problem
stringlengths 18
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|---|---|---|
Given the fraction $\frac{987654321}{2^{30}\cdot 5^5}$, calculate the minimum number of digits to the right of the decimal point needed to express the fraction as a decimal.
|
30
| 0.333333 |
A store owner bought 2000 markers at $0.20 each. To make a minimum profit of $200, if he sells the markers for $0.50 each, calculate the number of markers he must sell at least to achieve or exceed this profit.
|
1200
| 0.916667 |
Given that $q_1 (x)$ and $r_1$ are the quotient and the remainder, respectively, when the polynomial $x^9$ is divided by $x - 1$, and if $q_2 (x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x - 1$, find $r_2$.
|
9
| 0.416667 |
Positive integers $a$ and $b$ are each less than $8$. Find the smallest possible value for $3a-2ab$.
|
-77
| 0.416667 |
Given the fractions \(\frac{5}{7}\) and \(\frac{3}{4}\), find the value of \(\text{rem} \left(\frac{5}{7}, \frac{3}{4}\right)\).
|
\frac{5}{7}
| 0.666667 |
The area of the ring between two concentric circles is $50\pi$ square centimeters. Find the length of a chord of the larger circle which is tangent to the smaller circle.
|
10\sqrt{2}
| 0.916667 |
Given that each of $9$ standard dice, labeled from $1$ to $6$, are rolled, determine the sum that has the same probability as the sum of $15$ when all dice show their top faces.
|
48
| 0.583333 |
How many integer values of $x$ satisfy $|x| < 4.5\pi$?
|
29
| 0.916667 |
A garden pond contains 300 gallons of water. Each day, 1 gallon of water evaporates, and on every third day, it rains adding 2 gallons of water back into the pond. Determine the amount of water in the pond after 45 days.
|
285
| 0.916667 |
Soda is now sold in packs of 8, 18, and 30 cans. Find the minimum number of packs needed to buy exactly 144 cans of soda.
|
6
| 0.583333 |
The three-digit number $3a7$ is added to the number $414$ to give the three-digit number $7c1$. If $7c1$ is divisible by 11, then calculate the value of $a+c$.
|
14
| 0.75 |
A square diagram shows a uniform $11 \times 11$ grid of points including the points on the edges. Point $P$ is located at the center of the square. If point $Q$ is randomly picked from the remaining $120$ points, what is the probability that the line $PQ$ forms a line of symmetry for the square?
|
\frac{1}{3}
| 0.333333 |
Calculate the value of the expression: \[ \frac{2 \times 6}{12 \times 14} \times \frac{3 \times 12 \times 14}{2 \times 6 \times 3} \times 2 \]
|
2
| 0.666667 |
Given a circular park 20 feet in diameter with a straight walking path 5 feet wide passing through its center, calculate the area of the park covered by grass after the path is laid.
|
100\pi - 100
| 0.666667 |
Given positive integers \(a\) and \(b\) are each less than 10, find the smallest possible value for \(2 \cdot a - a \cdot b\).
|
-63
| 0.333333 |
Given points $P(-1, -3)$ and $Q(5, 3)$ are in the $xy$-plane, and point $R(2, n)$ lies on the line $y = 2x - 4$, find the value of $n$ such that the total distance $PR + RQ$ is minimized.
|
0
| 0.75 |
One third of Marcy's marbles are blue, one third are red, and ten of them are green. What is the smallest number of yellow marbles that Marcy could have?
|
0
| 0.833333 |
Three vertices of parallelogram $ABCD$ are $A(-1, 3), B(2, -1), D(7, 6)$ with $A$ and $D$ diagonally opposite. Calculate the product of the coordinates of vertex $C$.
|
40
| 0.083333 |
Before the district play, the Zebras had won $40\%$ of their soccer games. During district play, they won eight more games and lost three, to finish the season having won $55\%$ of their games. Calculate the total number of games the Zebras played.
|
24
| 0.833333 |
Given the product of the digits of a 3-digit positive integer is 30, find how many such integers exist.
|
12
| 0.25 |
In a math contest, 65 students are wearing blue shirts, and 67 students are wearing yellow shirts, making a total of 132 contestants. These students are grouped into 66 pairs. If in 28 of these pairs, both students are wearing blue shirts, determine the number of pairs that consist of students both wearing yellow shirts.
|
29
| 0.5 |
A half-sector of a circle of radius $6$ inches together with its interior is rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. Find the volume of the cone in cubic inches.
|
9\pi \sqrt{3}
| 0.833333 |
If a worker receives a 30% cut in wages, calculate the percentage of the raise he must obtain in order to regain his original pay.
|
42.86\%
| 0.583333 |
Given that $\sqrt{18}$ and $\sqrt{120}$ are the lower and upper limits, respectively, calculate the number of whole numbers between them.
|
6
| 0.833333 |
If $x$ and $y$ are non-zero numbers such that $x = 2 + \frac{1}{y}$ and $y = 2 + \frac{1}{x}$, find the value of $y$.
|
1 - \sqrt{2}
| 0.083333 |
Given an urn contains only coins and beads, and 30% of the objects in the urn are beads. Also, 30% of the coins in the urn are silver, find the percentage of the objects in the urn that are gold coins.
|
49\%
| 0.916667 |
The two spinners shown are spun once. Spinner 1 has sectors numbered 1, 4, 5, and Spinner 2 has sectors numbered 2, 3, 6. Determine the probability that the sum of the numbers in the two sectors falls between 6 and 10, inclusive.
|
\frac{2}{3}
| 0.583333 |
In the Metropolitan County, there are $25$ cities. From a given bar chart, the average population per city is indicated midway between $5,200$ and $5,800$. If two of these cities, due to a recent demographic survey, were found to exceed the average by double, calculate the closest total population of all these cities.
|
148,500
| 0.166667 |
Determine the ratio of the shorter side to the longer side of the rectangular park, given that by taking a shortcut along the diagonal, a boy saved a distance equal to $\frac{1}{3}$ of the longer side of the park.
|
\frac{5}{12}
| 0.666667 |
If \(16^{x+2} = 496 + 16^x\), solve for the value of \(x\).
|
\log_{16}(\frac{496}{255})
| 0.083333 |
Given that Paul owes Paula $45$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, $20$-cent coins, and $25$-cent coins, find the difference between the largest and smallest number of coins he can use to pay her.
|
7
| 0.5 |
Given that a triangle with integral sides is isosceles and has a perimeter of 12, find the area of the triangle.
|
4\sqrt{3}
| 0.416667 |
Given $a$, $b$, and $a-b\cdot\frac{1}{a}$ are non-zero values, evaluate $\frac{a^2-\frac{1}{b^2}}{b^2-\frac{1}{a^2}}$.
|
\frac{a^2}{b^2}
| 0.583333 |
Given a \(5 \times 5\) block of calendar dates, the numbers are reversed in the second, third, and fourth rows. Then, find the positive difference between the two diagonal sums.
|
0
| 0.416667 |
Given that real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$, calculate the probability that $\lfloor\log_3x\rfloor=\lfloor\log_3y\rfloor$.
|
\frac{1}{2}
| 0.5 |
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. Find the probability that $\lfloor\log_3x\rfloor = \lfloor\log_3y\rfloor$.
|
\frac{1}{2}
| 0.666667 |
Given the condition that a positive integer less than $1000$ is $7$ times the sum of its digits, determine the number of such integers.
|
4
| 0.583333 |
Given that Marla has a large white cube with an edge of 12 feet and enough green paint to cover 600 square feet, find the total area of the cube's faces that remain white after painting complete faces green and leaving exactly two adjacent faces untouched.
|
288
| 0.75 |
What is the minimum possible product of three different numbers from the set $\{-10, -7, -5, -3, 0, 2, 4, 6, 8\}$?
|
-480
| 0.166667 |
Given that Elmer's new car gives 60% better fuel efficiency, measured in kilometers per liter, than his old car, and the diesel fuel for his new car is 25% more expensive per liter than the gasoline for his old car, calculate the percentage by which Elmer will save money using his new car instead of his old car for a 100-kilometer trip.
|
21.875\%
| 0.75 |
Six semicircles are evenly arranged along the inside of a regular hexagon with a side length of 3 units. A circle is positioned in the center such that it is tangent to each of these semicircles. Find the radius of this central circle.
|
\frac{3 (\sqrt{3} - 1)}{2}
| 0.083333 |
Two pitchers, one with a capacity of 800 mL and the other 700 mL, are initially filled with apple juice: the first pitcher is 1/4 full and the second pitcher is 3/7 full. Water is then added to each pitcher to reach its full capacity, and both pitchers are then poured into a larger container. What fraction of the liquid in the larger container is apple juice?
|
\frac{1}{3}
| 0.916667 |
Calculate the value of $\text{rem} \left(\frac{5}{7}, \frac{3}{4}\right)$ and then multiply the result by $-2$.
|
-\frac{10}{7}
| 0.75 |
What is the difference between the sum of the first 1000 even counting numbers including 0, and the sum of the first 1000 odd counting numbers?
|
-1000
| 0.583333 |
A picture 5 feet across is hung on a wall that is 25 feet wide with a 2-foot offset from being centered toward one end of the wall. Determine the distance from the end of the wall to the nearest edge of the picture.
|
8
| 0.5 |
A point $P(a, b)$ in the $xy$-plane is first rotated counterclockwise by $180^\circ$ around the point $(2, 3)$ and then reflected about the line $y = x$. The final position of $P$ after these transformations is $(5, -4)$. Determine the value of $a - b$.
|
7
| 0.916667 |
Given that $\angle AOE = 40^\circ$, $\angle FOB = 60^\circ$, and points $F$ and $E$ are on the same side of diameter $\overline{AB}$, calculate the ratio of the area of the smaller sector $EOF$ to the area of the whole circle.
|
\frac{2}{9}
| 0.166667 |
A jar contains 6 different colors of gumdrops: 25% blue, 15% brown, 15% red, 10% yellow, 20% orange, and the rest are green. If there are 36 green gumdrops and one-third of the blue gumdrops are replaced with brown gumdrops, calculate the number of gumdrops that will be brown.
|
56
| 0.833333 |
Sara had walked one-third of the way from home to the library in 9 minutes, and she ran 5 times as fast as she walked. Determine the total time it took Sara to get from home to the library.
|
12.6
| 0.916667 |
Consider all triangles $ABC$ where $AB = AC$, $D$ is a point on $\overline{AC}$ such that $\overline{BD} \perp \overline{AC}$. Assume $AC$ and $CD$ are odd integers, and $BD^{2} = 65$. Find the smallest possible value of $AC$.
|
9
| 0.166667 |
Determine the measure of the angle $\angle BAC$ in $\triangle ABC$, which is scalene with $AB = AC$, given that point $P$ is on $BC$ such that $AP = PC$ and $BP = 2PC$.
|
120^\circ
| 0.5 |
A 3x3 magic square with the same row sums and column sums is given by:
\[
\left[
\begin{matrix}
2 & 7 & 6\\
9 & 5 & 1\\
4 & 3 & 8
\end{matrix}
\right]
\]
Compute the minimum number of entries that must be changed to make all three row sums and three column sums different from one another.
|
4
| 0.083333 |
Chloe chooses a real number uniformly at random from the interval $[0, 3000]$ and Laurent chooses a real number uniformly at random from the interval $[0, 4000]$. Determine the probability that Laurent's number is greater than Chloe's number.
|
\frac{5}{8}
| 0.083333 |
Let \(r\) be the number that results when both the base and the exponent of \(a^b\) are quadrupled, and \(r\) equals the product of \(a^b\) and \(x^{2b}\). Find the value of \(x\).
|
16 a^{3/2}
| 0.083333 |
Casper has exactly enough money to buy either 10 pieces of red candy, 18 pieces of green candy, 20 pieces of blue candy, or n pieces of purple candy, where a piece of purple candy costs 24 cents. What is the smallest possible value of n?
|
15
| 0.166667 |
A rectangular floor is 15 feet wide and 35 feet long, covered entirely by one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and last tiles, calculate the total number of tiles the bug visits.
|
45
| 0.916667 |
Given that there are $121$ grid points in a square, including the points at the edges, and point $P$ is in the center of the square, determine the probability that the line $PQ$ is a line of symmetry for the square when a point $Q$ is randomly chosen among the other $120$ points.
|
\frac{1}{3}
| 0.25 |
Given that Casper can buy either 10 pieces of orange candy, 16 pieces of yellow candy, 18 pieces of maroon candy, or n pieces of violet candy, where the price of each piece of violet candy is 18 cents. Find the smallest possible value of n.
|
40
| 0.5 |
A point is chosen at random from within a circle of radius 4. Calculate the probability that the point is closer to the center of the circle than it is to the boundary of the circle.
|
\frac{1}{4}
| 0.916667 |
Given the expression \(150(150-4) \div (150\cdot150 \cdot 2 -4)\), calculate the result.
|
\frac{21900}{44996}
| 0.083333 |
Given that there are 128 players in a chess tournament, with the strongest 32 players receiving a bye in the first round and the remaining players competing in the initial round, determine the total number of matches played until only one champion is crowned.
|
127
| 0.666667 |
At a local shop last month, packets of seeds were sold at 6 packets for $8. This month, they are on sale at 8 packets for $6. Determine the percent decrease in the price per packet during the sale.
|
43.75\%
| 0.833333 |
Given that Crystal modifies her running course by running due north for 2 miles, then east for 3 miles, and finally southwest until she returns to her starting point, determine the length of the last portion of her run.
|
\sqrt{13}
| 0.916667 |
An equilateral triangle $ABC$ with side length $4$ has a smaller equilateral triangle $DBE$ with side length $2$ cut from it, where $D,E$ lie on sides $AB$ and $BC$ respectively. Calculate the perimeter of the remaining quadrilateral $ACED$.
|
10
| 0.916667 |
Given the expression $3^{\left(1^{\left(2^8\right)}\right)} + \left(\left(3^1\right)^2\right)^8$, evaluate its value.
|
43046724
| 0.5 |
A circle of radius 3 is centered at A. An equilateral triangle with side 4 has a vertex at A. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle.
|
9\pi - 4\sqrt{3}
| 0.416667 |
Given a bag of popcorn kernels containing $\frac{3}{4}$ white kernels and $\frac{1}{4}$ yellow kernels, with $\frac{1}{4}$ of the kernels damaged and $\frac{3}{5}$ of the white kernels and $\frac{4}{5}$ of the yellow kernels good and capable of popping, calculate the probability that a kernel selected from the bag and which pops was white.
|
\frac{9}{13}
| 0.583333 |
Calculate $1,000,000,000,000 - 888,777,888,777$.
|
111,222,111,223
| 0.916667 |
Given that the angles of a triangle that is not right are $a^{\circ}$, $b^{\circ}$, and $c^{\circ}$, where $a$, $b$, and $c$ are prime numbers and $a > b > c$, determine the least possible value of $b$.
|
5
| 0.333333 |
In a classroom, $65$ students are wearing green shirts, and another $79$ students are wearing red shirts. The $144$ students are grouped into $72$ pairs. In exactly $27$ of these pairs, both students are wearing green shirts. Determine the number of pairs that consist of both students wearing red shirts.
|
34
| 0.666667 |
What is the remainder when $5^1 + 5^2 + 5^3 + \cdots + 5^{4020}$ is divided by 10?
|
0
| 0.916667 |
Find the number of pairs (m, n) of integers such that the equation $m^3 + 3m^2 + 2m = 8n^3 + 12n^2 + 6n + 1$ holds.
|
0
| 0.916667 |
How many subsets containing three different numbers can be selected from the set $\{ 90, 94, 102, 135, 165, 174 \}$ so that the sum of the three numbers is divisible by 5?
|
2
| 0.083333 |
If 60 students wear red shirts and 90 students wear green shirts, making a total of 150 students, and the students are grouped into 75 pairs, and 28 of these pairs contain students both wearing red shirts, determine the number of pairs that have students both wearing green shirts.
|
43
| 0.333333 |
Jessica wants to walk from her apartment to her friend Anna's apartment, which is located four blocks east and three blocks north of Jessica's apartment. Jessica needs to avoid a risky construction site, which is located two blocks east and one block north of her apartment. Calculate the number of ways Jessica can walk to Anna's apartment by walking a total of seven blocks.
|
17
| 0.833333 |
The radius of Earth at the equator is approximately 4200 miles. Suppose a jet flies once around this new Earth's equator at a speed of 525 miles per hour relative to Earth. If the flight path is negligibly above the equator, find the approximate number of hours of flight.
|
50
| 0.833333 |
A number is formed by repeating a two-digit number three times. Determine the smallest prime number by which any number of this form is always exactly divisible.
|
3
| 0.083333 |
An uncrossed belt is fitted without slack around two circular pulleys with radii of $10$ inches and $6$ inches. If the distance between the points of contact of the belt with the pulleys is $30$ inches, then calculate the distance between the centers of the pulleys in inches.
|
2\sqrt{229}
| 0.083333 |
Jacob wants to tile the floor of a 14-foot by 20-foot hallway. He plans to use 2-foot by 2-foot square tiles for a border along the edges of the hallway and fill in the rest of the floor with three-foot by three-foot square tiles. Calculate the total number of tiles he will use.
|
48
| 0.333333 |
Given an ellipse with semi-major axis 2 and semi-minor axis 1, find the number of points interior to the region such that the sum of the squares of the distances from these points to the foci of the ellipse is 5.
|
0
| 0.916667 |
Given the state income tax rate is $q\%$ for the first $\$30000$ of yearly income plus $(q + 1)\%$ for any amount above $\$30000$, and Samantha's state income tax amounts to $(q + 0.5)\%$ of her total annual income, determine Samantha's annual income.
|
60000
| 0.916667 |
A half-sector of a circle with a radius of $6$ inches, together with its interior, can be rolled up to form the lateral surface area of a right circular cone by taping along the two radii. Calculate the volume of the cone in cubic inches.
|
9\pi \sqrt{3}
| 0.916667 |
Given the quadratic polynomial $ax^2 + bx + c$ and a sequence of increasing, equally spaced $x$ values: $441, 484, 529, 576, 621, 676, 729, 784$, determine which function value is incorrect.
|
621
| 0.25 |
Given the sequence [..., p, q, r, s, 5, 8, 13, 21, 34, 55, 89...], where each term is the sum of the two terms to its left, find p.
|
1
| 0.75 |
What is the smallest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the square of the product of the first $n$ positive integers?
|
100
| 0.25 |
The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$, find the area expressed as $kd^2$ for some constant $k$.
|
\frac{10}{29} d^2
| 0.083333 |
Given a two-digit positive integer $N$, determine how many have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a prime number.
|
1
| 0.166667 |
One fifth of Sarah's marbles are blue, one sixth of them are red, and one tenth of them are green. What is the smallest number of yellow marbles Sarah could have?
|
16
| 0.916667 |
A shopkeeper buys $2000$ pens at $0.15$ each. If he sells them at $0.30$ each, determine the number of pens he must sell to make a profit of exactly $150.00$.
|
1500
| 0.5 |
Five blue beads, three green beads, and one red bead are placed in line in random order. Calculate the probability that no two blue beads are immediately adjacent to each other.
|
\frac{1}{126}
| 0.833333 |
Given that it is currently between 4:00 and 5:00 o'clock, and eight minutes from now, the minute hand of a clock will be exactly opposite to the position where the hour hand was six minutes ago, determine the exact time now.
|
4:45\frac{3}{11}
| 0.25 |
When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 2001 is greater than zero and is equal to the probability of obtaining a sum of $S$. Determine the smallest possible value of $S$.
|
337
| 0.75 |
Let the bisectors of the interior angles at $B$ and $C$ of triangle $ABC$ meet at point $E$. Express the measure of angle $BEC$ in terms of angle $A$.
|
90^\circ + \frac{A}{2}
| 0.25 |
A flower arrangement consists of yellow tulips, white tulips, yellow daisies, and white daisies. Given that half of the yellow flowers are tulips, two-thirds of the white flowers are daisies, and seven-tenths of the flowers are yellow, calculate the percentage of the flowers that are daisies.
|
55\%
| 0.916667 |
Determine the difference between the two smallest integers greater than $1$ that, when divided by any integer $k$ such that $2 \le k \le 13$, leave a remainder of $1$. Then, subtract $2^3\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13$ from this difference and find the result.
|
0
| 0.583333 |
Given the expressions $(2401^{\log_7 3456})^{\frac{1}{2}}$, calculate its value.
|
3456^2
| 0.416667 |
Find the sum of all numbers of the form $3k + 2$, where $k$ takes on integral values from $0$ to $n-1$.
|
\frac{3n^2 + n}{2}
| 0.083333 |
Let \(p\), \(q\), and \(r\) be the distinct roots of the polynomial \(x^3 - 27x^2 + 98x - 72\). It is given that there exist real numbers \(A\), \(B\), and \(C\) such that
\[
\frac{1}{s^3 - 27s^2 + 98s - 72} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r}
\]
for all \(s \not\in \{p, q, r\}\). What is \(\tfrac{1}{A} + \tfrac{1}{B} + \tfrac{1}{C}\)?
A) 255
B) 256
C) 257
D) 258
|
256
| 0.166667 |
Determine the value of $A + B + C$, where $A$, $B$, and $C$ are the dimensions of a three-dimensional rectangular box with faces having areas $40$, $40$, $90$, $90$, $100$, and $100$ square units.
|
\frac{83}{3}
| 0.75 |
Given the polynomial function $f$ of degree exactly 3, determine the number of possible values of $f(x)$ that satisfy $f(x^2) = [f(x)]^2 = f(f(x))$.
|
0
| 0.166667 |
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