prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Calculate the value of $(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193)$. | 167400 |
When $10^{93}-93$ is expressed as a single whole number, what is the sum of the digits? | 826 |
What number is directly above $142$ in this array of numbers? $$\begin{array}{cccccc}& & & 1 & &\\ & & 2 & 3 & 4 &\\ & 5 & 6 & 7 & 8 & 9\\ 10 & 11 & 12 &\cdots & &\\ \end{array}$$ Find the number directly above $142$. | 120 |
Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, at what time will her working day end? Please provide the answer in the format of AA:BB PM, where AA is the hour and BB is the minutes. Calculate... | 14 |
For how many three-digit whole numbers does the sum of the digits equal $25$? Provide your answer as an integer. | 6 |
Two children at a time can play pairball. For $90$ minutes, with only two children playing at a time, five children take turns so that each one plays the same amount of time. How many minutes does each child play? | 36 |
Let \(W, X, Y,\) and \(Z\) be four different digits selected from the set \(\{ 1,2,3,4,5,6,7,8,9\}.\) Find the smallest possible value of the sum \(\frac{W}{X} + \frac{Y}{Z}\). The original answer is in the format \(\frac{m}{n}\). Please provide the value of \(m + n\). | 97 |
A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green sq... | 6 |
Find the sum of the digits in the answer to $\underbrace{9999\cdots 99}_{94\mathrm{ nines}} \times \underbrace{4444\cdots 44}_{94\mathrm{ fours}}$ where a string of $94$ nines is multiplied by a string of $94$ fours. What is the sum of the digits in the result? | 846 |
Which operation has the same effect on a number as multiplying by \(\dfrac{3}{4}\) and then dividing by \(\dfrac{3}{5}\)? Provide your answer in the form of a fraction \(\frac{k}{m}\), and find the value of \(k + m\). | 9 |
Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. What is the perimeter of $III$? | 36 |
At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home? Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and give the value of \(k + m\). | 23 |
Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. Find the area of the rectangle. | 32 |
A lucky year is one in which at least one date, when written in the form month/day/year, has the following property: The product of the month times the day equals the last two digits of the year. For example, $1956$ is a lucky year because it has the date $7/8/56$ and $7\times 8 = 56$. Find the year from the following ... | 1994 |
The table below gives the percent of students in each grade at Annville and Cleona elementary schools: $$ \begin{array}{rccccccc} & \underline{\mathbf{K}} & \underline{\mathbf{1}} & \underline{\mathbf{2}} & \underline{\mathbf{3}} & \underline{\mathbf{4}} & \underline{\mathbf{5}} & \underline{\mathbf{6}} \\ \mathrm { An... | 15 |
Diana and Apollo each roll a standard die obtaining a number at random from \(1\) to \(6\). What is the probability that Diana's number is larger than Apollo's number? Provide your answer as a fraction in simplest form \(\frac{k}{m}\), and then calculate the value of \(k + m\). | 17 |
How many four-digit whole numbers are there such that the leftmost digit is odd, the second digit is even, and all four digits are different? Provide your answer as a whole number. | 1400 |
What is the smallest result that can be obtained from the following process? Choose three different numbers from the set $\{3,5,7,11,13,17\}$. Add two of these numbers. Multiply their sum by the third number. Find the smallest possible result. | 36 |
Six different digits from the set $\{ 1,2,3,4,5,6,7,8,9\}$ are placed in the squares in the figure shown so that the sum of the entries in the vertical column is $23$ and the sum of the entries in the horizontal row is $12$. What is the sum of the six digits used? | 29 |
How many subsets containing three different numbers can be selected from the set $\{ 89,95,99,132,166,173 \}$ so that the sum of the three numbers is even? | 12 |
Ahn chooses a two-digit integer, subtracts it from $200$, and doubles the result. What is the largest number Ahn can get? Provide your answer as an integer. | 380 |
Walter gets up at $6:30$ a.m., catches the school bus at $7:30$ a.m., has $6$ classes that last $50$ minutes each, has $30$ minutes for lunch, and has $2$ hours additional time at school. He takes the bus home and arrives at $4:00$ p.m. How many minutes has he spent on the bus? | 60 |
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. Find the ratio of the area of the inscribed square to the area of the large square. The original answer is in \(\dfrac{k}{m}\) format, please give the v... | 14 |
A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. Find the total number of diagonals in a cube. | 16 |
If the product $\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9$, find the sum of $a$ and $b$. | 35 |
A pair of $8$-sided dice have sides numbered $1$ through $8$. Each side has the same probability (chance) of landing face up. Find the probability that the product of the two numbers that land face-up exceeds $36$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 37 |
There are positive integers that have these properties: the sum of the squares of their digits is $50$, and each digit is larger than the one to its left. Find the product of the digits of the largest integer with both properties. | 36 |
All of the even numbers from $2$ to $98$ inclusive, excluding those ending in $0,$ are multiplied together. What is the rightmost digit (the units digit) of the product? | 6 |
Calculate the value of \(100 \times 19.98 \times 1.998 \times 1000\) and express your answer in the form of \(x^2\). What is the value of \(x\)? | 1998 |
Each of the letters $W$, $X$, $Y$, and $Z$ represents a different integer in the set $\{ 1,2,3,4\}$, but not necessarily in that order. If $\dfrac{W}{X} - \dfrac{Y}{Z}=1$, then what is the sum of $W$ and $Y$? | 7 |
Harry has $3$ sisters and $5$ brothers. His sister Harriet has $S$ sisters and $B$ brothers. What is the product of $S$ and $B$? | 12 |
An Annville Junior High School, $30\%$ of the students in the Math Club are in the Science Club, and $80\%$ of the students in the Science Club are in the Math Club. There are $15$ students in the Science Club. How many students are in the Math Club? | 40 |
Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion. Rule $1$: If the integer is less than $10$, multiply it by $9$. Rule $2$: If the integer is even and greater than 9, divide it by $... | 27 |
If the pattern in the diagram continues, what fraction of the eighth triangle would be shaded? Provide your answer in the form \frac{k}{m}, where the fraction is in simplest form. Then, calculate the value of k + m. | 23 |
A rectangular board of $8$ columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered $1$ through $8$, row two is $9$ through $16$, and so on. A student shades square $1$, then skips one square and shades square $3$, skips two squares and shades square $6$, skips $3$... | 120 |
Three generous friends, each with some money, redistribute the money as follows: Amy gives enough money to Jan and Toy to double each amount they have. Jan then gives enough to Amy and Toy to double their amounts. Finally, Toy gives enough to Amy and Jan to double their amounts. If Toy had $36$ dollars at the beginning... | 252 |
To make the statement $(6\ ?\ 3) + 4 - (2 - 1) = 5$ true, replace the question mark with the correct arithmetic operation. Convert the correct arithmetic operation to its ASCII code value and provide this value as your answer. | 47 |
Three flower beds overlap as shown. Bed $A$ has $500$ plants, bed $B$ has $450$ plants, and bed $C$ has $350$ plants. Beds $A$ and $B$ share $50$ plants, while beds $A$ and $C$ share $100$. What is the total number of plants? | 1150 |
Each of the five numbers $1, 4, 7, 10$, and $13$ is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. What is the largest possible value for the horizontal or vertical sum? | 24 |
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\}$, the second from $\{A,I,O\}$, and the third from $\{D,M,N,T\}$. When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added ... | 40 |
In a far-off land, three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? Provide your answer in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers, and give the value of \(k + m\). | 11 |
Square $ABCD$ has sides of length $3$. Segments $CM$ and $CN$ divide the square's area into three equal parts. If the length of segment $CM$ is \( \sqrt{x} \), what is the value of \( x \)? | 13 |
Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then find the total area of the shaded triangles rounded to the nearest in... | 6 |
In the year $2001$, the United States will host the International Mathematical Olympiad. Let $I, M,$ and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001$. What is the largest possible value of the sum $I + M + O$? | 671 |
The Fibonacci sequence $1,1,2,3,5,8,13,21,\cdots$ starts with two $1s$, and each term afterwards is the sum of its two predecessors. Find the last of the ten digits to appear in the units position of a number in the Fibonacci sequence. | 6 |
Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, find how many of the following four quantities change: $(a)$ the length of the segment $MN$ $(b)$ the perimeter of $\triangle PAB$ $(c)$ the area of $\triangle PAB$ $(d)$ the area of ... | 1 |
If $|x - 2| = p$, where $x < 2$, then find the value of $x - p$. The original answer is in the form $a - bp$, please provide the value of $a + b$. | 4 |
In rectangle \(ABCD\), \(AD=1\), \(P\) is on \(\overline{AB}\), and \(\overline{DB}\) and \(\overline{DP}\) trisect \(\angle ADC\). The perimeter of \(\triangle BDP\) is in the form \(k+\frac{m\sqrt{n}}{3}\), where \(k\), \(m\), and \(n\) are integers. Find the value of \(k + m + n\). | 9 |
What is the minimum possible product of three different numbers from the set \(\{-8,-6,-4,0,3,5,7\}\)? The original answer is in the form of -k, please give the value of k. | 280 |
Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? Provide your answer as an integer. | 20201 |
Three dice with faces numbered $1$ through $6$ are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). Find the total number of dots NOT visible in this view. | 41 |
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order)... | 80 |
The point $P = (1,2,3)$ is reflected in the $xy$-plane, then its image $Q$ is rotated by $180^\circ$ about the $x$-axis to produce $R$, and finally, $R$ is translated by 5 units in the positive-$y$ direction to produce $S$. What are the coordinates of $S$? Provide your answer in the form $(x, y, z)$, and calculate the ... | 7 |
The sides of a triangle with positive area have lengths $4$, $6$, and $x$. The sides of a second triangle with positive area have lengths $4$, $6$, and $y$. Find the smallest positive number that is not a possible value of $|x-y|$. | 8 |
Two non-zero real numbers, $a$ and $b,$ satisfy $ab = a - b$. Calculate the value of the expression $\frac {a}{b} + \frac {b}{a} - ab$ and provide the result as an integer. | 2 |
The number $64$ has the property that it is divisible by its unit digit. Find how many whole numbers between 10 and 50 have this property. | 17 |
Let A, M, and C be nonnegative integers such that A + M + C = 12. What is the maximum value of \( A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C \)? | 112 |
A block wall $100$ feet long and $7$ feet high will be constructed using blocks that are $1$ foot high and either $2$ feet long or $1$ foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to bui... | 353 |
Segments $AD=10$, $BE=6$, $CF=24$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $RS$, not intersecting the triangle. Points $D$, $E$, $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicular segment $GH$ drawn to $RS$ from the inte... | 43 |
There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg to be placed on a triangular peg board. Find the number of ways the pegs can be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color. | 1 |
One morning each member of Angela's family drank an $8$-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family? | 5 |
Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. The original answer is in the form -\frac{k}{m}, where the fraction is in simplest terms. Please provide the value of k + m. | 10 |
Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, find the perimeter of figure $ABCDEFG$. | 15 |
A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$, the second row $18,19,\ldots,34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13,$, the... | 555 |
A circle centered at \(O\) has radius \(1\) and contains the point \(A\). The segment \(AB\) is tangent to the circle at \(A\) and \(\angle AOB = \theta\). If point \(C\) lies on \(\overline{OA}\) and \(\overline{BC}\) bisects \(\angle ABO\), find the value of \(OC\). The original answer is in the form \(\frac{m}{n+\si... | 2 |
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is \(m\) times the area of the square. The original answer is in \(\frac{k}{nm}\) fo... | 5 |
In triangle $ABC$, $AB = 13$, $BC = 14$, $AC = 15$. Let $D$ denote the midpoint of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the bisector of angle $BAC$. Find the integer closest to the area of the triangle $ADE$. | 3 |
Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=10$. Find the maximum value of $A \cdot M \cdot C + A \cdot M + M \cdot C + C \cdot A$. | 69 |
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is \(m\) times the area of the square. The original answer is in the form \(\frac{k}... | 5 |
Keiko tosses one penny and Ephraim tosses two pennies. Find the probability that Ephraim gets the same number of heads that Keiko gets. Express your answer as a fraction in simplest form. The answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\). | 11 |
Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. The original answer is in the form of $-\frac{k}{m}$, where $k$ and $m$ are integers. Please give the value of $k + m$. | 10 |
If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\overarc {AC}$ and $\overarc{BC}$, and to $\overline{AB}$. If the length of $\overarc{BC}$ is $12$, find the circumference of the circle. | 27 |
The area of rectangle $ABCD$ is $72$ units squared. If point $A$ and the midpoints of $\overline{BC}$ and $\overline{CD}$ are joined to form a triangle, what is the area of that triangle? | 27 |
The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers? Provide your answer in the form of $aS + b$, where $a$ and $b$ are constants. Please give the value of $a + b$. | 14 |
Find the product of all positive odd integers less than \(10000\). The original answer is in the format \(\frac{10000!}{2^x \cdot y!}\). Please give the value of x + y. | 10000 |
On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is $1088$ feet per second and one mile is $5280$ feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning. If the answer is $x$ miles, what is the valu... | 2 |
A telephone number has the form ${ABC-DEF-GHIJ}$, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd ... | 8 |
A charity sells $140$ benefit tickets for a total of $＄2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money, in dollars, is raised by the full-price tickets? Please provide your answer as an integer. | 782 |
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? Express your answer in the form \(\frac{k}{m}\) and find the value of \(... | 8 |
Points $A$, $B$, $C$ and $D$ have these coordinates: $A(3,2)$, $B(3,-2)$, $C(-3,-2)$ and $D(-3, 0)$. Find the area of quadrilateral $ABCD$. | 18 |
How many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$? Provide your answer as an integer. | 801 |
The parabola with equation $p(x) = ax^2+bx+c$ and vertex $(h,k)$ is reflected about the line $y=k$. This results in the parabola with equation $q(x) = dx^2+ex+f$. Find the value of $a+b+c+d+e+f$. The original answer is in the form of $k*h$, please give the value of $k$. | 2 |
A charity sells $140$ benefit tickets for a total of $2001$ dollars. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. Find the amount of money, in dollars, raised by the full-price tickets. | 782 |
A street has parallel curbs $40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $15$ feet and each stripe is $50$ feet long. Find the distance, in feet, between the stripes. | 12 |
A square piece of paper, \(4\) inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. Find the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle. The original answ... | 11 |
A circle centered at \(A\) with a radius of \(1\) and a circle centered at \(B\) with a radius of \(4\) are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle can be expressed in the form \(\frac{k}{m}\) where \(\frac{k}{m... | 13 |
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter \(10\) and altitude \(12\), and the axes of the cylinder and cone coincide. Find the radius of the cylinder. The original answer is in the form \(\frac{k}{m}\), where the fraction is in simplest ... | 41 |
Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy: $$\begin{array}{rl} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{array}$$ Find the value of $a-d$. | 10 |
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $v$, $w$, $x$, $y$, and $z$. Find the value of $y + z$. $$\begin{array}{|c|c|c|}\hline v&24&w\\\hline18&x&y\\\hline25&z&21\\\hline\end{array}$$ | 46 |
In rectangle $ABCD$, points $F$ and $G$ lie on $AB$ so that $AF=FG=GB$ and $E$ is the midpoint of $\overline{DC}$. Also, $\overline{AC}$ intersects $\overline{EF}$ at $H$ and $\overline{EG}$ at $J$. The area of the rectangle $ABCD$ is $70$. Find the area of triangle $EHJ$. | 3 |
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? Provide your answer in the form \(\frac{k}{m}\) as a simplified fraction... | 8 |
In trapezoid $ABCD$, $\overline{AB}$ and $\overline{CD}$ are perpendicular to $\overline{AD}$, with $AB+CD=BC$, $AB<CD$, and $AD=7$. If the answer is $x$, what is the value of $100x$ for $AB\cdot CD$? | 1225 |
In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$. If the answer is $x^\circ$, find the value of $x$ for $\angle ACB$. | 75 |
Each half of this figure is composed of $3$ red triangles, $5$ blue triangles, and $8$ white triangles. When the upper half is folded down over the centerline, $2$ pairs of red triangles coincide, as do $3$ pairs of blue triangles. There are $2$ red-white pairs. How many white pairs coincide? Provide your answer as an ... | 5 |
Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence? Write your answer as an integer. | 4 |
There are $24$ four-digit whole numbers that use each of the four digits $2, 4, 5$ and $7$ exactly once. Only one of these four-digit numbers is a multiple of another one. Find this four-digit number. | 7425 |
The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \cdots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. Which digit is not contained in the number $M$? | 0 |
The function $f$ is given by the table $$\begin{array}{|c||c|c|c|c|c|}\hline\text{x}&1&2&3&4&5\\\hline\text{f(x)}&4&1&3&5&2\\\hline\end{array}$$ If $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \ge 0$, find the value of $u_{2002}$. | 2 |
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? Provide your answer as an integer. | 1 |
The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. Find the digit that $M$ doesn't contain. | 0 |
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