sungyub/dapo-math-17k-verl huggingface dataset:
user: Let $a$, $b$, and $c$ be positive integers satisfying the following equations:

$$a^4 + a^2b^2 + b^4 = 9633$$
$$2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$$

What is the sum of all distinct values of $a + b + c$?

Answer: 18
user: Find the number of 12-digit "words" that can be formed from the alphabet \( \{0,1,2,3,4,5,6\} \) if neighboring digits must differ by exactly 2.

Answer: 882
user: For each integer $k \ge 2$, the decimal expansions of the numbers $1024, 1024^2, \dots, 1024^k$ are concatenated, in that order, to obtain a number $X_k$. For example, $X_2 = 10241048576$. If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n \ge 2$ is an integer.

Answer: 5
user: In base $-2$ notation, digits are $0$ and $1$ only, and the places increase in powers of $-2$. For example, $11011$ stands for $(-2)^4 + (-2)^3 + (-2)^1 + (-2)^0$ and equals the number $7$ in base $10$.

If the decimal number $2019$ is expressed in base $-2$, how many non-zero digits does it contain?

Answer: 6
user: If you multiply all positive integer factors of $24$, you get $24^x$. Find $x$.

Answer: 4
user: The continent of Trianglandia is an equilateral triangle with a side length of $9$. It is divided into $81$ smaller equilateral triangles, each with a side length of $1$. Each smaller triangle (country) can choose to build a wall along at most one of its $3$ sides. Due to ongoing conflicts, no two adjacent countries can have their walls touch, even at a corner. What is the maximum number of walls that can be constructed in Trianglandia?

Answer: 45
user: Points $A$, $B$, $C$, $D$, and $E$ are on the same plane such that $A, E, C$ lie on a line in that order, $B, E, D$ lie on a line in that order, $AE = 1$, $BE = 4$, $CE = 3$, $DE = 2$, and $\angle AEB = 60^\circ$. Let $AB$ and $CD$ intersect at $P$. The square of the area of quadrilateral $PAED$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Answer: 967
user: Let $f(x) = x^2 + 4x + 2$. Let $r$ be the difference between the largest and smallest real solutions of the equation $f(f(f(f(x)))) = 0$. Then $r = a^{\frac{p}{q}}$ for some positive integers $a$, $p$, $q$ such that $a$ is square-free and $p,q$ are relatively prime positive integers. Compute $a + p + q$.

Answer: 35
user: Navi and Ozna are playing a game where Ozna starts first, and the two take turns making moves. A positive integer is written on the board. A move consists of either:

Subtracting any positive integer at most 2015 from it, or
If the integer on the board is divisible by $2014$, dividing it by $2014$.

The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$ such that $a \ge 2015$ and all numbers of the form $an + b$ will not be the starting integer, where $n$ is any positive integer.

Find the minimum number of starting integers where Navi wins.

Answer: 2013
user: Find the least prime number greater than $1000$ that divides $2^{1010} \cdot 23^{2020} + 1$.

Answer: 1013
user: Find the sum of all perfect squares that divide $2016$.

Answer: 210
user: Pete chooses $1004$ monic quadratic polynomials $f_1, \cdots, f_{1004}$ such that each integer from $0$ to $2007$ is a root of at least one of them. Vasya considers all equations of the form $f_i = f_j$ for $i \neq j$ and computes their roots. For each such root, Pete has to pay Vasya $1$ ruble. Find the least possible value of Vasya's income.

Answer: 0
user: Let $n = (10^{2020} + 2020)^2$. Find the sum of all the digits of $n$.

Answer: 25
user: Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.

Answer: 65
user: Initially, five variables are defined: $a_1 = 1, a_2 = 0, a_3 = 0, a_4 = 0, a_5 = 0$. On a turn, Evan can choose an integer $2 \le i \le 5$. Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2$, then now $a_1 = 1, a_2 = 0 + 1 = 1, a_3 = 0, a_4 = 0, a_5 = 0$. Find the minimum number of turns Evan needs to make $a_5$ exceed 1,000,000.

Answer: 127
user: Let $P(A)$ be the arithmetic mean of all elements of the set $A = \{ a_1, a_2, \ldots, a_n \}$, defined as $P(A) = \frac{1}{n} \sum^{n}_{i=1} a_i$. A subset $B$ of $A$ is called a "balanced subset" if $B$ is non-empty and $P(B) = P(A)$. Given the set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, find the number of all "balanced subsets" of $M$.

Answer: 51
user: Find the number of perfect squares of five digits whose last two digits are equal.

Answer: 31
user: There is an isosceles triangle $\triangle ABC$ with $\overline{AB} = \overline{AC} = 5$ and $\overline{BC} = 6$. Points $D$ and $E$ are on $\overline{AC}$ such that $\overline{AD} = 1$ and $\overline{EC} = 2$. If the area of $\triangle BDE$ is $S$, find $15S$.

Answer: 72
user: Find the number of ordered pairs $(x, y)$, where $x$ and $y$ are both integers between $1$ and $9$, inclusive, such that the product $x \times y$ ends in the digit $5$.

Answer: 9
user: Find the sum of all possible $n$ such that $n$ is a positive integer and there exist real numbers $a, b, c$ such that for every integer $m$, the quantity \( \frac{2013m^3 + am^2 + bm + c}{n} \) is an integer.

Answer: 2976
user: The following diagram shows an equilateral triangle $\triangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\triangle ABC$ a distance $\frac{1}{8} AB$ along a side of $\triangle ABC$ in the directions from $A$ to $B$, from $B$ to $C$, and from $C$ to $A$. The shaded region inside all four of the triangles has an area of $300$. Find the area of $\triangle ABC$.

Answer: 768
user: Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that:
- All four rectangles share a common vertex $P$.
- For each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$.
- For each $n = 1, 2, 3$, the opposite side of $R_n$ passes through a vertex of $R_{n-1}$ such that the center of $R_n$ is located counterclockwise of the center of $R_{n-1}$ with respect to $P$.

Compute the total area covered by the union of the four rectangles.

Answer: 30
user: Across the face of a rectangular post-it note, you idly draw lines that are parallel to its edges. Each time you draw a line, there is a $50\%$ chance it'll be in each direction and you never draw over an existing line or the edge of the post-it note. After a few minutes, you notice that you've drawn 20 lines. What is the expected number of rectangles that the post-it note will be partitioned into?

Answer: 116
user: Let $P(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$ be a polynomial where $a_i$ is a non-negative integer for each $i \in \{0, 1, 2, 3, \ldots, n\}$. If $P(1) = 4$ and $P(5) = 136$, what is the value of $P(3)$?

Answer: 34
user: A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$.

Answer: 2083
user: What is the sum of all two-digit odd numbers whose digits are all greater than $6$?

Answer: 528
user: Find the sum of all values of $a + b$, where $(a, b)$ is an ordered pair of positive integers such that $a^2 + \sqrt{2017 - b^2}$ is a perfect square.

Answer: 67
user: Three vertices of a parallelogram are \((2,-4), (-2,8),\) and \((12,7)\). Determine the sum of the three possible x-coordinates of the fourth vertex.

Answer: 12
user: Let $N$ be the set of natural numbers. Suppose $f: N \to N$ is a function satisfying the following conditions: 
$f(mn) = f(m)f(n)$ 
$f(m) < f(n)$ if $m < n$ 
$f(2) = 2$ 

What is the sum of $\sum_{k=1}^{20}f(k)$?

Answer: 210
user: Integers $a$, $b$, and $c$ satisfy the equations $a + b - c = 1$ and $a^2 + b^2 - c^2 = -1$. Determine the sum of all possible values of $a^2 + b^2 + c^2$.

Answer: 18
user: Non-degenerate quadrilateral $ABCD$ has integer side lengths with $AB = AD$ and $BC = CD$. The angles satisfy $\angle ABC = \angle BCD = \angle CDA$. Given that $AB = 3$ and $B \neq D$, determine the number of possible lengths for $BC$.

Answer: 8
user: Call an ordered triple $(a, b, c)$ of integers feral if $b - a$, $c - a$, and $c - b$ are all prime numbers. Find the number of feral triples where $1 \leq a < b < c \leq 20$.

Answer: 72
user: Let $a$, $b$, $c$ be positive integers such that $abc + bc + c = 2014$. Find the minimum possible value of $a + b + c$.

Answer: 40
user: The set of positive real numbers $x$ that satisfy $2 |x^2 - 9| \le 9 |x|$ is an interval $[m, M]$. Find $10m + M$.

Answer: 21
user: There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks?

Answer: 15
user: Let $a$, $b$, $c$, and $d$ be positive real numbers such that
\[ a^2 + b^2 - c^2 - d^2 = 0 \] and \[ a^2 - b^2 - c^2 + d^2 = \frac{56}{53}(bc + ad). \]
Let $M$ be the maximum possible value of $\frac{ab+cd}{bc+ad}$. If $M$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $100m + n$.

Answer: 4553
user: Let $ABC$ be a triangle such that $AB = 7$, $BC = 8$, and $CA = 9$. There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $\triangle ABC$. If $XA = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers, find $a + b$.

Answer: 61
user: What is the smallest positive integer $n$ such that $n!$ is divisible by $2012^{2012}$?

Answer: 1010527
user: Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$.

Answer: 41
user: Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.

Answer: 118
user: Let $O$ be the center of a circle with radius $26$, and let $A$ and $B$ be two distinct points on the circle. Let $M$ be the midpoint of segment $AB$. Consider a point $C$ such that $CO = 34$ and $\angle COM = 15^\circ$. Let $N$ be the midpoint of segment $CO$. Suppose that $\angle ACB = 90^\circ$. Find the length of $MN$.

Answer: 7
user: The degree-$6$ polynomial $f$ satisfies the following conditions: 
- $f(7) - f(1) = 1$, 
- $f(8) - f(2) = 16$, 
- $f(9) - f(3) = 81$, 
- $f(10) - f(4) = 256$, 
- $f(11) - f(5) = 625$. 

Compute $f(15) - f(-3)$.

Answer: 6723
user: Let $ABCD$ be an isosceles trapezoid with $AD = BC = 15$ such that the distance between its bases $AB$ and $CD$ is $7$. Suppose further that the circles with diameters $\overline{AD}$ and $\overline{BC}$ are tangent to each other. What is the area of the trapezoid?

Answer: 105
user: How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with the digit $1$, the resulting number is a multiple of $30$?

Answer: 162
user: Integers $1, 2, 3, \ldots, n$, where $n > 2$, are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed, and the average of the remaining numbers is found to be $17$. What is the maximum sum of the two removed numbers?

Answer: 51
user: Determine the value of \((101 \times 99) - (102 \times 98) + (103 \times 97) - (104 \times 96) + \ldots + (149 \times 51) - (150 \times 50)\).

Answer: 1275
user: Right triangle $ABC$ has a right angle at $C$. Point $D$ on side $\overline{AB}$ is the base of the altitude of $\triangle ABC$ from $C$. Point $E$ on side $\overline{BC}$ is the base of the altitude of $\triangle CBD$ from $D$. Given that $\triangle ACD$ has area $48$ and $\triangle CDE$ has area $40$, find the area of $\triangle DBE$.

Answer: 200
user: Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$.

Answer: 165
user: Solve in positive real numbers: $n + \lfloor \sqrt{n} \rfloor + \lfloor \sqrt[3]{n} \rfloor = 2014$

Answer: 1958
user: Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3 \times 3 \times 3$ cube such that each $1 \times 1 \times 3$ row or column contains exactly one white cube. Note that rotations are considered \textit{distinct}.

Answer: 12
user: You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$. At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$. What is $\lfloor E \rfloor$, the greatest integer less than or equal to $E$?

Answer: 695
user: Determine the sum of absolute values for the complex roots of \( 20x^8 + 7ix^7 - 7ix + 20 \).

Answer: 8
user: In the figure below, how many ways are there to select two squares which do not share an edge?

\[ \begin{asy} \text{size}(3\text{cm}); \text{for} \left( \text{int} \ t = -2; \ t \leq 2; \ t=t+1 \right) \{ \text{draw}( \text{shift}((t,0))*\text{unitsquare} ) ; \ \text{if} (t \neq 0) \text{draw}( \text{shift}((0,t))*\text{unitsquare} ); \} \end{asy} \]

Answer: 28
user: Using squares of side 1, a stair-like figure is formed in stages following a specific pattern. For example, the first stage uses 1 square, the second stage uses 5 squares, and so on. Determine the last stage for which the corresponding figure uses less than 2014 squares.

Answer: 32
user: Find the number of three-digit positive integers with three distinct digits such that the sum of the digits is an even number. For example, 925 and 824.

Answer: 328
user: The equation $166 \times 56 = 8590$ is valid in some base $b \ge 10$ (where $1, 6, 5, 8, 9, 0$ are digits in base $b$). Find the sum of all possible values of $b \ge 10$ that satisfy the equation.

Answer: 12
user: Let $\Omega_1$ and $\Omega_2$ be two circles in the plane. Suppose the common external tangent to $\Omega_1$ and $\Omega_2$ has length $2017$, while their common internal tangent has length $2009$. Find the product of the radii of $\Omega_1$ and $\Omega_2$.

Answer: 8052
user: Ali defines a pronunciation of any sequence of English letters to be a partition of those letters into substrings such that each substring contains at least one vowel. For example, $\text{A } | \text{ THEN } | \text{ A}$, $\text{ATH } | \text{ E } | \text{ NA}$, $\text{ATHENA}$, and $\text{AT } | \text{ HEN } | \text{ A}$ are all pronunciations of the sequence $\text{ATHENA}$. How many distinct pronunciations does $\text{YALEMATHCOMP}$ have? (Y is not a vowel.)

Answer: 45
user: Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.

Answer: 53
user: Suppose $x$, $y$, and $z$ are nonzero complex numbers such that $(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3$. Compute $$(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right).$$

Answer: 3
user: Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots$, find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$.

Answer: 1026
user: Let $x=\sqrt{37-20\sqrt{3}}$. Find the value of \(\frac{x^4-9x^3+5x^2-7x+68}{x^2-10x+19}\).

Answer: 7
user: Compute the only element of the set \[ \{1, 2, 3, 4, \ldots\} \cap \left\{\frac{404}{r^2-4} \;\bigg| \; r \in \mathbb{Q} \setminus \{-2, 2\}\right\}. \]

Answer: 2500
user: In $\triangle ABC$, three lines are drawn parallel to side $BC$, dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$?

Answer: 112
user: One test is a multiple choice test with $5$ questions, each having $4$ options. There are $2000$ candidates, each choosing only one answer for each question. Find the smallest possible integer $n$ such that a student's answer sheet, labeled $n$, has the following property: The answer sheet contains four sheets, and any two of these four sheets have exactly the same three answers.

Answer: 5
user: How many possible values are there for the sum $a + b + c + d$ if $a$, $b$, $c$, and $d$ are positive integers and $abcd = 2007$?

Answer: 4
user: Twelve friends participate in a tennis tournament, where each friend plays exactly one game against each of the other eleven friends. The winner of each game receives one point, while the loser receives zero points. There are no draws. The final points for the participants are denoted as $B_1, B_2, \ldots, B_{12}$. Determine the largest possible value of the sum $\Sigma_3 = B_1^3 + B_2^3 + \ldots + B_{12}^3$.

Answer: 4356
user: Find the greatest value of the positive integer $x$, such that the number \( A = 2^{182} + 4^x + 8^{700} \) is a perfect square.

Answer: 2008
user: Anthony denotes in sequence all positive integers divisible by $2$. Bertha denotes in sequence all positive integers divisible by $3$. Claire denotes in sequence all positive integers divisible by $4$. Orderly Dora denotes all numbers written by the other three. She puts them in order by size and does not repeat a number. What is the $2017^{th}$ number in her list?

Answer: 3026
user: Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{1} + \frac{1}{3}$.

Find the least positive integer $k$ making the number $k! \cdot S_{2016}$ an integer.

Answer: 7
user: Circles $P$, $Q$, and $R$ are externally tangent to one another. The external tangent of circles $P$ and $Q$ that does not intersect circle $R$ intersects circle $P$ at point $P_Q$ and circle $Q$ at point $Q_P$. Points $Q_R$, $R_Q$, $R_P$, and $P_R$ are defined similarly.

If the radius of circle $Q$ is $4$ and the line segment $\overline{Q_PP_Q}$ is parallel to $\overline{R_QQ_R}$, compute the length of $R_PP_R$.

Answer: 8
user: Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Answer: 17
user: On a $5 \times 5$ grid, we randomly place two \emph{cars}, each occupying a single cell and randomly facing one of the four cardinal directions. It is given that the two cars do not start in the same cell. In a \emph{move}, one chooses a car and shifts it one cell forward. The probability that there exists a sequence of moves such that, afterward, both cars occupy the same cell is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Answer: 1148
user: David draws a $2 \times 2$ grid of squares in chalk on the sidewalk outside NIMO HQ. He then draws one arrow in each square, each pointing in one of the four cardinal directions (north, south, east, west) parallel to the sides of the grid. In how many ways can David draw his arrows such that no two of the arrows are pointing at each other?

Answer: 194
user: Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$?

Answer: 11
user: Nine distinct light bulbs are arranged in a circle, each initially turned off. Determine the number of ways to turn on some of these bulbs such that no four consecutive bulbs remain off at the same time.

Answer: 367
user: The positive integer $n$ has the following property: if the last three digits of $n$ are removed, the number becomes $\sqrt[3]{n}$. Find $n$.

Answer: 32768
user: In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.

size(250);
 defaultpen (linewidth (0.7) + fontsize (10));
 draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
 label("$A$", (0,23), NW);
 label("$B$", (23, 23), NE);
 label("$C$", (23,0), SE);
 label("$D$", (0,0), SW);
 draw((0,6)--(23,6));
 draw((0,19)--(23,19));
 draw((5,0)--(5,23));
 draw((12,0)--(12,23));
 label("13", (17/2, 21));
 label("111",(35/2,25/2));
 label("37",(17/2,3));
 label("123",(2.5,12.5));
[/asy]

Answer: 180
user: Let $n = 2^{19} \times 3^{12}$. Let $M$ denote the number of positive divisors of $n^2$ which are less than $n$ but would not divide $n$. What is the number formed by taking the last two digits of $M$ (in the same order)?

Answer: 28
user: Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$. Find the maximum value of the following expression:
\[
P = \frac{\sqrt{b^2 + c^2}}{3-a} + \frac{\sqrt{c^2 + a^2}}{3-b} + a + b - 2022c
\]

Answer: 3
user: How many integer pairs $(a, b)$ with $1 < a, b \le 2015$ are there such that $\log_a b$ is an integer?

Answer: 2082
user: A T-tetromino is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square. Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes.

Answer: 4
user: Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Points $P$, $Q$, and $R$ are chosen on segments $BC$, $CA$, and $AB$, respectively, such that triangles $AQR$, $BPR$, and $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $\triangle PQR$. What is the perimeter of $\triangle PQR$?

Answer: 30
user: The parabolas $y = x^2 + 15x + 32$ and $x = y^2 + 49y + 593$ meet at one point $(x_0, y_0)$. Find $x_0 + y_0$.

Answer: -31
user: Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, and $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.

Answer: 2523
user: Let $A_1A_2A_3 \ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with center $O$. How many triangles $A_iA_jA_k$, $1 \leq i < j < k \leq 21$, contain the center point $O$ in their interior?

Answer: 385
user: For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?

Answer: 11
user: The sum of the following seven numbers is exactly 19: 
\[a_1=2.56,\ a_2=2.61,\ a_3=2.65,\ a_4=2.71,\ a_5=2.79,\ a_6=2.82,\ a_7=2.86.\] 
It is desired to replace each $a_i$ by an integer approximation $A_i$, where $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and the maximum of the "errors" $|A_i - a_i|$, denoted as $M$, is minimized. For this minimum $M$, what is $100M$?

Answer: 61
user: We say a positive integer $n$ is \emph{metallic} if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers?

Answer: 165