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stringlengths 18
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|---|---|---|
Consider those functions $f$ that satisfy $f(x+5)+f(x-5) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions.
|
30
| 0.166667 |
Given the sequence $\{a_1, a_2, a_3, \ldots\}$ where each positive even integer $k$ appears exactly $k+1$ times, and the sequence starts with 2, determine the sum of the constants $b$, $\alpha$, $\beta$, and $d$ in the formula $a_n = b\lfloor \alpha\sqrt{n} + \beta \rfloor + d$ such that the sequence retains its noted pattern for all positive integers $n$.
|
3
| 0.333333 |
Given that $720a$ is a square and $720b$ is a cube, where $a$ and $b$ are the two smallest positive integers, calculate the sum of $a$ and $b$.
|
305
| 0.833333 |
Given the expression $\frac{b - 1}{b + \frac{b}{b-1}}$, simplify the given expression, where $b \neq 1$.
|
\frac{(b-1)^2}{b^2}
| 0.916667 |
A rectangular piece of metal has dimensions 10 units by 6 units. A circular piece of maximum diameter is cut out from this rectangle (assuming the longest dimension is the diameter). Then, a square piece of maximum size is cut out from the circular piece. Calculate the area of the metal wasted.
|
42 \text{ units}^2
| 0.5 |
Let \( M = 42 \cdot 43 \cdot 75 \cdot 196 \). Find the ratio of the sum of the odd divisors of \( M \) to the sum of the even divisors of \( M \).
|
\frac{1}{14}
| 0.833333 |
Given a Ferris wheel with a radius of 30 feet, revolving at the constant rate of one revolution every 2 minutes, determine the time in seconds it takes for a rider to travel from the bottom of the wheel to a point 15 vertical feet above the bottom.
|
20
| 0.333333 |
Given that Ray's car averages 50 miles per gallon of gasoline, Tom's car averages 20 miles per gallon of gasoline, and Alice's car averages 25 miles per gallon, find the combined rate of miles per gallon for all three cars.
|
\frac{300}{11}
| 0.166667 |
Evaluate the expression $\sqrt{x + \sqrt{x + \sqrt{x}}}$ for $x \geq 0$.
**A)** $x\sqrt{x}$
**B)** $x\sqrt[4]{x}$
**C)** $\sqrt[8]{x+\left(x + x^{\frac{1}{2}}\right)^{\frac{1}{2}}}$
**D)** $\sqrt{2x}$
**E)** $\sqrt{x + \left(x + x^{\frac{1}{2}}\right)^{\frac{1}{2}}}$
|
\sqrt{x + \left(x + x^{\frac{1}{2}}\right)^{\frac{1}{2}}}
| 0.5 |
In an All-Area track meet, 320 sprinters participate in a 100-meter dash competition on a track with 8 lanes, allowing 8 sprinters to compete simultaneously. At the end of each race, the seven non-winners are eliminated, and the winner advances to compete in subsequent races. How many races are necessary to determine the champion sprinter?
|
46
| 0.75 |
What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<0.02$?
|
626
| 0.666667 |
Last year, 120 adult cats, 60 of whom were male, were brought into Rivertown Animal Shelter. 40% of the female cats, and 10% of the male cats were accompanied by a litter of kittens. The average number of kittens per litter was 5. Calculate the total number of cats and kittens received by the shelter last year.
|
270
| 0.916667 |
Given that Ron reversed the digits of a two-digit number $a$ and added 5 to the product with $b$, and the mistaken result was 266, determine the correct value of the product of $a$ and $b$.
|
828
| 0.25 |
In Markville, the sales tax rate is 7%. A dress originally priced at $150.00 receives a 25% discount during a store sale. Two assistants, Ann and Ben calculate the final cost differently. Ann calculates by adding the tax to the original price before applying the 25% discount. Ben discounts the price first and then adds the 7% tax. Additionally, Ann applies a special 5% service charge after all other calculations due to premium packaging. Determine the difference between the amount paid by Ann's customer and Ben's customer.
|
6.02
| 0.416667 |
A 4x4x4 cube is made of $64$ normal dice, where opposites sum to $7$. Calculate the smallest possible sum of all the values visible on the $6$ faces of the giant cube.
|
144
| 0.166667 |
Determine the number of integer values of $n$ for which $8000 \cdot \left(\frac{2}{5}\right)^n$ is an integer.
|
10
| 0.916667 |
Given that in $\triangle DEF$, $DE = 45$ and $DF = 75$, and a circle with center $D$ and radius $DE$ intersects $\overline{EF}$ at points $E$ and $Y$, with $\overline{EY}$ and $\overline{FY}$ having integer lengths, calculate the length of $EF$.
|
120
| 0.333333 |
The maximum number of the eight integers that can be larger than $20$ if their sum is $-20$.
|
7
| 0.666667 |
Given seven points on a circle numbered 1 through 7 in clockwise order, a bug jumps in a clockwise direction; if the point number is divisible by 3, it moves two points forward, otherwise, it moves three points forward. Determine the point where the bug will be after 2023 jumps if it starts on point 7.
|
1
| 0.25 |
In a botanical garden, three decorative plant beds are described, with Bed X containing 600 plants, Bed Y containing 500 plants, and Bed Z containing 400 plants. Beds X and Y share 80 plants, Beds X and Z share 120 plants, and Beds Y and Z share 70 plants. No plant is in all three beds. Calculate the total number of unique plants across all beds.
|
1230
| 0.75 |
Given a $5 \times 5$ block of calendar dates as shown below, first reverse the order of the numbers in the second, third, and fifth rows, and then find the sum of the numbers on each diagonal. What is the absolute difference between the two diagonal sums?
$$
\begin{tabular}{|c|c|c|c|c|}
\hline 1 & 2 & 3 & 4 & 5 \\
\hline 6 & 7 & 8 & 9 & 10 \\
\hline 11 & 12 & 13 & 14 & 15 \\
\hline 16 & 17 & 18 & 19 & 20 \\
\hline 21 & 22 & 23 & 24 & 25 \\
\hline
\end{tabular}
$$
|
4
| 0.083333 |
Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determine the average of the first and last numbers.
|
5.5
| 0.333333 |
Find the number of four-digit passwords that can be formed from digits $0$ to $9$, with repeated digits allowable, and excluding passwords that begin with the sequence $1,2,3$.
|
9990
| 0.833333 |
When $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $715$ terms that include all four variables $a$, $b$, $c$, and $d$, each to some positive power. Find the value of $N$.
|
N = 13
| 0.166667 |
Consider the set $\{1, 5, 9, 13, 17, 21, 25, 29\}$. Calculate the number of different integers that can be expressed as the sum of four distinct members of this set.
|
17
| 0.833333 |
A lemming starts at a corner of a square with a side length of 15 meters. It runs 9.3 meters along a diagonal toward the opposite corner, stops, makes a 90-degree right turn, and runs 3 more meters. Determine the average of the shortest distances from the lemming to each side of the square after it stops.
|
7.5
| 0.833333 |
Calculate the area of a rectangular garden which measures 2.5 meters in length and 0.48 meters in width.
|
1.2
| 0.916667 |
Points $A$ and $B$ are on a circle of radius $7$ and $AB=8$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$?
|
\sqrt{98 - 14\sqrt{33}}
| 0.25 |
Given that 132 is an even integer greater than 7, find the largest possible difference between two different prime numbers that sum to 132.
|
122
| 0.833333 |
Find the number of positive factors of 48 that are also multiples of 6.
|
4
| 0.916667 |
Given $x_{k+1} = x_k + \frac{1}{3}$ for $k=1, 2, \dots, n-1$ and $x_1=2$, calculate the sum of $x_1 + x_2 + \dots + x_n$.
|
\frac{n(n+11)}{6}
| 0.583333 |
Evaluate the expression: \(\frac{3^2+5^2+7^2}{2^2+4^2+6^2} - \frac{2^2+4^2+6^2}{3^2+5^2+7^2}\).
|
\frac{3753}{4648}
| 0.416667 |
What is the value of $\frac{3a^{-2} + \frac{a^{-3}}{3}}{a^2}$ when $a = 3$?
|
\frac{28}{729}
| 0.916667 |
Numbers between 0 and 1 are chosen as follows: A fair coin is flipped twice. If the first flip is heads and the second is tails, the number is 1. If the first flip is heads and the second is heads, the number is 0. If the first flip is tails, regardless of the second flip, the number is 0.5. Two independent numbers x and y are chosen using this method. Compute the probability that the absolute difference between x and y, |x-y|, is greater than 0.5.
|
\frac{1}{8}
| 0.166667 |
A group of students from Sequoia High School is volunteering for a local park cleaning event. The ratio of 9th-graders to 7th-graders is 3 : 2 while the ratio of 9th-graders to 6th-graders is 7 : 4. Determine the smallest number of students that could be volunteering in this event.
|
47
| 0.833333 |
Given that a circle is divided into 10 sectors, the central angles of these sectors, measured in degrees, increase arithmetically. Find the degree measure of the smallest possible sector angle.
|
9
| 0.166667 |
Determine the number of ordered pairs of positive integers $(A, B)$ that satisfy the equation $\frac{A}{8} = \frac{8}{B}$, given that $A$ and $B$ are even numbers.
|
5
| 0.833333 |
Evaluate $(y^2)^{(y^{(y^2))})$ at $y = 3$.
|
9^{19683}
| 0.75 |
Given that there are $12$ students in Mrs. Germain's class, $10$ students in Mr. Newton's class, and $11$ students in Mrs. Young's class taking the AMC 8, and $5$ students are in an advanced study group and are counted in both Mrs. Germain's and Mr. Newton's totals, calculate the number of unique mathematics students at Euclid Middle School taking the contest.
|
28
| 0.916667 |
A digital watch is set to a 24-hour format displaying both hours and minutes. Calculate the largest possible sum of the digits in the display when the watch displays a time between 12:00 and 23:59.
|
24
| 0.416667 |
Dana goes first and Carl's coin lands heads with probability $\frac{2}{7}$, and Dana's coin lands heads with probability $\frac{3}{8}$. Find the probability that Carl wins the game.
|
\frac{10}{31}
| 0.5 |
A half-sector of a circle of radius 6 inches is rolled up to form the lateral surface area of a right circular cone by taping together along the two radii. Calculate the volume of the cone.
|
9\pi \sqrt{3}
| 0.916667 |
A rectangular piece of metal has dimensions of 8 units by 10 units. Determine the total amount of metal wasted after a circular piece of maximum size is cut from this rectangle, and then a square piece of maximum size is cut from the circular piece.
|
48 \text{ units}^2
| 0.5 |
In how many ways can $420$ be written as the sum of an increasing sequence of two or more consecutive positive integers?
|
7
| 0.666667 |
A nine-digit number is formed by repeating a three-digit number three times; for example, $256256256$. Determine the common factor that divides any number of this form exactly.
|
1001001
| 0.166667 |
Determine the number of ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $10$, inclusive, that satisfy the equation $(\log_b a)^3 = 2 \log_b(a^{3})$.
|
27
| 0.583333 |
Given that the lighthouse is 60 meters high and the tank has a capacity of 150,000 liters, determine the height of a scale model of the lighthouse with a tank capacity of 0.15 liters.
|
0.6
| 0.25 |
The number of possible license plates under the old scheme is given by $26\cdot 10^5$, and the number of possible license plates under the new scheme is given by $26^2\cdot 10^4$. Calculate the ratio of the number of license plates under the new scheme to the number of license plates under the old scheme.
|
2.6
| 0.833333 |
Given that Annie runs 50% faster than Bonnie on a 500-meter circular track, determine the number of laps Annie will have run when she first laps Bonnie.
|
3
| 0.833333 |
Given that our number system has a base of eight, determine the fifteenth number in the sequence.
|
17
| 0.083333 |
Calculate the value of $(2501+2502+2503+\cdots+2600) - (401+402+403+\cdots+500) - (401+402+403+\cdots+450)$.
|
188725
| 0.916667 |
Determine $x$ in terms of $m$ and $n$ given that the equation $(x+m)^2 - 3(x+n)^2 = m^2 - 3n^2$.
|
x = 0 \text{ or } x = m - 3n
| 0.916667 |
How many pairs of positive integers \((a, b)\) with \(a+b \leq 150\) satisfy the equation \(\frac{a+b^{-1}}{a^{-1}+b} = 17?\)
|
8
| 0.583333 |
Let \( c_1, c_2, \ldots \) and \( d_1, d_2, \ldots \) be arithmetic progressions such that \( c_1 = 30, d_1 = 90 \), and \( c_{50} + d_{50} = 120 \). Find the sum of the first fifty terms of the progression \( c_1 + d_1, c_2 + d_2, \ldots \).
|
6000
| 0.916667 |
Alice starts her new job with a schedule of 4 work-days followed by 2 rest-days, and Bob's schedule is 5 work-days followed by 1 rest-day. Determine how many days in the first 800 days do both have rest-days on the same day.
|
133
| 0.916667 |
How many whole numbers between $200$ and $500$ contain the digit $3$?
|
138
| 0.333333 |
$\triangle ABC$ has a right angle at $C$ and $\angle A = 15^\circ$. $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$. Compute $\angle BDC$.
|
52.5^\circ
| 0.916667 |
A convex polyhedron $S$ has vertices $W_1,W_2,\ldots,W_m$, and $150$ edges. Each vertex $W_j$ is intersected by a plane $Q_j$ that cuts all edges connected to $W_j$. These cuts produce $m$ pyramids and a new polyhedron $T$. Assuming no two planes intersect inside or on $S$, determine the number of edges on polyhedron $T$.
|
450
| 0.416667 |
Given that Rani can cycle 20 miles in 2 hours and 45 minutes when she was a girl and 12 miles in 3 hours as an older woman, calculate the difference in time it takes for her to cycle a mile now compared to when she was a girl.
|
6.75
| 0.75 |
Calculate the area of each triangle with sides of length 15, 15, and 20, and with sides of length 15, 15, and 30, and determine the relationship between their areas.
|
0
| 0.083333 |
Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments.
|
9\pi
| 0.166667 |
Samantha lives 3 blocks north and 2 blocks east of the northeast corner of City Park. Her school is 3 blocks south and 3 blocks west of the southwest corner of City Park. On school days she bikes on streets to the northeast corner of City Park, then takes one of two diagonal paths through the park to the southwest corner, and then bikes on streets to school. Assuming she takes the shortest on-street routes, determine the total number of different routes she can take.
|
400
| 0.75 |
Given rectangle ACDE, where AC = 48 and AE = 36, point B is located one-fourth the length of AC from A, and F is located one-third the length of AE from A, calculate the area of quadrilateral ABDF.
|
504
| 0.5 |
Given a circle of radius $3$, determine the area of the region consisting of all line segments of length $6$ that are tangent to the circle at their midpoints.
|
9\pi
| 0.333333 |
Given that the total amount of money originally owned by Moe, Loki, and Nick was $72, and each of Loki, Moe, and Nick gave Ott$\, 4, determine the fractional part of the group's money that Ott now has.
|
\frac{1}{6}
| 0.416667 |
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 100?
|
10403
| 0.583333 |
Determine the radius $r$ of the circle with center $O$ tangent to one side of the equilateral triangle $ABC$ and to the coordinate axes.
|
1
| 0.166667 |
Jacqueline has 2 liters of soda. Liliane has 60% more soda than Jacqueline, and Alice has 40% more soda than Jacqueline. Calculate the percentage difference between the amount of soda Liliane has compared to Alice.
|
14.29\%
| 0.916667 |
Paul owes Paula 60 cents and has a pocket full of 5-cent coins, 10-cent coins, 25-cent coins, and a new 50-cent coin that he can use to pay her. Find the difference between the largest and smallest number of coins he can use to pay her exactly.
|
10
| 0.916667 |
Given the expression $2-(-3)-4-(-5) \times 2 -6-(-7)$, calculate its value.
|
12
| 0.75 |
Consider each positive integer $n$, let $g_1(n)$ be thrice the number of positive integer divisors of $n$ raised to the power of 2, and for $j \ge 2$, let $g_j(n) = g_1(g_{j-1}(n))$. Determine for how many values of $n \le 30$ is $g_{50}(n) = 243$.
|
0
| 0.583333 |
Given $\dfrac{15! - 14! - 13!}{11!}$, calculate the value.
|
30420
| 0.916667 |
Find the smallest positive integer $x$ such that $1260x = N^2$, where $N$ is an integer.
|
35
| 0.916667 |
Let $n$ be the smallest positive integer such that $n$ is divisible by $30$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. Find the number of digits of $n$.
|
9
| 0.833333 |
Let \( M = 57^{4} + 4\cdot57^{3} + 6\cdot57^{2} + 4\cdot57 + 1 \). Determine the number of positive integer factors of \( M \).
|
25
| 0.666667 |
Let $\angle ABC = 40^\circ$, $\angle ABD = 30^\circ$, and $\angle ABE = 15^\circ$. Given that point $E$ is on line segment $BD$, calculate the smallest possible degree measure for $\angle EBC$.
|
25^\circ
| 0.5 |
Given the two numbers have their difference, their sum, and their product in the ratio $1:8:30$, find the product of these two numbers.
|
\frac{400}{7}
| 0.583333 |
Determine the minimum number of fence posts required to fence a rectangular garden plot measuring 30 m by 50 m, where the fourth side is an existing 80 m wall.
|
12
| 0.083333 |
Bricklayer Brenda would take $8$ hours to build a chimney alone, and bricklayer Brandon would take $12$ hours to build it alone. When they work together, their efficiency is diminished, resulting in a decreased output of $15$ bricks per hour due to their chatting. Given that they complete the chimney in $6$ hours when working together, determine the total number of bricks in the chimney.
|
360
| 0.416667 |
Given that a store prices an item so that when 6% sales tax is added, the total cost is exactly $m$ dollars, where $m$ is a positive integer, determine the smallest value of $m$.
|
53
| 0.833333 |
Given a bouquet containing purple tulips, yellow tulips, purple lilies, and yellow lilies, where half of the purple flowers are tulips, two-thirds of the yellow flowers are lilies, and seven-tenths of the flowers are purple, calculate the percentage of flowers that are lilies.
|
55\%
| 0.833333 |
Evaluate the expression
\[(4+5)(4^2+5^2)(4^4+5^4)(4^8+5^8)(4^{16}+5^{16})(4^{32}+5^{32})(4^{64}+5^{64})\]
|
5^{128} - 4^{128}
| 0.5 |
Given a list of nine numbers, the average of the first five numbers is 7, and the average of the last five numbers is 10. If the average of all nine numbers is $8\frac{2}{9}$, find the number common to both sets of five numbers.
|
11
| 0.833333 |
The Tigers beat the Sharks 3 out of the 4 times they played. They then played N more times, and the Sharks ended up winning at least 90% of all the games played. Determine the minimum possible value for N.
|
26
| 0.083333 |
What is the smallest prime number dividing the expression $2^{12} + 3^{10} + 7^{15}$?
|
2
| 0.916667 |
Consider a circle with center \(O\) and a given radius. There are two tangents to the circle; one is at a distance \(r\) (radius of the circle) above the center, and the other is at distance \(2r\) below the center. Determine the number of points which are equidistant from the circle and both tangents.
|
2
| 0.416667 |
In an isosceles trapezoid $ABCD$, with $AB \parallel CD$, the sides $AB$ and $CD$ are equal, and the legs $BC$ and $DA$ are also equal. The sum of the lengths of the bases $AB$ and $CD$ is 24 units. If the length of each leg is 13 units, calculate the perimeter of trapezoid $ABCD$.
|
50
| 0.916667 |
What is the hundreds digit of $(17! - 12!)$?
|
4
| 0.5 |
Evaluate the expression $3 - (-3)^{-\frac{2}{3}}$.
|
3 - \frac{1}{\sqrt[3]{9}}
| 0.75 |
The conference has 12 teams. Each team plays every other team twice and an additional 6 games against non-conference opponents. Calculate the total number of games in a season involving the conference teams.
|
204
| 0.833333 |
You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits.
|
11.125
| 0.583333 |
Given that Lakeview Academy has 1500 students, each student takes 6 classes per day, and each class has 25 students, each class is taught by a main teacher and a co-teacher, and each teacher teaches 5 classes, determine the total number of teachers at Lakeview Academy.
|
144
| 0.333333 |
Let U be the set of the 3000 smallest positive multiples of 5, and let V be the set of the 3000 smallest positive multiples of 7. Determine the number of elements common to U and V.
|
428
| 0.916667 |
It takes Clea 120 seconds to walk up an escalator when it is not operating, and only 48 seconds to walk up the escalator when it is operating. Calculate the time it takes Clea to ride up the operating escalator when she just stands on it.
|
80
| 0.916667 |
Given the equation $a \cdot b + 125 = 30 \cdot \text{lcm}(a, b) + 24 \cdot \text{gcd}(a, b) + a \mod b$, where $\text{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a, b)$ denotes their least common multiple, assuming $a \geq b$, calculate the number of ordered pairs $(a, b)$ of positive integers that satisfy this equation.
|
0
| 0.333333 |
If $7 = k \cdot 3^r$ and $49 = k \cdot 9^r$, solve for $r$.
|
\log_3 7
| 0.416667 |
For how many positive integers $n$ is $n^3 - 9n^2 + 23n - 15$ a prime number?
|
1
| 0.916667 |
Seven points on a circle are given. Four of the chords joining pairs of the seven points are selected at random. What is the probability that the four chords form a convex quadrilateral?
|
\frac{1}{171}
| 0.333333 |
How many perfect cubes lie between $3^6+1$ and $3^{12}+1$, inclusive?
|
72
| 0.75 |
Evaluate the value of $(3(3(3(3(3(3+1)+2)+3)+4)+5)+6)$.
|
1272
| 0.333333 |
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