ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Number $125$ is written as the sum of several pairwise distinct and relatively prime numbers, greater than $1$. What is the maximal possible number of terms in this sum?
Answer: 8
Problem: Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Neymar 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 Neymar will never be able to write on the blackboard?
Answer: 2033
Problem: A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2 + kx + 2014$ and $x^2 + kx - 2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous.
Answer: 0
Problem: Jay is given $99$ stacks of blocks, such that the $i$th stack has $i^2$ blocks. Jay must choose a positive integer $N$ such that from each stack, he may take either $0$ blocks or exactly $N$ blocks. Compute the value Jay should choose for $N$ in order to maximize the number of blocks he may take from the $99$ stacks.
Answer: 4489
Problem: Compute the remainder when $10^{2021}$ is divided by $10101$.
Answer: 9091
Problem: Compute the number of ways to erase 24 letters from the string ``OMOMO\cdots OMO'' (with length 27), such that the three remaining letters are O, M, and O in that order. Note that the order in which they are erased does not matter.
Answer: 455
Problem: Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x} = \frac{y}{13} = \frac{x}{y}$, find the value of $x^3 + y^3$.
Answer: 1170
Problem: 16 chess players held a tournament among themselves where every two players played exactly one game. The scoring system was as follows: 1 point for a victory, 0.5 points for a draw, and 0 points for a defeat. It was found that exactly 15 chess players shared the first place. Determine how many points the sixteenth chess player could have scored.
Answer: 0
Problem: Let $a$, $b$, and $c$ be real numbers such that $0 \le a, b, c \le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$.
Answer: 25
Problem: Find all 4-digit numbers $\overline{abcd}$ that satisfy the following conditions:

1. The number is a multiple of $11$.
2. The 2-digit number $\overline{ac}$ is a multiple of $7$.
3. The sum of the digits $a + b + c + d$ equals $d^2$.
Answer: 3454
Problem: Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have.
Answer: 94
Problem: Let $a, b, c$ be real nonzero numbers such that $a + b + c = 12$ and \[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = 1. \] Compute the largest possible value of $abc - \left(a + 2b - 3c\right)$.
Answer: 56
Problem: A positive integer $n$ is said to be base-able if there exist positive integers $a$ and $b$, with $b > 1$, such that $n = a^b$. How many positive integer divisors of $729000000$ are base-able?
Answer: 90
Problem: Let $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1) = 91$ and $P(-1) = -121$, compute the maximum possible value of \[ \dfrac{ab+bc+ca}{abc+a+b+c}. \]
Answer: 7
Problem: Alexis noticed her brother Joshua measuring something on the floor. He was designing a sandcastle and needed to measure the base of a spiral staircase. The castle walls form a rectangle, and the center of the staircase is: 
- 9 inches from one corner, 
- 15 inches from another corner, 
- 16 inches from a third corner, 
- and some whole number of inches from the furthest corner. 

Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four distances are measured on the same plane (the ground).
Answer: 20
Problem: Complex number $\omega$ satisfies $\omega^5 = 2$. Find the sum of all possible values of $\omega^4 + \omega^3 + \omega^2 + \omega + 1$.
Answer: 5
Problem: What is the remainder when $2^{1023}$ is divided by $1023$?
Answer: 8
Problem: Bob's Rice ID number consists of six digits, each ranging from $1$ to $9$. Each digit can be repeated any number of times. The ID number satisfies the following divisibility conditions:

- The number formed by the first two digits is divisible by $2$.
- The number formed by the first three digits is divisible by $3$.
- The number formed by the first four digits is divisible by $4$.
- The number formed by the first five digits is divisible by $5$.
- The entire six-digit ID number is divisible by $6$.

One example of such an ID number is $123252$. Determine how many different possibilities exist for Bob's ID number.
Answer: 324
Problem: Consider the grid of points $X = \{(m,n) \mid 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $(c = a \pm 2$ and $d = b \pm 1)$ or $(c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is:
Answer: 48
Problem: The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum, expressed as an irreducible fraction.
Answer: 168
Problem: Ana has \(22\) coins. She can take from her friends either \(6\) coins or \(18\) coins, or she can give \(12\) coins to her friends. She can perform these operations as many times as she wants. Find the least number of coins Ana can have.
Answer: 4
Problem: Six people each flip a fair coin. Everyone who flipped tails then flips their coin again. Given that the probability that all the coins are now heads can be expressed as a simplified fraction $\frac{m}{n}$, compute $m+n$.
Answer: 4825
Problem: Find the number of ordered triplets $(a, b, c)$ of positive integers such that $abc = 2008$.
Answer: 30
Problem: Suppose $x$, $y$, and $z$ are integers that satisfy the system of equations:  
\[ x^2y + y^2z + z^2x = 2186 \]  
\[ xy^2 + yz^2 + zx^2 = 2188. \]  
Evaluate $x^2 + y^2 + z^2$.
Answer: 245
Problem: Consider a regular $2n + 1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.
Answer: 1
Problem: For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Define $g(n) = 3^{f(n)}$. Compute the sum:
\[ \sum_{i=1}^{1000} g(i). \]
Answer: 8000
Problem: In how many ways can Irena arrange the integers $1$ through $7$ in a line such that for any three consecutive numbers, the largest is not in the rightmost position?
Answer: 232
Problem: Let $a_n$ be the integer closest to $\sqrt{n}$. Find the sum $$\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{1980}}.$$
Answer: 88
Problem: In each of the $9$ small circles of the following figure, write positive integers less than $10$, without repetitions. Additionally, it is true that the sum of the $5$ numbers located around each one of the $3$ circles is always equal to $S$. Find the largest possible value of $S$.
Answer: 28
Problem: If $x^{x^4} = 4$, what is the value of $x^{x^2} + x^{x^8}$?
Answer: 258
Problem: Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying the following system of equations:
\[
\begin{align*}
  2a_1 + a_2 + a_3 + a_4 + a_5 &= 1 + \frac{1}{8}a_4, \\
  2a_2 + a_3 + a_4 + a_5 &= 2 + \frac{1}{4}a_3, \\
  2a_3 + a_4 + a_5 &= 4 + \frac{1}{2}a_2, \\
  2a_4 + a_5 &= 6 + a_1.
\end{align*}
\]
Compute $a_1 + a_2 + a_3 + a_4 + a_5$.
Answer: 2
Problem: How many five-letter "words" can you spell using the letters $S$, $I$, and $T$, if a "word" is defined as any sequence of letters that does not contain three consecutive consonants?
Answer: 123
Problem: Talithia throws a party on the fifth Saturday of every month that has five Saturdays. That is, if a month has five Saturdays, Talithia has a party on the fifth Saturday of that month, and if a month has four Saturdays, then Talithia does not have a party that month. Given that January $1$, $2010$ was a Friday, compute the number of parties Talithia will have in $2010$.
Answer: 4
Problem: In triangle $ABC$, $\angle A = 45^\circ$ and $M$ is the midpoint of $\overline{BC}$. The line $\overline{AM}$ intersects the circumcircle of $ABC$ for the second time at point $D$, and $AM = 2MD$. Find $\cos\angle AOD$, where $O$ is the circumcenter of $ABC$.
Answer: 0
Problem: Patrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to the park. One hour after Tanya left, Jose started bicycling along the same road from school to the park. Tanya ran at a constant rate of $2$ miles per hour faster than Patrick walked, Jose bicycled at a constant rate of $7$ miles per hour faster than Tanya ran, and all three arrived at the park at the same time. The distance from the school to the park is $\frac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 277
Problem: Moor owns $3$ shirts, one each of black, red, and green. Moor also owns $3$ pairs of pants, one each of white, red, and green. Being stylish, he decides to wear an outfit consisting of one shirt and one pair of pants that are different colors. How many combinations of shirts and pants can Moor choose?
Answer: 7
Problem: Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.
Answer: 31
Problem: Find a number with $3$ digits, knowing that:
1. The sum of its digits is $9$.
2. The product of its digits is $24$.
3. The number read from right to left is $\frac{27}{38}$ of the original.
Answer: 342
Problem: The trinomial $f(x)$ satisfies the equation $(f(x))^3 - f(x) = 0$, which has three real roots. Determine the y-coordinate of the vertex of the function $f(x)$.
Answer: 0
Problem: Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he "only" solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively, with the exception of December 3, 2013, where he solved $60$ problems out of boredom. Altogether, how many problems does Richard solve in 2013?
Answer: 1100
Problem: Set 
\[A_n=\sum_{k=1}^n \frac{k^6}{2^k}.\]
Find \(\lim_{n\to\infty} A_n.\)
Answer: 9366
Problem: Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?
Answer: 63
Problem: How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.)
Answer: 1225
Problem: Let $x$, $y$, $z$ be real numbers that satisfy the following equations:

\[
(x-y)^2 + (y-z)^2 + (z-x)^2 = 8, \\
x^3 + y^3 + z^3 = 1
\]

Find the minimum value of $x^4 + y^4 + z^4$.
Answer: 3
Problem: Find the smallest positive integer $k$ which can be expressed in the form $k = 19^n - 5^m$ for some positive integers $m$ and $n$.
Answer: 14
Problem: Point $P$ is in the interior of $\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The three feet of perpendicular lines from $P$ to sides $BC$, $CA$, $AB$ are $D$, $E$, $F$ respectively. Suppose the minimal value of $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ can be written as $\frac{a}{b}\sqrt{c}$, where $\gcd(a,b) = 1$ and $c$ is square-free, calculate $abc$.
Answer: 600
Problem: The **doubling sum** function is defined by:
\[ D(a,n) = a + 2a + 4a + 8a + \ldots \] 
This sum contains \( n \) terms. For example, we have:
\[ D(5,3) = 5 + 10 + 20 = 35 \]
and
\[ D(11,5) = 11 + 22 + 44 + 88 + 176 = 341. \]
Determine the smallest positive integer \( n \) such that for every integer \( i \) between 1 and 6, inclusive, there exists a positive integer \( a_i \) such that \( D(a_i,i) = n. \)
Answer: 9765
Problem: A triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $\frac{a}{b}$ is the length of the shortest altitude of the triangle. Find the value of $a+b$.
Answer: 2713
Problem: Find the smallest positive integer $k$ such that $k!$ ends in at least 43 zeroes.
Answer: 175
Problem: As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$ and $AB > 14$, compute the minimal area of $\triangle AEG$.
Answer: 98
Problem: Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Answer: 12
Problem: How many integers $x$ satisfy the following conditions?

1. $|x| + 5 < 7$
2. $|x - 3| > 2$
Answer: 2
Problem: You come across an ancient mathematical manuscript. It reads: 

"To determine if a number is divisible by seventeen, follow this procedure: 
- Take the number formed by the last two digits of the number.
- Subtract the number formed by the third- and fourth-to-last digits.
- Add the number formed by the fifth- and sixth-to-last digits, and so on.

The resulting number is divisible by seventeen if and only if the original number is divisible by seventeen."

What is the sum of the five smallest bases the ancient culture might have been using? 

(Note: "Seventeen" is the number represented by $17$ in base $10$, not $17$ in the base that the ancient culture was using. Express your answer in base $10$.)
Answer: 106
Problem: Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.
Answer: 37805
Problem: Two circles have radii $2$ and $3$, and the distance between their centers is $10$. Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$.

A common external tangent is a tangent line to two circles such that the circles are on the same side of the line, while a common internal tangent is a tangent line to two circles such that the circles are on opposite sides of the line.
Answer: 24
Problem: Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of positive integers.
Answer: 1
Problem: Compute the number of triangles of different sizes which contain the gray triangle in the figure below.

\[ \text{[asy]} \]

```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: Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A \cap B \neq \emptyset$ and $|A| \neq |B|$.
Answer: 23606
Problem: For any positive integer $a$, let $\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest possible value of $4\tau(n) - n$ over all positive integers $n$.
Answer: 12
Problem: Jonathan discovers that his ideal match is Sara Lark. To increase his chances of finding a girlfriend, he is open to dating any girl whose name is an anagram of "Sara Lark," provided that the name includes both a first and last name with at least one letter each. How many such anagrams are possible?
Answer: 23520
Problem: Determine the value of the sum \( \left| \sum_{1 \leq i < j \leq 50} ij(-1)^{i+j} \right|.\)
Answer: 21150
Problem: In Vila Par, all the truth coins weigh an even number of grams, while the false coins weigh an odd number of grams. The electronic device only provides the parity (odd or even) of the total weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, determine the smallest integer $k$ such that there exists a strategy to identify the two false coins using the electronic device at most $k$ times.
Answer: 21
Problem: Find the sum of the first 5 positive integers $n$ such that $n^2 - 1$ is the product of 3 distinct primes.
Answer: 104
Problem: Gorf the frog is standing on the first lily pad in a row of lily pads numbered from $1$ to $20$ from left to right. On a single jump, Gorf can jump either $1$, $2$, or $3$ lily pads to the right. Unfortunately, all the prime-numbered lily pads are contaminated with deadly poison. How many sequences of jumps allow Gorf to jump to the twentieth lily pad while avoiding the poison?
Answer: 25
Problem: Find the smallest natural number $n$ for which there exists a natural number $x$ such that
\[(x+1)^3 + (x+2)^3 + (x+3)^3 + (x+4)^3 = (x+n)^3.\]
Answer: 10
Problem: Let $f$ be a function such that $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$ and $f(1) = 100$. Calculate $\sum_{k = 1}^{10} f(k!)$.
Answer: 403791300
Problem: Determine the leftmost three digits of the number \[1^1 + 2^2 + 3^3 + \ldots + 999^{999} + 1000^{1000}.\]
Answer: 100
Problem: Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as:
\[
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\]
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
Answer: 4
Problem: The positive integers $v, w, x, y$, and $z$ satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute $10^4 v + 10^3 w + 10^2 x + 10 y + z$.
Answer: 12354
Problem: Compute the smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$.
Answer: 48
Problem: There is a rectangular field that measures $20\,\text{m}$ by $15\,\text{m}$. Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\,\text{m}$ sides such that he is $6\,\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly?
Answer: 17
Problem: Let $s(n)$ be the number of 1's in the binary representation of $n$. Find the number of ordered pairs of integers $(a, b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$.
Answer: 648
Problem: Let $x, y, z$ be positive real numbers that satisfy at least one of the inequalities: $2xy > 1$, $yz > 1$. Find the least possible value of:
$$xy^3z^2 + \frac{4z}{x} - 8yz - \frac{4}{yz}.$$
Answer: -8
Problem: Let $a_n = \sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}$. In other words, $a_n$ is the sum of $\frac{1}{2^{d+ \frac{n}{d}}}$ over all divisors $d$ of $n$. 

Find
$$\frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + \ldots}{a_1 + a_2 + a_3 + \ldots}$$
Answer: 4
Problem: Given 10 real numbers $a_1, a_2, \ldots, a_{10}$, 45 sums of two of these numbers are formed as $a_i + a_j$ for $1 \leq i < j \leq 10$. It is known that not all these sums are integers. Determine the minimum value of $k$ such that it is possible that among the 45 sums, there are $k$ sums that are not integers and $45-k$ sums that are integers.
Answer: 9
Problem: A $12 \times 12$ board is divided into $144$ unit squares by drawing lines parallel to the sides. Two rooks placed on two unit squares are said to be non-attacking if they are not in the same column or same row. Find the least number $N$ such that if $N$ rooks are placed on the unit squares, one rook per square, we can always find $7$ rooks such that no two are attacking each other.
Answer: 73
Problem: The Stromquist Comet is visible every 61 years. If the comet was visible in 2017, what is the next leap year when the comet will be visible?
Answer: 2444
Problem: Let $f(x) = (x-1)(x-2)(x-3)^2(x-4)(x-5)(x-6)$. Compute $f''(3) - f'(3) + f(3)$.
Answer: -24
Problem: A few (at least $5$) integers are placed on a circle such that each integer is divisible by the sum of its neighboring integers. If the sum of all the integers is positive, what is the minimal value of this sum?
Answer: 2
Problem: Let $ABC$ be a triangle with side lengths $13$, $14$, and $15$. The points on the interior of $\triangle ABC$ that are at least $1$ unit away from each side are shaded. The area of the shaded region can be expressed in simplest form as $\frac{m}{n}$. Find the value of $m+n$.
Answer: 193
Problem: Let $S = \{1, 2, \dots, 9\}$. Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S$, $f(f(f(s))) = s$ and $f(s) - s$ is not divisible by $3$.
Answer: 288
Problem: An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \in \overline{AB}$, $Q \in \overline{AC}$, and $N, P \in \overline{BC}$.

Suppose that $ABC$ is an equilateral triangle with side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. The area of $AMNPQ$ is given by $n - p \sqrt{q}$, where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n + 10p + q$.
Answer: 5073
Problem: Determine the sum of all possible positive integers $n$, such that the product of its digits equals $n^2 - 15n - 27$.
Answer: 17
Problem: A prime number $p$ is called a \textbf{moderate} number if for every $2$ positive integers $k > 1$ and $m$, there exist $k$ positive integers $n_1, n_2, \ldots, n_k$ such that:
\[ n_1^2 + n_2^2 + \ldots + n_k^2 = p^{k+m} \]

Determine the smallest prime $r$ which is not moderate, given that $q$ is the smallest \textbf{moderate} number and $q < r$.
Answer: 7
Problem: A group of 6 students decided to form "study groups" and "service activity groups" based on the following principle:

- Each group must have exactly 3 members.
- For any pair of students, there are the same number of study groups and service activity groups that both students are members of.

Given that there is at least one group and no three students belong to both the same study group and service activity group, find the minimum number of groups.
Answer: 8
Problem: A rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles and similar to the large rectangle. The sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures $1$. Determine the minimum possible value of the area of the large rectangle.
Answer: 2000
Problem: What is the smallest positive odd integer that has the same number of positive divisors as $360$?
Answer: 3465
Problem: Let $f$ be a function such that $f(x) + f(x+1) = 2^x$ and $f(0) = 2010$. Find the last two digits of $f(2010)$.
Answer: 51
Problem: Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x \rfloor$ is known as the $\textit{floor}$ function, which returns the greatest integer less than or equal to $x$.
Answer: 1973
Problem: Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that: 
- Each member in $S$ is a positive integer not exceeding $2002$.
- If $a, b \in S$ (not necessarily different), then $ab \not\in S$.
Answer: 1958