inclusionAI/Ring-lite-rl-data huggingface dataset:
$\underline{\text { Senderov V.A. }}$

A thousand non-overlapping arcs are placed on a circle, and two natural numbers are written on each of them. The sum of the numbers on each arc is divisible by the product of the numbers on the next arc in the clockwise direction. What is the greatest possible value of the largest of the written numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2001}
Pablo will decorate each of $6$ identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the $12$ decisions he must make. After the paint dries, he will place the $6$ balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. What is $m?$ (Recall that two events $A$ and $B$ are independent if $P(A \text{ and }B) = P(A) \cdot P(B).$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243

}
Given that the sequence $\left\{\frac{1}{b_{n}}\right\}$ is a "dream sequence" defined by $\frac{1}{a_{n+1}}- \frac{2}{a_{n}}=0$, and that $b_1+b_2+b_3=2$, find the value of $b_6+b_7+b_8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Let  $V=[(x,y,z)|0\le x,y,z\le 2008]$  be a set of points in a 3-D space.
If the distance between two points is either  $1, \sqrt{2}, 2$ , we color the two points differently.
How many colors are needed to color all points in  $V$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the number of unique closed paths on the edges of a given Platonic solid, where each edge can be traversed only once, and paths that are related by a rotation of the solid are considered identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Seventy-five percent of claims follow a normal distribution with a mean of 3,000 and a variance of 1,000,000. The remaining 25% of claims follow a normal distribution with a mean of 4,000 and a variance of 1,000,000. Calculate the probability that a randomly selected claim exceeds 5,000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0568}
Let $a \neq b$ be positive real numbers and $m, n$ be positive integers. An $m+n$-gon $P$ has the property that $m$ sides have length $a$ and $n$ sides have length $b$. Further suppose that $P$ can be inscribed in a circle of radius $a+b$. Compute the number of ordered pairs $(m, n)$, with $m, n \leq 100$, for which such a polygon $P$ exists for some distinct values of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{940}
At the base of a quadrilateral pyramid lies a rhombus $A B C D$, in which $\angle B A D=60^{\circ}$. It is known that $S A=S C$, $S D=S B=A B$. A point $E$ is taken on the edge $D C$ such that the area of triangle $B S E$ is the smallest among the areas of all sections of the pyramid containing the segment $B S$ and intersecting the segment $D C$. Find the ratio $D E: E C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Evaluate the line integral \( \int_C (xy) \, dx + \exp(y^2) \, dy \) using Green's Theorem, where \( C \) is the triangle with vertices \((0,0)\), \((2,0)\), and \((2,1)\). Ensure your answer is positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Seven dwarfs stood at the corners of their garden, each at one corner, and stretched a rope around the entire garden. Snow White started from Doc and walked along the rope. First, she walked four meters to the east where she met Prof. From there, she continued two meters north before reaching Grumpy. From Grumpy, she walked west and after two meters met Bashful. Continuing three meters north, she reached Happy. She then walked west and after four meters met Sneezy, from where she had three meters south to Sleepy. Finally, she followed the rope by the shortest path back to Doc, thus walking around the entire garden.

How many square meters is the entire garden?

Hint: Draw the shape of the garden, preferably on graph paper.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
In the trapezoid \(ABCE\), the base \(AE\) is 16, \(CE = 8\sqrt{3}\). A circle passing through points \(A\), \(B\), and \(C\) intersects the line \(AE\) again at point \(H\), where \(\angle AHB = 60^{\circ}\). Find \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the function \( f(x) = \frac{4 e^{x-1}}{x+1} + x^2 - 3ax + a^2 - 1 \), determine the minimum possible value of the real number \( a \) such that there exists \( x_0 > 0 \) with \( f(x_0) \leq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given vectors $\overrightarrow {a}$ = (4, -7) and $\overrightarrow {b}$ = (3, -4), find the projection of $\overrightarrow {a}$ - $2\overrightarrow {b}$ in the direction of $\overrightarrow {b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-2"]}
Determine the number of nontrivial cyclic subgroups in the group \( \Bbb Z_8 \oplus \Bbb Z_4 \oplus \Bbb Z_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Evaluate the rational map \( \hat{\phi} \) at the point \((1:1:0)\), where \( \phi:(t_0:t_1:t_2) \mapsto (\frac{1}{t_0}:\frac{1}{t_1}:\frac{1}{t_2}) \) and \( \hat{\phi} \) is the extension of \( \phi \) with domain \( \mathbb{P}^2-\{(1:0:0),(0:1:0),(0:0:1)\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0:0:1)}
Given $(3x-1)^{10} = a + a_1x + a_2x^2 + \ldots + a_{10}x^{10}$, find the value of $a_1 + a_2 + \ldots + a_{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1023}
Determine the optimal fraction \( q \) of your money to bet each time in order to grow your money by \( m \)-fold in the least amount of bets, given a biased coin with probability \( p \) of landing heads. Assume \( m \) is sufficiently large to ignore finite size effects. Is there an explicit formula for the optimal \( q \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2p - 1}
João managed to paint the squares of an \( n \times n \) board in black and white so that the intersections of any two rows and any two columns did not consist of squares with the same color. What is the maximum value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Point \( O \) is the center of the circle circumscribed around triangle \( ABC \) with sides \( AB = 8 \) and \( AC = 5 \). Find the length of side \( BC \) if the length of the vector \(\overrightarrow{OA} + 3 \overrightarrow{OB} - 4 \overrightarrow{OC}\) is 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $n \geq 2$ be an integer and consider the vector space $V=F^{2n}$ with the standard basis $e_1,\ldots,e_{2n}$. The second exterior power $\Lambda^2(V)$ contains the element $\omega=e_1 \wedge e_2 + e_3 \wedge e_4 + \cdots + e_{2n-1} \wedge e_{2n}$. Since $\Lambda^{2n}(V)$ is spanned by $x := e_1 \wedge \cdots \wedge e_{2n}$, the $n$-fold wedge product $\omega \wedge \cdots \wedge \omega$ is a scalar multiple of $x$. Compute the scalar $c$ such that $c x = \omega \wedge \cdots \wedge \omega$. What is the value of $c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
B4 In a cube with edge 6, there is a sphere with diameter 6. The sphere thus touches the six faces of the cube internally. We imagine the cube divided into 216 unit cubes of $1 \times 1 \times 1$. How many of these unit cubes lie entirely within the sphere? A unit cube with a vertex on the sphere's surface and otherwise inside the sphere is counted as lying entirely within the sphere.

The work of the Stichting Nederlandse Wiskunde Olympiade is made possible by financial contributions and support from:

The Ministry of Education, Culture and Science

The Dutch Association of Mathematics Teachers

The Mathematical Society

Eindhoven University of Technology

Citogroep

Utrecht University of Applied Sciences

Educatieve Partners Nederland

Woltersgroep Groningen

AKZO/NOBEL
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
What is the largest possible value of $8x^2 + 9xy + 18y^2 + 2x + 3y$ such that $4x^2 + 9y^2 = 8$, where $x, y$ are real numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Determine the minimum number of drawings required, with replacement, from an urn containing 18 differently-coloured balls, to achieve a 90% probability of drawing at least one ball of each colour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Compute the first cohomology group $H^1(\mathbb{Z}, \mathbb{Z}_2)$, where $\mathbb{Z}$ is the group of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_2}
Evaluate the limit as $x$ approaches 0 from the positive side: $$\lim_{x \to 0+}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/6}
Given the set \( A = \{1, 2, 3\} \) and a function \( f: A \rightarrow A \) which satisfies \( f(f(x)) \geq x \) for any \( x \in A \), how many such functions \( f \) exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the number of zeros, including multiplicities, of the polynomial \( f(z) = z^7 + 9z^4 - 7z + 3 \) that lie outside the disk \( \{ z : |z| \leq 2 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the maximal number of crosses, each consisting of 5 squares, that can be placed on an 8x8 grid without overlapping.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If six points are chosen sequentially at random on the circumference of a circle, what is the probability that the triangle formed by the first three is disjoint from that formed by the second three?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{10}}
Given the circles \( C_{1}: x^{2}+y^{2}-2 x+4 y-b^{2}+5=0 \) and \( C_{2}: x^{2}+y^{2}-2(a-6) x-2 a y +2 a^{2}-12 a+27=0 \), which intersect at two distinct points \( A(x_{1}, y_{1}) \) and \( B(x_{2}, y_{2}) \), find the value of \( a \) such that \( \frac{y_{1}+y_{2}}{x_{1}+x_{2}}+\frac{x_{1}-x_{2}}{y_{1}-y_{2}} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
At a round rotating table with 8 white and 7 black cups seated are 15 gnomes. The gnomes are wearing 8 white and 7 black hats. Each gnome takes a cup that matches the color of their hat and places it in front of them. After this, the table rotates randomly. What is the maximum number of color matches between the cups and hats that can be guaranteed after the table rotates (the gnomes can choose how to sit, but do not know how the table will rotate)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$, each once, on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made.

[i]Proposed by Andrew Wen[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 + n}
10 camels cost as much as 24 horses, 26 horses cost as much as 4 oxen, and 6 oxen cost as much as 4 elephants. The cost of 10 elephants is Rs. 170,000. What is the cost of a camel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4,184.62}
Determine the least number of colors that can be used to color all the hexagons on the given tiling such that no two adjacent hexagons share the same color. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
5. In a $4 \times 4$ grid, fill each cell with 0 or 1, such that the sum of the four numbers in each $2 \times 2$ subgrid is odd. There are $\qquad$ different ways to do this.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
Fred had 212 sheets of paper. He received another batch of papers from Jane. Jane initially planned to give him 307 reams of paper, with 500 sheets per ream. But she ended up giving him 50% more than she originally planned. After receiving these additional papers from Jane, Fred decided to give Charles 25% of the total paper sheets he now has. How many sheets of paper does Fred have left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{172847}
In the tetrahedron P-ABC, if PA=PB=BC=AC=5, PC=AB=4$\sqrt{2}$, then the surface area of its circumscribed sphere is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41\pi}
\section*{Problem 6B - 131236B}

Let \(M\) be the set of all points \(P(x, y)\) in a plane rectangular Cartesian coordinate system, where \(x, y\) are rational integers, and \(0 \leq x \leq 4\) and \(0 \leq y \leq 4\).

Determine the probability that the distance between two different points chosen arbitrarily from \(M\) is a rational integer (the unit of measurement being the unit of the coordinate system).

Hint: If \(n\) is the number of different ways to choose two points and \(m\) is the number of ways in which the distance is a rational integer, then the quotient \(\frac{m}{n}\) is the probability to be determined. Two selection possibilities are considered different if and only if the sets of points (each consisting of two points) selected in them are different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{25}}
Find the quotient: 0.284973 / 29
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.009827}
You have 50 dimes and 20 quarters. What percent of the value of your money is in quarters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
In an isosceles triangle \( ABC \) (\( AB = BC \)), a circle is inscribed. A line parallel to side \( BC \) and tangent to the circle intersects side \( AB \) at a point \( N \) such that \( AN = \frac{3}{8} AB \). Find the radius of the circle if the area of triangle \( ABC \) is 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Calculate the value of the expression \( \sqrt{\frac{16^{12} + 8^{15}}{16^5 + 8^{16}}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{2}}{4}}
An unpainted cone has radius 3 cm and slant height 5 cm. The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is 2 cm. When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as p/q where p and q are positive integers with no common divisor larger than 1. What is the value of p+q?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
Given in $\triangle ABC$, $\angle A \leq \angle B \leq \angle C$, and $2\angle B = 5\angle A$, the range of values for $\angle B$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0^\circ < \angle B \leq 75^\circ}
Seven people of seven diﬀerent ages are attending a meeting. The seven people leave the meeting one at a
time in random order. Given that the youngest person leaves the meeting sometime before the oldest
person leaves the meeting, the probability that the third, fourth, and ﬁfth people to leave the meeting do so in order of their ages (youngest to oldest) is $\frac{m}{n}$ , 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{25}
Seven people of seven diﬀerent ages are attending a meeting. The seven people leave the meeting one at a
time in random order. Given that the youngest person leaves the meeting sometime before the oldest
person leaves the meeting, the probability that the third, fourth, and ﬁfth people to leave the meeting do so in order of their ages (youngest to oldest) is $\frac{m}{n}$ , 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{25}
Given the function $f(x) = \ln(e^x + e^{-x}) + x^2$, find the range of $x$ for which $f(x) > f(2x-1)$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1}{3}, 1\right)}
Determine the number of ring homomorphisms from $\mathbb{Z}_{11}$ to $\mathbb{Z}_{13}$ that map $1$ to a non-zero element.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are rods of various lengths and quantities. The lengths are 20 cm, 21 cm, 22 cm, 23 cm, 24 cm, and 25 cm. The quantities are 23, 64, and 32 respectively. What is the average length of the rods?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21.08}
Around a circular table, there are 18 girls seated, 11 dressed in blue and 7 dressed in red. Each girl is asked if the girl to her right is dressed in blue, and each one responds with either yes or no. It is known that a girl tells the truth only when both of her neighbors, the one to her right and the one to her left, are dressed in the same color. How many girls will respond yes? If there is more than one possibility, state all.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $D$, $E$, and $F$ be the feet of the altitudes from vertices $A$, $B$, and $C$ of an acute triangle $\triangle ABC$, respectively. Given that $AF = 28$, $FB = 35$, and $BD = 45$. Let $P$ be a point on segment $BE$ such that $AP = 42$. Determine the length of $CP$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In the prism \(A B C A_1 B_1 C_1\), the medians of the bases \(A B C\) and \(A_1 B_1 C_1\) intersect at points \(O\) and \(O_1\) respectively. A line is drawn through the midpoint of the segment \(O O_1\) parallel to the line \(C A_1\). Find the length of the segment of this line that lies inside the prism, given that \(C A_1 = a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3} a}
Count the number of three-digit whole numbers for which the sum of the digits equals 25. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let  $S$  be the set of integers which are both a multiple of  $70$  and a factor of  $630{,}000$ .  A random element  $c$  of  $S$  is selected. If the probability that there exists an integer  $d$  with  $\gcd (c,d) = 70$  and  $\operatorname{lcm} (c,d) = 630{,}000$  is  $\frac mn$  for some relatively prime integers  $m$  and  $n$ , compute  $100m+n$ .

*Proposed by Eugene Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\arccos \sqrt{2 / 3}} \frac{\tan x + 2}{\sin^2 x + 2 \cos^2 x - 3} \, dx
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\ln2+\sqrt{2}\pi}{4}}
Let $S$ be the set of all rational numbers $r$ , $0<r<1$ , that have a repeating decimal expansion in the form $0.abcabcabc\ldots=0.\overline{abc}$ , where the digits $a$ , $b$ , and $c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators are required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{660}
Example 7 (China Mathematical Olympiad) Let $X$ be a 56-element set. Find the smallest positive integer $n$, such that for any 15 subsets of $X$, if the union of any 7 of them has at least $n$ elements, then there must exist 3 of these 15 subsets whose intersection is non-empty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tournament the total number of points gained by all the teams was 61. What is the maximum number of points that the tournament's winning team could have obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
A cube with edge length \( n \) is composed of white and black unit cubes such that each white cube shares a face with exactly three black cubes, and each black cube shares a face with exactly three white cubes. For which values of \( n \) is this possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \text{ is even}}
从一个正方体的八个顶点中取出三个.则以这三个点为顶点构成直角三角形的概率是$\underline{\quad}$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{13}
11. Let $a, b, c, d$ be four distinct real numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4$, and $a c=$ $b d$. Find the maximum value of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12}
How many homomorphisms are there from the dihedral group $D_4$ to $\mathbb{Z}_4$, given that $D_4 = \{I, R, R^2, R^3, F, RF, R^2F, R^3F\}$ and the hint suggests there are between one and five homomorphisms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the largest possible area of a rectangle whose vertices' coordinates satisfy the equation

\[ |y+1|\left(y^{2}+2y+28\right) + |x-2| = 9\left(y^{2}+2y+4\right) \]

with sides parallel to the coordinate axes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34.171875}
Jane can arrange 25 vases of flowers in a day. If there are 378 vases of flowers to be arranged and she has 17 days to complete the task, how many vases of flowers will she arrange on the last day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many values of $x$, where $-19 < x < 98$, satisfy the equation $\cos^2 x + 2\sin^2 x = 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
How do I find the value of the following integral: $$\int_{0}^{\infty} \frac{\ln(x)}{x^2+x+1} dx$$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$. Let $G$ be the GCD of all elements of $A$.
Then the value of $G$ is?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{798}