problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
A wooden block with dimensions $12 \times 10 \times 9$ is formed by gluing together unit cubes. Determine the greatest number of unit cubes that can be seen from a single point.
|
288
| 0.25 |
Given the digits 1, 2, 3, 7, 8, 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in one of the six boxes in the given addition problem, with the condition that each number must contain one digit from 1, 2, 3 and one digit from 7, 8, 9.
|
417
| 0.333333 |
It is now between 9:00 and 10:00 o'clock, and nine minutes from now, the minute hand of a watch will be exactly opposite the place where the hour hand was six minutes ago. Determine the exact time now.
|
9:06
| 0.083333 |
Chantal starts hiking from a trailhead to a fire tower 6 miles away, walking initially at 5 miles per hour on a flat portion for 3 miles. She then hikes uphill at 3 miles per hour for 3 miles, and descends at 4 miles per hour. If Chantal meets Jean exactly at the 3-mile point as Chantal descends, calculate Jean's average speed in miles per hour.
|
\frac{60}{47}
| 0.083333 |
Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and both adjacent vertices and the ends of each diagonal must have different colors. Calculate the total number of different colorings possible.
|
5040
| 0.083333 |
Let $n$ be the number of pairs of values of $a$ and $d$ such that the equations $ax+3y+d=0$ and $4x-ay+8=0$ represent the same line. Find $n$.
|
0
| 0.916667 |
Starting with the display "1," calculate the fewest number of keystrokes required to reach "300" by only allowing up to three consecutive [+1] operations before a [x2] must be pressed.
|
11
| 0.25 |
How many unique pairs of parallel edges does a regular octahedron have?
|
6
| 0.166667 |
What is the largest number of solid $2\text{-in} \times 3\text{-in} \times 1\text{-in}$ blocks that can fit in a $4\text{-in} \times 3\text{-in}\times 3\text{-in}$ box?
|
6
| 0.916667 |
The number $2024$ needs to be expressed in the form $\frac{a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$, where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_m + b_n$ is as small as possible. Find the value of $|a_m - b_n|$.
|
1
| 0.166667 |
A $4 \times 4$ square is partitioned into $16$ smaller squares. Each square is painted either red or blue randomly and independently with each color being equally likely. The square is rotated $90^\circ$ counterclockwise. After the rotation, any red square that has moved into a position that was occupied by a blue square is repainted blue. What is the probability that the entire grid is blue after this operation?
A) $\frac{1}{256}$
B) $\frac{1}{4096}$
C) $\frac{1}{65536}$
D) $\frac{1}{16384}$
E) $\frac{1}{1024}$
|
\frac{1}{65536}
| 0.833333 |
The product of the two 99-digit numbers 707,070,707,...,070,707 and 909,090,909,...,090,909 has a thousands digit A and a units digit B. Calculate the sum of A and B.
|
5
| 0.083333 |
Let $A,B,C$ and $D$ be four different digits selected from the set $\{1,2,3,4,5,6,7,8 \}$. Find the minimum value of the expression $\dfrac{A}{B} + \dfrac{C}{D}$.
|
\frac{11}{28}
| 0.25 |
Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 1004\}$. What is the probability that $abc + a + b + c$ is divisible by $4$?
A) $\frac{9}{64}$
B) $\frac{1}{8}$
C) $\frac{1}{64}$
D) $\frac{1}{16}$
E) $\frac{1}{4}$
|
\frac{9}{64}
| 0.5 |
If $(a + \frac{1}{a})^2 = 5$, find the value of $a^3 + \frac{1}{a^3}$.
|
2\sqrt{5}
| 0.333333 |
Given numbers $5, 6, 7, 8, 9, 10, 11, 12, 13$ are written in a $3\times3$ array, with the condition that two consecutive numbers must share an edge. If the sum of the numbers in the four corners is $32$, calculate the number in the center of the array.
|
13
| 0.166667 |
Given $\log 216$, evaluate this expression.
|
3(\log 2 + \log 3)
| 0.166667 |
Ridley designs nonstandard checkerboards with 32 squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. Calculate the total number of black squares on such a checkerboard.
|
512
| 0.916667 |
A square in the coordinate plane has vertices at $(0, 0), (20, 0), (20, 20),$ and $(0, 20)$. Find the radius $d$ such that the probability the point is within $d$ units of a lattice point is $\tfrac{3}{4}$, and determine $d$ to the nearest tenth.
|
0.5
| 0.166667 |
Let $C$ and $D$ be consecutive positive integers with $C < D$, and $C$, $D$, and $C+D$ represent number bases in the equation $231_C + 56_D = 105_{C+D}$. Determine the value of $C+D$.
|
7
| 0.416667 |
Given a two-digit positive integer is called $\emph{entangled}$ if it is equal to twice the sum of its nonzero tens digit and the cube of its units digit, how many two-digit positive integers are entangled?
|
0
| 0.666667 |
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$—a point halfway along segment $AB$—and $AC$ in $R$—a point halfway along segment $AC$, and touches the circle at $Q$. Given that $AB=24$, calculate the perimeter of $\triangle APR$.
|
48
| 0.25 |
In their base $10$ representations, the integer $a$ consists of a sequence of $1000$ sixes and the integer $b$ consists of a sequence of $1000$ sevens. What is the sum of the digits of the base $10$ representation of $9ab$?
A) 18876
B) 19986
C) 20000
D) 21000
|
19986
| 0.916667 |
Determine the value of \(\text{rem} \left(\frac{5}{7}, -\frac{3}{4}\right)\).
|
-\frac{1}{28}
| 0.083333 |
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{28}\cdot 5^3}$ as a decimal?
|
28
| 0.416667 |
For how many integers $n$ is $(n+2i)^5$ a real number?
|
0
| 0.25 |
Given a $3 \times 3$ square is placed inside a larger diagram, with circles centered at each corner with radius equal to the distance from the corner to the center of the square, construct a new square $EFGH$ such that each of its sides is exactly tangent to one of these circles, with sides parallel to the original square. Determine the area of square $EFGH$.
|
27 + 18\sqrt{2}
| 0.166667 |
The sum of the first $2k + 3$ terms of the arithmetic series with first term $k^2 + k + 1$ is computed.
|
2k^3 + 7k^2 + 10k + 6
| 0.333333 |
A rectangle $ABCD$ has its diagonal $DB$ divided into three segments by two lines $L$ and $L'$, which are parallel to each other and perpendicular to $DB$. Line $L$ passes through $A$, and $L'$ passes through $C$. The segments along $DB$ created by $L$ and $L'$ have lengths $DE = 2$, $EF = 1$, and $FB = 1$. Find the area of rectangle $ABCD$, rounded to one decimal place.
A) $6.8$
B) $6.9$
C) $7.0$
D) $7.1$
|
6.9
| 0.583333 |
Given a triangle with vertices at $(0,0)$, $(2,2)$, and $(8m,0)$, find the sum of all possible values of $m$ such that a line defined by $y = mx$ divides the triangle into two triangles of equal area.
|
-\frac{1}{4}
| 0.166667 |
Jenny and Jack run on a circular track. Jenny runs counterclockwise and completes a lap every 75 seconds, while Jack runs clockwise and completes a lap every 70 seconds. They start at the same place and at the same time. Between 15 minutes and 16 minutes from the start, a photographer standing outside the track takes a picture that shows one-third of the track, centered on the starting line. What is the probability that both Jenny and Jack are in the picture?
A) $\frac{23}{60}$
B) $\frac{12}{60}$
C) $\frac{13}{60}$
D) $\frac{46}{60}$
E) $\frac{120}{60}$
|
\frac{23}{60}
| 0.5 |
Given real numbers $x$ and $y$, find the least possible value of $(x^2y^2-1)^2 + (x^2+y^2)^2$.
|
1
| 0.833333 |
Using only pennies, nickels, dimes, quarters, and half-dollars, determine the smallest number of coins needed to pay any amount of money less than a dollar and a half.
|
10
| 0.25 |
Given that in triangle $\triangle ABC$, point D is on line segment AC such that $\angle ABD$ measures $70^\circ$ and the total sum of angles around point B is $200^\circ$, and $\angle CBD = 60^\circ$, find the measure of $\angle ABC$.
|
70
| 0.083333 |
Given that vertex $E$ of right $\triangle ABE$ (with $\angle ABE = 90^\circ$) is inside square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and hypotenuse $AE$, find the area of $\triangle ABF$ where the length of $AB$ is $\sqrt{2}$.
|
\frac{1}{2}
| 0.75 |
A school club buys 1200 candy bars at a price of four for $3 dollars, and sells all the candy bars at a price of three for $2 dollars, or five for $3 dollars if more than 50 are bought at once. Calculate their total profit in dollars.
|
-100
| 0.25 |
Let there be $k$ red balls and $N$ green balls arranged in a line, where $k$ is a positive integer and $N$ is a positive multiple of 10. Determine the least value of $N$ such that the probability that at least $\frac{3}{5}$ of the green balls are on the same side of all the red balls combined is less than $\frac{8}{10}$ when there are $2$ red balls, and calculate the sum of its digits.
|
1
| 0.083333 |
Given $12345 \times 5 + 23451 \times 4 + 34512 \times 3 + 45123 \times 2 + 51234 \times 1$, evaluate the expression.
|
400545
| 0.333333 |
In a round-robin tournament with 8 teams, where each team plays one game against each other team, resulting in a win or loss, determine the maximum number of teams that could be tied for the most wins at the end of the tournament.
|
7
| 0.333333 |
A clock is set correctly at 1 P.M. on March 15 and subsequently loses exactly 3 minutes per day. By what amount of time, in minutes, must the clock be adjusted on the morning of March 22 at 9 A.M. to reflect the correct time?
A) $17$ minutes
B) $18$ minutes
C) $19$ minutes
D) $20$ minutes
E) $21$ minutes
|
21
| 0.333333 |
Given the quadratic equation \( ax^2 + bx + c \) and the table of values \( 6300, 6481, 6664, 6851, 7040, 7231, 7424, 7619, 7816 \) for a sequence of equally spaced increasing values of \( x \), determine the function value that does not belong to the table.
|
6851
| 0.25 |
Let $d(m)$ denote the number of positive integers that divide $m$, including $1$ and $m$. Define the function $g(m)=\frac{d(m)}{\sqrt[4]{m}}$. Find the unique positive integer $M$ such that $g(M) < g(m)$ for all positive integers $m \ne M$, and calculate the product of the digits of $M$.
|
2
| 0.416667 |
A circular disc with a diameter of $2D$ is placed on a $6 \times 6$ checkerboard, aiming for the centers to coincide. Calculate the number of checkerboard squares which are completely covered by the disc.
|
16
| 0.083333 |
In their base $10$ representations, the integer $a$ consists of a sequence of $2000$ eights and the integer $b$ consists of a sequence of $2000$ fives. What is the sum of the digits of the base $10$ representation of $9ab$?
A) 17765
B) 17965
C) 18005
D) 18100
E) 18200
|
18005
| 0.916667 |
Define a positive real number as "modified special" if it has a decimal representation consisting entirely of digits $0$ and $3$. Find the smallest positive integer $n$ such that $1$ can be written as a sum of $n$ modified special numbers.
|
3
| 0.25 |
Determine the units digit of the decimal expansion of (17 + √224)^21 + (17 + √224)^85.
|
8
| 0.166667 |
Let $ABCD$ be a square with side length 3. Points $E$ and $F$ are on sides $AB$ and $AD$ respectively such that $AE=2AF$. Determine the maximum area of quadrilateral $CDFE$.
A) $\frac{1}{2}$
B) $\frac{3}{2}$
C) $\frac{5}{2}$
D) $\frac{7}{2}$
E) $\frac{9}{2}$
|
\frac{9}{2}
| 0.166667 |
In a badminton tournament, $2n$ women and $3n$ men participate, and each player competes exactly once against every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is $\frac{4}{3}$, find the value of $n$.
|
3
| 0.666667 |
Let $ABCD$ be a regular tetrahedron, and let $E$ be a point lying exactly in the middle of edge $AB$. Denote $s$ as the sum of the distances from $E$ to the planes $DAC$, $DBC$, and the plane containing the face $BCD$, and $S$ as the sum of the distances from $E$ to the remaining edges $AC$, $BC$, $CD$, $DA$, and $DB$. Calculate $\frac{s}{S}$.
A) $\frac{9}{10}$
B) $\frac{3}{2}$
C) $1$
D) $\frac{1}{3}$
E) $\frac{\sqrt{2}}{2}$
|
\frac{9}{10}
| 0.083333 |
The number $395$ is expressed in the form $395 = \frac{a_1!a_2!}{b_1!b_2!b_3!}$, where $a_1 \ge a_2$ and $b_1 \ge b_2 \ge b_3$ are positive integers, and $a_1 + b_1$ is as small as possible. Calculate the value of $|a_1 - b_1|$.
|
1
| 0.25 |
Let $b_1, b_2, \dots, b_{2018}$ be a strictly increasing sequence of positive integers such that
\[ b_1 + b_2 + \cdots + b_{2018} = 2018^3.\]
Determine the remnants of $b_1^3 + b_2^3 +\cdots + b_{2018}^3$ and $b_1^4 + b_2^4 + \cdots + b_{2018}^4$ when divided by 5.
A) Cube remainder 3, Fourth Power remainder 3
B) Cube remainder 3, Fourth Power remainder 1
C) Cube remainder 1, Fourth Power remainder 3
D) Cube remainder 2, Fourth Power remainder 1
|
B) Cube remainder 3, Fourth Power remainder 1
| 0.083333 |
In a modified game, each of $5$ players rolls a standard $6$-sided die. The winner is identified as the player who rolls the highest number. In case of a tie for the highest roll, the tied players roll again, repeating this process until a single player wins. Hugo is among these players. What is the probability that Hugo's first roll was a $6$, given that he won the game?
A) $\frac{1024}{7776}$
B) $\frac{3125}{7776}$
C) $\frac{2000}{7776}$
D) $\frac{5000}{7776}$
E) $\frac{625}{7776}$
|
\frac{3125}{7776}
| 0.5 |
Consider an expanded diamond-shaped grid where you need to spell the word "DIAMOND" by moving rightward, downward, or diagonally downward to the right between adjacent letters in the given diagram. For how many paths, consisting of a sequence of horizontal, vertical, or diagonal line segments, with each segment connecting a pair of adjacent letters, is the word DIAMOND spelled out as the path is traversed from beginning to end?
|
64
| 0.083333 |
Let $S=\{(x,y) : x\in \{0,1,2,3,4,5\}, y\in \{0,1,2,3,4,5,6\}\}$. Find the product of the values of $\tan(\angle{CBA})$ for all right triangles $t=\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$.
|
1
| 0.75 |
In $\triangle DEF$, $\angle E = 90^\circ$ and $DE = 10$. An equilateral triangle $DEF_{1}$ and a regular pentagon $EF_{2}F_{3}F_{4}F_{5}$ are constructed on the sides $DE$ and $EF$ outside the triangle, respectively. The points $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$ and $F_{5}$ lie on a circle. Compute the perimeter of triangle $DEF$.
|
30
| 0.5 |
My friend reads four times as fast as I do. If it takes me 3 hours to read a novel, and immediately after reading, I take another hour to write a summary, how many minutes will it take my friend to read the same novel and write a summary assuming writing speed is the same for both of us?
|
105 \text{ min}
| 0.916667 |
Terrell usually lifts two 25-pound weights 10 times. If he decides to lift one 20-pound weight instead, how many times must Terrell lift it in order to lift the same total weight?
|
25
| 0.75 |
The cost of 1 piece of gum is 2 cents. What is the cost of 500 pieces of gum, in cents and in dollars?
|
10.00
| 0.166667 |
If \( g(x) = \frac{6x + 2}{x - 2} \), find the value of \( g(8) \).
|
\frac{25}{3}
| 0.916667 |
I have 7 books, among which three are identical copies of a physics book, and the remaining books are all different from each other. In how many ways can I arrange these books on a shelf?
|
840
| 0.583333 |
A 16-slice pizza is made with only pepperoni and mushroom toppings, and every slice has at least one topping. Only nine slices have pepperoni, and exactly twelve slices have mushrooms. How many slices have only mushrooms?
|
7
| 0.916667 |
Find $n$ such that $(n+1)(n!) = 5040$.
|
6
| 0.916667 |
The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?
|
44
| 0.916667 |
Parrots absorb 40% of the seeds they consume. Today, a parrot absorbed 8 ounces of seeds. What were the total ounces of seeds consumed by the parrot today? Additionally, how many ounces would be twice the total amount of seeds consumed?
|
40
| 0.666667 |
What is the remainder when $1632 \cdot 2024$ is divided by $400$?
|
368
| 0.916667 |
A student needs to provide his waist size in centimeters for a customized lab coat, based on the measurements in inches. If there are $10$ inches in a foot and $25$ centimeters in a foot, then what size should the student specify in centimeters if his waist size is $40$ inches?
|
100
| 0.166667 |
The sum of two positive integers is 40 and their difference is 8. What is the product of the two integers and the value of the positive difference of the squares of these integers?
|
320
| 0.25 |
Given a modified operation $\nabla$ such that for $a>0$ and $b>0$, $$a \nabla b = \frac{a^2 + b^2}{1 + a^2b^2},$$ calculate $3 \nabla 4$.
|
\frac{25}{145}
| 0.083333 |
Compute: $(23+15)^2-(23-15)^2$.
|
1380
| 0.916667 |
The sides of a triangle have lengths of $8$, $15$, and $17$. Find the length of the shortest altitude.
|
\frac{120}{17}
| 0.916667 |
To calculate $49^2$, Sam mentally computes the value of $50^2$ and subtracts some quantity. Additionally, Sam adds 1. What number does Sam subtract before adding 1 to compute $49^2$ correctly?
|
100
| 0.333333 |
What is the least positive multiple of 25 that is greater than 475?
|
500
| 0.916667 |
In triangle $ABC$, $\angle C = 90^\circ$, $AC = 4$, and $AB = \sqrt{41}$. What is $\tan B$?
|
\frac{4}{5}
| 0.833333 |
What is the least integer value of \( x \) such that \( |3x + 10| \leq 25 \)?
|
-11
| 0.833333 |
What is the least positive integer that can be added to 525 to yield a multiple of 5?
|
5
| 0.25 |
Liam rolls a fair icosahedral die with numbers $1,2,3,...,20$ on its faces. What is the expected number of digits in the number Liam obtains? Express your answer as a decimal.
|
1.55
| 0.916667 |
Bertha has 5 daughters and no sons. Some of her daughters have 5 daughters, and the rest have none. Bertha has a total of 25 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters?
|
21
| 0.833333 |
In a rectangular coordinate system, what is the number of units in the distance from the origin to the point $(20, 21)$?
|
29
| 0.5 |
A sphere is inscribed in a cube. The edge of the cube measures 8 feet. Find the volume of the sphere in cubic feet, and express your answer in terms of \(\pi\).
|
\frac{256}{3}\pi
| 0.666667 |
In right triangle $ABC$, it is given that $\sin A = \frac{3}{5}$ and $\sin B = 1$. Find $\sin C$.
|
\frac{4}{5}
| 0.916667 |
Let $t(x) = \sqrt{4x+4}$ and $f(x) = 6 + t(x)$. What is $t(f(3))$?
|
\sqrt{44}
| 0.083333 |
On a high-performance racing bicycle, the front wheel has a diameter of 28 inches, and the back wheel has a diameter of 20 inches. If the bicycle does not slip, how many revolutions will the back wheel make while the front wheel makes 50 revolutions?
|
70
| 0.916667 |
If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$, what is the angle between the vectors $-2\mathbf{a}$ and $\mathbf{b}$?
|
120^\circ
| 0.833333 |
What is the value of $c$ if the lines with equations $y = \frac{5}{2}x + 5$ and $y = (3c)x + 3$ are parallel?
|
\frac{5}{6}
| 0.916667 |
Points $A$, $B$, $C$, and $T$ are in space such that each of $\overline{TA}$, $\overline{TB}$, and $\overline{TC}$ is perpendicular to the other two. If $TA = 8$, $TB = 15$, and $TC = 12$, what is the volume of pyramid $TABC$?
|
240
| 0.916667 |
Sarah constructs a conical sand sculpture for a beach competition. The sculpture has a height of 15 inches and a circular base with a radius of 8 inches. Sarah needs to find the smallest cube-shaped box to transport her sculpture upright without tilting. What is the volume of this cube-shaped box, in cubic inches?
|
4096
| 0.75 |
Evaluate the expression $7^3 + 3(7^2)(2) + 3(7)(2^2) + 2^3$.
|
729
| 0.916667 |
Define a new operation $a \text{ Z } b = a^3 - 3a^2b + 3ab^2 - b^3$. What is the value of $4 \text{ Z } 3$?
|
1
| 0.916667 |
Four concentric circles are drawn with radii of 2, 4, 6, and 8. The innermost circle is painted black, the next ring is painted white, the ring after that is black, and the outermost ring is white. What is the ratio of the black area to the white area? Express your answer as a common fraction.
|
\frac{3}{5}
| 0.916667 |
How many five-digit positive integers are there with thousands digit $3$ and ten-thousands digit $1$?
|
1000
| 0.75 |
$\textbf{Elisa's Stamp Album}$
Elisa sorts the stamps in her collection by country and decade of issuance. She paid the following prices: China and Japan, $7$ cents each, Italy $5$ cents each, and Germany $6$ cents each. (China and Japan are Asian countries; Italy and Germany are European countries.) The numbers of stamps per decade and country are given in the table below:
\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{Country} & \text{'50s} & \text{'60s} & \text{'70s} & \text{'90s} \\
\hline
China & 5 & 9 & 11 & 8 \\
Japan & 10 & 6 & 9 & 12 \\
Italy & 7 & 8 & 5 & 11 \\
Germany & 8 & 7 & 10 & 9 \\
\hline
\end{array}
\]
Calculate the total cost of her Asian stamps from before the '80s and the total number of European stamps from the '90s.
|
20
| 0.083333 |
A regular polygon has a perimeter of 180 cm and each side has a length of 15 cm. How many sides does this polygon have and what is its area if the apothem is 12 cm?
|
1080 \text{ cm}^2
| 0.916667 |
Makoto drove from his office to a client's location and back. The odometer read 82,435 when he left his office, and it read 82,475 when he returned. Makoto's car consumes fuel at a rate of 25 miles per gallon and the price of one gallon of gas is \$3.75. What was the cost of the gas used for Makoto's round trip? (Express your answer in dollars and round to the nearest cent.)
|
\$6.00
| 0.583333 |
Determine the area of the region bounded by the equation $x^2 + y^2 + 6x - 8y - 5 = 0$.
|
30\pi
| 0.916667 |
Chris, David, Eva, and Fiona weigh themselves in pairs. Together, Chris and David weigh 330 pounds, David and Eva weigh 290 pounds, and Eva and Fiona weigh 310 pounds. How many pounds do Chris and Fiona weigh together?
|
350
| 0.833333 |
Let $p$ and $q$ be angles such that $\cos p + \cos q = \frac{1}{3}$ and $\sin p + \sin q = \frac{5}{13}.$ Find
\[\tan \left( \frac{p + q}{2} \right).\]
|
\frac{15}{13}
| 0.916667 |
When a polynomial is divided by \(3x^3 - 4x^2 + 5x - 6\), what are the possible degrees of the remainder? Enter all the possible values, separated by commas.
|
0, 1, 2
| 0.75 |
Find the units digit of the following within the indicated number base: $63_9 + 74_9$
|
7
| 0.916667 |
A number is chosen at random from the set of consecutive natural numbers $\{1, 2, 3, \ldots, 120\}$. What is the probability that the number chosen is a factor of $5!$? Express your answer as a common fraction.
|
\frac{2}{15}
| 0.916667 |
Find the distance between the points (0,12) and (9,0).
|
15
| 0.083333 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.