ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.) Provide your answer as a number.
Answer: 24
Problem: Compute the remainder when $20^{(13^{14})}$ is divided by $11$.
Answer: 5
Problem: The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is the value of $b-a$?
Answer: 7
Problem: An ice ballerina rotates at a constant angular velocity at one particular point, meaning she does not translationally move. Her arms are fully extended as she rotates, and her moment of inertia is \(I\). Now, she pulls her arms in, and her moment of inertia becomes \(\frac{7}{10}I\). What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)? The original answer is in the format \(\frac{k}{m}\), please give the value of \(k + m\).
Answer: 17
Problem: Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping squares, respectively. If the pattern were continued, find the number of nonoverlapping squares in figure $100$.
Answer: 20201
Problem: Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1$? Provide your answer in the form of \frac{k}{m}, and calculate the value of k + m.
Answer: 5
Problem: By inserting parentheses, it is possible to give the expression $2 \times 3 + 4 \times 5$ several values. How many different values can be obtained?
Answer: 4
Problem: At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. How many pairs are there in which both students are wearing yellow shirts?
Answer: 32
Problem: A coin is altered so that the probability that it lands on heads is less than \(\frac{1}{2}\). When the coin is flipped four times, the probability of getting an equal number of heads and tails is \(\frac{1}{6}\). What is the probability that the coin lands on heads? Express your answer in the form \(\frac{k - \sqrt{m}}{n}\), where the fraction is in its simplest form. Please calculate the value of k + m + n.
Answer: 12
Problem: Hull Street is nine miles long. The first two miles of Hull Street are in Richmond, and the last two miles of the street are in Midlothian. If Kathy starts driving along Hull Street from a random point in Richmond and stops on the street at a random point in Midlothian, what is the probability that Kathy drove less than six miles? The original answer is in the form of \(\frac{k}{m}\), where both k and m are integers, and the fraction is in its simplest form. Please find the value of k + m.
Answer: 9
Problem: Adams plans a profit of $10\%$ on the selling price of an article and his expenses are $15\%$ of sales. If the rate of markup on an article that sells for $\$5.00$ is $x\%$, find the value of $3x$. Provide your answer as an integer without the percentage symbol.
Answer: 100
Problem: If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to $43$, find the probability that this number is divisible by $11$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 6
Problem: \(\overline{AB}\) is the hypotenuse of a right triangle \(ABC\). Median \(\overline{AD}\) has length \(7\) and median \(\overline{BE}\) has length \(4\). The length of \(\overline{AB}\) is in the form of a\sqrt{b}. Please give the value of a + b.
Answer: 15
Problem: A circle with center $O$ and radius $15$ is given. Let $P$ be a point such that $|OP|=9$. Find the number of chords of the circle that pass through $P$ and have integer length.
Answer: 12
Problem: In the sequence $2001, 2002, 2003, \ldots$, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000$. What is the value of the $2004^\text{th}$ term in this sequence?
Answer: 0
Problem: Consider the graphs $y = Ax^2$ and $y^2 + 3 = x^2 + 4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. Determine the number of points at which the two graphs intersect.
Answer: 4
Problem: At a party with $n \geq 4$ people, if every 3 people have exactly 1 common friend, find the number of different values that $n$ can take.
Answer: 4
Problem: Given \( \frac{\log{a}}{p} = \frac{\log{b}}{q} = \frac{\log{c}}{r} = \log{x} \), all logarithms to the same base and \( x \neq 1 \). If \( \frac{b^2}{ac} = x^y \), find the value of \( y \). The original answer is in the form \(aq - bp - cr\), please provide the value of a + b + c.
Answer: 4
Problem: If $y = 2x$ and $z = 2y$, then find the expression for $x + y + z$ in terms of $x$. The answer should be in the form $ax$, where $a$ is a constant. Please determine the value of $a$.
Answer: 7
Problem: How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$? Find the number of such coefficients.
Answer: 16
Problem: In triangle \(ABC\), \(\angle BAC = 60^\circ\), \(AB = 3a\), and \(AC = 4a\), where \(a > 0\). Let \(M\) be a point on segment \(AB\) such that \(AM = \frac{1}{3} AB\), and \(N\) be a point on side \(AC\) such that \(AN = \frac{1}{2} AC\). Let \(I\) be the midpoint of \(MN\). Determine the length of \(BI\). The original answer is in the form \(\frac{ka\sqrt{m}}{n}\), please give the value of \(k + m + n\).
Answer: 22
Problem: Find the integer that does not divide $n^{2225} - n^{2005}$ for every integer value of $n$.
Answer: 7
Problem: Let \(ABC\) be a triangle with incenter \(I\), centroid \(G\), and \(|AC| > |AB|\). If \(IG \parallel BC\), \(|BC| = 2\), and \(\text{Area}(ABC) = \frac{3\sqrt{5}}{8}\), then find the value of \(|AB|\). The original answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\).
Answer: 17
Problem: Find the number of primes $p$, such that $x^{3} - 5x^{2} - 22x + 56 \equiv 0 \pmod{p}$ has no three distinct integer roots in $[0, p)$.
Answer: 4
Problem: Point \(P\) is taken interior to a square with side-length \(a\) such that it is equally distant from two consecutive vertices and from the side opposite these vertices. If \(d\) represents the common distance, find the value of \(d\) in terms of \(a\). The original answer is in the form \(\frac{ka}{m}\), where the fraction is in its simplest form. Please find the value of \(k + m\).
Answer: 13
Problem: Let $E$ be a point inside the rhombus $ABCD$ such that $|AE| = |EB|$, $m(\widehat{EAB}) = 12^\circ$, and $m(\widehat{DAE}) = 72^\circ$. What is the value of $m(\widehat{CDE})$ in degrees?
Answer: 66
Problem: A man with mass $m$ jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance $H$ at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant $k$, and stretches from an original length of $L_0$ to a final length $L = L_0 + h$. The maximum tension in the bungee cord is four times the weight of the man.

Find the maximum extension of the bungee cord $h$ in terms of $H$. The original answer is in the format \frac{k}{m}, please give the value of k + m.
Answer: 3
Problem: For how many integers \( n \) between 1 and 100 does \( x^2 + x - n \) factor into the product of two linear factors with integer coefficients?
Answer: 9
Problem: What is the greatest integer $k$ which makes the statement "When we take any $6$ subsets with $5$ elements of the set \( \{1,2,\dots, 9\} \), there exist $k$ of them having at least one common element." true?
Answer: 4
Problem: Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. Find the number of different possible denominators.
Answer: 5
Problem: The geometric series \(a + ar + ar^{2} + \ldots\) has a sum of \(7\), and the terms involving odd powers of \(r\) have a sum of \(3\). The original answer is in the form \(\frac{k}{m}\). Find the value of \(a + r\) and give the sum of \(k + m\).
Answer: 7
Problem: Find the value of \(i + 2i^2 + 3i^3 + \ldots + 2002i^{2002}\). The original answer is in the form of \(k + mi\), please give the value of \(k + m\).
Answer: -1
Problem: Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the $4$ replaced by $3$ and the other die has the $3$ replaced by $4$. When these dice are rolled, find the probability that the sum is an odd number. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 14
Problem: If $x$ is a number satisfying the equation $\sqrt[3]{x+9} - \sqrt[3]{x-9} = 3$, find the value of $x^2$ and determine which interval it falls into among the following: $[55, 65]$, $[65, 75]$, $[75, 85]$, $[85, 95]$, $[95, 105]$. The original answer is in the form $[a, b]$, please give the value of $a + b$.
Answer: 160
Problem: Flights are arranged between 13 countries. For $k \ge 2$, the sequence $A_1, A_2, \ldots, A_k$ is said to be a cycle if there exists a flight from $A_1$ to $A_2$, from $A_2$ to $A_3$, $\ldots$, from $A_{k-1}$ to $A_k$, and from $A_k$ to $A_1$. Find the smallest possible number of flights such that, regardless of how the flights are arranged, there exists a cycle.
Answer: 79
Problem: A cone-shaped mountain has its base on the ocean floor and has a height of $8000$ feet. The top $\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet?
Answer: 4000
Problem: A wooden cube has edges of length $3$ meters. Square holes, with a side of $1$ meter, are centered in each face and cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. What is the total surface area, including the inside, in square meters?
Answer: 72
Problem: Let \(a\), \(b\), and \(c\) be the roots of the polynomial \(x^3 + 4x^2 - 7x - 1\). Find the polynomial that has roots \(ab\), \(bc\), and \(ac\). The original answer is in the form \(mx^3 + nx^2 + px + q\). Please give the value of \(m + n + p + q\).
Answer: 3
Problem: In the adjoining diagram, $BO$ bisects $\angle CBA$, $CO$ bisects $\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12$, $BC=24$, and $AC=18$, find the perimeter of $\triangle AMN$.
Answer: 30
Problem: Let $ABCD$ be a convex quadrilateral with $\angle C = \angle D = 90^\circ$. The circle $K$ passing through $A$ and $B$ is tangent to $CD$ at $C$. Let $E$ be the intersection of $K$ and $[AD]$. If $|BC| = 20$, $|AD| = 16$, then find the length of $|CE|$. The original answer is in the form $k\sqrt{m}$, please give the value of $k + m$.
Answer: 9
Problem: Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate?
Answer: 15
Problem: [BD] is a median of \( \triangle ABC \). \( m(\widehat{ABD}) = 90^\circ \), \(|AB| = 2\), and \(|AC| = 6\). Find the length of \(|BC|\). The original answer is in the form of \(k\sqrt{m}\). Please give the value of \(k + m\).
Answer: 8
Problem: Find the largest positive value attained by the function
\[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \ x\ \text{a real number} \]
If the answer is in the form $a\sqrt{b}$, where $a$ and $b$ are integers, find the value of $a$.
Answer: 2
Problem: For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer? Provide the number of such positive integers.
Answer: 7
Problem: On circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 30^\circ$, and $\angle DOB = 45^\circ$. Find the ratio of the area of the smaller sector $COD$ to the area of the circle. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 31
Problem: Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P \in [AC]$, $Q \in [BC]$, $R \in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K \in [BC]$, $M \in [AB]$, and $L \in [AC]$. Find $|AB|+|AC|$.
Answer: 12
Problem: Let \(D\) be a point on side \([BC]\) of \(\triangle ABC\) such that \(|AB|=3\), \(|CD|=1\), and \(|AC|=|BD|=\sqrt{5}\). If the \(B\)-altitude of \(\triangle ABC\) meets \(AD\) at \(E\), find the length of \(|CE|\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
Problem: If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ is on $\overline{AQ}$, and $\measuredangle QPC = 60^\circ$, find the ratio of the length of $PQ$ to the length of $AQ$. The original answer is in \(\frac{k\sqrt{3}}{m}\) format, please give the value of k + m.
Answer: 4
Problem: When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$?
Answer: 337
Problem: The chord $[CD]$ of the circle with diameter $[AB]$ is perpendicular to $[AB]$. Let $M$ and $N$ be the midpoints of $[BC]$ and $[AD]$, respectively. If $|BC|=6$ and $|AD|=2\sqrt{3}$, find the length of $|MN|$. The original answer is in the form \(\sqrt{m}\). Please find the value of \(m\).
Answer: 21
Problem: Medians \(AD\) and \(CE\) of triangle \(ABC\) intersect at point \(M\). The midpoint of \(AE\) is \(N\). The area of triangle \(MNE\) is \(k\) times the area of triangle \(ABC\). The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Answer: 13
Problem: Find the sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$. The original answer is in the form an(b + c), where a, b, and c are constants. Please give the value of a + b + c.
Answer: 4
Problem: Given three squares as in the figure, where the largest square has area 1, and the area $A$ is known. Find the area $B$ of the smallest square in terms of $A$. The original answer is in the form \frac{(k - A)^m}{n}, where k, m, and n are integers. Please calculate the value of k + m + n.
Answer: 7
Problem: If $a$ and $b$ are the roots of the equation $x^2 - 2cx - 5d = 0$, and $c$ and $d$ are the roots of the equation $x^2 - 2ax - 5b = 0$, where $a, b, c, d$ are distinct real numbers, find the value of $a + b + c + d$.
Answer: 30
Problem: If \(x \neq 0\) or \(4\) and \(y \neq 0\) or \(6\), then \(\frac{2}{x}+\frac{3}{y}=\frac{1}{2}\) is equivalent to the equation \(\frac{ay}{y-b}=x\). Please determine the sum of a and b.
Answer: 10
Problem: Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
Answer: 1680
Problem: Two medians of a triangle with unequal sides are $3$ inches and $6$ inches. Its area is $3 \sqrt{15}$ square inches. Find the length of the third median in inches. The original answer is in the form $k \sqrt{m}$, please give the value of $k + m$.
Answer: 9
Problem: Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one? Express your answer as a fraction in simplest form. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 10
Problem: For every $n \ge 2$, $a_n = \frac{\sqrt[3]{n^3 + n^2 - n - 1}}{n}$. Find the least value of positive integer $k$ satisfying $a_2a_3\cdots a_k > 3$.
Answer: 106
Problem: In the overlapping triangles $\triangle{ABC}$ and $\triangle{ABE}$ sharing common side $AB$, $\angle{EAB}$ and $\angle{ABC}$ are right angles, $AB = 4$, $BC = 6$, $AE = 8$, and $\overline{AC}$ and $\overline{BE}$ intersect at $D$. Find the difference between the areas of $\triangle{ADE}$ and $\triangle{BDC}$.
Answer: 4
Problem: The pattern of figures $\triangle$ $\bullet$ $\square$ $\blacktriangle$ $\circ$ is repeated in the sequence:
\[
\triangle, \bullet, \square, \blacktriangle, \circ, \triangle, \bullet, \square, \blacktriangle, \circ, \ldots
\]
Determine the 214th figure in the sequence. Write the position of the figure in the pattern as a number from 1 to 5.
Answer: 4
Problem: The equation \(z(z+i)(z+3i)=2002i\) has a zero of the form \(a+bi\), where \(a\) and \(b\) are positive real numbers. The original answer is in the form \(\sqrt{k}\). Please find the value of \(k\).
Answer: 118
Problem: How many integer triples $(x,y,z)$ are there satisfying the equation $x^3+y^3=x^2yz+xy^2z+2$?
Answer: 4
Problem: What is the sum of distinct real numbers \(x\) such that \((2x^2+5x+9)^2=56(x^3+1)\)? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 11
Problem: Find the shortest distance from an integral point to the line $y=\frac{5}{3}x+\frac{4}{5}$. The original answer is in the form \frac{\sqrt{k}}{m}, where k and m are integers. Please find the value of k + m.
Answer: 119
Problem: The hour part of a defective digital watch displays only the numbers from $1$ to $12$. After one minute from $n: 59$, although it must display $(n + 1): 00$, it displays $2n: 00$ (considering $\mod\, 12$). For example, after $7: 59$, it displays $2: 00$ instead of $8: 00$. If we set the watch to an arbitrary time, find the probability that the hour part displays $4$ after exactly one day. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 4
Problem: How many ordered triples of integers $(a, b, c)$ satisfy the following equations?
\[ |a + b| + c = 19 \]
\[ ab + |c| = 97 \]
Provide the number of such ordered triples.
Answer: 12
Problem: A man buys a house for $10000$ dollars and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep, pays $325$ dollars a year in taxes, and realizes $5\frac{1}{2}\%$ on his investment. If the monthly rent is $x$ dollars, what is the value of $100x$?
Answer: 8333
Problem: For all non-zero numbers $x$ and $y$ such that $x = \frac{1}{y}$, find the value of \[\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)\]. The original answer is in the form $ax^2 - by^2$. Please find the value of $a + b$.
Answer: 2
Problem: There are integers \( a \), \( b \), and \( c \), each greater than 1, such that \( \sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}} \) for all \( N > 1 \). Find the value of \( b \).
Answer: 3
Problem: If $a=\sqrt[3]{9}-\sqrt[3]{3}+1$, find the value of $\left(\frac{4-a}{a}\right)^6$.
Answer: 9
Problem: What is the sum of the real roots of the equation $(x\sqrt{x})^x = x^{x\sqrt{x}}$? Provide your answer in the form of a simplified fraction \frac{k}{m}, and then give the value of k + m.
Answer: 17
Problem: Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have an area of $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? Provide your answer as an integer.
Answer: 20
Problem: Emily has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any previous boxes. How many balls in total are in the boxes as a result of Emily's $2017$th step?
Answer: 9
Problem: Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream? The answer should be in the form \frac{k}{m}, where k and m are coprime. Please provide the value of k + m.
Answer: 7
Problem: For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which 
\[n(A) + n(B) + n(C) = n(A \cup B \cup C)\] 
and 
\[|A| = |B| = 100,\] 
then find the minimum possible value of $|A \cap B \cap C|$.
Answer: 97
Problem: Let $ABCD$ be a convex quadrilateral such that $|AB|=10$, $|CD|=3\sqrt{6}$, $m(\widehat{ABD})=60^\circ$, $m(\widehat{BDC})=45^\circ$, and $|BD|=13+3\sqrt{3}$. Find the length of $|AC|$.
Answer: 16
Problem: We have a square grid of $4 \times 4$ points. Find the number of triangles that can be formed with vertices on these points, given that the three vertices must not lie on a straight line.
Answer: 516
Problem: The zeroes of the function $f(x) = x^2 - ax + 2a$ are integers. Find the sum of all possible values of $a$.
Answer: 16
Problem: What is the remainder when the sum
\[\binom{2013}{1} + 2013\binom{2013}{3} + 2013^2\binom{2013}{5} + \dots + 2013^{1006}\binom{2013}{2013}\]
is divided by $41$?
Answer: 20
Problem: Five points on a circle are numbered 1, 2, 3, 4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle. If it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps, on which point will it be?
Answer: 4
Problem: How many natural numbers (i.e., 1, 2, 3,...) are equal to three times the sum of their digits? Find the number of such natural numbers.
Answer: 1
Problem: Let $ABC$ be an isosceles triangle with $|AB| = |AC| = 10$ and $|BC| = 12$. Points $P$ and $R$ are on $[BC]$ such that $|BP| = |RC| = 3$. Points $S$ and $T$ are the midpoints of $[AB]$ and $[AC]$, respectively. If $M$ and $N$ are the feet of the perpendiculars from $S$ and $R$ to $PT$, find the length of $|MN|$. The original answer is in the form \frac{k\sqrt{m}}{n}, please provide the value of k + m + n.
Answer: 36
Problem: Let $f(x) = x - \frac{1}{x}$. How many different solutions are there to the equation $f(f(f(x))) = 1$? Provide your answer as an integer.
Answer: 8
Problem: The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1$? The original answer is in the form of $k \pi \sqrt{m}$, please give the value of $k + m$.

[asy]
unitsize(20);
filldraw(circle((0,0),2*\sqrt{3}+1),rgb(0.5,0.5,0.5));
filldraw(circle((-2,0),1),white);
filldraw(circle((0,0),1),white);
filldraw(circle((2,0),1),white);
filldraw(circle((1,\sqrt{3}),1),white);
filldraw(circle((3,\sqrt{3}),1),white);
filldraw(circle((-1,\sqrt{3}),1),white);
filldraw(circle((-3,\sqrt{3}),1),white);
filldraw(circle((1,-1*\sqrt{3}),1),white);
filldraw(circle((3,-1*\sqrt{3}),1),white);
filldraw(circle((-1,-1*\sqrt{3}),1),white);
filldraw(circle((-3,-1*\sqrt{3}),1),white);
filldraw(circle((0,2*\sqrt{3}),1),white);
filldraw(circle((0,-2*\sqrt{3}),1),white);
[/asy]
Answer: 7
Problem: A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0)$, $(2020, 0)$, $(2020, 2020)$, and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) If the answer is $d$ to the nearest tenth, what is the value of $10d$?
Answer: 4