is 10% hydrochloric acid to get a solution that is 15% hydrochloric acid, if 3.52941176471 liters of pure hydrochloric acid is added?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
For all positive integers  $n$ , denote by  $\sigma(n)$  the sum of the positive divisors of  $n$  and  $\nu_p(n)$  the largest power of  $p$  which divides  $n$ .  Compute the largest positive integer  $k$  such that  $5^k$  divides \[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\] where  $N=6^{1999}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The product of a million whole numbers is equal to million. What can be the greatest possible value of the sum of these numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1999999}
21 Let $n$ be a given natural number, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$ as $m$. Find the maximum value of $m$ when
$$a_{1}^{2}+\cdots+a_{n}^{2}=1$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{12}{n\left(n^{2}-1\right)}}}
In a football tournament, 15 teams participated, each playing exactly once against every other team. A win earned 3 points, a draw earned 1 point, and a loss earned 0 points.

After the tournament ended, it turned out that some 6 teams scored at least $N$ points each. What is the highest possible integer value for $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}