ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Q. A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along with its interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is square-free. Find $m + n$.
Answer: 21
Problem: Rectangle $ABCD$ measures $70$ by $40$. Eighteen points (including $A$ and $C$) are marked on the diagonal $AC$, dividing the diagonal into $17$ congruent pieces. Twenty-two points (including $A$ and $B$) are marked on the side $AB$, dividing the side into $21$ congruent pieces. Seventeen non-overlapping triangles are constructed as shown. Each triangle has two vertices that are two of these adjacent marked points on the side of the rectangle, and one vertex that is one of the marked points along the diagonal of the rectangle. Only the left $17$ of the $21$ congruent pieces along the side of the rectangle are used as bases of these triangles. Find the sum of the areas of these $17$ triangles.
Answer: 600
Problem: Find the maximum possible value obtainable by inserting a single set of parentheses into the expression $1 + 2 \times 3 + 4 \times 5 + 6$.
Answer: 77
Problem: Let $(x, y)$ be an intersection of the equations $y = 4x^2 - 28x + 41$ and $x^2 + 25y^2 - 7x + 100y + \frac{349}{4} = 0$. Find the sum of all possible values of $x$.
Answer: 14
Problem: When Eva counts, she skips all numbers containing a digit divisible by 3. For example, the first ten numbers she counts are 1, 2, 4, 5, 7, 8, 11, 12, 14, 15. What is the $100^{\text{th}}$ number she counts?
Answer: 255
Problem: Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$?
Answer: 444
Problem: Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to form such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
Answer: 62
Problem: Let $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given that $RI=3$, $CE=8$, and $ER=7$, compute $IC$.
Answer: 4
Problem: Let $x_1, x_2, \dots, x_{100}$ be real numbers such that $|x_1| = 63$ and $|x_{n+1}| = |x_n + 1|$ for $n = 1, 2, \dots, 99$. Find the largest possible value of $(-x_1 - x_2 - \cdots - x_{100})$.
Answer: 2034
Problem: Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different suits. At the beginning, they are given $20$ cards each. 

Each turn, one player can either place a card on the table or remove cards from the table whose sum is $15$. 

At the end of the game:
- One player has a $5$ and a $3$ in their hand.
- On the table, there's a $9$.
- The other player has one card in their hand.

What is the value of this card?
Answer: 8
Problem: Let $T = (a,b,c)$ be a triangle with sides $a$, $b$, and $c$, and area $\triangle$. Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a$, $b' = h_b$, $c' = h_c$) and denote its area by $\triangle'$. Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$, and denote its area by $\triangle''$. Given that $\triangle' = 30$ and $\triangle'' = 20$, find $\triangle$.
Answer: 45
Problem: In the quadrilateral $ABCD$, angles $\angle B$ and $\angle C$ are equal to $120^\circ$. Given $AB = CD = 1$ and $CB = 4$, find the length of $AD$.
Answer: 5
Problem: A $6$-inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.
Answer: 12
Problem: How many ordered triples of nonzero integers $(a, b, c)$ satisfy the equation $2abc = a + b + c + 4$?
Answer: 6
Problem: Given the list \[A=[9,12,1,20,17,4,10,7,15,8,13,14],\] we would like to sort it in increasing order. To accomplish this, we will perform the following operation repeatedly: remove an element, then insert it at any position in the list, shifting elements if necessary. What is the minimum number of applications of this operation necessary to sort $A$?
Answer: 6
Problem: Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.
Answer: 13
Problem: Let $A, B, M, C, D$ be distinct points on a line such that $AB = BM = MC = CD = 6$. Circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ and radii $4$ and $9$ are tangent to line $AD$ at $A$ and $D$ respectively, such that $O_1$ and $O_2$ lie on the same side of line $AD$. Let $P$ be the point such that $PB \perp O_1M$ and $PC \perp O_2M$. Determine the value of $PO_2^2 - PO_1^2$.
Answer: 65
Problem: A geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd positive integer. How many possible values of $n$ are there?
Answer: 8
Problem: Suppose \( \{a_1,a_2,\ldots,a_7\} \) is a set of pair-wisely different positive integers. If \( a_1, 2a_2, \ldots, 7a_7 \) can form an arithmetic series (in this order), find the smallest positive value of \(|a_7-a_1|\).
Answer: 360
Problem: Let $S = 2^3 + 3^4 + 5^4 + 7^4 + \cdots + 17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$.
Answer: 93
Problem: An integer $n \geq 2$ is said to be *tuanis* if, when you add the smallest prime divisor of $n$ and the largest prime divisor of $n$ (these divisors can be the same), you obtain an odd result. Calculate the sum of all *tuanis* numbers that are less than or equal to $2023$.
Answer: 1021086
Problem: Bob chooses a $4$-digit binary string uniformly at random and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob's string is $0000$ and the stream of bits begins $101000001\ldots$, then $N = 7$.
Answer: 19
Problem: Let $w, x, y,$ and $z$ be nonzero complex numbers, and let $n$ be a positive integer. Suppose that the following conditions hold:

- $\frac{1}{w} + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3,$
- $wx + wy + wz + xy + xz + yz = 14,$
- $(w + x)^3 + (w + y)^3 + (w + z)^3 + (x + y)^3 + (x + z)^3 + (y + z)^3 = 2160,$
- $w + x + y + z + i\sqrt{n} \in \mathbb{R}.$

Compute $n$.
Answer: 47
Problem: There are seven cards in a hat, and on card $k$ there is a number $2^{k-1}$, where $k=1,2,...,7$. Solarin picks the cards at random from the hat, one card at a time, until the sum of the numbers on the cards in his hand exceeds $124$. What is the most probable sum he can get?
Answer: 127
Problem: If $f$ is a monic cubic polynomial with $f(0) = -64$, and all roots of $f$ are non-negative real numbers, what is the largest possible value of $f(-1)$? 

(Note: A polynomial is monic if it has a leading coefficient of $1$.)
Answer: -125
Problem: A rectangular grid with side lengths that are integers greater than $1$ is given. Smaller rectangles, each with an area equal to an odd integer and each side length also an integer greater than $1$, are cut out one by one. Finally, one single unit is left. Find the least possible area of the initial grid before the cuttings.
Answer: 121
Problem: In how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ such that $x < y < z$?
Answer: 784
Problem: How many positive integers $N$ satisfy all of the following three conditions?

1. $N$ is divisible by $2020$.
2. $N$ has at most $2020$ decimal digits.
3. The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.
Answer: 508536
Problem: Temerant is a spherical planet with a radius of $1000$ kilometers. The government wants to build twelve towers of the same height on the equator of Temerant, so that every point on the equator can be seen from at least one tower. The minimum possible height of the towers can be written, in kilometers, as $a\sqrt{b} - c\sqrt{d} - e$ for positive integers $a, b, c, d$, and $e$ (with $b$ and $d$ not divisible by the square of any prime). Compute $a + b + c + d + e$.
Answer: 3008
Problem: Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x \neq y$, we have $xy \notin A$. Determine the maximum possible size of $A$.
Answer: 999001
Problem: What is the remainder when $7^{8^9}$ is divided by $1000$?
Answer: 801
Problem: Let $X$ be the set consisting of twenty positive integers $n, n+2, \ldots, n+38$. Determine the smallest value of $n$ such that any three numbers $a, b, c \in X$, not necessarily distinct, can form the sides of an acute-angled triangle.
Answer: 92
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: If you multiply all positive integer factors of $24$, you get $24^x$. Find $x$.
Answer: 4
Problem: 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
Problem: 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
Problem: 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
Problem: 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:

1. Subtracting any positive integer at most 2015 from it, or
2. 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
Problem: Find the least prime number greater than $1000$ that divides $2^{1010} \cdot 23^{2020} + 1$.
Answer: 1013
Problem: Find the sum of all perfect squares that divide $2016$.
Answer: 210
Problem: 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
Problem: Let $n = (10^{2020} + 2020)^2$. Find the sum of all the digits of $n$.
Answer: 25
Problem: 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
Problem: 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
Problem: 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
Problem: Find the number of perfect squares of five digits whose last two digits are equal.
Answer: 31
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: What is the sum of all two-digit odd numbers whose digits are all greater than $6$?
Answer: 528
Problem: 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
Problem: 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
Problem: Let $N$ be the set of natural numbers. Suppose $f: N \to N$ is a function satisfying the following conditions:  
1. $f(mn) = f(m)f(n)$  
2. $f(m) < f(n)$ if $m < n$  
3. $f(2) = 2$  

What is the sum of $\sum_{k=1}^{20}f(k)$?
Answer: 210
Problem: 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
Problem: 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
Problem: 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
Problem: Let $a$, $b$, $c$ be positive integers such that $abc + bc + c = 2014$. Find the minimum possible value of $a + b + c$.
Answer: 40
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: What is the smallest positive integer $n$ such that $n!$ is divisible by $2012^{2012}$?
Answer: 1010527
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$.
Answer: 165
Problem: Solve in positive real numbers: $n + \lfloor \sqrt{n} \rfloor + \lfloor \sqrt[3]{n} \rfloor = 2014$
Answer: 1958
Problem: 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
Problem: 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
Problem: Determine the sum of absolute values for the complex roots of \( 20x^8 + 7ix^7 - 7ix + 20 \).
Answer: 8
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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