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What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 60? | 4087 | 0.833333 |
Susie buys 5 muffins and 4 bananas. Calvin spends three times as much as Susie for 3 muffins and 20 bananas. Express the cost ratio of a muffin to a banana as a fraction. | \frac{2}{3} | 0.916667 |
A cylindrical oil tank, standing vertically, has a height of 12 feet and an interior diameter of 4 feet. The oil covers the entire circular base and reaches a height of $h$ feet inside the tank. If the volume of the oil is 48 cubic feet, determine the height of the oil in the tank. | \frac{12}{\pi}\text{ feet} | 0.916667 |
In a round-robin tournament with 8 teams, where each team plays one game against each other team, and each game results in one team winning and one team losing, what is the maximum number of teams that could be tied for the most wins at the end of the tournament? | 7 | 0.333333 |
Given that $x$ and $y$ are distinct nonzero real numbers such that $x^2 + \frac{2}{x} = y + \frac{2}{y}$, find the value of $xy$. | 2 | 0.666667 |
Given the number of blue chips is at least one third of the white chips and at most one quarter of the red chips, and the number of chips that are either white or blue amounts to at least 72, determine the minimum number of red chips. | 72 | 0.916667 |
Find the remainder when the polynomial $x^4 + 3x^3 - 4$ is divided by the polynomial $x^2 - 1$. | 3x - 3 | 0.916667 |
Given that the sum of two numbers is 15 and their product is 36, find the sum of their reciprocals and the sum of their squares. | 153 | 0.25 |
Triangle PQR has vertices P = (1,2), Q = (-2,5), and R, where R is on the line x + y = 9. Calculate the area of triangle PQR. | 9 | 0.916667 |
Peter and his family ordered a 16-slice pizza. Peter ate two slices and shared a third of another slice equally with his sister, Lily. What fraction of the pizza did Peter eat? | \frac{13}{96} | 0.25 |
Given a $4\times 4$ block of dates starting from 5 and increasing by 1 sequentially, first reverse the order in the columns, then calculate the positive difference between the sums of the two diagonals. | 0 | 0.916667 |
A number y is given. x is p percent less than y and z is q percent less than y. Express p and q in terms of y given that x is 10 units less than y and z is 20 units less than y. | p = \frac{1000}{y}, q = \frac{2000}{y} | 0.75 |
Given that 20% of the students scored 60 points, 15% scored 75 points, 40% scored 85 points, and the remaining students scored 95 points, determine the difference between the mean and the median score of the students' scores on this test. | 4 | 0.333333 |
Given letters $A, B, C,$ and $D$ represent four different digits selected from $1, 2, \ldots, 9$, calculate the value of $A+B+1$, where $\frac{A+B+1}{C+D}$ is an integer that is as large as possible. | 18 | 0.833333 |
A circle passes through the vertices of a right triangle with side lengths $8, 15, 17$. What is the radius of this circle? | \frac{17}{2} | 0.083333 |
Two distinct numbers are selected from the set $\{1,2,3,4,\dots,38\}$ so that the sum of the remaining $36$ numbers equals the product of these two selected numbers plus one. Find the difference of these two numbers. | 20 | 0.75 |
In Mr. Lee's classroom, there are six more boys than girls among a total of 36 students. What is the ratio of the number of boys to the number of girls? | \frac{7}{5} | 0.916667 |
What is the value of \[\frac{3^{2015} - 3^{2013} + 3^{2011}}{3^{2015} + 3^{2013} - 3^{2011}}?\] | \frac{73}{89} | 0.25 |
Given trapezoid ABCD with side $\overline{AD}$ perpendicular to side $\overline{DC}$, lengths AD = AB = 3, and DC = 7. Point E is on side $\overline{DC}$ such that $\overline{BE}$ is parallel to $\overline{AD}$, $\overline{BE}$ equally divides DC into sections with segment DE = 3, calculate the area of $\triangle BEC$. | 6 | 0.833333 |
Given the two-digit number $10a+b$ and its reverse $10b+a$, where $a$ and $b$ are the tens and units digits respectively, the difference between the two numbers is $10b+a - (10a+b) = 9b-9a = 7(a+b)$. Find the sum of the two-digit number and its reverse. | 99 | 0.25 |
A $3 \times 3$ grid is populated randomly with the numbers $1,2,...,9$, where each number is used exactly once. Calculate the probability that the product of numbers in each row is odd. | 0 | 0.916667 |
Determine the probability that a 4 × 4 square grid becomes a single uniform color (all white or all black) after rotation. | \frac{1}{32768} | 0.833333 |
A tetrahedron has a triangular base with sides all equal to 2, and each of its three lateral faces are squares. A smaller tetrahedron is placed within the larger one so that its base is parallel to the base of the larger tetrahedron and its vertices touch the midpoints of the lateral faces of the larger tetrahedron. Calculate the volume of this smaller tetrahedron. | \frac{\sqrt{2}}{12} | 0.5 |
Solve for the number of distinct real solutions to the equation $|x - |3x - 2|| = 5$. | 2 | 0.833333 |
Given parallelogram ABCD has AB = 24, BC = 30, and the height from A to DC is 12. Find the area of the parallelogram. | 288 | 0.75 |
A rectangular yard has two flower beds in the form of congruent isosceles right triangles. The rest of the yard is a trapezoid. The parallel sides of the trapezoid measure $18$ and $30$ meters. Determine the fraction of the yard occupied by the flower beds. | \frac{1}{5} | 0.083333 |
Given $g(x) = x^2 + 4x + 4$, count the number of integers $t$ in the set $\{0, 1, 2, \dots, 30\}$ such that $g(t)$ has a remainder of zero when divided by $10$. | 3 | 0.916667 |
Given 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 $u$ and $v$, with $u<v$. Find the ratio $u:v$. | \frac{1}{2} | 0.416667 |
For any positive integer $N$, the notation $N!$ denotes the product of the integers $1$ through $N$. Determine the largest integer $n$ for which $5^n$ is a factor of the sum $120! + 121! + 122!$. | 28 | 0.916667 |
Given the equation $x^4 + y^2 = 6y - 3$, determine how many ordered pairs of integers $(x, y)$ satisfy the equation. | 0 | 0.666667 |
How many positive factors of 72 are also multiples of 9? | 4 | 0.916667 |
Four friends — Alex, Betty, Clara, and Dave — participated in a relay race by running in pairs, with one pair sitting out each race. Dave ran in 8 races, which was more than any other friend, and Betty ran in 3 races, which was fewer than any other friend. Determine the total number of races those pairs completed. | 10 | 0.25 |
In triangle $ABC$, $AC=2 \cdot AB$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, such that $\angle BAE = \angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose $\triangle AFE$ is isosceles with $AF=FE$, and that $\triangle CFE$ is equilateral. Find the measure of $\angle ACB$.
A) $30^\circ$
B) $45^\circ$
C) $60^\circ$
D) $75^\circ$
E) $90^\circ$ | 60^\circ | 0.833333 |
Determine the value of x that minimizes the expression \( y = (x - a)^2 + (x - b)^2 + c(x - d)^2 \). | \frac{a+b+cd}{2+c} | 0.916667 |
Determine the value of $\sqrt{x^2 + y^2}$, given that $8x + 15y = 120$ and both $x$ and $y$ must be non-negative. | \frac{120}{17} | 0.416667 |
Find a value of \( y \) that satisfies the equation \( y^2 + (2b)^2 = (3a - y)^2 \). | \frac{9a^2 - 4b^2}{6a} | 0.5 |
Tom ate $60\%$ of a chocolate cake. Jenny took one fourth of what was left. What portion of the cake was still not eaten? | 30\% | 0.333333 |
Casper started with a certain number of candies. On the first day, he ate $\frac{1}{2}$ of his candies and gave $3$ candies to his brother. On the second day, he ate another $\frac{1}{2}$ of his remaining candies and then gave $5$ candies to his sister. On the third day, he again ate half of the candies left and gave $2$ candies to a friend. On the fourth day, Casper ate the last $10$ candies. Determine the initial number of candies Casper had. | 122 | 0.5 |
Given the product of $x^5$, $x^2 + \frac{1}{x^2}$, and $1 + \frac{2}{x} + \frac{3}{x^2}$, determine the degree of the resulting polynomial. | 7 | 0.916667 |
Given a positive number $y$, determine the condition that satisfies the inequality $\sqrt{2y} < 3y$. | y > \frac{2}{9} | 0.666667 |
Given Jo and Blair take turns counting numbers where each player adds 2 to the last number said by the other person, find the 30th number said, given that Jo starts by saying 3. | 61 | 0.916667 |
Anthony earns $25 per hour, and there are two types of taxes deducted from his wage: 2% for federal taxes and 0.5% for state taxes. Calculate the total amount of cents per hour of Anthony's wages used to pay both taxes combined. | 62.5 | 0.583333 |
The perimeter of an equilateral triangle exceeds the perimeter of a regular hexagon by $2001 \ \text{cm}$, and the length of each side of the triangle exceeds the length of each side of the hexagon by $d \ \text{cm}$. Given that the hexagon has a perimeter greater than 0, determine the number of positive integers that are NOT a possible value for $d$. | 667 | 0.666667 |
Determine the values of $k$ such that the points $(1, -2)$, $(3, k)$, and $(6, 2k - 2)$ are collinear. | -10 | 0.916667 |
Given Jane lists the whole numbers $1$ through $50$ once and Tom copies Jane's numbers, replacing each occurrence of the digit $3$ by the digit $2$, calculate how much larger Jane's sum is than Tom's sum. | 105 | 0.416667 |
Given that Ryan took three tests, the first with 30 questions and a score of 85%, the second with 35 questions (20 math and 15 science) with scores of 95% and 80% respectively, and the third with 15 questions and a score of 65%, calculate the percentage of all problems Ryan answered correctly. | 82.81\% | 0.25 |
Given the graphs of $y = -|x-a| + b$ and $y = |x-c| + d$, which intersect at points $(3,6)$ and $(9,2)$, find the value of $a+c$. | 12 | 0.75 |
Express $y$ in terms of $x$, given that $x = 1 + 3^p$ and $y = 1 + 3^{-p}$. | \frac{x}{x-1} | 0.75 |
Given the expression $(3n + 2i)^6$, find the number of integers $n$ for which this expression is a pure integer. | 1 | 0.5 |
A circle is inscribed in a triangle with side lengths $9, 14$, and $20$. Let the segments of the side of length $14$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $\frac{r}{s}$? | \frac{3}{25} | 0.166667 |
A point is chosen at random within a rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{3}{4}$. Find the value of $d$ to the nearest tenth. | 0.5 | 0.166667 |
Given a rectangular grid sized at $5 \times 8$, consisting of alternating light and dark squares, calculate the difference between the number of dark squares and the number of light squares. | 0 | 0.333333 |
A standard deck now includes 4 red cards, 4 green cards, and 4 blue cards, each labeled A, B, C, D. Two cards are dealt from this deck. A winning pair is defined as two cards either of the same color or sharing the same label. Calculate the probability of drawing a winning pair. | \frac{5}{11} | 0.083333 |
Given the base exchange rate of 1 EUR = 0.85 GBP and a 5% fee on the total exchanged amount, calculate the amount of GBP the student will receive for exchanging 100 EUR. | 80.75 | 0.583333 |
A chef bakes a $24$-inch by $15$-inch pan of brownies. Each brownie is cut into pieces that measure $3$ inches by $2$ inches. Calculate the number of pieces of brownie the pan contains. | 60 | 0.416667 |
How many distinct integers can be expressed as the sum of three distinct members of the set $\{2, 5, 8, 11, 14, 17, 20\}$? | 13 | 0.25 |
A 4x4x4 cube is made of $64$ normal dice. Each die's opposite sides sum to $7$. Calculate the smallest possible sum of all of the values visible on the $6$ faces of the large cube. | 144 | 0.083333 |
Given a box containing $30$ red balls, $24$ green balls, $16$ yellow balls, $14$ blue balls, $12$ white balls, and $4$ purple balls, determine the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $12$ balls of a single color will be drawn. | 60 | 0.25 |
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,\pi/2)$. Calculate the probability that $\lfloor \sin x \rfloor = \lfloor \sin y \rfloor$. | 1 | 0.916667 |
Simplify the expression $(-\frac{1}{343})^{-2/3}$. | 49 | 0.916667 |
Suppose $d$ is a digit. For how many values of $d$ is $20.d05 > 20.05$? | 9 | 0.75 |
The area of the ring between two concentric circles is $16\pi$ square inches. Determine the length of a chord of the larger circle that is tangent to the smaller circle. | 8 | 0.916667 |
The volume of the top portion of the water tower is equal to the volume of a sphere, which can be calculated using the formula $V = \frac{4}{3}\pi r^3$. The top portion of the real tower has a volume of 50,000 liters, so we can solve for the radius:
$\frac{4}{3}\pi r^3 = 50,000$
Simplifying, we get $r^3 = \frac{50,000 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.
Now, we want to find the volume of the top portion of Logan’s model. It is given that this sphere should have a volume of 0.2 liters. Using the same formula, we can solve for the radius:
$\frac{4}{3}\pi r^3 = 0.2$
Simplifying, we get $r^3 = \frac{0.2 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.
Now, since the sphere is a model of the top portion of the water tower, the radius of the model is proportional to the radius of the real tower. Therefore, the ratio of the radius of the model to the radius of the real tower is equal to the ratio of the volume of the model to the volume of the real tower.
$\frac{r_{model}}{r_{real}} = \frac{0.2}{50000}$
We can now equate the two expressions for the radius and solve for the height of the model:
$\frac{r_{model}}{60} = \frac{0.2}{50000}$ | 0.95 | 0.333333 |
In an obtuse isosceles triangle, the product of the lengths of the two congruent sides is twice the product of the base length and the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle? | 150^\circ | 0.583333 |
Alex, Mel, and Chelsea play a game that involves 8 rounds. In each round, only one of them wins, and the outcomes of the rounds are independent. Each round, the probability of Alex winning is $\frac{1}{2}$. Mel and Chelsea have equal probabilities of winning any round. Calculate the probability that Alex wins four rounds, Mel wins three rounds, and Chelsea wins one round. | \frac{35}{512} | 0.75 |
A choir consisting of boys and girls is planning a fundraiser. Initially, $60\%$ of the choir are girls. Shortly thereafter, four girls leave and six boys join while two boys leave, and then $50\%$ of the choir are girls. Calculate the initial number of girls in the choir. | 24 | 0.75 |
The real value of $x$ such that $\frac{27^{x-2}}{9^{x-1}} = 81^{3x-1}$. | 0 | 0.916667 |
The Hawks scored a certain number of points, and the total points scored by both teams together is 82. If the difference between the points scored by the Eagles and the Hawks is 18, and the spectator claimed that the Hawks scored 40 points, then determine the actual number of points the Hawks scored. | 32 | 0.833333 |
Nine friends dined at a restaurant and agreed to split the bill equally. Unfortunately, Judi forgot her wallet and could only pay $5, and another friend, Tom, could only pay half of his share. The remaining friends covered these shortages by each paying an extra $3. Calculate the total bill. | 156 | 0.75 |
Given the quadratic equation $x^2 - 5x + k = 0$ where $r$ and $s$ are the roots, create a new quadratic equation $x^2 + ax + b = 0$ such that the roots of this new equation are $r^2 + 1$ and $s^2 + 1$, find the value of $a$ given that $k = 6$. | -15 | 0.916667 |
Given that Ron mistakenly reversed the digits of the two-digit number $a$, and the product of $a$ and $b$ was mistakenly calculated as $221$, determine the correct product of $a$ and $b$. | 527 | 0.333333 |
Mina drives 20 miles at an average speed of 40 miles per hour. Calculate the number of additional miles she must drive at 60 miles per hour to average 55 miles per hour for her entire trip. | 90 | 0.916667 |
The taller tree's height is 60 feet. | 60 | 0.583333 |
For how many integers $x$ does a triangle with side lengths $15, 20$ and $x$ have all its angles acute? | 11 | 0.916667 |
Given the equations $ab = 2c$, $bc = 2a$, and $ca = 2b$, determine the number of ordered triples $(a, b, c)$ of non-zero real numbers that satisfy these equations. | 4 | 0.833333 |
Two numbers have their difference, their sum, and their product related to one another as $1:5:15$. Find the product of these two numbers. | 37.5 | 0.25 |
Given the product of the digits of a 3-digit positive integer equals 30, find the number of such integers. | 12 | 0.333333 |
Let $r$ be the number that results when the base of $a^b$ is quadrupled and the exponent is doubled, where $a,b>0$. If $r$ equals the square of the product of $a^b$ and $x^b$ where $x>0$, find $x$. | 4 | 0.916667 |
The sum of the dimensions of a rectangular prism is the sum of the number of edges, corners, and faces, where the dimensions are 2 units by 3 units by 4 units. Calculate the resulting sum. | 26 | 0.166667 |
Find the value of \( d \) given that runner \( A \) can beat \( B \) by \( 15 \) meters, \( B \) can beat \( C \) by \( 30 \) meters, and \( A \) can beat \( C \) by \( 40 \) meters. | 90 | 0.583333 |
Given the product sequence $\frac{5}{3} \cdot \frac{6}{5} \cdot \frac{7}{6} \cdot \ldots \cdot \frac{a}{b} = 12$, determine the sum of $a$ and $b$. | 71 | 0.416667 |
Given that the tallest building in the campus is 120 meters high, and the top portion is a cone that holds 30,000 liters of water, and Ella's model of this building has a top portion that holds 0.03 liters of water, determine the height of the tallest building in Ella's model. | 1.2 | 0.666667 |
Let $c_1, c_2, \ldots$ and $d_1, d_2, \ldots$ be arithmetic progressions such that $c_1 = 10, d_1 = 90$, and $c_{50} + d_{50} = 500$. Find the sum of the first fifty terms of the progression $c_1 + d_1, c_2 + d_2, \ldots$ | 15000 | 0.75 |
Expand and simplify $(1+x^3)(1-x^4)^2$. | 1 + x^3 - 2x^4 - 2x^7 + x^8 + x^{11} | 0.666667 |
Consider the set of all four-digit rising numbers using the digits 1 through 7. Find the digit that the 35th number in the list from smallest to largest does not contain. | 3 | 0.083333 |
Find the sum of all possible values of $s$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 30^\circ, \sin 30^\circ), (\cos 45^\circ, \sin 45^\circ), \text{ and } (\cos s^\circ, \sin s^\circ)\] is isosceles and its area is greater than $0.1$.
A) 15
B) 30
C) 45
D) 60 | 60 | 0.166667 |
A palindrome between $1000$ and $10,000$ is chosen at random. Find the probability that it is divisible by $11$. | 1 | 0.75 |
If $a=\log_5 243$ and $b=\log_3 27$, find the relationship between $a$ and $b$. | a = \frac{5}{3}b | 0.25 |
In an isosceles triangle, one of the angles measures $60^\circ$. Determine the sum of the three possible values of another angle $y^\circ$ in the triangle. | 180^\circ | 0.916667 |
Determine the number of integer solutions (x, y) to the equation xy - x - y = 3. | 6 | 0.75 |
A box contains $30$ red balls, $25$ green balls, $18$ yellow balls, $15$ blue balls, $12$ purple balls, $10$ white balls, and $7$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $20$ balls of a single color will be drawn? | 101 | 0.166667 |
Let $a + 2 = b + 4 = c + 6 = d + 8 = a + b + c + d + 10$. Solve for the value of $a + b + c + d$. | -\frac{20}{3} | 0.5 |
Erin the ant starts at a given vertex of a tetrahedron and crawls along exactly three edges in such a way that she visits every vertex exactly once. Calculate the number of paths meeting these conditions. | 6 | 0.666667 |
Claire's car's odometer reading was 12321, a palindrome. She drove to a location where her car's odometer showed another palindrome. If Claire drove for 4 hours and her speed never exceeded 65 miles per hour, calculate her greatest possible average speed. | 50 | 0.083333 |
The mean of the 9 data values $70, 110, x, 55, 45, 220, 85, 65, x$ is $x$. Calculate the value of $x$ given that the sum of the 9 data values is $9x$. | \frac{650}{7} | 0.333333 |
Given that $5.17171717\ldots$ can be written as a fraction, find the sum of the numerator and the denominator of this fraction when reduced to lowest terms. | 611 | 0.916667 |
Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate that is faster than the ship. She counts 300 equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts 60 steps of the same size from the front of the ship to the back. Determine the length of the ship in terms of Emily's equal steps. | 100 | 0.583333 |
A 2x2x2 cube is made of $8$ normal dice. Each die's opposite faces sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the small cube? | 48 | 0.083333 |
Given a particle projected with an initial velocity of $180$ feet per second from the ground at an angle of $45^\circ$ to the horizontal, derive the expressions for the horizontal range $R$ and the maximum height $H$ by using the given expression $s = 180t \sin(45^\circ) - 16t^2$ where $s$ is the vertical displacement in feet as a function of time $t$ in seconds. | 253.125 | 0.166667 |
Determine the number of even positive $3$-digit integers that are divisible by $5$ but do not contain the digit $5$. | 72 | 0.583333 |
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