sungyub/dapo-math-17k-verl huggingface dataset:
user: In the MTRP district, there are $10$ cities. Bob the builder wants to construct roads between these cities such that there is exactly one unique path between any two cities. The government has provided him with a budget of Rs. $20$, and each road requires a positive integer amount (in Rs.) to build. In how many ways can he build such a network of roads, ensuring the full budget is utilized? Determine the last two digits of your answer.

Answer: 0
user: Let $ABCDEF$ be a regular hexagon, and let $P$ be a point inside quadrilateral $ABCD$. If the area of triangle $PBC$ is $20$, and the area of triangle $PAD$ is $23$, compute the area of hexagon $ABCDEF$.

Answer: 189
user: For the NEMO, Kevin needs to compute the product:
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.

Answer: 870
user: Compute $\left\lceil\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$. The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.

Answer: 2019
user: A tailor met a tortoise sitting under a tree. When the tortoise was the tailor's age, the tailor was only a quarter of his current age. When the tree was the tortoise's age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise?

Answer: 77
user: What is the largest positive integer $n$ for which there are no positive integers $a, b$ such that $8a + 11b = n$?

Answer: 88
user: Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}\,dx.$

Answer: 0
user: (MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

Answer: 1944
user: Given that a duck found that $5 - 2\sqrt{3}i$ is one of the roots of the polynomial $-259 + 107x - 17x^2 + x^3$, what is the sum of the real parts of the other two roots?

Answer: 12
user: Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC = 60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.

Answer: 408
user: On a $10 \times 10$ chessboard, several knights are placed. In any $2 \times 2$ square, there is at least one knight. What is the smallest number of cells these knights can threaten? (Note: A knight does not threaten the square on which it stands, but it does threaten the squares on which other knights are standing.)

Answer: 50
user: For real numbers $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$.

Answer: 999
user: The function $f: \mathbb{N} \to \mathbb{N}_0$ is defined such that:
- $f(2) = 0$
- $f(3) > 0$
- $f(6042) = 2014$
- For all $m, n \in \mathbb{N}$, $f(m + n) - f(m) - f(n) \in \{0, 1\}$

Determine $f(2014)$.

$\mathbb{N}_0 = \{0, 1, 2, \ldots\}$

Answer: 671
user: A square is inscribed in a circle with a radius of $6$. A quarter circle is inscribed in the square, as depicted in the diagram below. Given that the area of the region inside the circle but outside the quarter circle is $n\pi$ for some positive integer $n$, determine the value of $n$.

size(5 cm);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw(circle((1,1),1.41));
draw(arc((0,0),2,0,90));[/asy]

Answer: 18
user: Find the number of ordered triples of positive integers $(a, b, c)$ such that 
\[6a + 10b + 15c = 3000.\]

Answer: 4851
user: A tamer wants to line up five lions and four tigers. We know that a tiger cannot be placed after another tiger. How many ways can the animals be arranged if the tamer cannot distinguish between two animals of the same species?

Answer: 15
user: Let $a, b, c, d$ be four positive integers such that $a > b > c > d$. Given that $ab + bc + ca + d^2 \mid (a + b)(b + c)(c + a)$. Find the minimal value of $\Omega(ab + bc + ca + d^2)$. Here $\Omega(n)$ denotes the number of prime factors $n$ has. For example, $\Omega(12) = 3$.

Answer: 3
user: Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$.

Answer: 78
user: How many values of $x$, where $-19 < x < 98$, satisfy the equation $\cos^2 x + 2\sin^2 x = 1$?

Answer: 38
user: A four-digit number has the following properties:
The units digit equals the tens digit increased by $1$.
The hundreds digit equals twice the tens digit.
The thousands digit is at least twice the units digit.
The number is twice a prime number.

Determine this four-digit number.

Answer: 9634
user: Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i \le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many 5-digit numbers does Kelvin like?

Answer: 14034
user: At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61,629,875}$ and $n = \sqrt[7]{170,859,375}$. Given that $m$ and $n$ are both integers, compute $100m+n$.

Answer: 39515
user: We record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday. On the first Monday, the number we record is $10.5$. On every Tuesday and every Thursday, the number we record is one third of what it was on the previous workday. On every Monday, Wednesday, and Friday, the number we record is double what it was on the previous workday. This pattern continues indefinitely. What is the sum of all the numbers we will record?

Answer: 252
user: Let $\alpha = \cos^{-1} \left( \frac{3}{5} \right)$ and $\beta = \sin^{-1} \left( \frac{3}{5} \right)$. 

\[\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}\]

can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.

Answer: 15013
user: How many ways are there to write $657$ as a sum of powers of two, where each power of two is used at most twice in the sum? For example, $256 + 256 + 128 + 16 + 1$ is a valid sum.

Answer: 41
user: In $\triangle ABC$, $AB = 2019$, $BC = 2020$, and $CA = 2021$. Yannick draws three regular $n$-gons in the plane of $\triangle ABC$ so that each $n$-gon shares a side with a distinct side of $\triangle ABC$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$?

Answer: 11
user: $A, B, C$ are points in the plane such that $\angle ABC = 90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA = 20$ and $BC = 21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Answer: 449
user: Let $P$ be a polynomial satisfying $P(x + 1) + P(x - 1) = x^3$ for all real numbers $x$. Find the value of $P(12)$.

Answer: 846
user: Parallelogram $ABCD$ is given such that $\angle ABC = 30^\circ$. Let $X$ be the foot of the perpendicular from $A$ onto $BC$, and $Y$ the foot of the perpendicular from $C$ onto $AB$. If $AX = 20$ and $CY = 22$, find the area of the parallelogram.

Answer: 880
user: A $3 \times 3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.

Answer: 405
user: For each integer $1 \le j \le 2017$, let $S_j$ denote the set of integers $0 \le i \le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1 \le j \le 2017} \left(1 - \prod_{i \in S_j} x_i\right).\]
Compute the remainder when
\[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right) \in \{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\]
is divided by $2017$.

Answer: 1840
user: Find the 2000th positive integer that is not the difference between any two integer squares.

Answer: 7998
user: After moving his sticky toy spider one morning, Tony heads outside to play "pirates" with his pal Nick, who lives a few doors down the street from the Kubiks. Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold. Victorious over their foes, Tony and Nick claim the prize. However, they must split some of the gold with their crew, which they imagine consists of eight other bloodthirsty pirates. Each of the pirates receives at least one gold coin, but none receive the same number of coins, then Tony and Nick split the remainder equally. If there are $2000$ gold coins in the chest, what is the greatest number of gold coins Tony could take as his share? (Assume each gold coin is equally valuable.)

Answer: 982
user: Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?

Answer: 270
user: Compute the number of three-digit numbers such that all three digits are distinct, in descending order, and one of the digits is a $5$.

Answer: 36
user: Henry starts with a list of the first 1000 positive integers and performs a series of steps on the list. At each step, he erases any non-positive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?

Answer: 11
user: Consider two parallel lines $a$ and $b$. The circles $C, C_1$ are tangent to each other and to the line $a$. The circles $C, C_2$ are tangent to each other and to the line $b$. The circles $C_1, C_2$ are tangent to each other, with radii $R_1 = 9$ and $R_2 = 16$. What is the radius $R$ of the circle $C$?

Answer: 24
user: The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$, such that point $C$ lies on segment $\overline{AB}$. Point $M$ is on the circle centered at $A$, and point $N$ is on the circle centered at $B$, such that line $MN$ is a common internal tangent of those two circles. Find the distance $MN$.

Answer: 140
user: Find a sequence using all the digits from $1$ to $9$ without repetition, such that any two consecutive digits form a number divisible by $7$ or $13$.

Answer: 784913526
user: Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \neq A$. If $AB = 10$, $AD = 24$, $XA = 17$, and $XB = 21$, then the area of $\triangle DEY$ can be written in simplest form as $\frac{m}{n}$. Find $m + n$.

Answer: 421
user: Let $f(x) = x^2 + 4x + 2$. Determine the difference $r$ between the largest and smallest real solutions of the equation $f(f(f(f(x)))) = 0$. Express $r$ in the form $a^{\frac{p}{q}}$, where $a$ is square-free, and $p, q$ are relatively prime positive integers. Compute $a + p + q$.

Answer: 35
user: We define a number as "funny" if it is divisible by the sum of its digits plus one. For example, since $1 + 2 + 1 \mid 12$, the number $12$ is considered funny. What is the maximum number of consecutive funny numbers?

Answer: 2
user: Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell > m > n$ and $\{\frac{3^\ell}{10^4}\} = \{\frac{3^m}{10^4}\} = \{\frac{3^n}{10^4}\}$, where $\{x\} = x - \lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.

Answer: 3003
user: Archimedes drew a square with side length $36$ cm into the sand and he also drew a circle of radius $36$ cm around each vertex of the square. If the total area of the grey parts is $n \cdot \pi$ cm$^2$, what is the value of $n$?

Answer: 3888
user: Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron. This is done by cutting along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?

Answer: 974
user: Find the least natural number $n$, which has at least 6 different divisors $1 = d_1 < d_2 < d_3 < d_4 < d_5 < d_6 < \ldots$, such that:
$d_3 + d_4 = d_5 + 6$
$d_4 + d_5 = d_6 + 7$

Answer: 494
user: Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob?

Answer: 101
user: Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1$, $x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$.

Answer: 75
user: A quadrilateral is given, where two of its sides have lengths of $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral.

Answer: 11
user: Gabriel plays a game of drawing triangles using the vertices of a regular polygon with $2019$ sides, following these rules:

The vertices used by each triangle must not have been previously used.
The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn.

If Gabriel continues to draw triangles until it is no longer possible, determine the minimum number of triangles that he drew.

Answer: 673
user: Determine the last (rightmost) three decimal digits of $n$ where:
\[
n=1 \times 3 \times 5 \times 7 \times \ldots \times 2019.
\]

Answer: 875
user: Let $f(x, y) = \frac{x + y}{2}$, $g(x, y) = \sqrt{xy}$, $h(x, y) = \frac{2xy}{x + y}$, and let
$$ S = \{ (a, b) \in \mathbb{N} \times \mathbb{N} \mid a \neq b \text{ and } f(a, b), g(a, b), h(a, b) \in \mathbb{N} \} $$
where $\mathbb{N}$ denotes the positive integers. Find the minimum of $f$ over $S$.

Answer: 25
user: Ralph is standing along a road which heads straight east. If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike. At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 miles per hour. Pamela’s direction is chosen so that she will reach a point on the road where Ralph is running at exactly the same time Ralph reaches that same point. Let $M$ and $N$ be relatively prime positive integers such that \( \frac{M}{N} \) is the number of hours that it takes Pamela and Ralph to meet. Find $M+N$.

Answer: 29
user: $N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically, or diagonally. If there are $1020$ handshakes, what is $N$?

Answer: 280
user: A permutation of a finite set is a one-to-one function from the set onto itself. A cycle in a permutation $P$ is a nonempty sequence of distinct items $x_1, \ldots, x_n$ such that $P(x_1) = x_2$, $P(x_2) = x_3$, $\ldots$, $P(x_n) = x_1$. Note that we allow the 1-cycle $x_1$ where $P(x_1) = x_1$ and the 2-cycle $x_1, x_2$ where $P(x_1) = x_2$ and $P(x_2) = x_1$. Every permutation of a finite set splits the set into a finite number of disjoint cycles. If this number equals 2, then the permutation is called bi-cyclic. Compute the number of bi-cyclic permutations of the 7-element set formed by the letters of "PROBLEM".

Answer: 1764
user: If $x = \sqrt{2} + \sqrt{3} + \sqrt{6}$ is a root of the polynomial $x^4 + ax^3 + bx^2 + cx + d = 0$, where $a, b, c, d$ are integers, what is the value of $|a + b + c + d|$?

Answer: 93
user: One rainy afternoon, you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005^{th}$ digit that you write?

Answer: 5
user: Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move, one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$.

Answer: 16
user: Let $x$, $y$, and $z$ be positive integers satisfying the following system of equations:

\[
\begin{align*}
&x^2 + \frac{2023}{x} = 2y^2, \\
&y + \frac{2028}{y^2} = z^2, \\
&2z + \frac{2025}{z^2} = xy.
\end{align*}
\]

Find $x + y + z$.

Answer: 25
user: Find the sum of all positive integers $n$ such that exactly $2\%$ of the numbers in the set $\{1, 2, \ldots, n\}$ are perfect squares.

Answer: 7350
user: Find the largest positive integer value of $n < 1000$ such that $\phi(36n) = \phi(25n) + \phi(16n)$, where $\phi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$.

Answer: 996
user: Let $\triangle ABC$ be a triangle with $\angle ABC = 76^\circ$ and $\angle ACB = 72^\circ$. Points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively, such that $\angle ABQ = 22^\circ$ and $\angle ACP = 44^\circ$. Find the measure of $\angle APQ$.

Answer: 36
user: A $9 \times 9$ board has a number written on each square: all squares in the first row have $1$, all squares in the second row have $2$, $\ldots$, all squares in the ninth row have $9$. We will call a "special" rectangle any rectangle of $2 \times 3$, $3 \times 2$, $4 \times 5$, or $5 \times 4$ on the board. 

The permitted operations are:
- Simultaneously add $1$ to all the numbers located in a special rectangle.
- Simultaneously subtract $1$ from all numbers located in a special rectangle.

Demonstrate that it is possible to achieve, through a succession of permitted operations, that $80$ squares have $0$ (zero). What number is left in the remaining box?

Answer: 1
user: What is the remainder when $128^{2023}$ is divided by $126$?

Answer: 2
user: There is a real number $x$ between $0$ and $\frac{\pi}{2}$ such that:

$$\frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x} = \frac{12}{11}$$

and:

$$\sin x + \cos x = \frac{\sqrt{m}}{n}$$

where $m$ and $n$ are positive integers, and $m$ is not divisible by the square of any prime. Find $m + n$.

Answer: 8
user: Let $0 < \theta < 2\pi$ be a real number for which $\cos (\theta) + \cos (2\theta) + \cos (3\theta) + \cdots + \cos (2020\theta) = 0$ and $\theta =\frac{\pi}{n}$ for some positive integer $n$. Compute the sum of the possible values of $n \leq 2020$.

Answer: 1926
user: A positive integer $N$ has 20 digits when written in base 9 and 13 digits when written in base 27. How many digits does $N$ have when written in base 3?

Answer: 39
user: Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$.

Answer: 505
user: 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 $2017$.

Answer: 605
user: A sequence of integers $a_1$, $a_2$, $a_3$, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. 

What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?

Answer: 986
user: Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$.

Answer: 423
user: Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex on the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?

Answer: 81
user: Suppose we have $2013$ piles of coins, with the $i$th pile containing exactly $i$ coins. We wish to remove the coins in a series of steps. In each step, we are allowed to take away coins from as many piles as we wish, but we have to take the same number of coins from each pile. We cannot take away more coins than a pile actually has. What is the minimum number of steps we have to take?

Answer: 11
user: The one hundred U.S. Senators are standing in a line in alphabetical order. Each senator either always tells the truth or always lies. The $i$th person in line says:

"Of the $101-i$ people who are not ahead of me in line (including myself), more than half of them are truth-tellers."

How many possibilities are there for the set of truth-tellers on the U.S. Senate?

Answer: 101
user: Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.

Answer: 8
user: This year, February $29$ fell on a Sunday. In which year will February $29$ next fall on a Sunday?

Answer: 2032
user: How many 6-tuples \((a_1,a_2,a_3,a_4,a_5,a_6)\) are there such that each of \(a_1,a_2,a_3,a_4,a_5,a_6\) is from the set \(\{1,2,3,4\}\) and the six expressions 
\[
a_j^2 - a_j a_{j+1} + a_{j+1}^2
\] 
for \(j = 1,2,3,4,5,6\) (where \(a_7\) is to be taken as \(a_1\)) are all equal to one another?

Answer: 40
user: A box contains red, green, blue, and white balls, totaling 111 balls. If you take out 100 balls without looking, there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to ensure that among them there will always be balls of at least 3 different colors?

Answer: 88
user: For real numbers $a$, $b$, and $c$, the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots.

Answer: 67
user: Find the sum of all positive integers $B$ such that $(111)_B = (aabbcc)_6$, where $a, b, c$ represent distinct base $6$ digits, with $a \neq 0$.

Answer: 237
user: How many integers $n$ satisfy the conditions $1 \leq n \leq 2020$ and $n^n$ is a perfect square?

Answer: 1032
user: Consider a right cylinder with height $5\sqrt{3}$. A plane intersects each of the bases of the cylinder at exactly one point, and the cylindric section (the intersection of the plane and the cylinder) forms an ellipse. Find the product of the sum and the difference of the lengths of the major and minor axes of this ellipse.

Answer: 75
user: Points $A$, $B$, $C$, $D$, and $E$ are located on a plane. It is known that:
- $CA = 12$
- $AB = 8$
- $BC = 4$
- $CD = 5$
- $DB = 3$
- $BE = 6$
- $ED = 3$

Find the length of $AE$.

Answer: 10
user: Form a square with sides of length $5$. Triangular pieces from the four corners are removed to form a regular octagon. Find the area removed to the nearest integer.

Answer: 4
user: Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?

Answer: 45