ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Find all natural numbers $n$ such that $n$, $n^2 + 10$, $n^2 - 2$, $n^3 + 6$, and $n^5 + 36$ are all prime numbers.
Answer: 7
Problem: In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee made the following two claims:

1. Less than $12$ employees have a more difficult job.
2. At least $30$ employees receive a higher salary.

Assuming that an employee either always lies or always tells the truth, determine how many employees are there in the enterprise.
Answer: 42
Problem: Find the number of pairs $(a,b)$ of natural numbers such that:
1. $b$ is a 3-digit number.
2. $a+1$ divides $b-1$.
3. $b$ divides $a^{2} + a + 2$.
Answer: 16
Problem: In the figure, a semicircle is folded along the $AN$ string and intersects the $MN$ diameter at $B$. It is given that $MB:BN = 2:3$ and $MN = 10$. If $AN = x$, what is the value of $x^2$?
Answer: 80
Problem: Bob draws the graph of $y = x^3 - 13x^2 + 40x + 25$ and is dismayed to find out that it only has one root. Alice comes to the rescue, translating the axes so that the origin is at the point that used to be $(-20, 16)$. This new graph has three $x$-intercepts; compute their sum.
Answer: 73
Problem: In a triangle $ABC$, the median $AD$ divides $\angle BAC$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular to $AB$. Given that $BE=3$ and $BA=4$, find the integer nearest to $BC^2$.
Answer: 29
Problem: Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ 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
Problem: In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied.

One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$. Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.

The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
Answer: 102
Problem: Find the number of ordered pairs $(a, b)$ of positive integers such that $a < b$ and $100 \leq a, b \leq 1000$ satisfying $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$.
Answer: 20
Problem: The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, \ldots, 2020$. The students are initially divided into $20$ groups of $101$, with each division into groups equally likely. In each group, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we choose the champion uniformly at random. If the expected value of the champion’s number can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime integers, determine $a + b$.
Answer: 2123
Problem: Towers grow at points along a line. All towers start with height $0$ and grow at the rate of $1$ meter per second. As soon as any two adjacent towers are each at least $1$ meter tall, a new tower begins to grow at a point along the line exactly halfway between those two adjacent towers. Before time $0$, there are no towers, but at time $0$, the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time $10$ seconds.
Answer: 1033
Problem: Find the minimum value of $m$ such that any $m$-element subset of the set of integers \( \{1, 2, \ldots, 2016\} \) contains at least two distinct numbers \( a \) and \( b \) which satisfy \(|a - b| \le 3\).
Answer: 505
Problem: Suppose real numbers $a, b, c, d$ satisfy the following conditions:
- $a + b + c + d = 17$
- $ab + bc + cd + da = 46$

Determine the minimum possible value of $a^2 + b^2 + c^2 + d^2$. If this minimum value can be expressed as a rational number $\frac{p}{q}$ in its simplest form, find $p + q$.
Answer: 199
Problem: Anders is solving a math problem and encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$, where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q \cdot 15!$ for some rational number $q$. Find $q$.
Answer: 4
Problem: For some complex number $\omega$ with $|\omega| = 2016$, there is some real $\lambda>1$ such that $\omega$, $\omega^{2}$, and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\frac{a + \sqrt{b}}{c}$, where $a,b,$ and $c$ are positive integers and $b$ is squarefree. Compute $\sqrt{a+b+c}$.
Answer: 4032
Problem: Calculate the product of all positive integers less than $100$ that have exactly three positive divisors. Show that this product is a square.
Answer: 44100
Problem: Connor is thinking of a two-digit number $n$, which satisfies the following properties:

- If $n > 70$, then $n$ is a perfect square.
- If $n > 40$, then $n$ is prime.
- If $n < 80$, then the sum of the digits of $n$ is $14$.

What is Connor's number?
Answer: 59
Problem: Find the sum of the decimal digits of the number $(10^{2021} + 2021)^2$?
Answer: 36
Problem: Given a regular 2007-gon, find the minimal number $k$ such that among every $k$ vertices of the polygon, there always exist 4 vertices forming a convex quadrilateral where 3 sides of the quadrilateral are also sides of the polygon.
Answer: 1506
Problem: An isosceles triangle has a base with length $12$ and the altitude to the base has length $18$. Find the area of the region of points inside the triangle that are a distance of at most $3$ from that altitude.
Answer: 81
Problem: Jacob likes to watch Mickey Mouse Clubhouse! One day, he decides to create his own Mickey Mouse head with two circles $\omega_1$ and $\omega_2$ and a circle $\omega$, with centers $O_1$, $O_2$, and $O$, respectively. Let $\omega_1$ and $\omega$ meet at points $P_1$ and $Q_1$, and let $\omega_2$ and $\omega$ meet at points $P_2$ and $Q_2$. Point $P_1$ is closer to $O_2$ than $Q_1$, and point $P_2$ is closer to $O_1$ than $Q_2$. Given that $P_1$ and $P_2$ lie on $O_1O_2$ such that $O_1P_1 = P_1P_2 = P_2O_2 = 2$, and $Q_1O_1 \parallel Q_2O_2$, the area of $\omega$ can be written as $n \pi$. Find $n$.
Answer: 4
Problem: Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$, compute the maximum possible value of $a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}$.
Answer: 365
Problem: Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for military service. He managed to expose all but one draftee. (He would have exposed this one too if the lucky guy was not taken by a stroke at the very moment when the doctor yelled at him "Turn around!") How many malingerers were exposed by the vigilant doctor? 

Each digit substitutes a letter. The same digits substitute the same letters, while distinct digits substitute distinct letters.
Answer: 10999
Problem: Split a face of a regular tetrahedron into four congruent equilateral triangles. How many different ways can the seven triangles of the tetrahedron be colored using only the colors orange and black? (Two tetrahedra are considered to be colored the same way if you can rotate one so it looks like the other.)
Answer: 48
Problem: Define $f(x) = 2x + 3$ and suppose that $g(x+2) = f(f(x-1) \cdot f(x+1) + f(x))$. Find $g(6)$.
Answer: 259
Problem: A list of positive integers is called *good* if the maximum element of the list appears exactly once. A *sublist* is a list formed by one or more consecutive elements of a list. For example, in the list $10, 34, 34, 22, 30, 22$, the sublist $22, 30, 22$ is good, while $10, 34, 34, 22$ is not.

A list is *very good* if all its sublists are good. Find the minimum value of $k$ such that there exists a very good list of length $2019$ with $k$ different values in it.
Answer: 11
Problem: Rectangle $ABCD$ has $AB = 24$ and $BC = 7$. Let $d$ be the distance between the centers of the incircles of $\triangle ABC$ and $\triangle CDA$. Find $d^2$.
Answer: 325
Problem: A large rectangle is tiled by some $1 \times 1$ tiles. In the center, there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is five tiles wide. That red border is surrounded by a white border which is five tiles wide. Finally, the white border is surrounded by a red border which is five tiles wide. The resulting pattern is pictured below. In all, $2900$ red tiles are used to tile the large rectangle. Find the perimeter of the large rectangle.
Answer: 350
Problem: In a right triangle $ABC$ (where $\angle C = 90^\circ$), it is given that $AC = 4$ cm and $BC = 3$ cm. The points $A_1$, $B_1$, and $C_1$ are positioned such that $AA_1 \parallel BC$, $BB_1 \parallel A_1C$, and $CC_1 \parallel A_1B_1$. Additionally, $\angle A_1B_1C_1 = 90^\circ$ and $A_1B_1 = 1$ cm. Find the length of $B_1C_1$.
Answer: 12
Problem: Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n \ge 1$. Determine the largest power of $5$ that divides $x_{2014}^2 + 1$.
Answer: 2014
Problem: In the figure, points $A$, $B$, $C$, and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$, and segment $CD$ measures $4$. The circumferences with centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$?
Answer: 24
Problem: Compute the sum of the roots of the equation \((2 - x)^{2005} + x^{2005} = 0\).
Answer: 2004
Problem: What is the smallest value that the sum of the digits of the number $3n^2 + n + 1$, where $n \in \mathbb{N}$, can take?
Answer: 3
Problem: A number is considered *interesting* if it meets the following criteria:

1. It is a $6$-digit integer.
2. It contains no zeros.
3. The first $3$ digits are strictly increasing.
4. The last $3$ digits are non-increasing.

What is the average of all interesting numbers?
Answer: 308253
Problem: Among $100$ points in the plane, no three are collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least $50$.
Answer: 1037
Problem: The triangle $ABC$ has side lengths $AB = 39$, $BC = 57$, and $CA = 70$. The median $\overline{AD}$ is divided into three congruent segments by points $E$ and $F$. Lines $BE$ and $BF$ intersect side $\overline{AC}$ at points $G$ and $H$, respectively. Find the distance from $G$ to $H$.
Answer: 21
Problem: What is the maximal number of checkers that can be placed on an $8 \times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?
Answer: 32
Problem: How many 4-digit numbers \( \overline{abcd} \) are there such that \( a < b < c < d \) and \( b-a < c-b < d-c \)?
Answer: 7
Problem: For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$?
Answer: 42
Problem: Mustafa bought a big rug. The seller measured the rug with a ruler that was supposed to measure one meter. As it turned out to be $30$ meters long by $20$ meters wide, he charged Rs $120,000$. When Mustafa arrived home, he measured the rug again and realized that the seller had overcharged him by $9,408$ Rs. How many centimeters long is the ruler used by the seller?
Answer: 96
Problem: Determine the smallest integer $n > 1$ such that $n^2(n - 1)$ is divisible by 2009.
Answer: 42
Problem: Tyler has two calculators, both of which initially display zero. The first calculator has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke.

A positive integer $n$ is called \textit{ambivalent} if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive.
Answer: 34776
Problem: A square $ABCD$ has a side length of $1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $BC$, $CD$, $DA$, respectively. The area of square $PQRS$ is $a + b\sqrt{2}$, where $a$ and $b$ are integers. Find the value of $a + b$.
Answer: 1
Problem: What is the greatest integer not exceeding the sum $\sum_{n=1}^{1599} \frac{1}{\sqrt{n}}$?
Answer: 78
Problem: Euhan and Minjune are playing a game. They choose a number $N$ so that they can only say integers up to $N$. Euhan starts by saying $1$, and each player takes turns saying either $n+1$ or $4n$ (if possible), where $n$ is the last number said. The player who says $N$ wins. What is the smallest number larger than $2019$ for which Minjune has a winning strategy?
Answer: 2020
Problem: Let $S(x) = x + 1$ and $V(x) = x^2 - 1$. Find the sum of the squares of all real solutions to $S(V(S(V(x)))) = 1$.
Answer: 2
Problem: \( \triangle ABC \) has an area of \( 240 \). Points \( X, Y, Z \) lie on sides \( AB, BC, \) and \( CA, \) respectively. Given that \( \frac{AX}{BX} = 3 \), \( \frac{BY}{CY} = 4 \), and \( \frac{CZ}{AZ} = 5 \), find the area of \( \triangle XYZ \).
Answer: 122
Problem: What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?
Answer: 3
Problem: Seven people of seven different ages are attending a meeting. The seven people leave the meeting one at a time in random order. Given that the youngest person leaves the meeting sometime before the oldest person leaves the meeting, determine the probability that the third, fourth, and fifth people to leave the meeting do so in order of their ages (youngest to oldest). The probability is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 25
Problem: Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x + 1$; otherwise, he replaces $x$ with $\lfloor x/3 \rfloor$. If Amir performs the operation four times, let $\frac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m, n$ are relatively prime positive integers. Find $m + n$.
Answer: 27
Problem: How many distinct triangles $ABC$ are there, up to similarity, such that the magnitudes of the angles $A$, $B$, and $C$ in degrees are positive integers and satisfy the equation:

$$\cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1$$

for some positive integer $k$, where $kC$ does not exceed $360^{\circ}$?
Answer: 6
Problem: Consider all 6-digit numbers of the form \( abccba \) where \( b \) is odd. Determine the number of all such 6-digit numbers that are divisible by 7.
Answer: 70
Problem: Given that three roots of \( f(x) = x^4 + ax^2 + bx + c \) are \( 2 \), \( -3 \), and \( 5 \), what is the value of \( a + b + c \)?
Answer: 79
Problem: Compute the number of triangles of different sizes which contain the gray triangle in the figure below.

```asy
size(5cm);
real n = 4;
for (int i = 0; i < n; ++i) {
    draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i));
}
for (int i = 0; i < n; ++i) {
    draw((n-i,0)--((n-i)/2,(n-i)*0.866));
}
for (int i = 0; i < n; ++i) {
    draw((i,0)--((n+i)/2,(n-i)*0.866));
}
filldraw((1.5,0.866)--(2,2*0.866)--(2.5,0.866)--cycle, gray);
```
Answer: 4
Problem: The points $A$, $B$, $C$, $D$, $E$ lie on a line $\ell$ in this order. Suppose $T$ is a point not on $\ell$ such that $\angle BTC = \angle DTE$, and $\overline{AT}$ is tangent to the circumcircle of triangle $BTE$. If $AB = 2$, $BC = 36$, and $CD = 15$, compute $DE$.
Answer: 954
Problem: Eighteen people are standing in a socially-distanced line to enter a grocery store. Five people are wearing a black mask, $6$ are wearing a gray mask, and $7$ are wearing a white mask. Suppose that these $18$ people got in line in a random order. The expected number of pairs of adjacent people wearing different-colored masks can be given by $\frac{a}{b}$, where $\gcd(a, b) = 1$. Compute $a+b$.
Answer: 116
Problem: Alice, Bob, Cassie, Dave, and Ethan are going on a road trip and need to arrange themselves among a driver's seat, a passenger seat, and three distinguishable back row seats. Alice, Bob, and Cassie are not allowed to drive. Alice and Bob are also not allowed to sit in the front passenger seat. Find the number of possible seating arrangements.
Answer: 24
Problem: The Jones family has five children, and the Smith family has three children. Among these eight children, there are five girls and three boys. Let \( \frac{m}{n} \) represent the probability that at least one of the families has only girls. Given that \( m \) and \( n \) are relatively prime positive integers, find \( m + n \).
Answer: 67
Problem: $x$ is a base-10 number such that when the digits of $x$ are interpreted as a base-20 number, the resulting number is twice the value as when they are interpreted as a base-13 number. Find the sum of all possible values of $x$.
Answer: 198
Problem: A sequence starts with $2017$ as its first term, and each subsequent term is the sum of the cubes of the digits in the previous number. What is the $2017$th term of this sequence?
Answer: 217
Problem: Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that $x \leq 20$ and $y \leq 23$. (Philena knows that Nathan’s pair must satisfy $x \leq 20$ and $y \leq 23$.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair $(a, b)$ of positive integers and tells it to Nathan. Nathan says YES if $x \leq a$ and $y \leq b$, and NO otherwise. Find, with proof, the smallest positive integer $N$ for which Philena has a strategy that guarantees she can be certain of Nathan’s pair after at most $N$ rounds.
Answer: 9
Problem: A triangle $ABC$ with $AC = 20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance from $A$ to $t$ is $25$ and the distance from $C$ to $t$ is $16$. If $S$ denotes the area of triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$.
Answer: 10
Problem: Let $A$, $B$, $C$, $D$, $E$, and $F$ be 6 points around a circle, listed in clockwise order. We have $AB = 3\sqrt{2}$, $BC = 3\sqrt{3}$, $CD = 6\sqrt{6}$, $DE = 4\sqrt{2}$, and $EF = 5\sqrt{2}$. Given that $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent, determine the square of $AF$.
Answer: 225
Problem: Find the last three digits of the sum:
\[ 2005^{11} + 2005^{12} + \ldots + 2005^{2006} \]
Answer: 500
Problem: Find the largest positive integer $N$ satisfying the following properties:

- $N$ is divisible by $7$.
- Swapping the $i^{\text{th}}$ and $j^{\text{th}}$ digits of $N$ (for any $i$ and $j$ with $i \neq j$) gives an integer which is \textit{not} divisible by $7$.
Answer: 987546
Problem: In the plane, there are $10$ lines in general position, meaning that no two are parallel and no three are concurrent. At each intersection of two lines, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?
Answer: 4050
Problem: There is an interval $[a, b]$ that is the solution to the inequality $|3x - 80| \le |2x - 105|$. Find $a + b$.
Answer: 12
Problem: Given $\triangle ABC$, where $A$ is at $(0,0)$, $B$ is at $(20,0)$, and $C$ is on the positive $y$-axis. Cone $M$ is formed when $\triangle ABC$ is rotated about the $x$-axis, and cone $N$ is formed when $\triangle ABC$ is rotated about the $y$-axis. If the volume of cone $M$ minus the volume of cone $N$ is $140\pi$, find the length of $\overline{BC}$.
Answer: 29
Problem: Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183$ and $a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac{9}{5} (a + b)$.
Answer: 73
Problem: Let $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that:
- $P$ passes through $A$ and $B$,
- the focus $F$ of $P$ lies on $E$,
- the orthocenter $H$ of $\triangle FAB$ lies on the directrix of $P$.

If the major and minor axes of $E$ have lengths $50$ and $14$, respectively, compute $AH^2 + BH^2$.
Answer: 2402
Problem: Let $ABCDEF$ be a regular hexagon with a side length of $3$. Points $X$, $Y$, and $Z$ are located on segments $AB$, $CD$, and $EF$ respectively, such that $AX = CY = EZ = 1$. The area of triangle $XYZ$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $\gcd(a, c) = 1$. Find $100a + 10b + c$.
Answer: 2134
Problem: Consider \(13\) marbles that are labeled with positive integers such that the product of all \(13\) integers is \(360\). Moor randomly picks up \(5\) marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?
Answer: 24
Problem: The altitudes of a triangle are $12$, $15$, and $20$. What is the area of this triangle?
Answer: 150
Problem: Find the largest integer $n \ge 3$ for which there exists an $n$-digit number $\overline{a_1a_2a_3\ldots a_n}$ with non-zero digits $a_1$, $a_2$, and $a_n$, such that the number is divisible by $\overline{a_2a_3\ldots a_n}$.
Answer: 6
Problem: Let $ABCD$ be a square with $AB = 6$. A point $P$ inside the square is 2 units away from side $BC$ and 3 units away from side $CD$. What is the distance from $P$ to $A$?
Answer: 5
Problem: Let $p$ be the largest prime less than $2013$ for which \[ N = 20 + p^{p^{p+1}-13} \] is also prime. Find the remainder when $N$ is divided by $10^4$.
Answer: 1781
Problem: Consider the sequence $$1, 7, 8, 49, 50, 56, 57, 343, \ldots$$ which consists of sums of distinct powers of $7$, that is, $7^0$, $7^1$, $7^0 + 7^1$, $7^2$, $\ldots$ in increasing order. At what position will $16856$ occur in this sequence?
Answer: 36
Problem: Given that $a_n a_{n-2} - a_{n-1}^2 + a_n - n a_{n-2} = -n^2 + 3n - 1$ and initial conditions $a_0 = 1$, $a_1 = 3$, find $a_{20}$.
Answer: 1048596
Problem: For each set of five integers $S= \{a_1, a_2, a_3, a_4, a_5\}$, let $P_S$ be the product of all differences between two of the elements, namely:

$$P_S = (a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)$$

Determine the greatest integer $n$ such that given any set $S$ of five integers, $n$ divides $P_S$.
Answer: 288
Problem: John and Mary each select a natural number and tell them to Bill. Bill writes down their sum and product on two separate pieces of paper, hides one, and shows the other to John and Mary. John looks at the number, which is $2002$, and declares he cannot determine Mary's number. Knowing this, Mary also says she cannot determine John's number. What was Mary's number?
Answer: 1001
Problem: A set $A$ contains exactly $n$ integers, each greater than $1$, where every prime factor of these integers is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer.
Answer: 17
Problem: Evaluate the sum: \( \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_{20}} \), where \( a_n = \sqrt{1 + \left(1 - \frac{1}{n}\right)^2} + \sqrt{1 + \left(1 + \frac{1}{n}\right)^2} \) for \( n \ge 1 \).
Answer: 7