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stringlengths 18
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|---|---|---|
Given two boxes, each containing the chips numbered $1$, $2$, $4$, $5$, a chip is drawn randomly from each box. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$.
|
\frac{1}{2}
| 0.5 |
Consider a list where each integer $n$ from 1 to 300 appears exactly $n$ times. What is the median of this number list?
|
212
| 0.416667 |
Given that Country X has $a\%$ of the world's population and $b\%$ of the world's wealth, and Country Y has $b\%$ of the world's population and $a\%$ of its wealth, find the ratio of the wealth of a citizen of X to the wealth of a citizen of Y.
|
\frac{b^2}{a^2}
| 0.833333 |
How many primes less than $200$ have $3$ as the ones digit?
|
12
| 0.75 |
Determine the number of solution-pairs in the positive integers for the equation $4x + 7y = 975$.
|
35
| 0.916667 |
What is the probability that a randomly drawn positive factor of $90$ is less than $10$?
|
\frac{1}{2}
| 0.916667 |
Given that the length of $\overline{AB}$ is triple the length of $\overline{BD}$, and the length of $\overline{AC}$ is $7$ times the length of $\overline{CD}$, calculate the fraction of the length of $\overline{AD}$ that is the length of $\overline{BC}$.
|
\frac{1}{8}
| 0.166667 |
Given the four-digit number 5005, calculate the total number of different numbers that can be formed by rearranging its four digits.
|
3
| 0.583333 |
Tom, Dick, and Harry each flip a fair coin repeatedly until they get their first tail. Calculate the probability that all three flip their coins an even number of times and they all get their first tail on the same flip.
|
\frac{1}{63}
| 0.25 |
Given that $\triangle XYZ$ exists, and $P$ divides side $XZ$ in the ratio $2:3$, while $K$ is the midpoint of $YP$, and $Q$ is the point of intersection of $XK$ and $YZ$, determine the ratio in which $Q$ divides side $YZ$.
|
\frac{2}{5}
| 0.583333 |
How many whole numbers between 1 and 2000 do not contain the digit 2?
|
1457
| 0.166667 |
Given the number of terms in an arithmetic progression (A.P.) is even, the sum of the odd-numbered terms is 36, the sum of the even-numbered terms is 45, and the last term exceeds the first term by 15, and the common difference is a whole number, determine the number of terms in the A.P.
|
6
| 0.666667 |
Given a triangle with side lengths $15, 30$, and $x$, determine the number of integers $x$ for which all its angles are acute.
|
8
| 0.833333 |
Given there are 121 grid points in a square, arranged in an 11 x 11 grid, and point $P$ is at the center of the square, compute the probability that the line $PQ$ is a line of symmetry for the square, given that point $Q$ is randomly selected from the other 120 points.
|
\frac{1}{3}
| 0.583333 |
Given that $A > B > 0$ and $C = A + B$, if $C$ is $y\%$ greater than $B$, calculate the value of $y$.
|
100 \cdot \frac{A}{B}
| 0.25 |
Given the fraction $\frac{987654321}{2^{27}\cdot 5^3}$, determine the minimum number of digits to the right of the decimal point needed to express this fraction as a decimal.
|
27
| 0.25 |
In the $xy$-plane, find the number of lines whose $x$-intercept is a positive prime number greater than 5 and whose $y$-intercept is a positive integer that pass through the point $(6,5)$.
|
2
| 0.833333 |
Given John thought of a positive two-digit number, he multiplied it by $5$ and added $13$, then switched the digits of the result, obtaining a number between $82$ and $86$, inclusive. Determine John's original number.
|
11
| 0.166667 |
Determine how many pairs consist of both students wearing red shirts at a study group event where $65$ students are wearing green shirts and $85$ students are wearing red shirts, with a total of $150$ students divided into $75$ pairs, and exactly $30$ of these pairs consisting of both students wearing green shirts.
|
40
| 0.5 |
Point $P$ moves along a vertical line that intersects side $AB$ of $\triangle PAB$. Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ respectively. As $P$ moves vertically, calculate how many of the four quantities listed change: (i) the length of the segment $MN$, (ii) the perimeter of $\triangle PAB$, (iii) the area of $\triangle PAB$, (iv) the area of trapezoid $ABNM$.
|
3
| 0.916667 |
Let $S$ be a square one of whose diagonals has endpoints $(1/4,3/4)$ and $(-1/4,-3/4)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\leq x\leq 100$ and $0\leq y\leq 100$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly three points with integer coordinates in its interior?
A) $\frac{3}{50}$
B) $\frac{7}{100}$
C) $\frac{1}{50}$
D) $\frac{1}{25}$
E) $\frac{3}{100}$
|
\frac{3}{100}
| 0.25 |
Sam drove 150 miles in 120 minutes. His average speed during the first 30 minutes was 75 mph, and his average speed during the second 30 minutes was 70 mph. Calculate Sam's average speed in mph during the last 60 minutes.
|
77.5
| 0.916667 |
Given that there exist positive integers $A$, $B$, and $C$, with no common factor greater than $1$, such that $A \log_{50} 5 + B \log_{50} 2 = C$, calculate $A + B + C$.
|
4
| 0.916667 |
A digital clock is set to display time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits in this display.
|
24
| 0.083333 |
Given right triangle $ABC$ where angle $B$ is a right angle, point $D$ is the foot of the altitude from $B$ onto hypotenuse $AC$. If $AD = 5$ cm and $DC = 3$ cm, calculate the area of triangle $ABC$.
|
4\sqrt{15}
| 0.25 |
Given $y = x^2 + px + q$, and the least possible value of $y$ is 1, determine the value of $q$.
|
1 + \frac{p^2}{4}
| 0.166667 |
Given the equation \(\frac{9}{x^2} - \frac{6}{x} + 1 = 0\), find the value of \(\frac{2}{x}\).
|
\frac{2}{3}
| 0.583333 |
The bakery owner turns on his doughnut machine at $\text{8:00}\ {\small\text{AM}}$. By $\text{11:40}\ {\small\text{AM}}$, the machine has completed one-fourth of the day's job. At what time will the doughnut machine complete the entire job?
|
10:40\ \text{PM}
| 0.583333 |
A number $m$ is added to the set $\{2, 5, 8, 11\}$ to make the mean of the set of five numbers equal to its median. Determine the number of possible values of $m$ that satisfy this condition.
|
3
| 0.833333 |
Given Peter's family ordered a 16-slice pizza, Peter ate 2 slices alone, shared 1 slice equally with his brother Paul, and shared 1 slice equally with their sister Sarah, and brother Paul. What fraction of the pizza did Peter eat in total?
|
\frac{17}{96}
| 0.583333 |
The sum of the circumferences of the small circles constructed on the circumference of a given circle behaves as $n$ becomes very large.
|
2\pi^2 R
| 0.583333 |
Mia chooses a real number uniformly at random from the interval $[0, 3000]$. Independently, Jake chooses a real number uniformly at random from the interval $[0, 6000]$. Calculate the probability that Jake's number is greater than Mia's number.
|
\frac{3}{4}
| 0.5 |
Let $R_k$ denote an integer whose base-ten representation consists of $k$ ones. For instance, $R_3=111$ and $R_5=11111$. Consider the quotient $Q = R_{30}/R_5$. Calculate the sum of the digits of $Q$, which is an integer whose base-ten representation comprises of only ones and zeros.
|
6
| 0.416667 |
Four years ago, Jim was twice as old as his sister Eliza, and six years before that, Jim was three times as old as Eliza. Let x be the number of years until the ratio of their ages is 3 : 2. Determine the value of x.
|
8
| 0.166667 |
A circle is inscribed in a triangle with side lengths $9, 12$, and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $r$ and $s$, with $r<s$. Find the ratio $r:s$.
|
\frac{1}{2}
| 0.333333 |
There are $12$ horses, each taking unique prime minutes to complete a lap: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37$ minutes respectively. Determine the least time $T > 0$, in minutes, where at least $6$ horses are back at the starting point again together, and calculate the sum of the digits of $T$.
|
6
| 0.916667 |
Given the expression $12 - (3 \times 4)$, calculate Harry's answer, and then subtract Terry's answer, where Terry's answer is obtained by ignoring the parentheses and calculating $12 - 3 \times 4$.
|
-36
| 0.166667 |
Given that Jack purchased 15 pairs of socks for a total of $36, some of which cost $2 a pair, some $3 a pair, and some $4 a pair, and he bought at least one pair of each type, determine the number of pairs of $2 socks Jack bought.
|
10
| 0.916667 |
Determine the remainder when the polynomial $x^4 - 3x^2 + 1$ is divided by the polynomial $x^3 - x - 1$.
|
-2x^2 + x + 1
| 0.583333 |
Find the difference in the total value of Liam's and Mia's fifty-cent coins, where Liam has $3p + 2$ fifty-cent coins and Mia has $2p + 7$ fifty-cent coins, expressed in pennies.
|
50p - 250
| 0.916667 |
In a right trapezoid, one of the non-rectangular vertex angles is divided into angles $x^\circ$ and $y^\circ$, where both angles are prime numbers and the total angle is $90^\circ$, find the least possible value of $y^\circ$.
|
7
| 0.583333 |
Given Josie makes lemonade by using 150 grams of lemon juice, 200 grams of sugar, and 300 grams of honey, and there are 30 calories in 100 grams of lemon juice, 386 calories in 100 grams of sugar, and 304 calories in 100 grams of honey, determine the total number of calories in 250 grams of her lemonade.
|
665
| 0.666667 |
In a $3 \times 3$ grid, numbers $1$ to $9$ are arranged such that if two numbers are consecutive, they must share an edge. The numbers in the four corners add up to $20$. The sum of the numbers along one of the diagonals also equals $15$. Determine the number in the center square.
|
5
| 0.666667 |
Given that $f(x+5) + f(x-5) = f(x)$ for all real $x$, find the least common positive period $p$ for all such functions $f$.
|
30
| 0.416667 |
Given that Lin can climb a flight of stairs 1, 2, 3, or 4 steps at a time, find the total number of different ways Lin can climb 8 stairs.
|
108
| 0.916667 |
Ralph walked down a street and passed five houses in a row, each painted a different color: green, blue, orange, red, and yellow. He passed the orange house before the red house and after the green house. He also passed the blue house before the yellow house, which was not next to the blue house. Additionally, the blue house was not next to the orange house. Determine the total number of orderings of the colored houses.
|
3
| 0.416667 |
Karl's car uses a gallon of gas every 30 miles, and his gas tank holds 16 gallons when it is full. One day, Karl started with a full tank of gas, drove 360 miles, bought 10 gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. Determine the total distance Karl drove that day.
|
540
| 0.916667 |
What is the maximum value of $\frac{(3^t - 4t)t}{9^t}$ for real values of $t$?
|
\frac{1}{16}
| 0.083333 |
Given that $f(x+5) + f(x-5) = f(x)$ for all real $x$, find the least common positive period $p$ for all such functions $f$.
|
30
| 0.583333 |
Given the complex numbers $2+i$, $-1+2i$, and $-2-i$ are vertices of a square in the complex plane, find the fourth complex number which completes the square.
|
1-2i
| 0.833333 |
Given that the sum of a polygon's interior angles is $2843^\circ$ and one angle was accidentally left out, determine the degree measure of the missed angle.
|
37
| 0.75 |
Find the integer $d$ such that the remainder $r$ is the same when each of the numbers $1210, 1690$, and $2670$ is divided by $d$, and calculate the value of $d-4r$.
|
-20
| 0.916667 |
Given the numbers $1, 2, 2, 3, 4, 4, 4, 5$, find the sum of the mean, median, and mode of these numbers.
|
10.625
| 0.916667 |
An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 16, find the area of the hexagon.
|
24
| 0.75 |
If 2 pints of paint are needed to paint a statue 8 ft. high, calculate the number of pints it will take to paint 800 statues similar to the original but each only 2 ft. high.
|
100
| 0.833333 |
For how many integers x does a triangle with side lengths 7, 15, and x have all its angles acute?
|
3
| 0.75 |
Given that there are 10 children and a total of 26 wheels, determine the number of tricycles.
|
6
| 0.916667 |
Given that point $E$ is the midpoint of side $\overline{BC}$ in square $ABCD$, and $\overline{DE}$ meets diagonal $\overline{AC}$ at $F$, determine the area of square $ABCD$ if the area of quadrilateral $DFEC$ is $36$.
|
144
| 0.666667 |
Given that Shea is now 75 inches tall, having grown 25% from a common initial height with Mika, and Mika has grown 10% of her original height, determine Mika's current height in inches.
|
66
| 0.416667 |
Given that $\frac{3}{8}$ of the knights are red, and the rest are blue, and $\frac{1}{5}$ of all the knights are magical, the fraction of magical red knights is $3$ times the fraction of magical blue knights, calculate the fraction of red knights that are magical.
|
\frac{12}{35}
| 0.333333 |
Determine the number of factors the polynomial $x^{15} - x$ has when factored completely using polynomials and monomials with integral coefficients.
|
5
| 0.833333 |
Julia leaves Green Bay at 7:00 AM heading for Appleton on her scooter, traveling at a uniform rate of 15 miles per hour, and Mark leaves Appleton at 7:45 AM heading for Green Bay on his scooter, traveling at a uniform rate of 20 miles per hour. They travel on the same 85-mile route between Green Bay and Appleton. Determine the time at which they meet.
|
9:51
| 0.833333 |
A store announces a "40% off everything" sale. Furthermore, a coupon offers an additional 30% discount on the discounted sale price. Calculate the percentage off the original price that the final price represents.
|
58\%
| 0.916667 |
Let's consider a number $300^7$. Calculate the number of positive integer divisors of $300^7$ that are perfect squares or perfect cubes (or both).
|
313
| 0.75 |
Let x, y, and z be the three numbers such that their sum is 120. The ratio of the first to the second is 3/4 and the ratio of the second to the third is 3/5. Find the value of y.
|
\frac{1440}{41}
| 0.916667 |
Given that the erroneous product of two positive integers $a$ and $b$, where the digits of the two-digit number $a$ are reversed, is $189$, calculate the correct value of the product of $a$ and $b$.
|
108
| 0.083333 |
Given Paula initially had enough paint to paint 36 rooms, but lost four cans leaving her enough paint for only 28 rooms, determine the number of cans of paint she used for the 28 rooms.
|
14
| 0.916667 |
Given a rectangular box-shaped fort with dimensions 15 feet in length, 12 feet in width, and 6 feet in height, and walls and floor uniformly 1.5 feet thick, calculate the total number of one-foot cubical blocks used to build the fort.
|
594
| 0.5 |
Given Tom can make 6 pies in one batch, and each pie is a circle with a radius of 8 cm, determine the number of pies Bob can make in one batch, if Bob's pies are right-angled triangles with legs of 6 cm and 8 cm, using the same amount of dough as Tom.
|
50
| 0.583333 |
Consider a $3 \times 5$ rectangular grid, determine the maximum number of X's that can be placed such that no four X's align in a row vertically, horizontally, or diagonally.
|
9
| 0.833333 |
Simplify the expression $(x - y)^{-2}(x^{-1} - y^{-1})$ and express it with negative exponents.
|
-\frac{1}{xy(x - y)}
| 0.5 |
A circle centered at O is divided into 16 equal arcs labeled from $A$ through $P$. Points X and Y are chosen so that angle X at the circumference spans 1 arc and angle Y at the circumference spans 6 arcs. Calculate the sum of the degrees in the angles X and Y.
|
78.75^\circ
| 0.833333 |
Positive integers $a$ and $b$ are each less than $11$. Calculate the smallest possible value for $2 \cdot a - a \cdot b$.
|
-80
| 0.333333 |
A digital watch in a 24-hour format displays hours and minutes. What is the largest possible sum of the digits in this display?
|
24
| 0.083333 |
Given that Sarah and Jill start a swimming race from opposite ends of a 50-meter pool, and they cross paths two minutes after they start, determine the time it takes for them to cross paths for the second time.
|
6
| 0.166667 |
Eight semicircles are inscribed in a regular octagon such that their diameters coincide with the sides of the octagon. Each side of the octagon has a length of 3. Calculate the area of the region inside the octagon but outside all of the semicircles.
|
18(1+\sqrt{2}) - 9\pi
| 0.75 |
What is the value of $\sqrt{\log_4{8} + \log_8{4}}$?
|
\sqrt{\frac{13}{6}}
| 0.416667 |
Given the graphs of $y = -2|x-a| + b$ and $y = 2|x-c| + d$ intersect at points $(1,7)$ and $(7, 3)$, determine $a+c$.
|
8
| 0.583333 |
The maximum possible median number of cans of soda bought per customer on that day can be calculated if each of 70 customers purchased 5 cans and each of the other 50 customers purchased only 1 can.
|
5
| 0.75 |
Given that point D is a point on side BC such that AD = DC and angle DAC measures 50 degrees, calculate the degree measure of angle ADB.
|
100^\circ
| 0.583333 |
For each positive integer n > 1, let \( P(n) \) denote the greatest prime factor of \( n \). Determine how many positive integers n satisfy both \( P(n) = \sqrt{n} \) and \( P(n+50) = \sqrt{n+50} \).
|
0
| 0.833333 |
The probability that when all 10 dice are rolled, the sum of the numbers on the top faces is 15, equals the probability that the sum of the numbers on the top faces is what value.
|
55
| 0.583333 |
A month with 30 days has the same number of Sundays and Tuesdays. How many of the seven days of the week could be the first day of this month?
|
3
| 0.166667 |
How many even positive 3-digit integers are divisible by 5 but do not contain the digit 5?
|
72
| 0.75 |
If the ratio of the legs of a right triangle is 2:3, determine the ratio of the corresponding segments of the hypotenuse made by a perpendicular from the vertex.
|
\frac{4}{9}
| 0.5 |
Given quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at $O$, $BO=4$, $OD=5$, $AO=9$, $OC=2$, and $AB=7$, find the length of $AD$.
|
\sqrt{166}
| 0.5 |
Given the function $g(x) = bx^3 - 1$, determine the value of $b$ if $g(g(1)) = -1$.
|
1
| 0.833333 |
A triangle has an area of $50$ square units, one side of length $12$, and the median to that side of length $13$. Calculate the sine of the acute angle $\theta$ formed by that side and the median.
|
\frac{25}{39}
| 0.416667 |
The probability that the sum of the numbers on the top faces of the 8 dice is 12 is equal to the probability that the sum of the numbers on the top faces of the 8 dice is another value.
|
44
| 0.833333 |
Three different numbers are randomly selected from the set $\{-3, -2, -1, 1, 2, 3, 4\}$ and multiplied together. Calculate the probability that the product is negative.
|
\frac{19}{35}
| 0.5 |
Given $2^{3x} = 128$, calculate the value of $2^{-x}$.
|
\frac{1}{2^{\frac{7}{3}}}
| 0.166667 |
Given the set $\{-10, -7, -3, 0, 4, 6, 9\}$, find the minimum possible product of three different numbers from this set.
|
-540
| 0.166667 |
Given that Larry, Julius, and Nina are playing a game where they take turns throwing a ball at a bottle with probabilities of $\frac{1}{3}$ of knocking it off the ledge at each turn, calculate the probability that Larry wins the game.
|
\frac{9}{19}
| 0.916667 |
George walks $2$ miles to school at a normal speed of $4$ miles per hour. He walked the first $\frac{3}{4}$ mile at $3$ miles per hour and the next $\frac{3}{4}$ mile at $4$ miles per hour. Find the speed at which George must run the last $\frac{1}{2}$ mile to arrive just as school begins.
|
8
| 0.833333 |
Given the set ${2, 5, 8, 11, 14, 17, 20}$, determine the number of different integers that can be expressed as the sum of three distinct members of this set.
|
13
| 0.25 |
Handy Fiona helped her neighbor, spending 1 hour 30 minutes on Monday, 1 hour 15 minutes on Tuesday, from 9:10 AM to 12:20 PM on Wednesday, and 45 minutes on Thursday. She is paid $\textdollar4$ per hour. Calculate the total amount Handy Fiona earned for the week.
|
26.67
| 0.5 |
Given $T_n = 2 + (1 - 2) + (3 - 4) + \cdots + (-1)^{n-1}n$, evaluate $T_{19} + T_{34} + T_{51}$.
|
25
| 0.416667 |
In $\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ bisect the right angle. If the area of $\triangle CHA$ is $K$, calculate the area of $\triangle ABC$.
|
2K
| 0.166667 |
Determine for how many positive integers \( n \) the expression \( n^3 - 7n^2 + 17n - 11 \) is a prime number.
|
1
| 0.583333 |
Given $2^w \cdot 3^x \cdot 5^y \cdot 7^z = 13230$, calculate the value of $3w + 2x + 6y + 4z$.
|
23
| 0.916667 |
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