prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Mary typed a six-digit number, but the two $1$s she typed didn't show. What appeared was $2002.$ How many different six-digit numbers could she have typed? | 15 |
According to the standard convention for exponentiation, $2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.$ If the order in which the exponentiations are performed is changed, how many other values are possible? | 1 |
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. If the area of the shaded region is $x\pi$, find the value of $x$. | 2 |
Circles of radius $2$ and $3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region. If the answer is $x\pi$, what is the value of $x$? [asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real r1=3; real r2=2; real r3=5; pair... | 12 |
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. If the answer is $x\pi$, what is the value of $x$? | 2 |
For how many positive integers $m$ is $\frac{2002}{m^2 -2}$ a positive integer? Provide your answer as an integer. | 3 |
How many ordered triples of positive integers $(x,y,z)$ satisfy $(x^y)^z=64?$ | 9 |
Let \(AB\) be a segment of length \(26\), and let points \(C\) and \(D\) be located on \(AB\) such that \(AC=1\) and \(AD=8\). Let \(E\) and \(F\) be points on one of the semicircles with diameter \(AB\) for which \(EC\) and \(FD\) are perpendicular to \(AB\). Find the length of \(EF\). The original answer is in the fo... | 9 |
The function $f$ is given by the table $\begin{array}{|c||c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \hline \end{array}$ If $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \ge 0$, find the value of $u_{2002}$. | 2 |
Jamal wants to save $30$ files onto disks, each with $1.44~ MB$ space. $3$ of the files take up $0.8~ MB$, $12$ of the files take up $0.7 ~MB$, and the rest take up $0.4 ~MB$. It is not possible to split a file onto $2$ different disks. What is the smallest number of disks needed to store all $30$ files? | 13 |
Compute the sum of all the roots of \((2x+3)(x-4)+(2x+3)(x-6)=0\). Express your answer as a fraction in simplest form \(\frac{k}{m}\), and provide the value of \(k + m\). | 9 |
How many different integers can be expressed as the sum of three distinct members of the set $\{1,4,7,10,13,16,19\}$? | 13 |
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files? | 13 |
For how many integers $n$ is $\dfrac{n}{20-n}$ the square of an integer? | 4 |
Two different positive numbers \(a\) and \(b\) each differ from their reciprocals by 1. The original answer is in the form \(\sqrt{m}\). What is the value of \(m\)? | 5 |
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ such that $k^2_1 + k^2_2 + ... + k^2_n = 2002?$ | 17 |
The number $25^{64}\cdot 64^{25}$ is the square of a positive integer $N$. In decimal representation, what is the sum of the digits of $N$? | 14 |
Find \(i + 2i^2 + 3i^3 + \cdots + 2002i^{2002}\). The original answer is in the form \(a + bi\), where \(a\) and \(b\) are integers. Please give the value of \(a + b\). | -1 |
Using the digits $1, 2, 3, 4, 5, 6, 7,$ and $9,$ form $4$ two-digit prime numbers, using each digit only once. What is the sum of the $4$ prime numbers? | 190 |
How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$? Find the number of such four-digit numbers. | 7 |
There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find the absolute difference \(|P_s - P_d|\). The original answer is in \(\frac{k}{m}\) format, ... | 2002 |
Tina randomly selects two distinct numbers from the set \(\{ 1, 2, 3, 4, 5 \}\), and Sergio randomly selects a number from the set \(\{ 1, 2, \cdots, 10 \}\). What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina? Express your answer in the form \(\frac{k}{m}\) in simples... | 7 |
Juan rolls a fair regular octahedral die marked with the numbers $1$ through $8$. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of $3$? The original answer is in the form of \frac{k}{m}, where the fraction is in simplest form. Please give the value of k + ... | 3 |
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup ... | 7 |
A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\triangle ADG$. The original answer is in the form k + m\sqrt{2}, where k and m are integers. Please find the value of k + m. | 7 |
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? Provide your answer as an integer. | 207 |
A $3\times3\times3$ cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube? | 90 |
For how many positive integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime number? | 3 |
A point \(P\) is randomly selected from the rectangular region with vertices \((0,0),(2,0),(2,1),(0,1)\). What is the probability, expressed as a fraction \(\frac{k}{m}\), that \(P\) is closer to the origin than it is to the point \((3,1)\)? Please provide the value of \(k + m\). | 7 |
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach? If the area can be expressed as $x\pi$, find the value of $x$. | 3 |
Suppose that $\{a_n\}$ is an arithmetic sequence with $$a_1+a_2+\cdots+a_{100}=100 ~\text{and}~ a_{101}+a_{102}+\cdots+a_{200}=200.$$ If the answer is $x$, what is the value of $100x$? | 1 |
How many whole numbers between $99$ and $999$ contain exactly one $0$? Provide your answer as an integer. | 162 |
Points \(A,B,C,D,E\) and \(F\) lie, in that order, on \(\overline{AF}\), dividing it into five segments, each of length \(1\). Point \(G\) is not on line \(AF\). Point \(H\) lies on \(\overline{GD}\), and point \(J\) lies on \(\overline{GF}\). The line segments \(\overline{HC}, \overline{JE},\) and \(\overline{AG}\) ar... | 8 |
Let $\triangle XOY$ be a right-angled triangle with $m\angle XOY = 90^{\circ}$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN = 19$ and $YM = 22$, find the length of $XY$. | 26 |
Consider the sequence of numbers: $4,7,1,8,9,7,6,\cdots$ For $n>2$, the $n$-th term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. Find the smallest value of $n$ for which $S_n>10,000$. | 1999 |
For all positive integers $n$ less than $2002$, let $$\begin{aligned} a_n =\left\{ \begin{array}{lr} 11, & {if\ }n\ {is\ divisible\ by\ }13\ {and\ }14;\\ 13, & {if\ }n\ {is\ divisible\ by\ }14\ {and\ }11;\\ 14, & {if\ }n\ {is\ divisible\ by\ }11\ {and\ }13;\\ 0, & {otherwise}. \end{array} \right. \end{aligned}$$ Calcul... | 448 |
Under the new AMC $10, 12$ scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of... | 1885 |
A set of tiles numbered $1$ through $100$ is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with $1$. Find the number of times the operation must be performed to reduce the number of tiles in the set to one. | 18 |
Triangle $ABC$ is a right triangle with $\angle ACB$ as its right angle, $m\angle ABC = 60^\circ$, and $AB = 10$. Let $P$ be randomly chosen inside $ABC$, and extend $\overline{BP}$ to meet $\overline{AC}$ at $D$. Find the probability that $BD > 5\sqrt{2}$. The original answer is in the form \(\frac{k-\sqrt{m}}{n}\), w... | 9 |
For all integers $n$ greater than $1$, define $a_n = \frac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Find the integer value of $b - c$. | -1 |
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches, including the top, bottom, and sides. | 26 |
The sum \( \frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\cdots+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}} \) is a root of the quadratic \( x^2+x+c \). The original answer is in the form \(-\frac{x*n}{y}\). Please find the value of x + y. | 3 |
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 \(\sqrt{m}\). Please find the value of \(m\). | 118 |
Let $a=\frac{1^2}{1} + \frac{2^2}{3} + \frac{3^2}{5} + \; \dots \; + \frac{1001^2}{2001}$ and $b=\frac{1^2}{3} + \frac{2^2}{5} + \frac{3^2}{7} + \; \dots \; + \frac{1001^2}{2003}.$ Find the integer closest to $a-b$. | 501 |
In triangle \(ABC\), side \(AC\) and the perpendicular bisector of \(BC\) meet at point \(D\), and \(BD\) bisects \(\angle ABC\). If \(AD=9\) and \(DC=7\), what is the area of triangle \(ABD\)? The original answer is in the form \(k\sqrt{m}\). Please provide the value of \(k + m\). | 19 |
In \(\triangle ABC\), we have \(AB = 1\) and \(AC = 2\). Side \(\overline{BC}\) and the median from \(A\) to \(\overline{BC}\) have the same length. If the length of \(BC\) is in the form \(\sqrt{k}\), please find the value of \(k\). | 2 |
Tina randomly selects two distinct numbers from the set $\{ 1, 2, 3, 4, 5 \}$, and Sergio randomly selects a number from the set $\{ 1, 2, ..., 10 \}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina? The answer should be expressed as a fraction in simplest form \fr... | 7 |
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, \ldots, k_n$ such that $k^2_1 + k^2_2 + \ldots + k^2_n = 2002?$ | 17 |
Find the number of ordered pairs of real numbers $(a,b)$ such that $(a+bi)^{2002} = a-bi$. What is the value of this number? | 2004 |
Let \(ABCD\) be a regular tetrahedron and let \(E\) be a point inside the face \(ABC.\) Denote by \(s\) the sum of the distances from \(E\) to the faces \(DAB, DBC, DCA,\) and by \(S\) the sum of the distances from \(E\) to the edges \(AB, BC, CA.\) Find the value of \(\frac{s}{S}\). The original answer is in the form ... | 5 |
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from $3$ pears and $8$ ounces of orange juice from $2$ oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What is the percentage of the blend that is ... | 40 |
In trapezoid $ABCD$ with bases $AB$ and $CD$, we have $AB = 52$, $BC = 12$, $CD = 39$, and $DA = 5$. Find the area of $ABCD$. | 210 |
When $15$ is appended to a list of integers, the mean is increased by $2$. When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$. How many integers were in the original list? | 4 |
Let \(a\) and \(b\) be real numbers such that \(\sin{a} + \sin{b} = \frac{\sqrt{2}}{2}\) and \(\cos {a} + \cos {b} = \frac{\sqrt{6}}{2}.\) The original answer is in the form \(\frac{\sqrt{k}}{m}\). Please find the value of \(k + m\). | 5 |
Loki, Moe, Nick, and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money, and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have? The original answer ... | 5 |
Moe uses a mower to cut his rectangular $90$-foot by $150$-foot lawn. The swath he cuts is $28$ inches wide, but he overlaps each cut by $4$ inches to make sure that no grass is missed. He walks at the rate of $5000$ feet per hour while pushing the mower. If the answer is $x$ hours, what is the value of $100x$? | 135 |
The sum of the two $5$-digit numbers $AMC10$ and $AMC12$ is $123422$. What is the value of $A+M+C$? | 14 |
The symbolism $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$. For example, $\lfloor 3 \rfloor = 3,$ and $\lfloor \frac{9}{2} \rfloor = 4$. Compute the sum \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{16} \rfloor.\] | 38 |
Penniless Pete's piggy bank has no pennies in it, but it has $100$ coins, all nickels, dimes, and quarters, whose total value is $8.35$. It does not necessarily contain coins of all three types. Find the difference between the largest and smallest number of dimes that could be in the bank. | 64 |
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8)=8$ and $\clubsuit(123)=1+2+3=6$. Find the number of two-digit values of $x$ for which $\clubsuit(\clubsuit(x))=3$. | 10 |
Simplify \(\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}\) and express your answer in the form of \(\sqrt[k]{x^p}\). Please find the value of k + p. | 3 |
Problems $8, 9$ and $10$ use the data found in the accompanying paragraph and figures Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: $\circ$ Roger's cookies are rectangles: $\circ$ Paul's... | 40 |
Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increased? Provide your answer as a fraction in th... | 36 |
Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of \(3:2:1\), respectively. Due to some confusion, they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct sh... | 23 |
Several figures can be made by attaching two equilateral triangles to the regular pentagon $ABCDE$ in two of the five positions shown. Find the number of non-congruent figures that can be constructed in this way. | 2 |
The sum of the two $5$-digit numbers $AMC10$ and $AMC12$ is $123422$. What is $A+M+C$? | 14 |
Cassandra sets her watch to the correct time at noon. At the actual time of $1:00$ PM, she notices that her watch reads $12:57$ and $36$ seconds. Assuming that her watch loses time at a constant rate, calculate the total time discrepancy in seconds between her watch and the actual time when her watch first reads $10:00... | 1500 |
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? | 12 |
What is the largest integer that is a divisor of \((n+1)(n+3)(n+5)(n+7)(n+9)\) for all positive even integers \(n\)? | 15 |
Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers $d$, $e$, and $10d+e$, where $d$ and $e$ are single digits. Find the sum of the digits of $n$. | 12 |
Given that $3^8\cdot5^2=a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b$. | 407 |
Points \(K, L, M,\) and \(N\) lie in the plane of the square \(ABCD\) such that \(AKB\), \(BLC\), \(CMD\), and \(DNA\) are equilateral triangles. If \(ABCD\) has an area of \(16\), find the area of \(KLMN\). The original answer is in the form \(k + m\sqrt{3}\). Please give the value of \(k + m\). | 48 |
In rectangle \(ABCD\), \(AB=5\) and \(BC=3\). Points \(F\) and \(G\) are on \(\overline{CD}\) so that \(DF=1\) and \(GC=2\). Lines \(AF\) and \(BG\) intersect at \(E\). The area of \(\triangle AEB\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\). | 27 |
In this addition problem, each letter stands for a different digit. $$\begin{array}{cccc}&T&W&O\\+&T&W&O\\\hline F&O&U&R\end{array}$$ If $T = 7$ and the letter $O$ represents an even number, what is the only possible value for $W$? | 3 |
A regular octagon \(ABCDEFGH\) has an area of one square unit. What is the area of the rectangle \(ABEF\)? Provide your answer in the form \(\frac{k}{m}\) as a simplified fraction, and give the value of \(k + m\). | 3 |
A point \(P\) is chosen at random in the interior of equilateral triangle \(ABC\). What is the probability that \(\triangle ABP\) has a greater area than each of \(\triangle ACP\) and \(\triangle BCP\)? Express your answer in the form \(\frac{k}{m}\), where \(k\) and \(m\) are coprime positive integers. Find the value ... | 4 |
Three semicircles of radius \(1\) are constructed on diameter \(\overline{AB}\) of a semicircle of radius \(2\). The centers of the small semicircles divide \(\overline{AB}\) into four line segments of equal length, as shown. Find the area of the shaded region that lies within the large semicircle but outside the small... | 18 |
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $75\%$ of the volume of the frozen ice cream. What is the ratio of the cone's height to its rad... | 3 |
Square \(ABCD\) has sides of length \(4\), and \(M\) is the midpoint of \(\overline{CD}\). A circle with radius \(2\) and center \(M\) intersects a circle with radius \(4\) and center \(A\) at points \(P\) and \(D\). Find the distance from \(P\) to \(\overline{AD}\). The original answer is in \(\frac{k}{m}\) format, pl... | 21 |
If \(\log (xy^3) = 1\) and \(\log (x^2y) = 1\), find the value of \(\log (xy)\). The answer should be in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in its simplest form. Please provide the value of \(k + m\). | 8 |
What is the largest integer that is a divisor of $(n+1)(n+3)(n+5)(n+7)(n+9)$ for all positive even integers $n$? | 15 |
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. Find the number of values of $n$ for which $q+r$ is divisible by $11$. | 8181 |
Three semicircles of radius $1$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $2$. The centers of the small semicircles divide $\overline{AB}$ into four line segments of equal length, as shown. Find the area of the shaded region that lies within the large semicircle but outside the smaller semic... | 18 |
Let \(S\) be the set of permutations of the sequence \(1,2,3,4,5\) for which the first term is not \(1\). A permutation is chosen randomly from \(S\). The probability that the second term is \(2\), in lowest terms, is \(\frac{a}{b}\). Please find the value of \(a + b\). | 19 |
In rectangle \(ABCD\), \(AB=5\) and \(BC=3\). Points \(F\) and \(G\) are on \(\overline{CD}\) so that \(DF=1\) and \(GC=2\). Lines \(AF\) and \(BG\) intersect at \(E\). The area of \(\triangle AEB\) is in the form \(\frac{k}{m}\). Please give the value of \(k+m\). | 27 |
How many $15$-letter arrangements of $5$ A's, $5$ B's, and $5$ C's have no A's in the first $5$ letters, no B's in the next $5$ letters, and no C's in the last $5$ letters? Provide your answer as a sum of binomial coefficients in the form \(\sum\limits_{k=0}^{n}\binom{m}{k}^{p}\), and then calculate the value of m + n ... | 13 |
A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements? Express your answer in the form of \frac{k}{m}, where k and m are co... | 41 |
The area of trapezoid $ABCD$ is $164\ \text{cm}^2$. The altitude is $8\ \text{cm}$, $AB$ is $10\ \text{cm}$, and $CD$ is $17\ \text{cm}$. What is the length of $BC$, in centimeters? | 10 |
In rectangle $ABCD$, we have $AB=8$, $BC=9$, $H$ is on $BC$ with $BH=6$, $E$ is on $AD$ with $DE=4$, line $EC$ intersects line $AH$ at $G$, and $F$ is on line $AD$ with $GF \perp AF$. Find the length of $GF$. | 20 |
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if... | 1507509 |
A regular octagon \(ABCDEFGH\) has an area of one square unit. What is the area of the rectangle \(ABEF\)? The answer is in the form \(\frac{k}{m}\), where the fraction is in its simplest form. Please find the value of \(k + m\). | 3 |
How many perfect squares are divisors of the product $1! \cdot 2! \cdot 3! \cdot \cdots \cdot 9!$? Provide your answer as an integer. | 672 |
The first four terms in an arithmetic sequence are \(x+y\), \(x-y\), \(xy\), and \(\frac{x}{y}\), in that order. What is the fifth term? The original answer is in the form \(\frac{k}{m}\). Please find the value of k + m. | 163 |
Positive integers $a,b,$ and $c$ are chosen so that $a<b<c$, and the system of equations $2x + y = 2003 \quad$ and $\quad y = |x-a| + |x-b| + |x-c|$ has exactly one solution. What is the minimum value of $c$? | 1002 |
How many distinct four-digit numbers are divisible by $3$ and have $23$ as their last two digits? Provide your answer as an integer. | 30 |
Let $f(x)= \sqrt{ax^2+bx}$. Find the number of real values of $a$ for which there is at least one positive value of $b$ such that the domain of $f$ and the range of $f$ are the same set. | 2 |
Three points are chosen randomly and independently on a circle. Find the probability that all three pairwise distances between the points are less than the radius of the circle. The probability is expressed in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\). | 13 |
How many two-digit positive integers have at least one $7$ as a digit? Fill in the blank with the correct number of such integers. | 18 |
What is the value of \(x\) if \(|x-1|=|x-2|\)? Provide your answer in the form of \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Calculate the value of \(k + m\). | 5 |
A standard six-sided die is rolled, and $P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $P$? | 12 |
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