sungyub/dapo-math-17k-verl huggingface dataset:
user: An up-right path from lattice points $P$ and $Q$ on the $xy$-plane is a path in which every move is either one unit right or one unit up. The probability that a randomly chosen up-right path from $(0,0)$ to $(10,3)$ does not intersect the graph of $y=x^2+0.5$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 248
user: A triangle $\triangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point in the plane such that lines $AD$ and $BE$ are parallel. Suppose that $\angle EBC = 25^\circ$, $\angle BCA = 32^\circ$, and $\angle CAB = 60^\circ$. Find the smallest possible value of $\angle DAB$ in degrees.

Answer: 63
user: Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?

Answer: 6
user: Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.

Answer: 41
user: The license plate of each automobile in Iran consists of a two-digit and a three-digit number, as well as a letter from the Persian alphabet. The digit $0$ is not used in these numbers. To each license plate, we assign the product of the two numbers on it. For example, if the two numbers are $12$ and $365$ on a license plate, the assigned number would be $12 \times 365 = 4380$. What is the average of all the assigned numbers to all possible license plates?

Answer: 30525
user: Colin has $900$ Choco Pies. He realizes that for some integer values of $n \le 900$, if he eats $n$ pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of $n$ are there?

Answer: 27
user: Find the sum of all positive integers $n$ for which $|2^n + 5^n - 65|$ is a perfect square.

Answer: 6
user: Let $A = (2, 0)$, $B = (0, 2)$, $C = (-2, 0)$, and $D = (0, -2)$. Compute the greatest possible value of the product $PA \cdot PB \cdot PC \cdot PD$, where $P$ is a point on the circle $x^2 + y^2 = 9$.

Answer: 97
user: Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$

Answer: 25
user: Julius has a set of five positive integers whose mean is 100. If Julius removes the median of the set, the mean of the remaining four numbers increases by 5, and the median decreases by 5. Find the maximum possible value of the largest of the five numbers.

Answer: 269
user: Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors.

Answer: 135
user: Circle $\omega_1$ and $\omega_2$ have centers $(0,6)$ and $(20,0)$, respectively. Both circles have radius $30$, and intersect at two points $X$ and $Y$. The line through $X$ and $Y$ can be written in the form $y = mx+b$. Compute $100m+b$.

Answer: 303
user: We are given a series of similar triangles with areas $1^2, 3^2, 5^2, \ldots, 49^2$. If the smallest triangle has a perimeter of 4, what is the sum of the perimeters of all the triangles?

Answer: 2500
user: Let $\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. The altitude from $A$ intersects $BC$ at $D$.

Let $\omega_1$ and $\omega_2$ be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. The common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersects $AD$ at $E$. Compute the length of $AE$.

Answer: 7
user: Determine the number of rational numbers whose decimal expansion is of the form $0.\overline{abcdef}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are digits. Each of the digits $2$, $0$, $1$, and $5$ must appear at least once in the sequence $abcdef$.

Answer: 23160
user: Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly 1987 zeros.

Answer: 7960
user: If $n$ is a positive even number, find the last two digits of $(2^{6n} + 26) - (6^{2n} - 62)$.

Answer: 88
user: The positive real numbers $a, b, c$ satisfy the following equations:

$$\frac{a}{2b+1} + \frac{2b}{3c+1} + \frac{3c}{a+1} = 1$$

$$\frac{1}{a+1} + \frac{1}{2b+1} + \frac{1}{3c+1} = 2$$

What is the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$?

Answer: 12
user: If \[ \frac{1}{1+2} + \frac{1}{1+2+3} + \ldots + \frac{1}{1+2+\ldots+20} = \frac{m}{n} \] where $m$ and $n$ are positive integers with no common divisor, find $m + n$.

Answer: 40
user: The diagram below shows $\triangle ABC$ with area $64$, where $D, E$, and $F$ are the midpoints of $BC, CA$, and $AB$, respectively. Point $G$ is the intersection of $DF$ and $BE$. Find the area of quadrilateral $AFGE$.

Answer: 24
user: Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.

Answer: 9
user: Triangle $ABC$ has side lengths $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD=178$ and $CD=153$. Compute the distance from $B$ to the midpoint of segment $AD$.

Answer: 208
user: Let $f(x) = \sin(\sin(x))$. Evaluate:
\[
\lim_{h \to 0} \frac{f(x+h) - f(h)}{x}
\] at $x = \pi$.

Answer: 0
user: Find the sum of the $2007$ roots of 
\[(x-1)^{2007} + 2(x-2)^{2006} + 3(x-3)^{2005} + \cdots + 2006(x-2006)^2 + 2007(x-2007).\]

Answer: 2005
user: Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$.

Answer: 35
user: A prime number $q$ is called a 'Kowai' number if $q = p^2 + 10$, where $q$, $p$, $p^2-2$, $p^2-8$, and $p^3+6$ are all prime numbers. We know that at least one 'Kowai' number can be found. Find the summation of all 'Kowai' numbers.

Answer: 59
user: 1296 CMU students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the 'score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students?

Answer: 94605
user: What is the remainder of the number $1 \binom{2008}{0} + 2 \binom{2008}{1} + \ldots + 2009 \binom{2008}{2008}$ when divided by $2008$?

Answer: 256
user: An ordered pair $(n,p)$ is defined as "juicy" if the following conditions are satisfied for a positive integer $n$ and an odd prime $p$:

$n^{2} \equiv 1 \pmod{p^{2}}$
$n \equiv -1 \pmod{p}$

Determine the number of such "juicy" pairs $(n,p)$ where $n, p \leq 200$.

Answer: 36
user: Let $LOVER$ be a convex pentagon such that $LOVE$ is a rectangle. Given that $OV = 20$ and $LO = VE = RE = RL = 23$, compute the radius of the circle passing through $R$, $O$, and $V$.

Answer: 23
user: Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$, respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$, respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$. If $O_1O_2=25$, find the area of quadrilateral $O_1AO_2B$.

Answer: 84
user: Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of \(\frac{\text{gcd}(m,n)}{\text{lcm}(m,n)}\) over all ordered pairs of positive integers \((m, n)\) where \(mn = k\). If $S$ can be written in simplest form as \(\frac{r}{s}\), compute $r + s$.

Answer: 895
user: The natural number $n > 1$ is called "heavy" if it is coprime with the sum of its divisors. What is the maximal number of consecutive "heavy" numbers?

Answer: 4
user: Let $ABCD$ be a square, and let $l$ be a line passing through the midpoint of segment $AB$ that intersects segment $BC$. Given that the distances from $A$ and $C$ to $l$ are $4$ and $7$, respectively, compute the area of $ABCD$.

Answer: 185
user: We say that a polynomial $p$ is respectful if $\forall x, y \in \mathbb{Z}$, $y - x$ divides $p(y) - p(x)$, and $\forall x \in \mathbb{Z}$, $p(x) \in \mathbb{Z}$. We say that a respectful polynomial is disguising if it is nonzero, and all of its non-zero coefficients lie between $0$ and $1$, exclusive. Determine $\sum \deg(f)\cdot f(2)$, where the sum includes all disguising polynomials $f$ of degree at most $5$.

Answer: 0
user: Consider the $2 \times 3$ rectangle below. We fill in the small squares with the numbers $1, 2, 3, 4, 5, 6$ (one per square). Define a tasty filling to be one such that each row is not in numerical order from left to right and each column is not in numerical order from top to bottom. If the probability that a randomly selected filling is tasty is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$?

Answer: 157
user: Compute \( \sum_{k=1}^{420} \gcd(k,420) \).

Answer: 4680
user: $x, y$ are real numbers such that \[x^2 + y^2 = 1, \quad 20x^3 - 15x = 3\] Find the value of \(|20y^3 - 15y|\).

Answer: 4
user: Evaluate \[ \frac{1}{729} \sum_{a=1}^{9} \sum_{b=1}^9 \sum_{c=1}^9 \left( abc+ab+bc+ca+a+b+c \right). \]

Answer: 215
user: Evaluate the integral:

$$ \int_0^1 \frac{6x + 5a\sqrt{x}}{4\sqrt{x + a\sqrt{x}}}\, dx $$

where $a = 2007 \times 2009$.

Answer: 2008
user: For $0 \leq d \leq 9$, we define the numbers 
\[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]
Find the last digit of the number 
\[S_{0}+S_{1}+\cdots+S_{9}.\]

Answer: 8
user: Let $N = 2014! + 2015! + 2016! + \dots + 9999!$. How many zeros are at the end of the decimal representation of $N$?

Answer: 501
user: Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. 

You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.

Answer: 178
user: There are three men and eleven women taking a dance class. In how many different ways can each man be paired with a woman partner and then have the eight remaining women be paired into four pairs of two?

Answer: 103950
user: Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\frac{a}{b}$ for coprime positive integers $a$ and $b$. Find $a + b$.

Answer: 39
user: Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$

Answer: 216
user: In how many ways can you rearrange the letters of "Alejandro" such that it contains one of the words "ned" or "den"?

Answer: 5040
user: Find the distance $\overline{CF}$ in the diagram below where $ABDE$ is a square, and angles and lengths are as given:

\[\text{[asy]}\]
\[\text{markscalefactor=0.15;}\]
\[\text{size(8cm);}\]
\[\text{pair A = (0,0);}\]
\[\text{pair B = (17,0);}\]
\[\text{pair E = (0,17);}\]
\[\text{pair D = (17,17);}\]
\[\text{pair F = (-120/17,225/17);}\]
\[\text{pair C = (17+120/17, 64/17);}\]
\[\text{draw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle);}\]
\[\text{label("$A$", A, S);}\]
\[\text{label("$B$", B, S);}\]
\[\text{label("$C$", C, dir(0));}\]
\[\text{label("$D$", D, N);}\]
\[\text{label("$E$", E, N);}\]
\[\text{label("$F$", F, W);}\]
\[\text{label("$8$", (F+E)/2, NW);}\]
\[\text{label("$15$", (F+A)/2, SW);}\]
\[\text{label("$8$", (C+B)/2, SE);}\]
\[\text{label("$15$", (D+C)/2, NE);}\]
\[\text{draw(rightanglemark(E,F,A));}\]
\[\text{draw(rightanglemark(D,C,B));}\]
\[\text{[/asy]}\]

The length $\overline{CF}$ is of the form $a\sqrt{b}$ for integers $a, b$ such that no integer square greater than $1$ divides $b$. What is $a + b$?

Answer: 25
user: Consider the following increasing sequence $1, 3, 5, 7, 9, \ldots$ of all positive integers consisting only of odd digits. Find the $2017^{th}$ term of the above sequence.

Answer: 39953
user: Find the remainder when \( \left \lfloor \frac{149^{151} + 151^{149}}{22499} \right \rfloor \) is divided by \(10^4\).

Answer: 7800
user: Let $T=TNFTPP$. Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.

Answer: 44
user: If $x, y, z \in \mathbb{R}$ are solutions to the system of equations:
\[
\begin{cases}
 x - y + z - 1 = 0 \\
 xy + 2z^2 - 6z + 1 = 0
\end{cases}
\]
What is the greatest value of $(x - 1)^2 + (y + 1)^2$?

Answer: 11
user: Two identical blue blocks, two identical red blocks, two identical green blocks, and two identical purple blocks are placed next to each other in a row. Find the number of distinct arrangements of these blocks where no blue block is placed next to a red block, and no green block is placed next to a purple block.

Answer: 36
user: 2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?

Answer: 1000
user: Unit square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. The point $P$ is the intersection of $EF$ and $GH$. It is given that $BF = \frac{1}{4}$ and $BF + DH = FH$. Calculate the nearest integer to the degree measure of $\angle FAH$.

Answer: 45
user: Find the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$.

Answer: 8
user: Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 - x$. Let $$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)} $$ for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.

Answer: 3984
user: If $a$, $b$ are integers and $s = a^3 + b^3 - 60ab(a + b) \geq 2012$, find the least possible value of $s$.

Answer: 2015
user: Find the sum of the real roots of the polynomial:
\[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \]

Answer: 330
user: In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$, say $C_1$, and then the midpoint of $A_1D$, which is $A_2$. He continues in this fashion indefinitely. The total length of his path, if $AB=5$ and $BC=12$, is of the form $a + b\sqrt{c}$. Find $\frac{abc}{4}$.

Answer: 793
user: Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ have no common points. The line $AB$ is a common internal tangent, and the line $CD$ is a common external tangent to these circles, where $A, C \in k_1$ and $B, D \in k_2$. Knowing that $AB=12$ and $CD=16$, find the value of the product $r_1r_2$.

Answer: 28
user: Find all the real numbers $k$ such that for any non-zero real numbers $a$ and $b$, at least one of the numbers $a$, $b$, or $\frac{5}{a^2} + \frac{6}{b^3}$ is less than or equal to $k$.

Answer: 2
user: All vertices of a convex polyhedron are endpoints of exactly four edges. Find the minimal possible number of triangular faces of the polyhedron.

Answer: 8
user: Jerry buys a bottle containing 150 pills. Using a standard 12-hour clock, he notices that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what time will the clock show when he takes the last pill from the bottle?

Answer: 1
user: Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, and $252$, in no particular order. Determine the maximum possible value of $x + y$.

Answer: 804
user: Consider a $2008 \times 2008$ chessboard. Let $M$ be the smallest number of rectangles that can be drawn on the chessboard such that the sides of every cell of the board are contained within the sides of one of the rectangles. Find the value of $M$. (For example, for a $2 \times 3$ chessboard, the value of $M$ is 3.)

Answer: 2009
user: In a rectangle $ABCD$, let $E$ be the midpoint of $AB$, and $F$ be a point on $AC$ such that $BF$ is perpendicular to $AC$, and $FE$ is perpendicular to $BD$. Suppose $BC = 8\sqrt{3}$. Find $AB$.

Answer: 24
user: Let $\tau(n)$ denote the number of positive integer divisors of $n$. For example, $\tau(4) = 3$. Find the sum of all positive integers $n$ such that $2 \tau(n) = n$.

Answer: 20
user: Given the following system of equations: 
\[ a+4b+9c+16d+25e+36f+49g=1 \] 
\[ 4a+9b+16c+25d+36e+49f+64g=12 \] 
\[ 9a+16b+25c+36d+49e+64f+81g=123 \] 
Determine the value of \( 16a+25b+36c+49d+64e+81f+100g \).

Answer: 334
user: Let $S$ be a randomly selected four-element subset of \( \{1, 2, 3, 4, 5, 6, 7, 8\} \). Let \( m \) and \( n \) be relatively prime positive integers so that the expected value of the maximum element in \( S \) is \( \frac{m}{n} \). Find \( m + n \).

Answer: 41
user: Call a day a *perfect* day if the sum of the digits of the month plus the sum of the digits of the day equals the sum of the digits of the year. For example, February $28$, $2028$ is a perfect day because $2+2+8=2+0+2+8$. Find the number of perfect days in $2018$.

Answer: 36
user: Let $a, b,$ and $c$ be real numbers such that $a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ and $abc=5$. The value of \(\left(a-\frac{1}{b}\right)^3+\left(b-\frac{1}{c}\right)^3+\left(c-\frac{1}{a}\right)^3\) can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Answer: 77
user: Find $n$ such that $20^{2009} = 10^{2000} \cdot 40^9 \cdot 2^n$.

Answer: 1991
user: Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Answer: 5
user: Let $[ABCD]$ be a convex quadrilateral with an area of $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have an area of $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$.

Answer: 2013
user: Evaluate $ \sqrt{\pi \int_0^{2008} x|\sin \pi x|\, dx}$

Answer: 2008
user: The side lengths of a triangle are natural number multiples of $7$ and are less than $40$. How many such triangles satisfy these conditions?

Answer: 22
user: Find the smallest natural $k$ such that among any $k$ distinct and pairwise coprime naturals smaller than $2018$, a prime can be found.

Answer: 16
user: Compute the expected sum of elements in a subset of $\{1, 2, 3, \ldots, 2020\}$ (including the empty set) chosen uniformly at random.

Answer: 1020605
user: A circle intersects an equilateral triangle $\triangle XYZ$ at points $A$, $B$, $C$, $D$, $E$, and $F$ such that the points $X$, $A$, $B$, $Y$, $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. Given:

- $AC^2 + CE^2 + EA^2 = 1900$
- $BD^2 + DF^2 + FB^2 = 2092$

Compute the positive difference between the areas of triangles $\triangle ACE$ and $\triangle BDF$.

Answer: 96
user: Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ with $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. Determine the smallest $N$ such that $a_{N} > 2017$.

Answer: 95
user: Determine the number of ordered pairs of positive integers $(m,n)$ with $1 \leq m \leq 100$ and $1 \leq n \leq 100$ such that \[ \gcd(m+1,n+1) = 10\gcd(m,n). \]

Answer: 52
user: Determine the minimum value of \( n \) such that in a group of 1991 participants, where each participant knows at least \( n \) other participants (with mutual friendship), there exists a subset of 6 participants in which every pair of participants knows each other.

Answer: 1593
user: Let $a, b, c, x, y,$ and $z$ be complex numbers such that \[ a = \frac{b+c}{x-2}, \ b = \frac{c+a}{y-2}, \ c = \frac{a+b}{z-2}. \] If $xy + yz + xz = 67$ and $x + y + z = 2010$, find the value of $xyz$.

Answer: -5892
user: Given 2005 distinct numbers $a_1, a_2, \dots, a_{2005}$. By one question, we may take three different indices $1 \le i < j < k \le 2005$ and find out the set of numbers $\{a_i, a_j, a_k\}$ (unordered, of course). Find the minimal number of questions necessary to determine all numbers $a_i$.

Answer: 1003
user: Point $P_1$ is located 600 miles West of point $P_2$. At 7:00 AM, a car departs from $P_1$ and drives East at a speed of 50 mph. At 8:00 AM, another car departs from $P_2$ and drives West at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of $x$?

Answer: 60
user: Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $$\{10a+b \mid 1 \leq a \leq 5, 1 \leq b \leq 5\}$$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$?

Answer: 47
user: Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le 3$, let $D_i$, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$.

Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.

Answer: 917
user: Let $P$ be the product of all non-zero digits of the positive integer $n$. For example, $P(4) = 4$, $P(50) = 5$, $P(123) = 6$, $P(2009) = 18$. 
Find the value of the sum: $P(1) + P(2) + \ldots + P(2008) + P(2009)$.

Answer: 194763
user: There is a sequence defined as follows:
- $a(2) = 0$
- $a(3) = 1$
- For $n \geq 4$, $a(n) = a\left(\left\lfloor \frac{n}{2} \right\rfloor\right) + a\left(\left\lceil \frac{n}{2} \right\rceil\right)$

Find $a(2014)$.

$\left\lfloor \frac{n}{2} \right\rfloor$ and $\left\lceil \frac{n}{2} \right\rceil$ denote the floor function (largest integer $\leq \frac{n}{2}$) and the ceiling function (smallest integer $\geq \frac{n}{2}$), respectively.

Answer: 34
user: Let $a_k$ be the number of ordered $10$-tuples $(x_1, x_2, \ldots, x_{10})$ of nonnegative integers such that:
\[ x_1^2 + x_2^2 + \ldots + x_{10}^2 = k. \]
Define $b_k = 0$ if $a_k$ is even and $b_k = 1$ if $a_k$ is odd. Determine the value of:
\[ \sum_{i=1}^{2012} b_{4i}. \]

Answer: 1006
user: A right triangle has the property that its sides are pairwise relatively prime positive integers and that the ratio of its area to its perimeter is a perfect square. Find the minimum possible area of this triangle.

Answer: 30
user: Let \( x \) be the answer to this question. Find the value of \( 2017 - 2016x \).

Answer: 1
user: Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament. They agreed on the following rules of attendance: 

There should always be at least one person present on each day. 
On no two days should the same subset of politicians attend. 
The members present on day $N$ should include, for each $K<N$ (where $K \ge 1$), at least one member who was present on day $K$. 

**Question:** For how many days can the parliament sit before one of the rules is broken?

Answer: 128