problem
stringlengths 18
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Keiko walks once around a track at exactly the same constant speed every day. The track has straight sides and the ends are semi-circular. The track has a width of $6$ meters, and it takes her $48$ seconds longer to walk around the outer edge of the track than around the inner edge. Calculate Keiko's speed in meters per second.
|
\frac{\pi}{4}
| 0.833333 |
Given that $x \neq 1$, $y \neq 2$, and $z \neq 3$, compute the value in simplest form of the expression $\frac{x-1}{3-z} \cdot \frac{y-2}{1-x} \cdot \frac{z-3}{2-y}$.
|
-1
| 0.833333 |
A solid cube of side length $4$ has a smaller cube of side length $1.5$ removed from each corner. Calculate the total number of edges of the remaining solid.
|
36
| 0.833333 |
Given parallelogram EFGH has side lengths EF = 40 cm, FG = 30 cm, and the longer diagonal EH = 50 cm, calculate the area of parallelogram EFGH.
|
1200
| 0.083333 |
Given that two angles of an isosceles triangle measure $60^\circ$ and $x^\circ$, find the sum of the three possible values of $x$.
|
180
| 0.916667 |
A rectangle has a ratio of length to width of 5:2. If the diagonal of the rectangle is 13 units, determine the value of $k$ such that the area of the rectangle may be expressed as $kd^2$, where $d$ is the diagonal.
|
\frac{10}{29}
| 0.75 |
An item's price is reduced by 15%. What percentage increase is required on this new price to bring it back to its original value?
|
17.65\%
| 0.916667 |
Given the equation $x^{4} + y^2 = 4y$, determine the number of ordered pairs of integers $(x, y)$ that satisfy this equation.
|
2
| 0.75 |
Johann has 64 fair coins. He flips all the coins, and any coin that lands on tails is tossed again, with the process repeating up to three more times for each coin, for a total of four tosses per coin. Calculate the expected number of coins that show heads after these tosses.
|
60
| 0.833333 |
Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $AC$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{2}{5}DA$. Determine the ratio of the area of $\triangle DFE$ to the area of $\triangle ABE$.
|
\frac{2}{5}
| 0.5 |
Given $6 \cdot 3^x = 7^{y + 3}$, find the value of $x$ when $y = -3$.
|
\log_3\left(\frac{1}{6}\right)
| 0.083333 |
Given that N is a two-digit positive integer, find the number of N such that the sum of N and the number obtained by reversing the order of the digits of N is a perfect cube.
|
0
| 0.333333 |
In the list where each integer $n$ appears $n$ times for $1 \leq n \leq 300$, find the median of the numbers.
|
212
| 0.666667 |
Let $M = 36 \cdot 36 \cdot 85 \cdot 128$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$.
|
\frac{1}{4094}
| 0.416667 |
A bug starts crawling from position $-3$ on a number line. It first moves to $-7$, then reverses direction to head towards $8$, and finally turns around again and stops at $2$. Calculate the total distance the bug has crawled.
|
25
| 0.916667 |
Given hexadecimal numbers use the digits $0$ to $9$ and the letters $A$ to $F$ for values $10$ to $15$, determine how many of the first $512$ decimal numbers only contain numeric digits in their hexadecimal representation, and find the sum of the digits of this quantity.
|
2
| 0.25 |
For how many positive integer values of \(n\) are both \(\frac{n}{5}\) and \(5n\) four-digit whole numbers?
|
0
| 0.833333 |
Find the smallest integer n such that (x+y+z)^2 ≤ n(x^2+y^2+z^2) for all real numbers x, y, and z.
|
3
| 0.666667 |
Starting with the display "1," calculate the fewest number of keystrokes needed to reach "240" using only the keys [+1] and [x2].
|
10
| 0.5 |
Consider the first four terms of an arithmetic sequence given by $a, x, b, 3x$, and determine the ratio of $a$ to $b$.
|
0
| 0.916667 |
Given $M = 57^{5} + 5\cdot 57^{4} + 10\cdot 57^{3} + 10\cdot 57^{2} + 5\cdot 57 + 1$, calculate the number of positive integers that are factors of $M$.
|
36
| 0.583333 |
Farmer Pythagoras has now expanded his field, which remains a right triangle. The lengths of the legs of this field are $5$ units and $12$ units, respectively. He leaves an unplanted rectangular area $R$ in the corner where the two legs meet at a right angle. This rectangle has dimensions such that its shorter side runs along the leg of length $5$ units. The shortest distance from the rectangle $R$ to the hypotenuse is $3$ units. Find the fraction of the field that is planted.
A) $\frac{151}{200}$
B) $\frac{148}{200}$
C) $\frac{155}{200}$
D) $\frac{160}{200}$
|
\frac{151}{200}
| 0.5 |
Shelby drives her scooter at a speed of 40 miles per hour if it is sunny, and 25 miles per hour if it is raining. One particular day, she drove in the sun for the first part of the journey and in the rain for the last part, covering a total of 25 miles in 45 minutes. Calculate the time, in minutes, she drove in the rain.
|
20
| 0.916667 |
Given the function g defined on the set of positive rational numbers by g(x \cdot y) = g(x) + g(y) for all positive rational numbers x and y, and g(n) = n^2 for every prime number n, calculate g(x) for x = \frac{25}{21}.
|
-8
| 0.833333 |
Given that $65^7$ has positive integer divisors that are perfect squares or perfect cubes (or both), calculate the number of such divisors.
|
21
| 0.916667 |
Determine the number of pairs of positive integers $(x,y)$ which satisfy the system of equations $x^2 + y^2 = x^4$ and $x > y$.
|
0
| 0.75 |
A rectangular piece of metal with dimensions $2a$ by $b$ (where $b \leq 2a$) is used to cut out a circular piece of maximum size, and then a square piece of maximum size is cut out of the circular piece. Find the total amount of metal wasted.
|
2ab - \frac{b^2}{2}
| 0.75 |
After the year 2002, which is a palindrome, identify the next year where the sum of the product of its digits is greater than 15. Find the sum of the product of the digits of that year.
|
16
| 0.25 |
Given Paige calculates the sum of the interior angles of a convex polygon, mistakenly omitting two angles this time, and arrives at a sum of $3240^\circ$, calculate the combined degree measure of the two forgotten angles.
|
180^\circ
| 0.583333 |
The price of an item is decreased by 20%. To bring it back to its original value and then increase it by an additional 10%, the price after restoration must be increased by what percentage.
|
37.5\%
| 0.166667 |
Given a list of $3000$ positive integers with a unique mode occurring exactly $12$ times, calculate the least number of distinct values that can occur in the list.
|
273
| 0.166667 |
Julia drives from her home to the theatre to watch a show. She drives 40 miles in the first hour, but realizes that she will be 1.5 hours late if she continues at this speed. She increases her speed by 20 miles per hour for the rest of the way to the theatre and arrives 1 hour early. Let $d$ be the distance from her home to the theatre; calculate $d$.
|
340
| 0.333333 |
A committee has $60 to buy flowers for a conference room. Orchids cost $4 each, and lilies cost $3 each. Find the number of different combinations of orchids ($o$) and lilies ($l$) that can be purchased for exactly $60, given that the committee must buy at least 10 flowers in total.
|
6
| 0.833333 |
Find the solution of the equation $\sqrt{5x^3-1} + \sqrt{x^3-1} = 4$.
|
\sqrt[3]{2}
| 0.75 |
If $\theta$ is an acute angle and $\cos 2\theta = b$, find the value of $\sin\theta + \cos\theta$.
|
\sqrt{1 + \sqrt{1 - b^2}}
| 0.5 |
Given that three generations of the Patel family are attending a theater play, with three members from each generation, and where the members of the youngest generation enjoy a $40\%$ discount as students, the members of the middle generation pay the full price, and the members of the oldest generation receive a $20\%$ discount as senior citizens, and a discounted ticket for Mr. Patel costs $10.00, determine the total amount Mr. Patel must pay in order to cover all the ticket costs.
|
90
| 0.416667 |
Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187} - \cdots$. Evaluate the sum of this series.
|
\frac{15}{26}
| 0.083333 |
Let $x$ be a real number selected uniformly at random between 100 and 300. If $\lfloor {\sqrt{x}} \rfloor = 15$, find the probability that $\lfloor {\sqrt{100x}} \rfloor = 150$.
|
\frac{301}{3100}
| 0.833333 |
Calculate the total amount of metal wasted by cutting a circular piece of maximum size from a rectangle with sides $a$ and $b$ (where $a < b$), and then a square piece of maximum size from this circular piece.
|
ab - \frac{a^2}{2}
| 0.333333 |
In quadrilateral $EFGH$, the internal angles form an arithmetic sequence. Furthermore, triangles $EFG$ and $HGF$ are similar with $\angle EFG = \angle HGF$ and $\angle EGF = \angle HFG$. Each of these triangles' angles also forms an arithmetic sequence. In degrees, what is the largest possible sum of the largest and smallest angles of $EFGH$?
|
180
| 0.75 |
If $x_{k+1} = x_k + \frac{1}{3}$ for $k=1, 2, \dots, n-1$ and $x_1=2$, find $x_1 + x_2 + \dots + x_n$.
|
\frac{n(n+11)}{6}
| 0.5 |
The maximum value of $\frac{(3^t - 4t)t}{9^t}$ for real values of $t$.
|
\frac{1}{16}
| 0.166667 |
Marvin had a birthday on Friday, May 27 in the non-leap year 2007. In what year will his birthday next fall on a Thursday?
|
2017
| 0.166667 |
A sequence contains one red ball, one blue ball, and $N$ green balls where $N$ is a positive multiple of 7. Calculate the probability $P(N)$ such that at least $\frac{2}{3}$ of the green balls are on the same side as the red ball but on the opposite side of the blue ball. Determine the least value of $N$ for which $P(N) < \frac{7}{10}$, and find the sum of its digits.
|
7
| 0.333333 |
Given the 9 data values are $70, 110, x, 50, x, 210, 100, 85, 40$, and both the mean, median, and mode of these values are all equal to $x$. What is the value of $x$?
|
95
| 0.833333 |
What is the largest integer n for which 7^n is a factor of the sum 47! + 48! + 49!?
|
10
| 0.833333 |
Given that 20% of the participants scored 60 points, 25% scored 75 points, 15% scored 85 points, 30% scored 90 points, and the rest scored 95 points, calculate the difference between the mean and median score of the participants' scores on this competition.
|
5
| 0.25 |
Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n^2+1):(4n^2+27)$ for all $n$, find the ratio of the fifteenth term of the first series to the fifteenth term of the second series.
|
\frac{7}{4}
| 0.083333 |
A particle moves such that its time for the third and subsequent miles varies directly as the square of the integral number of miles already traveled. For each subsequent mile the time is constant. If the third mile is traversed in $8$ hours, determine the expression for the time, in hours, needed to traverse the $n$th mile.
|
2(n-1)^2
| 0.333333 |
Let \( p \) be a positive integer. Carlos and Alice play a game on a whiteboard where they interact with numbers derived from powers of integers. Carlos starts by writing the smallest power of 2 (exponentiation, not square) with \( p+1 \) digits. Every time Carlos writes a number, Alice erases the last \( p \) digits of it. Carlos then writes the next higher power of 2, Alice erases the last \( p \) digits, and this process continues until the last two numbers that remain on the board differ by at least 3. Let \( h(p) \) be the smallest positive integer not written on the board. Determine \( h(1) \).
|
2
| 0.583333 |
Given that in triangle ABC with ∠C = 120°, M is the midpoint of AB, D is a point on BC such that BD:DC = 3:2, and E is the midpoint of AC, determine the area of triangle BME if the area of triangle ABC is 36 square units.
|
9
| 0.416667 |
Calculate the number of diagonals that can be drawn in a regular polygon with 150 sides, where no diagonal that can divide the polygon into two polygons with equal number of sides is considered.
|
10950
| 0.416667 |
Let $x$ and $y$ be two-digit positive integers with mean $70$. Find the maximum value of the ratio $\frac{x}{y}$.
|
\frac{99}{41}
| 0.25 |
Given that three primes $p, q,$ and $r$ satisfy $p+q=r$, $1 < p < q$, and $r > 10$, find the smallest value of $p$.
|
2
| 0.666667 |
Given a circle of radius 3 centered at B(3,3) is dilated to a circle of radius 5, and the new center of the circle is at B'(7,10), determine the distance the origin O(0,0) moves under this transformation.
|
\sqrt{29}
| 0.166667 |
Find the smallest positive integer greater than 3000 that is neither prime nor square and has no prime factor less than 60.
|
4087
| 0.5 |
Given the graph showing Suzanna's variable rate of 12 mph, with each subsequent 5-minute interval resulting in a 1 mph decrease, calculate the total distance she would ride in half an hour.
|
4.75
| 0.833333 |
Calculate the area enclosed by a pentagon on a geoboard with vertices at coordinates (4,1), (2,6), (6,7), (9,5), and (7,2).
|
25.5
| 0.916667 |
A rectangular table with dimensions $9'\times 12'$ needs to be moved from one corner to another inside a rectangular room. The shorter wall of the room is denoted by $W$ feet and the longer wall by $L$ feet. What is the smallest integer value for $W$, assuming $L \geq W$, such that the table can be relocated within the room without tilting or disassembling?
|
15
| 0.75 |
Given that \( n! \) is evenly divisible by \( 1 + 2 + \cdots + n \), find the number of positive integers \( n \) less than or equal to 50.
|
36
| 0.083333 |
Given the expression $(xy-z)^2 + (x+y+z)^2$, find the least possible value for this expression for real numbers $x$, $y$, and $z$.
|
0
| 0.916667 |
Given the equations \(3x - 4y - 2z = 0\) and \(x - 2y + 5z = 0\) with \(z \neq 0\), find the numerical value of \(\frac{2x^2 - xy}{y^2 + 4z^2}\).
|
\frac{744}{305}
| 0.166667 |
Find the number of 3-digit positive integers whose digits multiply to 30.
|
12
| 0.083333 |
Given the definition of a rising number as a positive integer each digit of which is larger than each of the digits to its left, find the smallest digit in the $103^{\text{rd}}$ five-digit rising number.
|
2
| 0.666667 |
How many four-digit whole numbers are there such that the leftmost digit is a prime number, the second digit is even, and all four digits are different?
|
1064
| 0.25 |
Square $IJKL$ has one vertex on each side of square $WXYZ$. Point $I$ is on $WZ$ with $WI = 3 \cdot IZ$. What is the ratio of the area of $IJKL$ to the area of $WXYZ$?
A) $\frac{1}{8}$
B) $\frac{1}{16}$
C) $\frac{1}{32}$
D) $\frac{1}{64}$
E) $\frac{1}{128}$
|
\frac{1}{8}
| 0.25 |
Set A has 30 elements, and set B has 25 elements. Set C has 10 elements. Calculate the smallest possible number of elements in the union A ∪ B ∪ C.
|
30
| 0.416667 |
Given $P(x) =(x-1^2)(x-2^2)\cdots(x-50^2)$, determine the number of integers $n$ such that $P(n)\leq 0$.
|
1300
| 0.083333 |
Given the circle with a circumference of 18 units, a tangent drawn from an external point P, and a secant that divides the circle into arcs of lengths m and n, where m = 2n, and the tangent's length, t, is the geometric mean between m and n, find the number of integer values that t can take.
|
0
| 0.75 |
A set of 36 square blocks is arranged into a 6 × 6 square. How many different combinations of 4 blocks can be selected from that set so that no two blocks are in the same row or column?
|
5400
| 0.916667 |
Given a two-digit integer, determine how many two-digit positive integers equal the sum of their nonzero tens digit and the cube of their units digit.
|
0
| 0.583333 |
Given a pizza shared with $12$ equally-sized slices, where Mark ate all the pepperoni slices and two plain slices. Mark preferred pepperoni on one-third of the pizza. Mark paid an additional $3 for the pepperoni topping. The cost of a plain pizza was $12. Find how much more Mark paid than Anne.
|
3
| 0.333333 |
The number $15!$ expressed in the base $18$ system, determine the value of $k$ such that the result ends with exactly $k$ zeros.
|
3
| 0.833333 |
Calculate the height of Camille’s model water tower, given that the real water tower is 50 meters high and can contain 200,000 liters of water, while her model's version contains only 0.05 liters of water.
|
0.315
| 0.166667 |
Given that $r$, $s$, and $t$ are the roots of the cubic equation $ax^3 + bx^2 + cx + d = 0$, find the value of $\frac{1}{r^2} + \frac{1}{s^2} + \frac{1}{t^2}$.
|
\frac{c^2 - 2bd}{d^2}
| 0.916667 |
Starting with the number 1 on the display of a calculator with only the [+1] and [x3] keys, calculate the fewest number of keystrokes needed to reach 243.
|
5
| 0.833333 |
An uncrossed belt is fitted without slack around two circular pulleys with radii of 10 inches and 6 inches respectively. If the distance between the points of contact of the belt with the pulleys is 30 inches, determine the distance between the centers of the pulleys in inches.
|
2\sqrt{229}
| 0.166667 |
In triangle $\triangle ABC$, $\angle C = 90^\circ$ and $\angle A = 30^\circ$. If $BD$ (with $D$ on $\overline{AC}$) is the angle bisector of $\angle ABC$, find $\angle BDC$.
|
60^\circ
| 0.5 |
Given positive integers $A$, $B$, and $C$, with no common factor greater than 1, such that $ A \log_{100} 5 + B \log_{100} 2 = C$, calculate the value of $A + B + C$.
|
5
| 0.5 |
Given six different awards are to be given to four students, with each student receiving at least one award. Calculate the number of different ways the awards can be distributed.
|
1560
| 0.666667 |
Given the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers, determine the number of fractions with the property that if both numerator and denominator are increased by 1, the value of the fraction is increased by $20\%$.
|
0
| 0.25 |
A triangle has a base of 18 inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three areas with a ratio of 1:2:1, from the top to the bottom respectively. Find the length of the middle segment parallel to the base.
|
9\sqrt{3} \text{ inches}
| 0.083333 |
Given the dimensions of a metallic rectangular sheet are 6 inches by 4 inches, and each dimension should be at least $x - 1.0$ inch and at most $x + 1.0$ inch from the reported measurement, and the final processed sheet dimensions shrink to 90% of its original size, calculate the minimum possible area of the rectangular sheet after processing.
|
12.15
| 0.833333 |
Let $a + 2 = b + 3 = c + 4 = d + 5 = a + b + c + d + 7$. Calculate $a + b + c + d$.
|
-\frac{14}{3}
| 0.5 |
Calculate the sum of all numbers of the form $3k + 2$ plus the sum of all numbers of the form $3k$, where $k$ takes integral values from $0$ to $n$.
|
(n+1)(3n+2)
| 0.833333 |
Three standard 6-sided dice are rolled once. The sum of the numbers rolled determines the radius of a circle. Calculate the probability that the numerical value of the circumference of the circle is greater than the numerical value of twice the area of the circle.
|
0
| 0.916667 |
Given that a basketball player made 7 baskets during a game, and each basket was worth either 1, 2, or 3 points, and the player made no more than 4 three-point baskets, determine the total number of different numbers that could represent the total points scored by the player.
|
12
| 0.666667 |
If two poles are 30'' and 50'' high and are 150'' apart, find the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole.
|
18.75
| 0.583333 |
Given that Josie jogs parallel to a canal along which a boat is moving at a constant speed in the same direction and counts 130 steps to reach the front of the boat from behind it, and 70 steps from the front to the back, find the length of the boat in terms of Josie's steps.
|
91
| 0.5 |
Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$.
|
27
| 0.083333 |
Given that the product of the ages of Kiana and her two younger twin sisters is $72$, find the sum of their three ages.
|
14
| 0.75 |
Given that there are 25 students participating in an after-school program, 12 taking yoga, 15 taking bridge, 11 taking painting, 10 taking at least two classes, and 7 taking exactly two classes, determine the number of students taking all three classes.
|
3
| 0.833333 |
Evaluate the sum: $\sum^{50}_{i=1} \sum^{150}_{j=1} (2i + 3j)$.
|
2081250
| 0.583333 |
Two spinners are spun once. Spinner 1 has sectors numbered: 1, 2, 4, while Spinner 2 has sectors numbered: 1, 3, 5, 7. Calculate the probability that the sum of the numbers on which the spinners land is both odd and prime.
|
\frac{1}{2}
| 0.833333 |
Let $z = a + bi$ be a complex number such that $z + |z| = 1 + 12i$. Calculate $|z|^2$.
|
\frac{21025}{4}
| 0.75 |
A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $2$ feet around the painted rectangle and occupies half of the area of the entire floor. Find the number of possible ordered pairs $(a,b)$.
|
3
| 0.916667 |
Given that five students from Maplewood school worked for 6 days, six students from Oakdale school worked for 4 days, and eight students from Pinecrest school worked for 7 days, and the total amount paid for the students' work was 1240 dollars, determine the total amount earned by the students from Oakdale school, ignoring additional fees.
|
270.55
| 0.583333 |
A point is chosen at random within the rectangle in the coordinate plane whose vertices are $(0, 0), (1000, 0), (1000, 2000),$ and $(0, 2000)$. The probability that the point lies within $d$ units of a lattice point is $\frac{3}{4}$. Calculate the value of $d$ to the nearest tenth.
|
0.5
| 0.333333 |
If $x = \frac{a}{b}$ with $a \neq b$ and $b \neq 0$, and $y = \frac{b}{a}$, calculate $\frac{a + 2b}{a - 2b}$.
|
\frac{x+2}{x-2}
| 0.916667 |
What is the sum of the digits of the square of $\text 999999999$?
|
81
| 0.166667 |
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