inclusionAI/Ring-lite-rl-data huggingface dataset:
In an isosceles trapezoid with an acute angle $\alpha$ at the base, a circle constructed on the lateral side as a diameter touches the other lateral side.

In what ratio does it divide the larger base of the trapezoid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin2\alpha:1}
Point $D$ is on side $AC$ of triangle $ABC$, $\angle ABD=15^{\circ}$ and  $\angle DBC=50^{\circ}$. What is the measure of angle $BAD$, in degrees?

[asy]draw((-43,0)--(43,0)--(43,40)--cycle);
draw((-4,0)--(43,40));
draw((39,4)--(39,0));
draw((39,4)--(43,4));
draw((-1,36)--(22.5,26),Arrow);
label("$15^{\circ}$",(-1,36),W);
label("$50^{\circ}$",(36,30),S);
label("D",(-4,0),S);
label("A",(-43,0),W);
label("B",(43,40),N);
label("C",(43,0),E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25^\circ}
Mason is cleaning out all the junk in his attic. Initially, the attic had 20% useful items, 10% valuable heirlooms, and 70% junk. Since then, he has given away 4 useful items to his neighbors and has sold 20 valuable items online. Now, his attic has 16 useful items remaining. Considering the new distribution and the items he has given away or sold, how many junk items does his attic currently have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
10 Let $x, y, z \in \mathbf{R}^{+}$, and satisfy $x y z + x + z = y$, find the maximum value of $p = \frac{2}{x^{2}+1} - \frac{2}{y^{2}+1} + \frac{3}{z^{2}+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3}}
Given three points \(A, B, C\) forming a triangle with angles \(30^{\circ}\), \(45^{\circ}\), and \(105^{\circ}\). Two of these points are chosen, and the perpendicular bisector of the segment connecting them is drawn. The third point is then reflected across this perpendicular bisector to obtain a fourth point \(D\). This procedure is repeated with the resulting set of four points, where two points are chosen, the perpendicular bisector is drawn, and all points are reflected across it. What is the maximum number of distinct points that can be obtained as a result of repeatedly applying this procedure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the minimum value for \(a, b > 0\) of the expression

$$
\frac{(3 a b - 6 b + a(1 - a))^{2} + \left(9 b^{2} + 2 a + 3 b(1 - a)\right)^{2}}{a^{2} + 9 b^{2}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Fill the numbers $1, 2, \cdots, 9$ into a $3 \times 3$ grid such that each cell contains one number, each row's numbers increase from left to right, and each column's numbers decrease from top to bottom. How many different arrangements are possible? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
In a certain football competition, a victory is worth 3 points, a draw is worth some points, and a defeat is worth 0 points. Each team plays 20 matches. A team scored 14 points after 5 games. The team needs to win at least 6 of the remaining matches to reach the 40-point mark by the end of the tournament. How many points is a draw worth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Art, Bella, Carla, and Dan each bake cookies of different shapes with the same thickness. Art's cookies are rectangles: 4 inches by 6 inches, while Bella's cookies are equilateral triangles with a side length of 5 inches. Carla's cookies are squares with a side length of 4 inches, and Dan's cookies are circles with a radius of 3 inches. If Art makes 18 cookies in one batch, how many cookies will be in one batch of Dan's cookies?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Carl's typing speed varies throughout the week. On Monday, Wednesday, and Friday, he types at a speed of 50 words per minute for 5 hours with 10 minutes of rest after every hour. On Tuesday and Thursday, his typing speed decreases to 40 words per minute, and he types for 3 hours with 15 minutes of rest after every hour. On Saturday, he types at a speed of 60 words per minute for 4 hours with no breaks. On Sunday, he rests and does not type at all. Calculate the total number of words Carl types in a week.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65,400}
If $$I_1=\int _0 ^1 \frac {dx}{1+\sqrt {x }}, \hspace{5mm} I_2=\int _0 ^1 \frac {dx}{1+\frac {1}{1+\sqrt {x}}}, \hspace{5mm} I_3=\int _0 ^1 \frac {dx}{1+\frac {1}{1+\frac {1}{1+\sqrt {x}}}} $$ then $$ \lim _{n \to \infty} I_n \hspace{5mm}\text{ is equal to ?} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\sqrt{5}-1}{2}\)}
On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, we have $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{494}
若 $\triangle A B C$ 的内角 $A, B, C$ 满足 $\sin A=\cos B=\tan C$, 求 $\cos ^{3} A+\cos ^{2} A-\cos A$ 的值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Circle \(\omega\) is inscribed in rhombus \(H M_{1} M_{2} T\) so that \(\omega\) is tangent to \(\overline{H M_{1}}\) at \(A\), \(\overline{M_{1} M_{2}}\) at \(I\), \(\overline{M_{2} T}\) at \(M\), and \(\overline{T H}\) at \(E\). Given that the area of \(H M_{1} M_{2} T\) is 1440 and the area of \(E M T\) is 405, find the area of \(A I M E\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{540}
Calculate: $\sqrt{8}\div 2{2}^{\frac{1}{2}}=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the equation (z+2i)(z-2i) = 2, solve for z. Express your answer in terms of i (imaginary unit), including the ± symbol if applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±\sqrt{2}i}
Given that $F\_1$ is the left focus of the hyperbola $C$: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, point $B$ has coordinates $(0, b)$, and the line $F\_1B$ intersects with the two asymptotes of hyperbola $C$ at points $P$ and $Q$. If $\overrightarrow{QP} = 4\overrightarrow{PF\_1}$, find the eccentricity of the hyperbola $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
5. The maximum value of the orthogonal projection area of a regular tetrahedron with edge length 1 on a horizontal plane is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
If $n$ is an odd number, calculate the remainder of $7^n+C_n^17^{n-1}+C_n^27^{n-2}+\ldots+C_n^{n-1}7$ when divided by 9. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
There are $2024$ people, consisting of knights and liars, and some of them are friends. Each person was asked how many friends they have, and the possible answers were $0, 1, \ldots, 2023$. Every knight answered truthfully, while every liar altered the real answer by exactly $1$. What is the minimal number of liars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
On an $m × n$ rectangular piece of graph paper there are several squares whose sides are on the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square is situated inside another one. What is the maximal number of such squares? Please provide the value when $m=3,n=4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let  $\Omega$  be a circle centered at  $O$ . Let  $ABCD$  be a quadrilateral inscribed in  $\Omega$ , such that  $AB = 12$ ,  $AD = 18$ , and  $AC$  is perpendicular to  $BD$ . The circumcircle of  $AOC$  intersects ray  $DB$  past  $B$  at  $P$ . Given that  $\angle PAD = 90^\circ$ , find  $BD^2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1980}{7}}
Find the largest prime number \( p \) such that there exist positive integers \( a \) and \( b \) satisfying
\[ p = \frac{b}{2} \sqrt{\frac{a - b}{a + b}} .\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Two circles touch internally. A line passing through the center of the larger circle intersects it at points \( A \) and \( D \), and intersects the smaller circle at points \( B \) and \( C \). Find the ratio of the radii of the circles, given that \( AB : BC : CD = 3 : 7 : 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Two circles intersect at the points  $C$  and  $D$ . The straight lines  $CD$  and  $BYXA$  intersect at the point  $Z$ . Moreever, the straight line  $WB$  is tangent to both of the circles. Suppose  $ZX=ZY$  and  $AB\cdot AX=100$ . Find the value of  $BW$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let point P be a moving point on the curve $y=x^2$, and let the tangent line to the curve at point P be $l$. A line passing through point P and perpendicular to line $l$ intersects the curve $y=x^2$ at another point Q. The minimum value of PQ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {3 \sqrt {3}}{2}}
Let \( p \) be an odd prime number. How many \( p \)-element subsets \( A \) of \(\{1, 2, \ldots, 2p\}\) are there, the sum of whose elements is divisible by \( p \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \frac{\binom{2p}{p} - 2}{p}}
## Task A-3.3.

On a board of dimensions $20 \times 19$, tiles of dimensions $3 \times 1$ are placed such that they cover exactly three squares of the board, and they do not overlap or touch each other, not even at the corners.

Determine the maximum possible number of $3 \times 1$ tiles on this board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given an arithmetic-geometric sequence ${a_{n}}$ with the sum of the first $n$ terms denoted as $S_{n}$, and $S_{2}=1$, $S_{4}=3$, find the value of $S_{6}$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $A B C$ be a triangle with $A B=5, B C=8$, and $C A=7$. Let $\Gamma$ be a circle internally tangent to the circumcircle of $A B C$ at $A$ which is also tangent to segment $B C. \Gamma$ intersects $A B$ and $A C$ at points $D$ and $E$, respectively. Determine the length of segment $D E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{40}{9}$}
For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\lceil \sqrt{s} \rceil \mid s \in S\}$. Find the number of sets $T$ such that $|f(T)| = 2$ and $f(f(T)) = \{2\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5043}
How many 3-digit positive integers have the property that the sum of their digits is greater than the product of their digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199}
At a killers' convention, 1000 participants attended, each receiving a registration number from 1 to 1000. By the end of the convention, it turned out that all the killers except for number 1 were dead. It is known that each killer could only kill killers with larger numbers, and the number of their victims could not exceed their own number. What is the minimum number of killers who could have participated in the killings at the convention?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the integral $ \int_{-\infty}^{\infty}\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x $ without using the change of variables $x = \log(t)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi^3}{4}\)}
Among the coordinates  $(x,y)$   $(1\leq x,y\leq 101)$ , choose some points such that there does not exist  $4$  points which form a isoceles trapezium with its base parallel to either the  $x$  or  $y$  axis(including rectangles). Find the maximum number of coordinate points that can be chosen.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
A point is taken arbitrarily inside a triangle, and lines parallel to the sides of the triangle are drawn through this point. These lines divide the area of the triangle into six parts. What is the area of the given triangle if the areas of the three resulting triangles are given as: $t_{1}, t_{2}, t_{3}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{t_{1}}+\sqrt{t_{2}}+\sqrt{t_{3}})^{2}}
9. Use the digits $1 \sim 9$ to form three three-digit numbers, each digit used only once, and all three numbers are divisible by 4. Then the largest of these three numbers is at least $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{548}
For real numbers $x$ and $y$, a new operation is defined as follows: $x*y = ax + by - 1$, where $a$ and $b$ are constants. If $1*2=4$ and $\left(-2\right)*3=10$, then $a*b = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In a table with $88$ rows and $253$ columns, each cell is colored either purple or yellow. Suppose that for each yellow cell $c$, the following condition holds:  
$$x(c) \cdot y(c) \geq 184,$$  
where $x(c)$ is the number of purple cells in the same row as $c$, and $y(c)$ is the number of purple cells in the same column as $c$.  
Find the least possible number of cells that are colored purple.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\] and \[y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\] the area of the triangle can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{107}
I'm sorry, I misunderstood your previous message. Here's a more difficult version of the problem without generating an answer:

One night, 27 percent of the female officers on a police force were on duty. If 275 police officers were on duty that night and the ratio of male to female officers was 4:3, how many female officers were on the police force?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{437}
$ABCD$  is a rectangle with  $AB = CD = 2$ . A circle centered at  $O$  is tangent to  $BC$ ,  $CD$ , and  $AD$  (and hence has radius  $1$ ). Another circle, centered at  $P$ , is tangent to circle  $O$  at point  $T$  and is also tangent to  $AB$  and  $BC$ . If line  $AT$  is tangent to both circles at  $T$ , find the radius of circle  $P$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - 2\sqrt{2}}
In a recent election, there were three candidates: Geoff, Laura and Jim. Geoff received 0.5 percent of the 6,000 votes cast, and Laura got twice as many votes as Geoff. The remaining votes went to Jim. To win the election, a candidate needed to receive more votes than any other. If Geoff needed exactly 3,000 more votes to win the election, what is the minimum percentage of total votes Jim had to have to guarantee his victory?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50.52\%}
$A_1, A_2, \ldots, A_n$ are subsets of a set $S$ with $|S| = 2019$ such that the union of any three of them equals $S$, but the union of any two does not equal $S$. Find the maximum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative" sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e., more people in the set voted for Drum than Hilton). Compute the remainder when $N$ is divided by $2017$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{605}
The distance from \(A\) to \(B\) is 999 km. Along the road, there are kilometer markers indicating the distances to \(A\) and \(B\): 0।999, 1।998, \(\ldots, 999।0. How many of these markers have only two different digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
How many ring homomorphisms are there from $\Bbb Z \times \Bbb Z\times \Bbb Z$ to $\Bbb Z$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine all possible values of the parameter \(a\) such that for all \(x\) and \(y\), the inequality \(2a x^{2} + 2a y^{2} + 4a x y - 2 x y - y^{2} - 2 x + 1 \geq 0\) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \geq \frac{1}{2}}
Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{972538}
$\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has $3$ numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2^n}{(n+1)!}}
Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$. Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1002}
Find the value of the algebraic expression $\frac{3 \sqrt{\sqrt{7+\sqrt{48}}}}{2(\sqrt{2}+\sqrt{6})}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{3}{4}$}
A chess piece called "the four-liner" attacks two vertical and two horizontal squares adjacent to the square on which it stands. What is the maximum number of non-attacking four-liners that can be placed on a $10 \times 10$ board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Let function $G(n)$ denote the number of solutions to the equation $\cos x = \sin nx$ on the interval $[0, 2\pi]$. For each integer $n$ greater than 2, what is the sum $\sum_{n=3}^{100} G(n)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10094}
In the derivation of Von Mangoldt's explicit formula, what is the residue of the integrand at $s=0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\log(2\pi)}
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x,y$ and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots\]
 What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2002}+1)^{\frac{10}{7}}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{428}
Calculate the limit

$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} 2^{x}}{1+x^{2} 5^{x}}\right)^{1 / \sin ^{3} x}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{5}}
In set theory, consider set S = {1, 2, 3, 4, 5} and subset A ⊆ S × S defined as A = {(1, 1), (1, 4), (2, 1), (2, 2), (2, 4), (3, 5), (4, 3), (4, 4), (5, 2)}. Given that B ⊆ S × S is a transitive relation such that A ⊆ B, what is the smallest possible cardinality of B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
In the park, there are 10,000 trees planted in a square grid (100 rows of 100 trees each). What is the maximum number of trees that can be cut down so that the following condition is met: if you stand on any stump, you will not see any other stump? (The trees can be considered thin enough.)

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
A mischievous child mounted the hour hand on the minute hand's axle and the minute hand on the hour hand's axle of a correctly functioning clock. The question is, how many times within a day does this clock display the correct time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
We are given a triangle  $ABC$ . Points  $D$  and  $E$  on the line  $AB$  are such that  $AD=AC$  and  $BE=BC$ , with the arrangment of points  $D - A - B - E$ . The circumscribed circles of the triangles  $DBC$  and  $EAC$  meet again at the point  $X\neq C$ , and the circumscribed circles of the triangles  $DEC$  and  $ABC$  meet again at the point  $Y\neq C$ . Find the measure of  $\angle ACB$  given the condition  $DY+EY=2XY$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Evaluate the limit $\displaystyle \lim\limits_{n\to\infty}\frac{I_n}{n^n}$, where the sequence $I_n$ is defined as $I_n = \int_0^n x^n\arctan\left(x^\frac{1}{n}\right)\,dx$ for $n \ge 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{4}\)}
Given two points O(0,0) and Q(a,b), point P1 is the midpoint of the line segment OQ, point P2 is the midpoint of the line segment QP1, P3 is the midpoint of the line segment P1P2, and so on, Pn+2 is the midpoint of the line segment PnPn+1. Find the limit position of point Pn as n approaches infinity. Express your answer as a coordinate pair (x, y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \left( \frac{2a}{3}, \frac{2b}{3} \right) }
Denmark wants to order pizza.  For toppings, he has 3 cheese, 4 meat and 5 vegetable options, one of which is peppers. He can have only one selection from each topping category (one cheese, one meat and one vegetable). However, if he chooses to have pepperoni, he cannot have peppers. How many topping combinations does he have total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57}
How can one solve $$\int_0^\infty \frac{1}{1-t^4} \, dt$$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{\pi}{4}$$}
Two chords \(AB\) and \(CD\) of a circle with center \(O\) each have a length of 10. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) respectively intersect at point \(P\), with \(DP = 3\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL : LC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/13}
5 Real numbers $a_{1}, a_{2}, \cdots, a_{n}(n>3)$ satisfy: $a_{1}+a_{2}+\cdots+a_{n} \geqslant n$, and $a_{1}^{2}+a_{2}^{2}+\cdots+$ $a_{n}^{2} \geqslant n^{2}$. Find the minimum value of $\max \left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$. (28th United States of America Mathematical Olympiad)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a plane, there are 7 points, with no three points being collinear. If 18 line segments are connected between these 7 points, then at most how many triangles can these segments form?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}