problem
stringlengths 18
4.46k
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stringlengths 1
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float64 0.08
0.92
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|---|---|---|
Determine the number of significant digits in the measurement of the side of a square whose computed area is $3.2400$ square inches to the nearest ten-thousandth of a square inch.
|
5
| 0.75 |
Given a triangle with a base of 24 inches, a line is drawn parallel to the base, terminating in the other two sides and dividing the triangle into two equal areas, calculate the length of this parallel line.
|
12\sqrt{2}\text{ inches}
| 0.833333 |
Given that a 4-digit positive integer has only even digits (0, 2, 4, 6, 8) and is divisible by 4, calculate the number of such integers.
|
300
| 0.75 |
When $10^{93} - 95$ is expressed as a single whole number, calculate the sum of its digits.
|
824
| 0.5 |
Let $(1-x+x^2)^n = b_0 + b_1x + b_2x^2 + \cdots + b_{2n}x^{2n}$ be an identity in $x$. Find the sum $t = b_0 + b_2 + b_4 + \cdots + b_{2n}$.
|
\frac{1 + 3^n}{2}
| 0.916667 |
The common ratio of the geometric progression is $\frac{\sqrt[6]{2}}{\sqrt[4]{2}}=\frac{\sqrt[12]{2}}{\sqrt[6]{2}}$, so a general term of the sequence is $(\sqrt[4]{2})\left(\frac{\sqrt[6]{2}}{\sqrt[4]{2}}\right)^{n-1}=(\sqrt[4]{2})\left(\sqrt[12]{2}\right)^{2n-3}=\sqrt[4]{2}\cdot 2^{(2n-3)/12}=\sqrt[4]{2}\cdot 2^{n-3/4}$, so find the fourth term
|
1
| 0.333333 |
For how many integer values of $n$ is the value of $8000 \cdot \left(\frac{3}{4}\right)^n$ an integer?
|
4
| 0.5 |
Given a square initially painted red, with each middle ninth of the square turning white every time the process is executed, calculate the fractional part of the original area of the square that stays red after four changes.
|
\frac{4096}{6561}
| 0.916667 |
A top hat contains 4 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 4 reds are drawn or until both green chips are drawn. Calculate the probability that all 4 red chips are drawn.
|
\frac{1}{3}
| 0.083333 |
Oliver has a training loop for his weekend cycling. He starts by cycling due north for 3 miles. He then cycles northeast, making a 30° angle with the north for 2 miles, followed by cycling southeast, making a 60° angle with the south for 2 miles. He completes his loop by cycling directly back to the starting point. Find the distance of this final segment of his ride.
|
\sqrt{11 + 6\sqrt{3}}
| 0.083333 |
If Jo and Blair take turns counting, starting from $1$ with each number being two more than the last number, but repeating the number every 5th turn, find the 34th number said.
|
55
| 0.083333 |
The ratio of the interior angles of two regular polygons with sides of unit length is $4:3$, and one of them has fewer than 15 sides. Determine the number of such pairs of polygons.
|
4
| 0.083333 |
A lemming is placed at a corner of a square with a side length of 8 meters. It runs 6.8 meters diagonally towards the opposite corner. After reaching this point, it makes a 90-degree right turn and runs another 2.5 meters. Calculate the average of the shortest distances from the lemming's final position to each side of the square.
|
4
| 0.833333 |
Determine the values of $p$ and $q$ for which the equation $\log p + \log q^2 = \log (p + q^2)$ holds.
|
p = \frac{q^2}{q^2 - 1}
| 0.5 |
If $x$ men working $x$ hours a day for $x$ days produce $x^2$ articles and each man's efficiency decreases by $\frac{1}{2}$ for every additional 10 men beyond 10 men in the workforce, calculate the number of articles produced by $y$ men working $y$ hours a day for $y$ days if $y = 20$.
|
400
| 0.333333 |
Determine the upcoming year in which Andrea's birthday next falls on a Tuesday, given that Andrea's birthday in 2015 fell on a Friday, December 18.
|
2018
| 0.583333 |
For how many positive integers $m$ is $\frac{2310}{m^2 - 2}$ a positive integer?
|
3
| 0.75 |
Given that the squares of the positive integers $m$ and $n$ differ by 144, determine the number of ordered pairs $(m,n)$ such that $m \ge n$.
|
4
| 0.75 |
A square with a side length of $1$ is divided into one triangle and three trapezoids by joining the center of the square to points on each side. These points divide each side into segments such that the length from a vertex to the point is $\frac{1}{4}$ and from the point to the center of the side is $\frac{3}{4}$. If each section (triangle and trapezoids) has an equal area, find the length of the longer parallel side of the trapezoids.
|
\frac{3}{4}
| 0.5 |
Given that Jo and Blair take turns counting from 1, incrementing the previous number by 2 during each turn, find the $30^{\text{th}}$ number said.
|
59
| 0.916667 |
Let $N$ be a two-digit number. We subtract the number with the digits reversed from $N$ and find that the result is a positive perfect fourth power. What is the digit $N$ can end?
|
0
| 0.75 |
Find the $1234$th digit to the right of the decimal point in $x = .123456789101112...498499500$.
|
4
| 0.5 |
A cube with side length $2$ is sliced by a plane that passes through a vertex $A$, the midpoint $M$ of an adjacent edge, and the midpoint $P$ of the face diagonal of the top face, not containing vertex $A$. Find the area of the triangle $AMP$.
|
\frac{\sqrt{5}}{2}
| 0.5 |
Find the number of distinct points in the xy-plane common to the graphs of $(x+y-7)(3x-2y+6)=0$ and $(x-y-2)(4x+y-10)=0$.
|
4
| 0.916667 |
Two circles, one with radius 2 and the other with radius 3, are to be placed as follows. The center of the circle with radius 2 is to be chosen uniformly and at random from the line segment joining (0,0) to (5,0). The center of the other circle is placed at (3,2). Determine the probability that the two circles intersect.
|
1
| 0.333333 |
Rectangle ABCD with AB = 7 and AD = 10 is joined with right triangle DCE so that DC is common to both shapes. The areas of ABCD and DCE are equal. If both ABCD and DCE form a pentagon, find the length of DE.
|
\sqrt{449}
| 0.25 |
Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files.
|
12
| 0.083333 |
Given Gilda starts with a bag of marbles, she gives $30\%$ of them to Pedro, $10\%$ of the remaining to Ebony, $15\%$ of the remaining to Jimmy, and $5\%$ of the remaining to Maria. Calculate the percentage of her original bag of marbles that Gilda has left.
|
50.87\%
| 0.416667 |
Given the first term of a geometric progression be $s$ and the common ratio be $(r+1)$, where $-1 < r < 0$, find the limit of the sum of the cubes of the terms in this progression as the number of terms approaches infinity.
|
\frac{s^3}{1 - (r+1)^3}
| 0.666667 |
Two boys start at the same point A on an elliptical track and move in opposite directions. The speeds of the boys are 7 ft. per second and 11 ft. per second, respectively. The elliptical track has a major axis of 100 ft and a minor axis of 60 ft. Calculate the number of times they meet, excluding the start and finish, assuming they start at the same time and finish when they first meet at point A again.
|
16
| 0.083333 |
The number $17!$ has a certain number of positive integer divisors. What is the probability that one of them is odd?
|
\frac{1}{16}
| 0.166667 |
A lemming sits at a corner of a rectangle with side lengths 12 meters and 8 meters. The lemming runs 7.5 meters along a diagonal toward the opposite corner. It then makes a $90^{\circ}$ right turn and runs 3 meters, followed by another $90^{\circ}$ right turn running an additional 2 meters. Calculate the average of the shortest distances from the lemming's final position to each side of the rectangle.
|
5
| 0.833333 |
Given a two-digit positive integer, determine how many integers satisfy the condition of being equal to the sum of its nonzero tens digit and the cube of its units digit.
|
0
| 0.25 |
Given an equilateral triangle $\triangle B_1B_2B_3$, for each positive integer $k$, point $B_{k+3}$ is the reflection of $B_k$ across the line $B_{k+1}B_{k+2}$, determine the measure of $\measuredangle B_{47}B_{45}B_{46}$.
|
60^{\circ}
| 0.916667 |
Find the smallest positive integer n for which (n-17)/(7n+8) is a non-zero reducible fraction.
|
144
| 0.833333 |
Given the digits $A$, $B$, $C$, and $D$, in the equations:
$$
\begin{array}{cc}
& A\ B \\
+ & C\ D \\
\hline
& A\ E \\
\end{array}
$$
and
$$
\begin{array}{cc}
& A\ B \\
- & D\ C \\
\hline
& A\ F \\
\end{array}
$$
Find the value of the digit $E$.
|
9
| 0.333333 |
Given that $\underbrace{9999\cdots 99}_{80\text{ nines}}$ is multiplied by $\underbrace{7777\cdots 77}_{80\text{ sevens}}$, calculate the sum of the digits in the resulting product.
|
720
| 0.083333 |
Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. If all three coins show heads, roll an additional fourth die. Determine the probability that the sum of the die rolls is odd.
|
\frac{7}{16}
| 0.333333 |
Starting from "1," using a calculator with only the [ +1 ] and [ x2 ] keys, find the fewest number of keystrokes needed to reach "400," ensuring the display shows "50" at least once during the sequence.
|
10
| 0.083333 |
Given that each of 24 balls is tossed independently and at random into one of the 4 bins, calculate $\frac{p'}{q'}$, where $p'$ is the probability that one bin ends up with 6 balls, and the other three bins end up with 6 balls each, and $q'$ is the probability that each bin ends up with exactly 6 balls.
|
1
| 0.083333 |
Given a $60$-question multiple-choice math exam where students are scored $5$ points for a correct answer, $0$ points for an unpicked answer, and $-2$ points for a wrong answer, determine the maximum number of questions that Maria could have answered correctly, given that her total score on the exam was $150$.
|
38
| 0.916667 |
For every $n$, the sum of n terms of an arithmetic progression is $5n + 4n^3$. Find the $r$th term of the progression.
|
12r^2 - 12r + 9
| 0.666667 |
The circle having (2,2) and (10,8) as the endpoints of a diameter intersects the x-axis at a second point. Find the x-coordinate of this point.
|
6
| 0.166667 |
With all three valves open, the tank fills in 1.2 hours, with only valves X and Z open it takes 2 hours, and with only valves Y and Z open it takes 3 hours, calculate the number of hours required with only valves X and Y open.
|
1.2
| 0.416667 |
Given that Four times Dick's age plus twice Tom's age equals three times Harry's age, and Three times the square of Harry's age is equal to twice the square of Dick's age added to four times the square of Tom's age, and their respective ages are relatively prime to each other, find the sum of the cubes of their ages.
|
349
| 0.25 |
Given that the dimensions of a box are $4\text{-in} \times 3\text{-in}\times2\text{-in}$ and the dimensions of a solid block are $3\text{-in} \times 1\text{-in}\times1\text{-in}$, calculate the maximum number of blocks that can fit in the box.
|
8
| 0.333333 |
Given the expression \( \left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\ldots\left(1-\frac{1}{12^{2}}\right) \), compute its value.
|
\frac{13}{24}
| 0.416667 |
Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(1,1)$. What is the area of the triangle enclosed by these two lines and the line $x + y = 8$.
|
9
| 0.833333 |
Given a set $T$ of $8$ integers taken from $\{1,2,\dots,20\}$ with the property that if $c$ and $d$ are elements of $T$ with $c<d$, then $d$ is not a multiple of $c$, find the greatest possible value of an element in $T$.
|
20
| 0.333333 |
Distinct points A, B, C, and D lie on a line, with AB=BC=CD=1. Points E, F, and G lie on a second line, parallel to the first, with EF=FG=1. Determine the number of possible values for the area of a triangle with positive area that has three of the six points as its vertices.
|
3
| 0.416667 |
Given the base $9$ representation of a perfect square that is also a multiple of $3$ is $ab4c$, where $a \ne 0$, find the value of $c$.
|
0
| 0.25 |
For all real values of $x$, simplify the expression $\sqrt{x^6 + x^4 + 1}$.
|
\sqrt{x^6 + x^4 + 1}
| 0.25 |
Determine the maximum number of integers among $a, b, c, d$ that can be negative if $2^a 3^b + 5^c 7^d = 6^e 10^f+4$.
|
0
| 0.25 |
A magician's hat contains 4 red chips and 3 green chips. Chips are drawn randomly, one at a time without replacement, until all of the green chips are drawn or all of the red chips are drawn. Determine the probability that all the red chips are drawn before all the green chips.
|
\frac{3}{7}
| 0.166667 |
Evaluate the expression \(3 + \sqrt{3} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3}\).
|
3 + \frac{2\sqrt{3}}{3}
| 0.916667 |
Given that Raashan, Sylvia, and Ted each start with $1, a bell rings every 15 seconds, and each player who has money gives $1 to another player chosen at random, what is the probability that after the bell has rung 2019 times, each player will have $1?
|
\frac{1}{4}
| 0.083333 |
Find the number of pairs (m, n) of integers that satisfy the equation m^3 + 4m^2 + 3m = 8n^3 + 12n^2 + 6n + 1.
|
0
| 0.583333 |
A traveler starts from the origin, walks 4 miles due west, then turns $120^\circ$ to his left and walks another 5 miles in the new direction. After that, he turns $60^\circ$ to his left again and walks another 2 miles. Find the distance between his starting point and the endpoint.
|
\sqrt{19}
| 0.083333 |
Given a two-digit positive integer, define it as spiky if it equals the sum of its tens digit and the cube of its units digit subtracted by twice the tens digit. How many two-digit positive integers are spiky?
|
0
| 0.916667 |
Find the smaller root of the equation $\left(x-\frac{1}{3}\right)^2 + \left(x-\frac{1}{3}\right)\left(x+\frac{1}{6}\right) = 0$.
|
\frac{1}{12}
| 0.916667 |
The line $10x + 8y = 80$ forms a triangle with the coordinate axes. Calculate the sum of the lengths of the altitudes of this triangle.
|
18 + \frac{40\sqrt{41}}{41}
| 0.666667 |
Determine the number of distinct pairs of integers $(x, y)$ such that $0 < x < y$ and $\sqrt{2500} = \sqrt{x} + 2\sqrt{y}$.
A) $0$
B) $1$
C) $2$
D) $3$
E) $4$
|
0
| 0.166667 |
The dimensions of a rectangular box in inches are positive integers with one dimension being an even number. The volume of the box is $1806$ in$^3$. Find the minimum possible sum of the three dimensions.
|
56
| 0.25 |
Lisa celebrated her 20th birthday on Monday, August 15, 2010, which was not a leap year. Find the next year when her birthday will fall on a Thursday.
|
2012
| 0.5 |
A circle of radius 3 is centered at a point O. An isosceles right triangle with hypotenuse 6 has its right-angled vertex at O. 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 - 1)
| 0.083333 |
What is the greatest power of $2$ that is a factor of $10^{1003} - 4^{502}$?
|
2^{1003}
| 0.083333 |
Determine the number of points common to the graphs of $(x-2y+3)(4x+y-5)=0$ and $(x+2y-3)(3x-4y+6)=0$.
|
3
| 0.666667 |
Evaluate the tens digit of \(4032^{4033} - 4036\).
|
9
| 0.916667 |
During a journey from the United States to France, Lucas took $d$ U.S. dollars. At the airport, he exchanged all his money to Euros at a rate where $5$ U.S. dollars yielded $8$ Euros. After spending $80$ Euros on souvenirs, Lucas found that he had exactly $d$ Euros left. Find the sum of the digits of $d$.
|
7
| 0.833333 |
A 4x4x4 cube is made of $64$ normal dice, where each die's opposite sides sum to $7$. Calculate the smallest possible sum of all the values visible on the $6$ faces of the large cube.
|
144
| 0.083333 |
Ann now wants to extend her toothpick staircase to a 6-step version. She already has a 4-step staircase which used 28 toothpicks. Determine how many additional toothpicks are needed to complete the 6-step staircase if the pattern observed in increases continues.
|
26
| 0.666667 |
In a circle, diameter $\overline{EB}$ is parallel to chord $\overline{DC}$. If the angles $AEB$ and $ABE$ are in the ratio of $2 : 3$, determine the degree measure of angle $BCD$.
|
36^\circ
| 0.083333 |
Given two integers have a sum of 29. When two more integers are added to the first two, the sum becomes 47. Finally, when three more integers are added to the sum of the previous four integers, the sum becomes 66. Determine the minimum number of even integers among the 7 integers.
|
1
| 0.083333 |
Two circles have centers at $(1,3)$ and $(10,6)$ with radii $3$ and $6$, respectively. Determine the y-intercept of a common external tangent to the circles if the tangent line can be expressed in the form $y=mx+b$ with $m>0$.
A) $\frac{5}{4}$
B) $\frac{6}{4}$
C) $\frac{7}{4}$
D) $\frac{8}{4}$
E) $\frac{9}{4}$
|
\frac{9}{4}
| 0.416667 |
Given Mr. and Mrs. Alpha want to name their baby Alpha so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated, and the first initial must be 'A', find the number of such monograms possible.
|
300
| 0.833333 |
Cagney can frost a cupcake in 15 seconds and Lacey can frost a cupcake in 25 seconds. After a 1-minute warm-up, their frosting speeds increase by 20%. Calculate the total number of cupcakes they can frost together in 10 minutes.
|
76
| 0.25 |
A right triangle has sides 5, 12, and 13, with 13 being the hypotenuse. A square with side length $x$ is inscribed in this triangle so that one side of the square lies along one of the legs of the triangle (length 12). Another square with side length $y$ is inscribed in the same triangle so that one side of the square lies along the hypotenuse. What is the ratio $\frac{x}{y}$?
A) $\frac{5}{13}$
B) $\frac{17}{18}$
C) $\frac{11}{12}$
D) $\frac{12}{13}$
E) $\frac{13}{12}$
|
\frac{12}{13}
| 0.666667 |
Given that the larger rectangular fort is $15$ feet long, $12$ feet wide, and $6$ feet high, with the floor, ceillings, and all four walls made with blocks that are one foot thick, determine the total number of blocks used to build the entire fort.
|
560
| 0.083333 |
Given the equation \[ x^{4} + y^2 = 4y + 4,\] determine the number of ordered pairs of integers \( (x, y) \) that satisfy the equation.
|
0
| 0.166667 |
Given two positive numbers that can be inserted between 4 and 16, such that the sequence forms two different arithmetic progressions, the first three numbers form an arithmetic progression, and the last three numbers form a different arithmetic progression, find the sum of these two numbers.
|
20
| 0.75 |
A coin has a probability of $3/4$ of turning up heads. Given that this coin is tossed 60 times, calculate the probability that the number of heads obtained is divisible by 3.
|
\frac{1}{3}
| 0.583333 |
Given the dimensions of a $2\text{-in} \times 1\text{-in} \times 2\text{-in}$ solid block and a $3\text{-in} \times 4\text{-in} \times 2\text{-in}$ box, calculate the largest number of blocks that can fit in the box.
|
6
| 0.666667 |
Given that right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $2$, and $\angle CAD = 30^{\circ}$, find $\sin(2\angle BAD)$.
|
\frac{1}{2}
| 0.916667 |
Given a square in the coordinate plane with vertices at $(0, 0), (3030, 0), (3030, 3030),$ and $(0, 3030)$, a point is chosen at random within the square. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Determine $d$ to the nearest tenth.
|
0.3
| 0.166667 |
Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line such that the length of segment $PA$ remains constant, determine how many of the following quantities change: the length of the segment $MN$, the perimeter of $\triangle PAB$, the area of $\triangle PAB$, the area of trapezoid $ABNM$.
|
3
| 0.75 |
Consider a square grid composed of $121$ points arranged in an $11 \times 11$ formation, including all the points on the edges. Point $P$ is situated at the center of this square. Calculate the probability that a line drawn from $P$ to any other randomly chosen point $Q$ from the remaining $120$ points acts as a line of symmetry for the square.
|
\frac{1}{3}
| 0.583333 |
Choose three different numbers from the set $\{2, 5, 8, 11, 14\}$. Add two of these numbers and multiply their sum by the third number. What is the smallest result that can be obtained through this process?
|
26
| 0.25 |
Given a two-digit positive integer, define it as $\emph{charming}$ if it is equal to the sum of its nonzero tens digit and the cube of its units digit. Determine how many two-digit positive integers are charming.
|
0
| 0.166667 |
Given that there are $6$-pound sapphires worth $\$$15 each, $3$-pound rubies worth $\$$9 each, and $2$-pound diamonds worth $\$$5 each, and there are at least $10$ of each type available, determine the maximum value of the gemstones Ellie can carry, given a weight limit of $24$ pounds.
|
\$72
| 0.333333 |
The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors. Determine the last digit to appear in the units position of a number in the Fibonacci sequence when considered modulo 12.
|
11
| 0.416667 |
How many different real numbers x satisfy the equation (2x^2 - 7)^2 - 8x = 48?
|
4
| 0.666667 |
Given $\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt{3}$, define $T$ as the set of all points in the complex plane of the form $a + b\omega + c\omega^2$, where $0 \leq a, b, c \leq 2$. Find the area of $T$.
|
6\sqrt{3}
| 0.083333 |
The greatest possible sum of the digits in the base-eight representation of a positive integer less than $5000$.
|
28
| 0.583333 |
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30} \cdot 5^3}$ as a decimal?
|
30
| 0.166667 |
Given two circles are internally tangent at a point, with circles centered at points $A$ and $B$ having radii $7$ and $4$ respectively, find the distance from point $B$ to the point where an internally tangent line intersects ray $AB$ at point $C$.
|
4
| 0.5 |
Given a lawn measuring 100 feet by 140 feet, a mower swath of 30 inches wide with a 6-inch overlap, and a walking speed of 4500 feet per hour, determine the closest estimate for the number of hours it will take to mow the entire lawn.
|
1.6
| 0.25 |
The numbers $-3, 1, 5, 8$, and $10$ are rearranged according to these rules:
1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't in the second or fourth position.
4. The sum of the first and last numbers is greater than 12. Find the average of the first and last numbers.
|
6.5
| 0.5 |
Given the population of Brazil in 2010 as approximately 195,000,000 and the area of Brazil as 3,288,000 square miles, determine the average number of square feet per person in Brazil.
|
470,000
| 0.083333 |
Given the school store sells 9 pencils and 10 notebooks for $\mathdollar 5.06$ and 6 pencils and 4 notebooks for $\mathdollar 2.42$, determine the cost of 20 pencils and 14 notebooks.
|
8.31
| 0.583333 |
Given a three-digit positive integer and a two-digit positive integer with all five digits in these numbers being different, the sum of the two numbers is a three-digit number. Find the smallest possible value for the sum of the digits of this three-digit sum.
|
3
| 0.083333 |
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