meetforever/GRPO_CL huggingface dataset:
user: At a strange party, each person knew exactly $22$ others.

1. For any pair of people $X$ and $Y$ who knew one another, there was no other person at the party that they both knew.

2. For any pair of people $X$ and $Y$ who did not know each other, there were exactly six other people that they both knew.

How many people were at the party?
Answer: 100
user: How many polynomials of degree exactly $5$ with real coefficients map the set \( \{1, 2, 3, 4, 5, 6\} \) to a permutation of itself?
Answer: 718
user: Given a positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Compute the largest positive integer $n$ such that:

$$n = s(n)^2 + 2s(n) - 2.$$
Answer: 397
user: Consider all positive integers written in a row: $123456789101112131415\ldots$ Find the $206788$-th digit from the left.
Answer: 7
user: Let $f(x) = 2^x + 3^x$. For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$?
Answer: 11
user: Eight football teams play matches against each other such that no two teams meet twice and no three teams play all three possible matches. What is the largest possible number of matches?
Answer: 16
user: Compute the smallest positive integer $a$ for which 
$$\sqrt{a + \sqrt{a + \ldots}} - \frac{1}{a + \frac{1}{a+\ldots}} > 7$$
Answer: 43
user: For a positive integer $n$, let $s(n)$ be the sum of the digits of $n$. If $n$ is a two-digit positive integer such that $\frac{n}{s(n)}$ is a multiple of $3$, compute the sum of all possible values of $n$.
Answer: 162
user: The number $16^4 + 16^2 + 1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.
Answer: 264
user: Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6$, $DB = 2$, and the ratio $\frac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5$. Then, if $OA$ is the length of the circumradius, then $OA^2$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Answer: 29
user: Find $k$ such that $k\pi$ is the area of the region of points in the plane satisfying $$\frac{x^2+y^2+1}{11} \le x \le \frac{x^2+y^2+1}{7}.$$
Answer: 18
user: Evaluate the integer part of the number
\[ H = \sqrt{1 + 2011^2 + \frac{2011^2}{2012^2}} + \frac{2011}{2012}. \]
Answer: 2012
user: We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with 
\[ 1 = d_1 < d_2 < \dots < d_k = n \quad (k \ge 6). \]
Find all positive integers $n$ such that 
\[ n = d_5^2 + d_6^2. \]
Answer: 500
user: David has a collection of 40 rocks, 30 stones, 20 minerals, and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
Answer: 30
user: Find the number of lattice points that the line $19x + 20y = 1909$ passes through in Quadrant I.
Answer: 5
user: Find all prime numbers \( p \) such that \( p^2 \) can be written as the sum of cubes of two positive integers, i.e., \( p^2 = a^3 + b^3 \) where \( a \) and \( b \) are positive integers.
Answer: 3
user: 64 numbers (not necessarily distinct) are placed on the squares of a chessboard such that the sum of the numbers in every $2\times2$ square is 7. What is the sum of the four numbers in the corners of the board?
Answer: 7
user: A subset $S$ of the set of integers from 0 to 99 is said to have property $A$ if it is impossible to fill a crossword puzzle with 2 rows and 2 columns using numbers from $S$ (where 0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ that has property $A$.
Answer: 25
user: Let $ABCD$ be a square with side length $4$. Points $E$ and $F$ are chosen on sides $BC$ and $DA$, respectively, such that $EF = 5$. Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$.
Answer: 16
user: Which digit must be substituted instead of the star so that the following large number \( \underbrace{66\ldots66}_{2023} \star \underbrace{55\ldots55}_{2023} \) is divisible by \( 7 \)?
Answer: 6
user: Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is mediocre if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of \{418, 481, 148, 184, 814, 841\}.

Determine the largest mediocre number.
Answer: 629
user: What is the smallest 6-digit number that uses each of the digits 1 through 6 exactly once and has the property that every pair of adjacent digits has a sum that is prime?
Answer: 143256
user: Let $ABCD$ be an isosceles trapezoid with $\overline{AD} \parallel \overline{BC}$. The incircle of $\triangle ABC$ has center $I$ and is tangent to $\overline{BC}$ at $P$. The incircle of $\triangle ABD$ has center $J$ and is tangent to $\overline{AD}$ at $Q$. If $PI = 8$, $IJ = 25$, and $JQ = 15$, compute the greatest integer less than or equal to the area of $ABCD$.
Answer: 1728
user: Let $1, 7, 19, \ldots$ be the sequence of numbers such that for all integers $n \ge 1$, the average of the first $n$ terms is equal to the $n$th perfect square. Compute the last three digits of the $2021$st term in the sequence.
Answer: 261
user: An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a $1 \times 2$ or $2 \times 1$ rectangle), and flip their colors: black becomes white and white becomes black. We call this operation a "step." If $C$ is the original coloring, let $S(C)$ be the least number of steps required to make all the unit squares black. Find, with proof, the greatest possible value of $S(C)$.
Answer: 32
user: Consider a polynomial $P(x) \in \mathbb{R}[x]$, with degree $2023$, such that $P(\sin^2(x)) + P(\cos^2(x)) = 1$ for all $x \in \mathbb{R}$. If the sum of all roots of $P$ is equal to $\frac{p}{q}$ with $p, q$ coprime, then what is the product $pq$?
Answer: 4046
user: The center of the circle passing through the midpoints of the sides of an isosceles triangle $ABC$ lies on the circumcircle of triangle $ABC$. If the larger angle of triangle $ABC$ is $\alpha^\circ$ and the smaller one is $\beta^\circ$, then what is the value of $\alpha - \beta$?
Answer: 90
user: For how many positive integers $n$, $1 \leq n \leq 2008$, can the set $\{1, 2, 3, \ldots, 4n\}$ be divided into $n$ disjoint 4-element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset?
Answer: 1004
user: All the $7$ digit numbers containing each of the digits $1, 2, 3, 4, 5, 6, 7$ exactly once, and not divisible by $5$ are arranged in increasing order. Find the $200^{th}$ number in the list.
Answer: 1425673
user: When $126$ is added to its reversal, $621$, the sum is $126 + 621 = 747$. Find the greatest integer which, when added to its reversal, yields $1211$.
Answer: 952
user: A number is called *trilegal* if its digits belong to the set \(\{1, 2, 3\}\) and if it is divisible by \(99\). How many trilegal numbers with \(10\) digits are there?
Answer: 2025
user: Let there be $A = 1^{a_1} \cdot 2^{a_2} \cdot \ldots \cdot 100^{a_{100}}$ and $B = 1^{b_1} \cdot 2^{b_2} \cdot \ldots \cdot 100^{b_{100}}$, where $a_i, b_i \in \mathbb{N}$ and $a_i + b_i = 101 - i$ for $i = 1, 2, \ldots, 100$. Find the last 1124 digits of $P = A \times B$.
Answer: 0
user: Work in base 3, where each digit can be 0, 1, or 2.

A "good" number of size $n$ is defined as a number in which there are no consecutive $1$s and no consecutive $2$s. 

How many good 10-digit numbers exist?
Answer: 8119
user: Consider the equations $x^2 + y^2 = 16$ and $xy = \frac{9}{2}$. Find the sum, over all ordered pairs $(x, y)$ satisfying these equations, of $|x+y|$.
Answer: 20
user: Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Esmeralda chooses a divisor $d > 1$ of $n$, erases $n$, and writes $n + d$. If the initial number on the board is $2022$, what is the largest composite number that Esmeralda will never be able to write on the blackboard?
Answer: 2023
user: Triangle $ABC$ has $\angle A = 90^\circ$, $\angle C = 30^\circ$, and $AC = 12$. Let the circumcircle of this triangle be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle CAD = 60^\circ$. Define points $E$ and $F$ to be the feet of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively. Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$ intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c}-\sqrt{d})$ for positive integers $a,b,c,d$, where $a,b$ are relatively prime, and the sum of $a,b,c,d$ is minimal. Find $a+b+c+d$.
Answer: 23
user: If $p$, $p^2 + 2$ are both primes, how many divisors does $p^5 + 2p^2$ have?
Answer: 6
user: The list of all numbers formed using the digits \(1, 2, 3, 4,\) and \(5\), without repetition, is created. These numbers are then ordered from least to greatest. Find the number that occupies the \(100^{th}\) position in this ordered list.
Answer: 51342
user: There is a graph with 30 vertices. If any 26 of its vertices, along with their outgoing edges, are deleted, the remaining graph is a connected graph with 4 vertices. What is the smallest number of edges in the initial graph with 30 vertices?
Answer: 405
user: If you roll six fair dice, let $p$ be the probability that exactly five different numbers appear on the upper faces of the six dice. If $p = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Answer: 133
user: A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled, and $n$ cards are dealt to each player $A$ and $B$. The players, $A$ and $B$, alternately discard a card face up, starting with $A$. The game ends when the sum of the discarded cards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither player makes a mistake?
Answer: 0
user: Holding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point $A$ is $A'$. She notices that $A'C = \frac{1}{3} BD$. If the area of $ABCD$ is $27\sqrt{2}$, find $BD$.
Answer: 9
user: Find the number of positive divisors $d$ of $15! = 15 \cdot 14 \cdot \cdots \cdot 2 \cdot 1$ such that $\gcd(d, 60) = 5$.
Answer: 36
user: Let $S$ be the set of integer triplets $(a, b, c)$ with $1 \le a \le b \le c$ that satisfy $a + b + c = 77$ and:
\[\frac{1}{a} +\frac{1}{b}+\frac{1}{c}= \frac{1}{5}.\]
What is the value of the sum $\sum_{a,b,c \in S} a\cdot b \cdot c$?
Answer: 15230
user: For a positive integer $n$, let $\theta(n)$ denote the number of integers $0 \leq x < 2010$ such that $x^2 - n$ is divisible by $2010$. Determine the remainder when $\sum_{n = 0}^{2009} n \cdot \theta(n)$ is divided by $2010$.
Answer: 335
user: Suppose that $a_1, a_2, a_3, \ldots$ is an infinite geometric sequence such that for all $i \ge 1$, $a_i$ is a positive integer. Suppose furthermore that $a_{20} + a_{21} = 20^{21}$. If the minimum possible value of $a_1$ can be expressed as $2^a 5^b$ for positive integers $a$ and $b$, find $a + b$.
Answer: 24
user: Find the number of different quadruples \((a, b, c, d)\) of positive integers such that \(ab = cd = a + b + c + d - 3\).
Answer: 10
user: Find the remainder when $8^{2014} + 6^{2014}$ is divided by 100.
Answer: 0
user: Given a regular dodecagon (a convex polygon with 12 congruent sides and angles) with area 1, there are two possible ways to dissect this polygon into 12 equilateral triangles and 6 squares. Let $T_1$ denote the union of all triangles in the first dissection, and $S_1$ the union of all squares. Define $T_2$ and $S_2$ similarly for the second dissection. Let $S$ and $T$ denote the areas of $S_1 \cap S_2$ and $T_1 \cap T_2$, respectively. If $\frac{S}{T} = \frac{a+b\sqrt{3}}{c}$ where $a$ and $b$ are integers, $c$ is a positive integer, and $\gcd(a,c)=1$, compute $10000a+100b+c$.
Answer: 40003
user: Joshua and Wendy are cleaning the storage shed to earn some spending money for the family vacation. They set aside five boxes to take to their bedrooms. Each box is a cube. The four smaller boxes are of equal size and, when stacked, match the height of the larger box.

Given that the volume of one smaller box is $216$ cubic inches, find the sum of the volumes of all five boxes (in cubic inches).
Answer: 14688
user: Somebody placed digits $1, 2, 3, \ldots, 9$ around the circumference of a circle in an arbitrary order. Reading clockwise, three consecutive digits form a $3$-digit whole number. There are nine such $3$-digit numbers altogether. Find their sum.
Answer: 4995
user: Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. For example, her expression could be $4 + 3 \times 3$, with a value of 13. Each positive digit is equally likely, each arithmetic sign ("$+$" or "$\times$") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?
Answer: 50
user: Suppose that $x^{10} + x + 1 = 0$ and $x^{100} = a_0 + a_1x + \ldots + a_9x^9$. Find $a_5$.
Answer: 252
user: The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient 1, meets the x-axis at the points $(1, 0)$, $(2, 0)$, $(3, 0)$, $\ldots$, $(2020, 0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $\frac{a!}{b}$, where $a$ and $b$ are positive integers and $a$ is as small as possible. Compute $a + b$.
Answer: 2023
user: Find the product of all possible real values for $k$ such that the system of equations
\[
x^2 + y^2 = 80
\]
\[
x^2 + y^2 = k + 2x - 8y
\]
has exactly one real solution $(x, y)$.
Answer: 960
user: A sequence of positive integers with $n$ terms satisfies $\sum_{i=1}^{n} a_i = 2007$. Find the least positive integer $n$ such that there exist some consecutive terms in the sequence with their sum equal to $30$.
Answer: 1018
user: For any finite non-empty set $X$ of integers, let $\max(X)$ denote the largest element of $X$ and $|X|$ denote the number of elements in $X$. If $N$ is the number of ordered pairs $(A, B)$ of finite non-empty sets of positive integers, such that
\[
\begin{aligned}
& \max(A) \times |B| = 12, \\
& |A| \times \max(B) = 11
\end{aligned}
\]
and $N$ can be written as $100a + b$ where $a, b$ are positive integers less than 100, find $a + b$.
Answer: 43
user: In the diagram of rectangles below, with lengths as labeled, let $A$ be the area of the rectangle labeled $A$, and so on. Find $36A + 6B + C + 6D$.

\[
\text{[asy]}
\text{size(3cm);}
\text{real[]} A = \{0,8,13\};
\text{real[]} B = \{0,7,12\};

\text{for (int i = 0; i < 3; ++i) \{}
    \text{draw((A[i],0)--(A[i],-B[2]));}
    \text{draw((0,-B[i])--(A[2],-B[i]));}
\}

\text{label("8", (4,0), N);}
\text{label("5", (10.5,0),N);}
\text{label("7", (0,-3.5),W);}
\text{label("5", (0,-9.5),W);}

\text{label("$A$", (4,-3.5));}
\text{label("$B$", (10.5,-3.5));}
\text{label("$C$", (10.5,-9.5));}
\text{label("$D$", (4,-9.5));}
\text{[/asy]}
\]
Answer: 2491
user: Let $S_P$ be the set of all polynomials $P$ with complex coefficients, such that $P(x^2) = P(x)P(x-1)$ for all complex numbers $x$. Suppose $P_0$ is the polynomial in $S_P$ of maximal degree such that $P_0(1) \mid 2016$. Find $P_0(10)$.
Answer: 12321
user: An alien from the planet OMO Centauri writes the first ten prime numbers in arbitrary order as $U$, $W$, $XW$, $ZZ$, $V$, $Y$, $ZV$, $ZW$, $ZY$, and $X$. Each letter represents a nonzero digit. Each letter represents the same digit everywhere it appears, and different letters represent different digits. Also, the alien is using a base other than base ten. The alien writes another number as $UZWX$. Compute this number (expressed in base ten, with the usual, human digits).
Answer: 5390
user: At the end of a soccer tournament, each pair of teams played exactly once, with no draws. It was observed that for any three teams $A$, $B$, and $C$, if $A$ defeated $B$ and $B$ defeated $C$, then $A$ defeated $C$. Each team calculated the positive difference between the number of games it won and the number of games it lost. The sum of all these differences was $5000$. How many teams played in the tournament? Find all possible answers.
Answer: 100
user: Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$, the following inequality holds:
\[ \frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2} \geq C. \]
Determine when equality occurs.
Answer: 18
user: Given a positive integer $k$, let $||k||$ denote the absolute difference between $k$ and the nearest perfect square. For example, $||13||=3$ since the nearest perfect square to $13$ is $16$. Compute the smallest positive integer $n$ such that \( \frac{||1|| + ||2|| + \cdots + ||n||}{n} = 100 \).
Answer: 89800
user: Let $a$ and $b$ be real numbers satisfying $2(\sin a + \cos a) \sin b = 3 - \cos b$. Find $3 \tan^2 a + 4 \tan^2 b$.
Answer: 35
user: Astrophysicists have discovered a minor planet with a radius of $30$ kilometers, whose surface is completely covered in water. A spherical meteor hits this planet and is submerged in the water, causing an increase of $1$ centimeter to the height of the water on the planet. What is the radius of the meteor in meters?
Answer: 300
user: Calculate the number of ways to choose $4$ numbers from the set $\{1, 2, \ldots, 11\}$ such that at least $2$ of the numbers are consecutive.
Answer: 260
user: Given the set $S = \{ xy(x + y) \mid x, y \in \mathbb{N} \}$. Let $a$ and $n$ be natural numbers such that $a + 2^k \in S$ for all $k = 1, 2, 3, \ldots, n$. Find the greatest value of $n$.
Answer: 3
user: A car has a 4-digit integer price, which is displayed in digital numbers (like those on a digital watch). While the salesman isn't watching, the buyer turns the price upside down and manages to purchase the car for $1626$ less. What was the original price of the car?
Answer: 6815
user: For $k=1, 2, \ldots, 2011$, we denote $S_k = \frac{1}{k} + \frac{1}{k+1} + \cdots + \frac{1}{2011}$. Compute the sum $S_1 + S_1^2 + S_2^2 + \cdots + S_{2011}^2$.
Answer: 4022
user: How many ordered pairs $(a, b)$ of positive integers are there such that \[\gcd(a, b)^3 = \mathrm{lcm}(a, b)^2 = 4^6\] is true?
Answer: 2
user: In a school tennis tournament with $m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants $m$ for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.
Answer: 6
user: Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point. Note that his first roll does not gain him a point. Find the expected number of points that Albert receives.
Answer: 5
user: Square $ABCD$ is cut by a line segment $EF$ into two rectangles $AEFD$ and $BCFE$. The lengths of the sides of each of these rectangles are positive integers. It is known that the area of the rectangle $AEFD$ is $30$ and it is larger than the area of the rectangle $BCFE$. Find the area of square $ABCD$.
Answer: 36
user: Some vertices of a regular $2024$-gon are marked such that for any regular polygon, all of whose vertices are vertices of the $2024$-gon, at least one of its vertices is marked. Find the minimal possible number of marked vertices.
Answer: 506
user: The distinct positive integers $a$ and $b$ have the property that $\frac{a+b}{2}$, $\sqrt{ab}$, and $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ are all positive integers. Find the smallest possible value of $|a-b|$.
Answer: 30
user: Let $a$, $b$, $c$, $d$ be complex numbers satisfying the following equations:

\[
\begin{align*}
  5 &= a + b + c + d \\
  125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\
  1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\
  25 &= a^4 + b^4 + c^4 + d^4
\end{align*}
\]

Compute $abcd$.
Answer: 70
user: Let $f$ be a non-constant polynomial such that 
\[ f(x-1) + f(x) + f(x+1) = \frac{f(x)^2}{2013x} \] 
for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
Answer: 6039
user: What is the units (i.e., rightmost) digit of \[ \left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ? \]
Answer: 3
user: A number is between $500$ and $1000$ and satisfies the following conditions: it leaves a remainder of $6$ when divided by $25$, and a remainder of $7$ when divided by $9$. Find the only odd number that meets these requirements.
Answer: 781
user: 25 little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colors of the rainbow to each donkey, so that successive donkeys receive balloons of different colors, and so that at least one balloon of each color is given to some donkey. Eeyore wants to give to each of the 24 remaining donkeys a pot of one of six colors of the rainbow (except red), so that at least one pot of each color is given to some donkey (but successive donkeys can receive pots of the same color). Which of the two friends has more ways to get his plan implemented, and how many times more?
Answer: 7
user: A circle $C_0$ is inscribed in an equilateral triangle $XYZ$ with a side length of 112. For each positive integer $n$, circle $C_n$ is inscribed in the region bounded by $XY$, $XZ$, and an arc of circle $C_{n-1}$. This forms an infinite sequence of circles that are tangent to sides $XY$ and $XZ$ and approach vertex $X$. If these circles collectively have an area of $m\pi$, find $m$.
Answer: 1176
user: In triangle $ABC$, points $D$ and $E$ lie on the interior of segments $AB$ and $AC$, respectively, such that $AD = 1$, $DB = 2$, $BC = 4$, $CE = 2$, and $EA = 3$. Let $DE$ intersect $BC$ at $F$. Determine the length of $CF$.
Answer: 2
user: There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Answer: 1009
user: 162 pluses and 144 minuses are placed in a $30 \times 30$ table such that each row and each column contains at most 17 signs. (No cell contains more than one sign.) For every plus, count the number of minuses in its row, and for every minus, count the number of pluses in its column. Find the maximum of the sum of these numbers.
Answer: 2592
user: Find the sum of all positive integers $1 \leq k \leq 99$ such that there exist positive integers $a$ and $b$ with the property that 
\[ x^{100} - ax^k + b = (x^2 - 2x + 1)P(x) \] 
for some polynomial $P(x)$ with integer coefficients.
Answer: 117
user: Let $a, b, c$, and $d$ be real numbers such that:
\[ a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018. \]
Evaluate the expression:
\[ 3b + 8c + 24d + 37a. \]
Answer: 1215
user: Find the smallest positive integer $n$ which cannot be expressed as \( n = \frac{2^a - 2^b}{2^c - 2^d} \) for some positive integers \( a, b, c, \) and \( d \).
Answer: 11