Given $\frac{n}{5}$ and $5n$ are four-digit whole numbers, determine the number of positive integer values of $n$.


Let $f(x)$ denote the sum of the digits of the positive integer $x$, where $x$ is a two-digit number. Determine the number of values of $x$ for which $f(f(x))=4$.


Sarah drove the first one-fourth of her journey on a mountain path, the next 30 miles on a paved road, and the remaining one-sixth on a highway. Determine the total length of Sarah's journey in miles.


Given that Larry has test scores of $75, 65, 85, 95,$ and $60$, determine the minimum test score he would need to raise his average test score by $5$ points with his next test.


Given $510^4$, determine the number of its positive integer divisors that are perfect squares or perfect cubes (or both).


Given that the probability that Alex wins each round is $\frac{3}{7}$ and Mel is three times as likely to win as Chelsea, find the probability that Alex wins four rounds, Mel wins two rounds, and Chelsea wins one round.


Given that $\text{P}$ can go to $\text{R}$ or $\text{S}$, $\text{R}$ can go to $\text{T}$ or directly to $\text{Q}$, $\text{S}$ can go only to $\text{T}$, and $\text{T}$ has a direct path to $\text{Q}$, determine the total number of different routes from point $\text{P}$ to point $\text{Q}$.


Keiko walks once around a track at a constant speed every day. The track has straight sides and semicircular ends, with a width of 8 meters. It takes her 48 seconds longer to walk around the outside edge of the track than around the inside edge. Calculate Keiko's speed in meters per second.


Given Paula the painter initially had enough paint to cover 36 rooms and after losing 4 cans of paint, her capacity reduced to cover 28 rooms, determine how many cans of paint Paula used for the 28 rooms.


A set of three standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.


Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be arithmetic progressions such that $a_1 = 50, b_1 = 100$, and $a_{50} + b_{50} = 850$. Find the sum of the first fifty terms of the progression $a_1 + b_1, a_2 + b_2, \ldots$


The binomial expansion of \((\frac{a}{x^2} + \frac{x^2}{a})^7\) needs to be calculated to determine its fourth term.


Determine the number of distinct terms in the simplified expansion of $[(x+5y)^3(x-5y)^3]^{3}$.


For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abc}$, a repeating decimal with a period of $3$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{xy}$, a repeating decimal with a period of $2$. Determine the smallest possible value of $n$.


Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots, and have coefficients $b$ and $c$ selected from the set of integers $\{1, 2, 3, 7, 8, 9\}$, where $c$ is also an even number?


Given that $d$ is a digit, determine the number of values of $d$ for which $3.0d03 > 3.003$.


Two spinners are shown: Spinner A has sectors numbered 1, 2, 3, and Spinner B has sectors numbered 2, 3, 3, 4. Calculate the probability that the sum of the numbers on which the spinners land is a multiple of 4.


Given a ≠ b, a^3 - b^3 = 27x^3, and a - b = 3x, express the possible values of a in terms of x.


Given $12 \cdot \left( \frac{1}{3} + \frac{1}{4} + \frac{1}{6} + \frac{1}{12} \right)^{-1}$, calculate the expression.


How many $3$-digit positive integers have digits whose product equals $36$?


For how many positive integer values of $n$ are both $\frac{n}{4}$ and $4n$ four-digit whole numbers?


Given that Kai has 5 unique marbles, determine the number of ways to arrange them in a line on a window such that neither the Moonstone nor the Earth marble is positioned at either end of the row.


Given Alex leads by 70 points at the halfway point of a 120-shot tournament, where a bullseye scores 12 points and other possible scores are 10, 5, and 3 points, and Alex scores at least 5 points on each shot, determine the minimum value for $n$ such that $n$ consecutive bullseyes guarantee Alex's victory.


A 20-quart container is fully filled with water. Five quarts are removed and replaced with pure antifreeze liquid. Then, five quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated three more times (for a total of five times). Determine the fractional part of the final mixture that is water.


Suppose \(P = \frac{s}{\sqrt{(1 + k)^n}}\), calculate the value of \(n\).

Solution: \boxed{\frac{2\log\left(\frac{s}{P}\right)}{\log(1 + k)}}


A fair coin is tossed 4 times. What is the probability of getting exactly three consecutive heads?


In her multiplication table from \(0 \times 0\) to \(15 \times 15\), calculate the fraction of numbers that are odd, rounded to the nearest hundredth.


Given the expression \(3^{3^{3^{3}}}\), calculate the number of distinct values this expression can represent by changing the order of exponentiations.


Given Miki has 15 apples and 10 bananas, She uses her juicer to extract 9 ounces of apple juice from 3 apples and 10 ounces of banana juice from 2 bananas. She makes a banana-apple juice blend using 5 apples and 4 bananas. Calculate the percent of the blend that is apple juice.


Suppose $x$ and $y$ are nonzero real numbers such that $\frac{4x+2y}{2x-8y} = 3$. Determine the value of $\frac{2x+8y}{8x-2y}$.


Consider two lines \( L_1 \) and \( L_2 \) with equations \( y = px \) and \( y = qx \) respectively. It is given that \( L_1 \) makes three times the angle with the horizontal x-axis as \( L_2 \) does and that \( L_1 \) has a negative slope that is half the magnitude of the slope of \( L_2 \). If \( q \) is positive and both lines are not vertical, determine the value of \( pq \).


Given that six students are to be seated in three rows of two seats each, with one seat reserved for a student council member who is Abby, calculate the probability that Abby and Bridget are seated next to each other in any row.


Find the coefficient of $x^6$ in the expansion of $\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^9$.


Given that $\log_{3x}729 = x$, find the real value of $x$.


Given a price reduction of $15\%$, a reduction of $10\%$, and a reduction of $5\%$, calculate the single equivalent discount.


Consider the modified sequence of positive integers in which the $n^{th}$ positive integer appears $n+1$ times. Determine the remainder when the $2023^{rd}$ term is divided by 5.


In a small town with a total population of $600$, the population is divided into four parts, and one part represents males, one part represents females, and the other two parts represent children. If there are twice as many children as males, find the number of males in the town.


Two circles with centers at $A$ and $B$ have $\overline{AB} = 50$. Points $P$ and $Q$ lie on both circles, and segment PQ is a common chord of length 40. The radius of circle $A$ is $\tfrac{2}{3}$ times the radius of circle $B$. Let $R$ be the midpoint of segment $\overline{PQ}$. Calculate $\overline{AR} + \overline{BR}$.


Given that the original value of the house is $12000, Mr. A sells it at a 15% loss, and then Mr. B sells it back to Mr. A at a 20% gain after holding it for one year and accruing a 5% annual interest, calculate the total amount Mr. A spends to buy back the house including the interest accrued.


A worker's wage is reduced by 30%. To restore their original salary, what percentage raise is required on their new reduced wage?


Given an isosceles trapezoid $ABCD$, the bases $AB = 10$ units and $CD = 18$ units, and the non-base sides $AD$ and $BC$ are equal in length. The height from $AB$ to $CD$ is $4$ units. Determine the perimeter of $ABCD$.

Solution: \boxed{28 + 8\sqrt{2}}


Cagney can frost a cupcake every 25 seconds, Lacey can frost a cupcake every 35 seconds, and Casey can frost a cupcake every 45 seconds. Working together, calculate the number of cupcakes they can frost in 10 minutes.


Given the first three terms of a geometric progression are $\sqrt[3]{5}$, $\sqrt[5]{5}$, and $\sqrt[15]{5}$, determine the fourth term.


Given a semipro basketball league with teams of 15 players, find the maximum possible salary, in dollars, for a single player, given that each player must be paid at least $20,000 and the total of all players' salaries cannot exceed $800,000.


The point $(x, -3)$ lies on the straight line joining the points $(2, 10)$ and $(6, 2)$ in the xy-plane, find the value of x.


At Greenfield High, 120 students participated in a national math competition, and their mean score was 150. The number of non-seniors was 75% more than the number of seniors. The mean score of the seniors was twice as high as that of the non-seniors. Calculate the mean score of the seniors.


Given a rhombus with one diagonal that is three times the length of its other diagonal, express the side of the rhombus in terms of $K$, where $K$ is the area of the rhombus in square inches.


Moe has a new, larger rectangular lawn measuring 120 feet by 180 feet. He uses a mower with a swath width of 30 inches. However, he overlaps each cut by 6 inches to ensure no grass is missed. Moe walks at a rate of 6000 feet per hour while pushing the mower. What is the closest estimate of the number of hours it will take Moe to mow the lawn?


Given $g(x) = 3x^3 - 4x + 5$, calculate the expression $g(x + h) - g(x)$.

Solution: \boxed{h(9x^2 + 9xh + 3h^2 - 4)}


Given the set of numbers $\{ 85, 91, 99, 132, 166, 170, 175 \}$, calculate the number of subsets containing three different numbers such that the sum of the three numbers is odd.


A woman invests in a property for $12,000 with the aim of receiving a $6\%$ return on her investment after covering all expenses including taxes and insurance. She pays $360 annually in taxes and $240 annually for insurance. She also keeps aside $10\%$ of each month's rent for maintenance. Calculate the monthly rent.


Let \( r \) be the number that results when both the base and the exponent of \( 2^b \) are doubled, where \( b > 0 \) is an integer. If \( r \) equals the product of \( 2^b \) and \( x^b \) where \( x > 0 \), find the value of \( x \).


Chubby creates a new nonstandard checkerboard with dimensions 29 squares on each side. The checkerboard maintains a black square in every corner and alternates red and black squares along every row and column. Calculate the number of black squares on this checkerboard.


Given that a light bulb is located 20 centimeters below the ceiling, the ceiling is 2.4 meters above the floor, Alice is 1.5 meters tall, and she can reach 50 centimeters above the top of her head, determine the height of a stool that will allow her to reach the light bulb.


How many whole numbers between $200$ and $600$ contain the digit $3$?


Seven points on a circle are numbered 1 through 7 in a clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves two points, and if it is on an even-numbered point, it moves three points. Starting on point 7, determine the bug's final position after 2023 jumps.


Given the numbers $12534, 25341, 53412, 34125$, calculate their sum.


Given two concentric circles, where one circle has a radius $r$ and the other has a radius $2r$, determine the number of common tangents that these two circles can have.


For how many positive integer values of $n$ are both $\frac{n}{4}$ and $4n$ three-digit whole numbers?


In a class quiz, $20\%$ of the students scored $70$ points, $20\%$ scored $80$ points, $25\%$ scored $85$ points, $25\%$ scored $90$ points, and the rest scored $100$ points. Find the difference between the mean and median score of the students' scores on this quiz.


The probability that the sum of the numbers on the top faces of 8 dice is 12. What other sum occurs with the same probability as q?

Solution: \boxed{2 \times 28 - 12 = 56 - 12 = 44}


A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 88 and the floor forms a perfect square with an even side length, find the total number of tiles that cover the floor.


Consider a function \( f(x) \) formulated as \( ax^3 + bx^2 + cx + d \). Suppose \( f(x+2) = 2x^3 - x^2 + 5x + 3 \). Determine the value of \( a + b + c + d \).


What is the maximum number of possible points of intersection of a circle and a square?


Right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<50$. Calculate the number of such right triangles.


How many whole numbers between 1 and 2000 do not contain the digits 1 or 2?


The complex number $z$ satisfies $z + |z| = 3 + 12i$. What is $|z|^{2}$?


Calculate the number of 4-digit integers where all digits are different, the first digit is not zero, the integer is a multiple of 5, the last digit is 5, and 6 is the largest digit allowed in the number.


Suppose that $p$ and $q$ are positive integers such that $108p = q^3$. What is the minimum possible value of $p + q$?


Given that \( a \) is a positive real number and \( b \) is an integer between \( 2 \) and \( 500 \), inclusive, find the number of ordered pairs \( (a,b) \) that satisfy the equation \( (\log_b a)^{1001}=\log_b(a^{1001}) \).


Let \( C_1 \) be a circle with radius 2, and \( C_2 \) a circle with radius 3, both positioned in the same plane and tangent to each other externally at one point. How many circles of radius 1 are in this plane and externally tangent to both \( C_1 \) and \( C_2 \)?


Let n be the number of points P inside a circle of radius 1, such that the sum of squares of the distances from P to the endpoints of a given diameter is 5. Find the value of n.


Calculate the product of $\frac{5}{3} \times \frac{6}{5} \times \frac{7}{6} \times \cdots \times \frac{2010}{2009}$.


Let \(\lfloor x\rfloor\) denote the greatest integer not exceeding \(x\), and define the function \[ f(x) = \lfloor x\rfloor(2013^{x-\lfloor x\rfloor}-1). \] Find the sum of the lengths of all intervals for \(x\) such that \(1 \leq x < 2013\) and \(f(x) \leq 1\).


Given circle O, points E and F are on the same side of diameter AB, ∠AOE = 40°, and ∠FOB = 60°. Find the ratio of the area of the smaller sector EOF to the area of the circle.


Given the quadratic equation $2Ax^2 + 3Bx + 4C = 0$ with roots $r$ and $s$, find the value of $p$ such that the quadratic equation $x^2 + px + q = 0$ has roots $r^2$ and $s^2$.

Solution: \boxed{\frac{16AC - 9B^2}{4A^2}}


Calculate the probability that a randomly chosen divisor of $15!$ is divisible by $3$.


Given the equation $x^4+y^2=4y$, determine the number of ordered pairs of integers $(x, y)$.


Three fair coins are tossed once. For each head that results, one fair die is rolled. Determine the probability that the sum of the results of the die rolls is odd.


Given that Luke has written down one integer three times and another integer four times, and the sum of the seven numbers is 150, with one of the numbers being 15, determine the other number.


In a science class, there are $n > 15$ students who have an overall average score of $75$ on a recent test. A group of $15$ of these students, however, scored an average of $85$. Calculate the average score of the remaining students in terms of $n$.


A half-sector of a circle of radius 6 inches is transformed into a cone by joining the two radii. Find the volume of the cone thus formed.

Solution: \boxed{9\pi \sqrt{3}}


Given a pyramid-like structure with a rectangular base consisting of $4$ apples by $7$ apples, each apple above the first level resting in a pocket formed by four apples below, and the stack topped off with a single row of apples, determine the total number of apples in the stack.


Jeremy's father drives him to school in 20 minutes during rush hour, and in 15 minutes when driving 20 miles per hour faster.


Alice and Bob play a game on a circle whose circumference is divided by 15 equally-spaced points, numbered clockwise from 1 to 15. Both start from point 15. Alice moves 7 points clockwise per turn, while Bob moves 11 points counterclockwise per turn. Determine how many turns it will take for Alice and Bob to meet at the same point for the first time.


Two concentric circles have a common center $C$. Chord $\overline{EF}$ is tangent to the inner circle at point $G$, $EC$ is $12$, and the chord $\overline{EF}$ has length $20$. Determine the area between the two circles.

Solution: \boxed{100 \pi}


Joe walked one third of the way from home to school, waited for 3 minutes, and then walked another one third before running the final third to school. He ran 4 times as fast as he walked. If Joe took a total of 12 minutes to walk two thirds of the way to school, determine the total time it took Joe to get from home to school.


The average of the first four numbers is 7, and the average of the last four numbers is 10. If the average of all seven numbers is 8, determine the number common to both sets of four numbers.


Evaluate $|5 - e|$, where $e$ is the base of the natural logarithms.

Solution: \boxed{5 - e}


Given that Alice won 5 games and lost 3 games, Bob won 4 games and lost 4 games, and Carol lost 5 games, determine the number of games Carol won.


Given that a scout troop buys 800 candy bars at a price of six for $3 dollars and incurs an additional expense of $50 on transportation, and they sell all the candy bars at the price of three for $2 dollars, calculate their profit.


A merchant buys goods at 20% off the list price. He desires to mark the goods so that he can give a discount of 25% on the marked price and still clear a profit of 30% on the selling price. What percent of the list price must he mark the goods?


A class of 30 students has taken three tests. The table below shows the distribution of grades for students who received the same grade across all three tests.

| Grade | Students |
|-------|----------|
| A     | 3        |
| B     | 6        |
| C     | 2        |
| D     | 2        |

Calculate the percentage of students who received the same grade on all three tests.


Given on hypotenuse $AB$ of a right triangle $ABC$, another right triangle $ABD$ is constructed with $AB$ as the hypotenuse, and $\overline{BC}=1$, $\overline{AC}=3$, $\overline{AD}=1$, find the length of $\overline{BD}$.


The postage for a package weighing $3.7$ ounces is calculated using the rate of $45$ cents for the first ounce and $25$ cents for each additional ounce, find the total postage cost.


Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from more than half of the other 7 balls.


What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 60?


Given that I put 10 gold coins in each treasure chest, but 4 treasure chests were left empty, and then I put 7 gold coins in each treasure chest, but had 5 gold coins left over, find the total number of gold coins I had.


A sphere with center $O$ has radius $5$. A triangle with sides of length $13, 13, and 10$ is situated in space such that each of its sides is tangent to the sphere. Calculate the distance from $O$ to the plane formed by the triangle.


An uncrossed belt runs without slack around two circular pulleys with radii of $10$ inches and $6$ inches. If the distance between the points of contact of the belt with the pulleys is $26$ inches, calculate the distance between the centers of the pulleys in inches.


Determine the number of ordered pairs of integers \((x, y)\) that satisfy the equation $x^{6} + y^2 = y + 6$.


Given Professor Lopez has eight different language books lined up on a bookshelf: three Russian, two French, and three Italian books, calculate the number of ways to arrange these eight books on the shelf, keeping the Russian books together and keeping the French books together.


Given that Jones purchases an automobile for $210$ monetary units (abbreviated m.u) and gives the salesman a $500$ m.u bill, and receives $260$ m.u in change, determine the value of the base $r$ in the Land of Mathesis.


Given a set $T$ of points in the $xy$-plane symmetric about the origin, the coordinate axes, the line $y=x$, and the line $y=-x$, with the point $(3,4)$ in $T$, determine the smallest number of points in $T$.


Chloe chooses a real number uniformly at random from the interval $[0, 100]$. Laurent, knowing the number Chloe chose, picks a real number uniformly at random from the interval $[0, 3x]$, where $x$ is the number Chloe picked. What is the probability that the number Laurent picks is greater than Chloe's number?


In a single tennis tournament with 104 players, the top 24 players get a bye in the first round. The rest of the players (80) are paired to play in the first round. How many matches are played in total to find the winner?


Walter gets $4$ dollars for doing his chores or $6$ dollars for doing them exceptionally well. Additionally, if Walter does his chores exceptionally well for more than 5 consecutive days, he receives a $10$ dollar bonus. After $12$ days of doing his chores daily, Walter has received a total of $58$ dollars. Determine the number of days Walter did his chores exceptionally well.


How many ordered pairs (a, b) such that a is a positive real number and b is an integer between 3 and 300, inclusive, satisfy the equation (log_b a)^{1009} = log_b (a^{1009})?


The number of solutions in positive integers of $3x + 2y = 841$.


Given that the odometer displayed $27472$ miles initially and the next higher palindrome three hours later, calculate Megan's average speed during this 3-hour period.


Arrange the numbers 2, 3, 5, 7, and 11 in any order. An iterative average is computed by finding the mean of the first two numbers, then computing the mean of that result with the third number, and continuing this method until the mean with the fifth number is computed. Calculate the difference between the largest and smallest possible values that can be obtained through this method.


Consider rectangle ABCD with an area of 96 square meters. Points E and G are the midpoints of sides AB and CD respectively, and points F and H are the midpoints of sides BC and DA respectively. Find the area of trapezoid EFGH in square meters.


Calculate the value of $\dfrac{13! - 11! + 9!}{10!}$.


Given twelve points are selected on the positive x-axis, and six points are selected on the positive y-axis, find the maximum possible number of points of intersection of the line segments that could lie in the interior of the first quadrant.


Two integers have a sum of 22. When two more integers are added to the first two integers, the sum is 36. Finally, when two more integers are added to the sum of the previous four integers, the sum is 50. What is the minimum number of odd integers among these 6 integers.


The length of rectangle PQRS is 8 inches and its width is 6 inches. Diagonal PR is divided into four equal segments by points T, U, and V. Find the area of triangle QTU.


Given Charles has $7q + 2$ quarters and Richard has $3q + 8$ quarters, calculate the difference in their money in nickels.

Solution: \boxed{20q - 30}


From the set $\{-9,-5,-3,0,4,5,8\}$, find the maximum possible product of two different numbers.


Jane ordered 5 pairs of yellow socks and some additional pairs of red socks. The price of each pair of yellow socks was half that of a red pair. When the socks were delivered, the numbers of yellow and red pairs had been exchanged, resulting in a total bill that was 60% higher than originally expected. What was the ratio of the number of pairs of yellow socks to the number of pairs of red socks in Jane's initial order?


Given that $a * b$ means $4a - 2b$, solve the equation $3 * (6 * x) = -2$ for $x$.


Three flower beds, X, Y, and Z, contain X = 600, Y = 480, and Z = 420 plants, respectively. The plants are shared as follows: Beds X and Y share 60 plants, Beds Y and Z share 70 plants, and Beds X and Z share 80 plants. Also, there are 30 plants common to all three beds. Find the total number of unique plants.


Keiko walks around a track where the ends are semicircles, and the sides are straight. The width of the track is now 8 meters, and it takes her 48 seconds longer to walk around the outside edge of the track than around the inside edge. Determine Keiko's speed in meters per second.


Given the real number $y$ selected uniformly at random between 200 and 300, find the probability that $\lfloor \sqrt{50y} \rfloor = 226$.


Given that $x = \log_{10}1000$, evaluate the expression $[\log_{10}(7x)]^2$.


How many subsets containing four different numbers can be selected from the set $\{42, 55, 78, 103, 144, 157, 198\}$ so that the sum of the four numbers is even?


A cowboy is 6 miles south of a stream which flows due east. He is also 10 miles west and 5 miles north of his cabin. Water his horse at the stream and then return to his cabin. Determine the shortest distance he must travel to accomplish this.


Seven positive consecutive integers start with $a$. What is the average of $7$ consecutive integers that start with this computed average, $b$?

Solution: \boxed{a + 6}


A sphere with center $O$ has a radius of $10$. A triangle with sides of length $13, 13$, and $10$ is situated in space so that each of its sides is tangent to the sphere. Find the distance between $O$ and the plane determined by the triangle.


A rectangular garden that is 12 feet wide and 18 feet long is tiled with 216 one-foot square tiles. A bug starts walking from one corner of the garden to the directly opposite corner in a straight line. Including the first and the last tile, calculate the number of tiles the bug visits.


A fair die is rolled eight times. What is the probability of rolling at least a five at least six times?


If $y$ cows give $y+2$ cans of milk in $y+3$ days, calculate the number of days it will take $y+4$ cows to give $y+6$ cans of milk.


Triangle $DEF$ is inscribed in a circle. The measure of the non-overlapping minor arcs $DE$, $EF$, and $FD$ are, respectively, $x+83^{\circ}$, $2x-15^{\circ}$, and $3x+4^{\circ}$. Calculate the measure of one interior angle of the triangle.


Linda drove the first quarter of her journey on a dirt road, the next 30 miles on a highway, and the remaining one-seventh on a city street. What is the total length of Linda's journey in miles?


A bag of popping corn contains $\frac{1}{2}$ white kernels, $\frac{1}{3}$ yellow kernels, and $\frac{1}{6}$ blue kernels. $\frac{3}{4}$ of the white kernels will pop, $\frac{1}{2}$ of the yellow ones will pop, and $\frac{1}{3}$ of the blue ones will pop. Calculate the probability that the kernel selected was white.


Given that $\frac{x}{2} = y^2$ and $\frac{x}{5} = 3y$, solve for $x$.


Given the expression $a+(b-(c+(d-f)))$, and having correctly evaluated it by coincidence while ignoring parentheses, find the value of $f$ that yields the same result when Larry substituted $2$, $3$, $4$, and $5$ for $a$, $b$, $c$, and $d$ respectively.


Five years ago, Tim was three times as old as his sister Sarah, and three years before that, Tim was five times as old as Sarah. Determine the number of years it will take for the ratio of their ages to be 3 : 2.


Given the equation $A \log_{100} 5 + B \log_{100} 2 = C$, find the values of positive integers $A$, $B$, and $C$ which have no common factor greater than 1, and calculate $A + B + C$.


Let $x$ be a real number selected uniformly at random between 150 and 300. If $\lfloor \sqrt{x} \rfloor = 16$, find the probability that $\lfloor \sqrt{10x} \rfloor = 160$.


The two concentric circles have the same center $C$. Chord $\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $12$, and chord $\overline{AD}$ has length $20$. Determine the area between the two circles.


Given the total population of birds at Town Lake after a 20% increase is 120 birds, and the distribution is 40% geese, 20% swans, 20% herons, and 20% ducks, calculate the percentage of birds that were not swans and were geese.


Find the largest product obtained by multiplying three different numbers from the set $\{ -4, -3, -2, 0, 3, 5 \}$.


In the xy-plane, determine how many lines whose x-intercept is a positive odd number and whose y-intercept is a positive prime number pass through the point (4,3).


Given a square grid diagram with 49 uniformly spaced points including points on the edges, point P is located at the center. If point Q is randomly selected from the remaining 48 points, determine the probability that the line PQ is a line of symmetry of the square.


Given that for two numbers $a$ and $b$, the operation $*$ between them is defined as $a * b = 4a + 2b$, solve the equation $3 * (4 * y) = -2$ for the value of $y$.


John and Peter went dining and tipped their server different amounts. John tipped $4$ dollars, which is $20\%$ of his bill. Peter tipped $3$ dollars, which is $15\%$ of his bill. What was the difference, in dollars, between their bills?


A palindrome between $100$ and $1000$ is chosen at random. Find the probability that it is divisible by $5$.


How many lattice points are on the line segment whose endpoints are $(7, 23)$ and $(67, 391)$?


Given that Julia goes for a jog around a circular track that covers a total of 28 miles and completes her run in 4 hours, calculate Julia's average speed in miles per hour.


The mean of three numbers is $20$ more than the least of the numbers and $30$ less than the greatest, and the median is $10$. Determine the sum of the three numbers.


Four integers, added three at a time, give the sums $160, 185, 205,$ and $230$. Find the largest of these four integers.


Given that $\frac{1}{4}$ of ninth graders are assigned exactly one sixth-grade buddy and this group constitutes $\frac{1}{3}$ of all sixth graders, determine the fraction of the total number of sixth graders and ninth graders who have a buddy.


Given the number $58$, find the number of ways it can be written as the sum of two prime numbers.


Given a rectangular array with $50$ rows and $40$ columns, determine the value of $\frac{C}{D}$.


Tom and Jerry are fishing 2 miles offshore when their boat begins to leak, admitting water at a steady rate of 15 gallons per minute. If their boat can sustain a maximum of 60 gallons of water before sinking, and Tom rows towards the shore at a speed of 3 miles per hour, determine the minimum rate at which Jerry must bail water, in gallons per minute, to avoid sinking before reaching the shore.

Solution: \boxed{13.5 \text{ gallons per minute}}


Find the units digit of the decimal expansion of (7 + \sqrt{50})^{21} + (7 - \sqrt{50})^{21}.


A class collects $60$ dollars to buy flowers for a teacher. Each tulip costs $4$ dollars and each sunflower costs $3$ dollars. They decide to buy more sunflowers than tulips. Determine the number of different combinations of tulips and sunflowers that could be purchased for exactly $60$ dollars.


Given that the average score for the morning group is 90 and the evening group is 80, and the ratio of the number of students in the morning group to the evening group is $\frac{4}{5}$, calculate the overall average score of the students from both groups.


Calculate \( \left| x - 3 - \sqrt{(x - 4)^2} \right| \) for \( x < 3 \).

Solution: \boxed{7 - 2x}


A $4 \times 4$ square is divided into $16$ unit squares. Each unit square is painted either white or black, each with a probability of $\frac{1}{2}$, independently. The square is then rotated $180^\circ$ about its center. After rotation, any white square that occupies a position previously held by a black square is repainted black; other squares retain their original color. Determine the probability that the entire grid is black after this process.


What is the maximum number of consecutive integers that can have a sum of $36$?


Let $a + 2 = b + 3 = c + 4 = d + 5 = a + b + c + d + 6$. Calculate the value of $a + b + c + d$.


Given the function $g$ defined by the table, where $x$ is mapped to $g(x)$ as follows: $\begin{tabular}{|c||c|c|c|c|c|}  \hline   x & 1 & 2 & 3 & 4 & 5 \\   \hline  g(x) & 3 & 4 & 2 & 1 & 5 \\  \hline \end{tabular}$, find $v_{2023}$ given that $v_0 = 3$ and $v_{n+1} = g(v_n)$ for $n \ge 0$.


Given that a cell phone protection plan costs $25$ dollars each month, plus $3$ cents per text message for the first $120$ messages and $2$ cents for each additional message, and $15$ cents for each minute used beyond $25$ hours of talk time, calculate the total monthly cost for a user who sent $140$ text messages and talked for $27$ hours.


The "Academy League" basketball conference has 8 teams. Each team plays every other team in the conference twice, and 6 games against non-conference opponents. Calculate the total number of games in a season involving the "Academy League" teams.


Elena is planning to spend a portion of her savings to purchase several notebooks that are all priced equally. She spends one fourth of her savings to purchase one half of the total number of notebooks she wants. What fraction of her savings will she have left after purchasing all the notebooks?


Given that the first digit is fixed as 1, the second digit is 5, and all other digits are distinct and odd, calculate the number of seven-digit numbers.


Connie subtracts a number by $8$ and gets $32$ as her answer. However, she should have added $8$ to get the correct answer. What is the correct answer?


How many four-digit integers have a remainder of 3 when divided by 7, a remainder of 6 when divided by 10, and a remainder of 9 when divided by 13?


Given a circle divided into 18 equally-spaced points, where Alice moves 7 points clockwise and Bob moves 13 points counterclockwise each turn, starting at point 18, determine the turn number at which they meet on the same point.


Given a rhombus $ABCD$ on a plane where $AB=BC=CD=DA=1$, points $\overrightarrow{AA^\prime}$, $\overrightarrow{BB^\prime}$, $\overrightarrow{CC^\prime}$, and $\overrightarrow{DD^\prime}$ are vertical rays in the same direction. If $AA^{\prime} = 12$, $BB^{\prime}= 20$, $CC^\prime = 12$, and $DD^\prime = 20$, find the length of $PQ$, where $P$ and $Q$ are the midpoints of $A^{\prime} C^{\prime}$ and $B^{\prime}D^{\prime}$, respectively.


Lucas read an average of 42 pages per day for the first four days and 50 pages per day for the next two days, then he read 30 pages on the final day. Calculate the total number of pages in the novel.


Given a cell phone plan that costs $25$ dollars each month, plus $10$ cents per text message sent, plus $15$ cents for each minute used over $20$ hours, calculate the total cost for John, who sent $150$ text messages and talked for $22$ hours in January.


Given that there are $1$ red tile, $2$ blue tiles, $2$ green tiles, and $4$ yellow tiles, calculate the number of distinguishable arrangements of the tiles in a row from left to right.


If 40% of the birds were flamingos and 20% were parrots, calculate the percent of the remaining birds that were flamingos after excluding the parrots.


Suppose a calculator has a special key that replaces the number \(x\) displayed with the number given by the formula \(f(x) = \frac{x+1}{x-1}\). Starting with an initial display of 3, find the number that will be displayed after 200 successive operations.


Given that Odell and Kershaw run for 30 minutes on a circular track, Odell runs clockwise at 260 m/min in the inner lane with a radius of 52 meters and Kershaw runs counterclockwise at 310 m/min in the outer lane with a radius of 62 meters, starting on the same radial line as Odell, calculate the number of times after the start they pass each other.


Given the equation $2 - 4\tan\theta + 3\cot2\theta = 0$, find the number of values of $\theta$ in the interval $0 < \theta < \pi$ that satisfy the equation.


In a garden with three intersecting flower beds, beds X and Y contain 600 plants and 500 plants respectively, beds Y and Z contain 500 and 400 plants respectively, and beds X and Z contain 600 and 400 plants respectively. Additionally, 100 plants are shared between beds X and Y, 80 plants are shared between beds Y and Z, and 120 plants are shared between beds X and Z. Furthermore, 30 plants are shared among all three beds. Calculate the total number of distinct plants present in these beds.


Determine the number of distinct terms in the expansion of $[(a+5b)^2 (a-5b)^2]^3$ when simplified.


Given that Al takes one green pill, one pink pill, and one blue pill each day for three weeks, a green pill costs $\$1$ more than a pink pill, while a blue pill costs $\$2$ less than a pink pill, and the total cost of Al's pills for the three weeks is $\$819$, determine the cost of one green pill.


Given a square with side length determined by the sum of the numbers rolled from a standard 6-sided dice, calculate the probability that the area of the square is less than its perimeter.


What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 70?


Find the number of solutions in positive integers to the equation $3x + 5y = 805$.


Given that the state income tax where Alex lives is levied at the rate of $q\%$ of the first $\textdollar 50000$ of annual income plus $(q + 3)\%$ of any amount above $\textdollar 50000$, and that the state income tax he paid amounted to $(q + 0.5)\%$ of his annual income, find Alex's annual income.


The Lions defeated the Eagles 3 out of 5 times they played initially. They then played $N$ more times, and the Eagles ended up winning at least 90% of all the games played. What is the minimum possible value for $N$?


Let \( x = -2023 \). Calculate the value of $\Bigg\vert\Big\vert |x|-x\Big\vert-|x|\Bigg\vert-x$.


The difference between a two-digit number and the number obtained by reversing its digits is 7 times the sum of the digits of either number. Calculate the sum of the two-digit number and its reverse.


The merchant buys goods at 30% off the list price. He wishes to mark the goods so that he can offer a discount of 20% on the marked price and still achieve a profit of 30% on the selling price. What percent of the list price must he mark the goods?


Calculate the sum of the arithmetic series: $22 + 22\frac{3}{7} + 22\frac{6}{7} + \cdots + 73$.


The numbers on each diagonal of a $5\times5$ block of calendar dates are added, where the order of the numbers in the second and the fifth rows are first reversed. Find the positive difference between the two diagonal sums.


If a number nine times as large as $x$ is decreased by three, then one third of the result equals $\frac{1}{3}\left(9x-3\right)$.

Solution: \boxed{3x - 1}


Given the numbers 294 and 490, calculate the ratio of the least common multiple to the greatest common factor.


Four fair dice are tossed. What is the probability that the numbers turned up can be arranged to form an arithmetic progression with a common difference of one?


You walk for 60 minutes at a speed of 3.5 mph and then run for 45 minutes at a speed of 8 mph. Calculate the total distance traveled in miles after a 15 minute break between walking and running.


The Tigers beat the Sharks 3 out of 5 times they initially played. Then, they played $N$ more games, and the Sharks ended up winning more than 90% of all the games played. Find the minimum possible value for $N$.


For how many integers \( x \) does a triangle with side lengths 12, 20, and \( x \) have all its angles acute?


Given a line with the equation $2x - 3y + 6 = 0$ and points $(p, q)$ and $(r, s)$ lying on this line, calculate the distance between these two points in terms of $p, r,$ and the coefficients of the line.

Solution: \boxed{\frac{\sqrt{13}}{3}|r - p|}


In the expression $\left(x\times y\right)-\left(z\times w\right)$, the variables $x, y, z$, and $w$ are to be filled with one of the digits $1,2,3,$ or $5,$ with each digit being used once. Calculate the number of different values that can be obtained.


Given the fraction $\frac{987654321}{2^{24}\cdot 5^6}$, determine the minimum number of digits to the right of the decimal point needed to express it as a decimal.


What is the value of the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 10}{1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2}$?


Consider the infinite series \(1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187} - \cdots\). Let \(T\) be the sum of this series. Determine \(T\).


In a right triangle, one of the acute angles is a prime number while the other can be either composite or prime. The angle that is prime is denoted by $p^{\circ}$ and the non-prime or optionally prime angle is denoted by $q^{\circ}$ with $p > q$. What is the least possible value of $q$?


The value of $p$ when the third and fourth terms of the expansion of $(x+y)^{11}$ have equal values when evaluated at $x=p$ and $y=1-p$.


Determine the degree of the polynomial when $\left(x^5\right)$, $\left(x + \frac{1}{x}\right)$, and $\left(1 + \frac{3}{x} + \frac{4}{x^2} + \frac{5}{x^3}\right)$ are multiplied together.


Points P and Q are 12 units apart, Q and R are 7 units apart, and R and S are 2 units apart. Determine the minimum distance between points P and S.


Given that Connie multiplies a number by $3$ and gets $90$ as her result, but should have subtracted $30$ from the number to get the correct answer, determine the correct number.


Given the competition begins at 3:15 PM and lasts for 1825 minutes, determine the end time of the competition.

Solution: \boxed{9:40 \text{ PM}}


Two concentric circles have radii of 2 and 4. Given two points chosen independently and uniformly at random on the outer circle, calculate the probability that the chord joining the two points intersects the inner circle.


Given $\bigtriangleup ABC$, $AB = 75$, and $AC = 100$. A circle with center $A$ and radius $75$ intersects $\overline{BC}$ at points $B$ and $Y$ and also intersects $\overline{AC}$ at another point $Z$ besides $A$, find the length of $BC$ if $\overline{BY}$ and $\overline{CY}$ are integer lengths.


Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$.


What time was it 1453 minutes after midnight on January 1, 2021?

Solution: \boxed{12:13 \text{ AM}}


Soda is sold in packs of 8, 15, and 18 cans. If a minimum of 95 cans are to be bought using no more than 4 packs of each type, determine the minimum number of packs needed.


If it is known that $\log_3(a) + \log_3(b) \geq 5$ and $a \geq b$, determine the least value that can be taken on by $a+b$.


Calculate the sum $S_{15} + S_{28} + S_{39}$, where $S_n=1-2+3-4+\cdots+(-1)^{n-1}n$ for integer values of $n$.


Given that the car ran exclusively on its battery for the first $60$ miles and then on gasoline at a rate of $0.03$ gallons per mile for the remainder of the trip, with an average of $50$ miles per gallon for the whole trip, determine the total length of the trip in miles.


Given Professor Lee has ten different mathematics books on a shelf, consisting of three calculus books, four algebra books, and three statistics books, determine the number of ways to arrange the ten books on the shelf keeping all calculus books together and all statistics books together.


Penniless Pete's piggy bank has no pennies in it, but it now contains 120 coins, all nickels, dimes, and half-dollars, with a total value of $10.50. It does not necessarily contain coins of all three types. Determine the difference between the largest and smallest number of dimes that could be in the bank.


What is the smallest sum of two 3-digit numbers that can be obtained by placing each of the six digits 1, 2, 3, 7, 8, and 9 in one of the six boxes in this addition problem?


Johann has $100$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time, and if still tails, they are tossed a fourth time. What is the expected number of coins that are now heads?


If Joe ran 2 times as fast as he walked and it took him 10 minutes to walk halfway to the store, determine the total time it took Joe to get from home to the store.


$60$ students are wearing blue shirts and $84$ students are wearing yellow shirts, and the $144$ students are assigned into $72$ pairs. In exactly $25$ of these pairs, both students are wearing blue shirts. Determine the number of pairs in which both students are wearing yellow shirts.


Given that there are 120 ninth graders in total, 85 students are taking a science class, and 65 students are taking an art class, determine the number of ninth graders who are taking only an art class and not a science class.


Given that $f(x) = 3x^2 + 5x + 4$, calculate $f(x+h) - f(x)$.

Solution: \boxed{h(6x + 3h + 5)}


A liquid $Y$, which does not mix with water, forms a circular film of 0.2 cm thickness when poured onto a large body of water. A cylindrical canister with a radius of 5 cm and a height of 10 cm is completely filled with liquid $Y$. Calculate the radius of the resulting circular film on the water surface.


Miki has 15 oranges and 15 pears. She extracts 12 ounces of orange juice from 3 oranges and 10 ounces of pear juice from 4 pears. Miki then makes a juice blend using 5 pears and 4 oranges. Calculate the percentage of the blend that is pear juice.


A fair coin is tossed 4 times. Calculate the probability of getting at least three consecutive heads.


Given that the total distance from $0$ to $40$ consists of ten steps, and Mark moves forward eight steps but then takes one step back, determine Mark's final position.


A large circular pizza dough has a radius of 4 inches. Nine mini-pizzas of radius 1 inch each are cut from it, all touching each other and the edge, except the central mini-pizza. What is the radius of the reshaped mini-pizza?


Determine the number of revolutions a wheel, with a fixed center and with an outside diameter of 8 feet, would require to cause a point on the rim to travel one mile.


Given the repeating decimal $0.\overline{714285}$, write it in simplest fractional form and determine the sum of the numerator and the denominator.


Lily and John shared a pizza with $12$ equally-sized slices. Lily wanted a plain pizza, but John wanted pepperoni on one-third of the pizza. The cost of a plain pizza was $12$ dollars, and adding pepperoni to one-third of the pizza cost an additional $3$ dollars. John ate all the pepperoni slices and two plain slices. Lily ate the remaining slices. Each paid for what they had eaten. Calculate the difference in the amount of money paid by John and Lily.


A rhombus has one diagonal that is three times the length of the other diagonal. Express the side of the rhombus in terms of $K$, where $K$ is the area of the rhombus in square inches.


In ∆PQR, S is a point on side PQ such that RS = SQ and ∠RSQ measures 60°. What is the degree measure of ∠RSP?


Find the coefficient of $x^9$ in the expansion of $\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^{10}$.


Given Alice has 30 apples, find the number of ways she can share them with Becky, Chris, and Dan so that each of Alice, Becky, and Chris gets at least three apples, and Dan gets at least one apple.


A bag contains $\frac{1}{2}$ white kernels, $\frac{1}{3}$ yellow kernels, and $\frac{1}{6}$ red kernels. Half of the white kernels will pop, two-thirds of the yellow kernels will pop, and one-third of the red kernels will pop. Determine the probability that the kernel selected was white.


What is the largest quotient that can be formed using two even numbers chosen from the set $\{-30, -4, -2, 2, 4, 10\}$, where the numerator is taken as the negative of the selected number and the denominator remains positive?


Given a cylindrical container has an initial radius of $10$ inches and a height of $5$ inches, find the value of $x$ such that the increase in volume due to a $2x$ inch increase in radius is equal to the increase in volume due to a $3x$ inch increase in height.


A deck consists of four red cards, four green cards, and four blue cards. Each color has cards labeled $A$, $B$, $C$, $D$. A winning pair is defined as two cards of the same color or two cards with the same label. Determine the probability of drawing a winning pair.


Determine the number of circular pipes with an inside diameter of 2 inches and a height of 3 feet, which will carry the same amount of water as a pipe with an inside diameter of 4 inches and a height of 6 feet.


Maria begins her work at 7:45 A.M. instead of 7:25 A.M. She still takes her lunch break at noon, but now her lunch lasts 1 hour and 15 minutes. Her working day is 9 hours long. Determine the end time of her working day.

Solution: \boxed{6:00 \text{ P.M.}}


In how many ways can $10003$ be written as the sum of two primes?


In a math test, $15\%$ of the students scored $65$ points, $25\%$ scored $75$ points, $40\%$ scored $85$ points, and the rest scored $95$ points. Calculate the difference between the mean and median score of this test.


Prisha writes down one integer three times and another integer two times, with their sum being $105$, and one of the numbers is $15$. Calculate the other number.


Given that $x$ and $y$ are distinct nonzero real numbers such that $x - \tfrac{2}{x} = y - \tfrac{2}{y}$, determine the product $xy$.


Given two cubical dice, one with the numbers $1, 2, 3, 7, 8, 9$ and the other with the numbers $4, 5, 6, 10, 11, 12$, are rolled, calculate the probability that the sum of the numbers on the top faces is exactly $13$.


Given that there is a list of nine numbers, the average of the first five numbers is $7$, and the average of the last five numbers is $10$, and the average of all nine numbers is $8\frac{2}{9}$, determine the number common to both sets of five numbers.


The first three terms of an arithmetic sequence are $2x - 3$, $3x + 1$, and $5x + k$ respectively. Find the value of $k$ that ensures these terms form an arithmetic sequence.

Solution: \boxed{5 - x}


Solution: \boxed{h(6x + 3h - 4)}


What is the smallest positive integer n such that $\sqrt{n}-\sqrt{n-1}<.02$?


Given a rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FG}$, where $DA = 20$ and $FD = AG = 5$, find the area of rectangle $ABCD$.


Given Patty has a total of 30 coins made up of quarters and dimes, and if her quarters were swapped for dimes and her dimes for quarters, the total value of her coins would increase by $1.20, determine the original worth of her coins.


A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads.


Find the smallest integer $n$ such that $(x^2+y^2+z^2+w^2)^2 \leq n(x^4+y^4+z^4+w^4)$ for all real numbers $x, y, z$, and $w$.


Given that two 8-sided dice are rolled and the sum of the numbers rolled determines the diameter of a circle, calculate the probability that the numerical value of the area of the circle is less than the numerical value of the circle’s circumference.


In a class of $20$ students, $20\%$ are juniors and $80\%$ are seniors. The overall average score on the test was $78$. Each junior scored points in such a way that their scores form a consecutive arithmetic sequence with a common difference of $2$, and the average score of the seniors was $75$. What is the score of the highest-scoring junior?


Given the expression \( 3^{3^{3^3}} \), which evaluates to \( 3^{(3^{27})} \) under standard conventions, how many other distinct values can be obtained by rearranging the order of operator calculations?


Find the difference between the two smallest integers greater than 1 which, when divided by any integer $k$ such that $2 \le k \le 6$, have a remainder of $1$.


All numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are filled into a 3 × 3 grid so that consecutive numbers occupy squares that share an edge. This time, the numbers in the four corners have a sum of 20, find the possible number in the center.


Given a group of students, initially $60\%$ are girls. Subsequently, three girls leave and three boys arrive, and it is then found that $50\%$ of the group are girls. Determine the initial number of girls in the group.


A basketball player made 8 baskets during a game. Each basket was worth either 1, 2, or 3 points. Determine the number of different total scores the player could have achieved.


Find the sum of the fourth powers of all real numbers satisfying the equation $x^{243} - 243^{81} = 0$.


Given that $a * b$ means $4a - 2b$, find the value of $x$ if $3 * (4 * x) = 10$.


If $y$ cows can produce $y+2$ cans of milk in $y+1$ days, determine the number of days it will take $y+4$ cows to produce $y+6$ cans of milk.


Given the operations \( a@b = ab - b^2 + b^3 \) and \( a\#b = a + b - ab^2 + ab^3 \), calculate the value of \( \frac{7@3}{7\#3} \).


Given a lawn with a rectangular shape of $120$ feet by $200$ feet, a mower with a $32$-inch swath width, and a $6$-inch overlap between each cut, and a walking speed of $4000$ feet per hour, calculate the time it will take John to mow the entire lawn.


Given that the connection from $P$ leads to $R$ or $S$, with $R$ offering paths to $T$ or $U$, and with $T$ and $U$ leading directly to $Q$, $S$ offering paths to $T$, $U$, and directly to $Q$, and $P$ offering a direct path to $Q$, determine the total number of different routes from $P$ to $Q$.


Given that all three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=2x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis, calculate the length of $\overline{BC}$ given the area of the triangle is $128$.


Calculate the value of the expression: $[x + (y-z)] - [(x+z) - y]$.

Solution: \boxed{2y - 2z}


A square board consists of an alternating pattern of light and dark squares similar to a chessboard. Determine the difference in the number of dark squares and the number of light squares in a $9 \times 9$ grid.


Two pitchers, one with a capacity of 800 mL and the other with 500 mL, contain orange juice. The first pitcher is 1/4 full of orange juice and the second pitcher is 3/5 full. Water is added until both pitchers are full. Both pitchers are then poured into one large container. Calculate the fraction of the mixture in the large container that is orange juice.


How many positive integers $b$ have the property that $\log_{b} 512$ is a positive integer.


Given that points $E$ and $F$ are on the same side of diameter $\overline{GH}$ in circle $P$, $\angle GPE = 60^\circ$, and $\angle FPH = 90^\circ$, find the ratio of the area of the smaller sector $PEF$ to the area of the circle.


In a right triangle, the lengths of the legs a and b maintain a ratio of 3 : 4, and the hypotenuse is c. A perpendicular is dropped from the vertex of the right angle to the hypotenuse, dividing it into two segments: r adjacent to a and s adjacent to b. Find the ratio of r to s.


Given $\triangle ABC$, $AB = 75$, and $AC=100$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. Calculate the length of $BC$.


Given that there are 18 players on each team, each player receives a minimum salary of $12,000, and the total salary cap for each team is $620,000, calculate the maximum possible salary for the highest-paid player in a team.


Solution: \boxed{\frac{x^2 + 5x + 6}{2}}


Given the repeating decimal $0.\overline{57}$, write it in its simplest fractional form and find the sum of the numerator and denominator.


Anh traveled 75 miles on the interstate and 15 miles on a mountain pass. The speed on the interstate was four times the speed on the mountain pass. If Anh spent 45 minutes driving on the mountain pass, determine the total time of his journey in minutes.


Max's house has 4 bedrooms, each with dimensions 15 feet by 12 feet by 9 feet. Each bedroom has walls that occupy 80 square feet in area, excluding doorways and windows. Calculate the total square feet of walls to be painted.


Thirty percent less than 90 is one-half more than what number?


Given the numbers from $000$ to $999$, calculate how many have three digits in non-decreasing or non-increasing order, including cases where digits can repeat.


Calculate the expansion for the expression $(1+x^2)(1-x^4)$.

Solution: \boxed{1 + x^2 - x^4 - x^6}


Pablo had four times as many coins as Sofia, who in turn had three times as many coins as Mia. Each of them ends up with an equal number of coins after Pablo distributes some of his coins. Determine the fraction of his coins that Pablo distributes to Sofia and Mia together.


Find the times between $8$ and $9$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$.


The diameter of a circle is divided into $n$ equal parts. On each part a semicircle is constructed. Find the value to which the sum of the areas of these semicircles approaches as $n$ becomes very large.


Find the arithmetic mean of the first 35 successive positive integers, starting from 5 and with a common difference of 2.


Given that a set $T$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y = x$, and that the point $(1,4)$ is in $T$, find the smallest number of points in $T$.


If a stationery shop owner purchased 2000 pencils at $0.20 each and sells them for $0.40 each, find how many he must sell to make a profit of exactly $160.00.


Let's find the smallest positive integers $x$ and $y$ for which the product of $180$ and $x$ is a fourth power and the product of $180$ and $y$ is a sixth power. Find the sum of $x$ and $y$.

Solution: \boxed{4500 + 4050000 = 4054500}


What is the least possible value of $(xy-2)^2 + (x-1+y)^2$ for real numbers $x$ and $y$?


Let $a$ and $b$ be distinct real numbers for which $\frac{a}{b} + \frac{a+5b}{b+5a} = 2$. Find $\frac{a}{b}$.


Given that the even number 138 can be expressed as the sum of two different prime numbers, calculate the largest possible difference between the two primes.


Find the area of a polygon with vertices at $(2, 1)$, $(4, 3)$, $(7, 1)$, and $(4, 6)$.


Given that Alice is 1.6 meters tall and can reach 50 centimeters above the top of her head, and the ceiling is 3 meters above the floor, find the height of the stool in centimeters if standing on it allows her to just reach a light bulb 15 centimeters below the ceiling while stretching her arm around a hanging decorative item 20 centimeters below the ceiling.


Given whole numbers between $200$ and $500$ contain the digit $3$, determine the number of such numbers.


Given that $\log_a b = c^3$ and $a + b + c = 100$, find the number of ordered triples of integers $(a,b,c)$ with $a \ge 2$, $b\ge 1$, and $c \ge 0$.


How many primes less than $200$ have $3$ as the ones digit?


Given $0\le x_0<1$, let
\[x_n=\left\{ \begin{array}{ll} 2x_{n-1} &\text{ if }2x_{n-1}<1 \\ 2x_{n-1}-1 &\text{ if }2x_{n-1}\ge 1 \end{array}\right.\]
for all integers $n>0$. Determine the number of $x_0$ values for which $x_0=x_6$.


Determine the difference between the number of dark squares and the number of light squares in a $9 \times 9$ grid with alternating colors, where each row starts with the opposite color of the last square from the preceding row.


Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 35 seconds. They work together for 7 minutes, but there is a 1-minute period where only Cagney is frosting because Lacey takes a break. What is the number of cupcakes they can frost in these 7 minutes?


Determine how many students are enrolled only in the math class, not in science, given that there are 120 students in the eighth grade, 85 students taking a math class, and 65 students taking a science class.


The product of the two $101$-digit numbers $707,070,707,...,070,707$ and $909,090,909,...,090,909$ has tens digit $C$ and units digit $D$. Find the sum of $C$ and $D$.


Each of two boxes contains three chips numbered $1$, $4$, $5$. Calculate the probability that the product of the numbers on the two chips drawn is a multiple of $3$.


$\frac{900^2}{306^2-294^2}$ can be simplified to a positive value.


Given the sticker price of a gadget, calculate the difference in cost of purchasing the gadget at store A, where a 20% discount is applied and two rebates of $60 and $20 are applied subsequently, and store B, where a 35% discount is applied, and if the difference in cost is $20, determine the sticker price of the gadget.


If \( a, b > 0 \) and the triangle in the first quadrant bounded by the coordinate axes and the graph of \( ax + by = 12 \) has an area of 9, find the value of \( ab \).


Given an isosceles triangle $ABC$ with a perimeter of $24$ cm and equal sides $a$, find the maximum area of triangle $ABC$.

Solution: \boxed{16\sqrt{3} \text{ cm}^2}


If a child has $3$ red cubes, $3$ blue cubes, and $4$ green cubes, and wants to build towers measuring $9$ cubes in height, calculate the number of different towers she can make.


Given the expression $\frac{x + 2}{x - 2}$, each $x$ is replaced by $\frac{x + 2}{x - 2}$. Calculate the resulting expression for $x = 2$.


Given a quadratic equation \( Dx^2 + Ex + F = 0 \) with roots \( \alpha \) and \( \beta \), find the value of \( p \) such that the roots of the equation \( x^2 + px + q = 0 \) are \( \alpha^2 + 1 \) and \( \beta^2 + 1 \).

Solution: \boxed{\frac{2DF - E^2 - 2D^2}{D^2}}


Given that $\left(64\right)^{0.375}\left(64\right)^{0.125}$, calculate the value of this expression.


Given two circles with radii 5 units and 7 units, determine the maximum number of different values for the number of lines that can be tangent to both circles simultaneously.


Determine the total number of different selections possible for five donuts when choosing from four types of donuts (glazed, chocolate, powdered, and jelly), with the additional constraint of purchasing at least one jelly donut.


Given that Esha can cycle 18 miles in 2 hours and 15 minutes, and she can now stroll 12 miles in 5 hours, calculate the difference in minutes between her current strolling speed and her previous cycling speed.


Given the set $\{1,2,3,4,\dots,44,45\}$, two distinct numbers are selected so that the sum of the remaining $43$ numbers is the product of these two numbers. Find the difference between these two numbers.


A merchant purchases products at a discount of $30\%$ from the list price and wants to set a new marked price. They plan to offer a $20\%$ discount on this marked price but still aim to make a $30\%$ profit on the selling price. Determine the percentage of the list price at which the goods must be marked.


Determine the value(s) of $k$ such that the points $(1,4)$, $(3,-2)$, and $(6, k/3)$ lie on the same straight line.


A triangular array of $3003$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. Calculate the sum of the digits of $N$.


Choose four different numbers from the set $\{3, 5, 7, 9, 11, 13\}$, add the first two and the last two of these numbers together, and then multiply the two sums. What is the smallest result that can be obtained from this process?


A standard 12-sided die is rolled once. The number rolled is the diameter of a circle in units. Calculate the probability that the numerical value of the circumference of the circle is less than ten times the numerical value of the area of the circle.


The product of the two $101$-digit numbers $707,070,707,...,070,707$ and $909,090,909,...,090,909$ has tens digit $C$ and units digit $D$. Find the sum of $C$ and $D$.


Triangle DEF has vertices D = (4,0), E = (0,4), and F, where F is on the line x + y = 10. Determine the area of triangle DEF.


A 9 × 9 board is composed of alternating dark and light squares, with the upper-left square being dark. How many more dark squares are there than light squares?


Two cyclists, X and Y, start at the same time to ride from Huntington to Montauk, a distance of 80 miles. Cyclist X travels 6 miles an hour slower than cyclist Y. Cyclist Y reaches Montauk and immediately turns back, meeting X 16 miles from Montauk. Determine the rate of cyclist X.


Given that there are 65 students wearing blue shirts and 85 students wearing yellow shirts, and they are paired into 75 pairs, with 30 pairs consisting of two blue shirts, find the number of pairs consisting of two yellow shirts.


Given a structure formed by connecting eight unit cubes, calculate the ratio of the volume in cubic units to the surface area in square units.


A store owner bought $2000$ pencils at $0.08$ each. If he sells them for $0.20$ each, determine the number of pencils he must sell to make a profit of exactly $160.00$.


If \(5(3y + 7\pi) = Q\), calculate the value of \(10(6y + 14\pi)\).


Given positive real numbers \( x \) and \( y \), define the operation \( x*y = \frac{x \cdot y}{x+y} \). If \( z = x + 2y \), calculate the value of the operation \( x*z \).

Solution: \boxed{\frac{x^2 + 2xy}{2(x + y)}}


John drove the first quarter of his journey on a highway, then 30 miles through a city, and the remaining one-sixth on country roads. Calculate the total length of John's journey in miles.


Jo and Blair take turns counting starting from $2$, with each number said being double the previous number said. What is the $53^{\text{rd}}$ number said?


Given that a rectangular room is 15 feet long and 108 inches wide, calculate the area of the new extended room after adding a 3 feet wide walkway along the entire length of one side, in square yards, where 1 yard equals 3 feet and 1 foot equals 12 inches.


The perimeter of an equilateral triangle exceeds the perimeter of a square by 2016 cm. The length of each side of the triangle exceeds the length of each side of the square by d cm. The square has a perimeter greater than 0. Find the number of positive integers that are NOT possible values for d.


Estimate the value of $\sqrt{90} - \sqrt{88}$.


A large equilateral triangle with a side length of $20$ units is intended to be filled with non-overlapping equilateral triangles whose side length is $2$ units each, but $10$ of these small triangles are missing from the filling due to damage. Calculate the number of small undamaged triangles needed to fill the remainder of the large triangle.


Joe had walked three-fourths of the way from home to school in 3 minutes, and ran the remaining one-fourth of the way to school at a speed 4 times his walking speed.


Solve for the number of roots satisfying the equation $\sqrt{9 - x} = 2x\sqrt{9 - x}$.


A circle of radius 3 is centered at $A$. An equilateral triangle with side 6 has a vertex at $A$. Find 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.

Solution: \boxed{9(\pi - \sqrt{3})}


Alice has \(8p + 2\) quarters and Bob has \(3p + 6\) quarters. Calculate the difference in their money in terms of half-dollars.

Solution: \boxed{2.5p - 2}


Given that Tom, Dick, and Harry are playing a game with a biased coin with probabilities of flipping heads as \(\frac{1}{3}\) and tails as \(\frac{2}{3}\), find the probability that all three will have flipped their coins the same number of times before getting their first head.


Suppose the radius of Earth at the equator is about 3950 miles. A jet flies around the Earth along the equator at a speed of 550 miles per hour. Assuming a negligible height above the equator, estimate the number of hours the flight would take.


Determine how many integers n result in $(n+i)^6$ being an integer.


The number of revolutions of a wheel, with fixed center and with an outside diameter of $10$ feet, required to cause a point on the rim to go two miles.


Given that Alice and Bob's real numbers are chosen uniformly at random from the intervals [0, 1] and $\left[\frac{1}{3}, \frac{2}{3}\right]$ respectively, determine the number that Carol should choose to maximize her chance of winning.


In a cartesian coordinate system, a region termed "rhino's horn segment" is studied. This area is bounded by the quarter-circle in the first quadrant centered at origin $(0,0)$ with radius $4$, the half-circle in the first quadrant centered at $(0,2)$ with radius $2$, and the line segment connecting $(0,0)$ to $(4,0)$. Calculate the area of the "rhino's horn segment".


Given that soda is sold in packs of 8, 15, and 32 cans, determine the minimum number of packs needed to buy exactly 120 cans of soda.


A laser is placed at the point $(2,3)$. The laser beam travels in a straight line. The objective is for the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, and finally hit the point $(8,3)$. Calculate the total distance the beam will travel along this path.


A regular polygon of n sides is inscribed in a circle of radius \( \sqrt{2} \). The area of the polygon is \( 6 \). Determine the value of n.


Two joggers, 8 miles apart, begin at the same point but head in different directions. They would meet each other in 2 hours if they jogged towards each other, but they would take 4 hours to be 8 miles apart again if they jogged away from each other in the same direction at the beginning. What is the ratio of the speed of the faster jogger to the slower jogger?


Given that Mindy made four purchases for $\textdollar 2.47$, $\textdollar 7.51$, $\textdollar 11.56$, and $\textdollar 4.98$, find her total, to the nearest dollar.


Consider a large equilateral triangle with a side length of 4 units. There's a smaller equilateral triangle whose area is one-third the area of the larger triangle. Find the median of the trapezoid formed by these triangles.

Solution: \boxed{2 + \frac{2\sqrt{3}}{3}}


Igor has received test scores of $88, 92, 75, 83,$ and $90$. He hopes to reach an average score of at least $87$ with his next test. What is the minimum score he would need on the upcoming test to meet his goal.


Determine the fraction of the area of the regular octagon $ABCDEFGH$ that is shaded given that the shaded region includes the entire triangles $\triangle BCO$ and $\triangle CDO$, and two-thirds of triangle $\triangle ABO$.


Consider a sequence where each term is the sum of the two previous terms. The sequence starts with two numbers. If the last three terms of an 8-term sequence are 21, 34, and 55, determine the first term of the sequence.


Points A and B are on a circle with a radius of 5 and AB = 6. Given that C is the midpoint of the major arc AB, calculate the length of line segment AC.


The average of the first 300 terms of the sequence $2, -4, 6, -8, 10, -12, \ldots,$ can be found by calculating the sum of the first 300 terms and dividing by 300, where the nth term of the sequence is given by $(-1)^{n+1}\cdot 2n$.


Let $M$ be the greatest five-digit number whose digits have a product of $90$. Find the sum of the digits of $M$.


Consider a $3 \times 3$ array where each row and each column is an arithmetic sequence with three terms. The first term of the first row is $3$, and the last term of the first row is $15$. Similarly, the first term of the last row is $9$, and the last term of the last row is $33$. Determine the value of the center square, labeled $Y$.


How many 4-digit positive integers, where each digit is odd, are divisible by 3?


If the sum of all angles except one of a convex polygon is $2970^{\circ}$, calculate the number of sides of the polygon.


Given a basketball player made $7$ baskets during a game, each either a $2$-point or a $3$-point shot, calculate the number of different total scores the player could have achieved.


Given a rectangular storage with length 20 feet, width 15 feet, and height 10 feet, and with the floor and each of the four walls being two feet thick, calculate the total number of one-foot cubical blocks needed for the construction.


Given six test scores have a mean of $85$, a median of $88$, and a mode of $90$. Calculate the sum of the two lowest test scores.


A cube has faces numbered with consecutive integers, including negative numbers. The sum of the numbers on each pair of opposite faces is the same. If the smallest number is -3, find the sum of all six numbers on this cube.


Given that point B is the intersection of three lines, forming angles $\angle \text{ABC}$, $\angle \text{ABD}$, and $\angle \text{CBD}$, and that $\angle \text{CBD}$ is a right angle and $\angle \text{ABD} = 30^\circ$, find the measure of $\angle \text{ABC}$ given that the sum of the angles around point B is $180^\circ$.


How many $3$-digit positive integers have digits whose product equals $36$?


A cell phone plan costs $20$ dollars each month, plus $0.07$ dollars per text message sent, plus $0.15$ dollars for each minute used over 30 hours. In February, Alex sent 150 text messages and talked for 32 hours. Calculate the total amount he paid.


Given that 6-pound rocks are valued at $18 each, 3-pound rocks are valued at $9 each, and 2-pound rocks are valued at $6 each, and that Alice can carry at most 21 pounds and has at least 15 of each size available, determine the maximum value of the rocks Alice can carry out of the cave.


A container holds $26$ orange balls, $21$ purple balls, $20$ brown balls, $15$ gray balls, $12$ silver balls, and $10$ golden balls. Calculate the minimum number of balls that must be drawn from the container without replacement to ensure that at least $17$ balls of one color are retrieved.


Given that $\log_{2}{m}= 2b-\log_{2}{(n+1)}$, solve for $m$.


Given the data set $[4, 21, 34, 34, 40, 42, 42, 44, 52, 59]$, calculate the number of outliers.


Jayden purchased four items for $\textdollar 2.93$, $\textdollar 7.58$, $\textdollar 12.49$, and $\textdollar 15.65$. Find the total amount spent, rounded to the nearest ten cents.


Given a list of seven numbers, the average of the first four numbers is $6$, and the average of the last four numbers is $9$. If the average of all seven numbers is $7\frac{6}{7}$, find the value of the number common to both sets of four numbers.


During her six-day workweek, Mary buys a beverage either coffee priced at $60$ cents or tea priced at $80$ cents. Twice a week, she also buys a $40$-cent cookie along with her beverage. Her total cost for the week amounts to a whole number of dollars. How many times did she buy tea?


Consider a sequence of hexagons where each new hexagon adds two layers of dots instead of one. The first hexagon has 1 dot. The second hexagon has 1 central dot, 6 dots in the first layer, and 12 dots in the second layer, making a total of 19 dots. Determine the total number of dots in the third hexagon.


Determine the largest number by which the expression $n^4 - n^2$ is divisible for all possible integral values of $n$.


Compute the value of $\frac{8}{4 \times 25}$.


The length of rectangle $XYZW$ is 8 inches and its width is 6 inches. Diagonal $XW$ is divided in the ratio 2:1 by point $P$. Calculate the area of triangle $YPW$.


Elena drives 45 miles in the first hour, but realizes that she will be 45 minutes late if she continues at the same speed. She increases her speed by 20 miles per hour for the rest of the journey and arrives 15 minutes early. Determine the total distance from Elena's home to the convention center.


Given the expression $n$ represents a positive integer, determine the number of integer solutions for which both $\frac{n}{4}$ and $4n$ are four-digit whole numbers.


Four runners start at the same point on a 400-meter circular track and run clockwise with constant speeds of 3 m/s, 3.5 m/s, 4 m/s, and 4.5 m/s, respectively. Determine the time in seconds they will run before they meet again anywhere on the course.


Let $a + 2 = b + 4 = c + 6 = d + 8 = a + b + c + d + 10$. Calculate the value of $a + b + c + d$.


Given that x is a perfect square, find the second next larger perfect square.

Solution: \boxed{x + 4\sqrt{x} + 4}


There are two integers greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 13$, has a remainder of $1$. Find the difference between the two smallest such integers.


What is the degree measure of the angle formed by the hands of a clock at 1:30?


Let S be the set of the 1500 smallest positive multiples of 5, and let T be the set of the 1500 smallest positive multiples of 7. Find the number of elements common to S and T.


Cheenu used to cycle 24 miles in 2 hours as a young man and now he can jog 18 miles in 3 hours as an older man. Calculate the difference in minutes per mile between his cycling speed and his jogging speed.


A man buys a house for $15,000 and wants to achieve a $6\%$ return on his investment while incurring a yearly tax of $450$, along with an additional $200$ yearly for owner's insurance. The percentage he sets aside from monthly rent for maintenance remains $12\frac{1}{2}\%$. Calculate the monthly rent.


Given a polynomial \(g(x)\) with leading coefficient 1, whose four roots are the reciprocals of the four roots of \(f(x) = x^4 + ax^3 + bx^2 + cx + d\), find \(g(1)\) in terms of \(a, b, c,\) and \(d\).

Solution: \boxed{\frac{1 + a + b + c + d}{d}}


Given a device purchased at $40 less a 15% discount, and a goal of achieving a 25% selling profit based on the cost after allowing a 10% discount on the final sales price, determine the marked price of the device in dollars.


Two friends, Alice and Bob, start cycling towards a park 80 miles away. Alice cycles 3 miles per hour slower than Bob. Upon reaching the park, Bob immediately turns around and starts cycling back, meeting Alice 15 miles away from the park. Find the speed of Alice.


Given that $x$ is a positive real number, simplify the expression $\sqrt[4]{x\sqrt[3]{x}}$.


Two subsets of the set $T = \{w, x, y, z, v\}$ need to be chosen so that their union is $T$ and their intersection contains exactly three elements. How many ways can this be accomplished, assuming the subsets are chosen without considering the order?


Given a rectangle $R_1$ with one side of length $3$ inches and an area of $18$ square inches, and a similar rectangle $R_2$ with a diagonal of length $20$ inches, calculate the area of $R_2$.


Given that $P(n)$ denotes the greatest prime factor of $n$ and $P(n+36) = \sqrt{n+36}$, determine the number of positive integers $n$ for which $P(n) = \sqrt{n}$.


Consider the expression $\sqrt{\frac{4}{5}} - \sqrt[3]{\frac{5}{4}}$. Evaluate its value.

Solution: \boxed{\frac{2\sqrt{5}}{5} - \frac{\sqrt[3]{5}}{\sqrt[3]{4}}}


$5y$ varies inversely as the cube of $x$. When $y=4, x=2$. Find the value of $y$ when $x=4$.


Given the number 3080, determine the sum of its prime factors.


Given that there are 2023 boxes in a line, with the box in the kth position containing k white marbles, determine the smallest value of n for which the probability that Isabella stops after drawing exactly n marbles is less than 1/2023.


Find the number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4 y^4 - 16x^2 y^2 + 15 = 0$.


Each of two boxes contains four chips numbered $1$, $2$, $3$, and $4$. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$.


Alice, Bob, and Chris went on a fishing trip and decided to split all costs equally. Alice paid $135, Bob paid $165, and Chris paid $225. To equalize their shares, Alice gave Chris $x$ dollars, and Bob gave Chris $y$ dollars. Calculate $x - y$.


A large cube $n$ units on each side is fully painted blue on all six faces and then cut into $n^3$ smaller cubes. Exactly one-third of the total number of faces of these smaller cubes are blue. What is $n$?


Given that $i^2 = -1$, for how many integers $n$ is $(n+i)^5$ an integer?


Calculate the product of the expressions $x^4$, $x - \frac{1}{x}$, and $1 + \frac{1}{x} + \frac{1}{x^2}$ and find its degree.


A checkerboard of 15 rows and 15 columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered 1,2,...,15, the second row 16,17,...,30, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1,2,...,15, the second column 16,17,...,30 and so on across the board, find the sum of the numbers in the squares that have the same numbers in both numbering systems.


What is the sum of the digits of the greatest prime number that is a divisor of 32,767?


Determine how many lattice points are located on the line segment whose endpoints are $(5, 11)$ and $(35, 221)$.


Let $y = 3(x-a)^2 + (x-b)^2 + kx$, where $a, b$ are constants and $k$ is a new constant. Find the value of $x$ at which $y$ is a minimum.

Solution: \boxed{\frac{6a + 2b - k}{8}}


Solution: \boxed{-\frac{3}{2} \cdot 2^{-3k}}


If a jar contains $5$ different colors of gumdrops, where $40\%$ are blue, $15\%$ are brown, $10\%$ are red, $20\%$ are yellow, and the rest are green, and there are $50$ green gumdrops in total, calculate the number of yellow gumdrops after a third of the red gumdrops are replaced with yellow gumdrops.


Given $(9x)^{18} = (18x)^9$, determine the non-zero real value for $x$.


Given that the Sunshine Café sold 310 cups of coffee to 120 customers in one morning, and each customer purchased at least one cup of coffee, determine the maximum possible median number of cups of coffee bought per customer that morning.


A circle with a radius of 7 is inscribed in a rectangle, and the ratio of the rectangle's length to its width is 3:1, whereas a square is inscribed inside the same circle. Calculate the total area of the rectangle and the square.


A positive integer $n$ has $72$ divisors and $5n$ has $90$ divisors. Find the greatest integer $k$ such that $5^k$ divides $n$.


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.


A 20-quart radiator is filled completely with water. Five quarts are removed and replaced with pure antifreeze liquid. Then five quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated three more times, for a total of five replacements. Calculate the fractional part of the final mixture that is water.


A conical container with a base radius of $14 \mathrm{cm}$ and height of $20 \mathrm{cm}$ is completely filled with water. This water is then carefully poured into a cylindrical container with a radius of $28 \mathrm{cm}$. If 10% of the water is spilled during transfer, calculate the height of the remaining water in the cylindrical container.


Consider a rhombus with diagonals such that one diagonal is three times the length of the other. Express the side length of the rhombus in terms of the area K.


Given that Crystal runs due north for 2 miles, then northwest for 1 mile, and southwest for 1 mile, find the distance of the last portion of her run that returns her directly to her starting point.


Given that Big Al, the ape, ate 140 bananas from May 1 through May 6, with each day's total being five more than the previous day, but on May 4 he did not eat any, calculate the number of bananas Big Al ate on May 6.


Six people are sitting in a row of 8 seats with the first and last seats remaining unoccupied. Each person must reseat themselves in either their original seat or the seat immediately adjacent to their original seat. Determine the number of ways the people can be reseated.


Three fair dice are thrown, and each die has six faces, numbered 1 through 6. Calculate the probability that the numbers shown can be arranged to form an arithmetic progression with a common difference of two.


A convex polyhedron S has vertices U1, U2, …, Um, and 120 edges. This polyhedron is intersected by planes Q1, Q2, …, Qm, where each plane Qk intersects only those edges that are connected to vertex Uk. No two planes intersect within the volume or on the surface of S. As a result, m pyramids are formed along with a new polyhedron T. Determine the number of edges that polyhedron T now possesses.


Given bed A has 600 plants, bed B has 500 plants, bed C has 400 plants, beds A and B share 60 plants, beds A and C share 80 plants, beds B and C share 40 plants, and beds A, B, and C share 20 plants collectively, calculate the total number of unique plants when considering just beds A, B, and C.


Given that $A$ and $B$ together can complete a job in 3 days, $B$ and $C$ can complete it in 3 days, and $A$ and $C$ together can complete it in 4.8 days, find the number of days required for $A$ to do the job alone.


A merchant buys an item for a discount of 30% of the list price. He seeks to set a new marked price that allows him to provide a discount of 20% on that marked price and still obtain a profit of 30% on the selling price. Calculate the percent of the list price at which he should mark the item.


Given the expression \[\frac{k(P+Q)}{k(P-Q)} - \frac{k(P-Q)}{k(P+Q)},\] where \( P = a + b \) and \( Q = a - b \), and \( k \) is a non-zero constant, simplify this expression.

Solution: \boxed{\frac{a^2 - b^2}{ab}}


Four runners start running simultaneously from the same point on a 600-meter circular track. They each run clockwise around the track at constant speeds of 5 m/s, 6.5 m/s, 7.5 m/s, and 8 m/s. Determine the time in seconds before the runners regroup somewhere on the track.


The taxi fare in New Metro City is $3.00 for the first 0.75 mile and additional mileage charged at the rate of $0.25 for each additional 0.05 mile. You decide to give the driver a $3 tip. Determine the total distance in miles that you can ride for $15.


An equilateral triangle is originally painted black. Each time the triangle is changed, the middle ninth of each black triangle turns white. After three changes, what fractional part of the original area of the black triangle remains black.


Given that a positive multiple of $4$ is less than $3000$, is a perfect square, and is a multiple of $5$, calculate how many such numbers exist.


Randy drove the first quarter of his trip on a gravel road, the next 30 miles on pavement, and the remaining one-sixth on a dirt road. Find the total distance of Randy's entire trip.


Let $x$ be the weight of the heavier package and $y$ be the weight of the lighter package. The total weight of two packages is 7 times the difference in their weights. Find the ratio of the weight of the heavier package to the lighter package.


If $x$ women working $x$ hours a day for $x$ days produce $x^2$ units of a product, determine the number of units produced by $z$ women working $z$ hours a day for $z$ days.


Given that Larry and Julius are playing a game where Larry throws first, and the probability of Larry knocking the bottle off the ledge is $\frac{1}{3}$ and the probability that Julius knocks the bottle off the ledge is $\frac{1}{4}$, calculate the probability that Larry wins the game.


John drove 150 miles in 3 hours. His average speed during the first hour was 45 mph, and he stopped for 30 minutes. His average speed during the next 45 minutes was 50 mph, and the remaining duration was driven at an unknown average speed v mph.


Given $\sqrt{\left(5-4\sqrt{2}\right)^2}+\sqrt{\left(5+4\sqrt{2}\right)^2}$, evaluate its value.


Let $g(x) = x^2 + 4x + 3$ and let $T$ be the set of integers $\{0, 1, 2, \dots, 20\}$. Determine the number of members $t$ of $T$ such that $g(t)$ has remainder zero when divided by $5$.


A box contains $36$ red balls, $24$ green balls, $18$ yellow balls, $15$ blue balls, $12$ white balls, and $10$ 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.


Given a circle of radius $4$ and a circle of radius $6$, find the total number of possible values of $m$, where $m$ is the number of lines that are simultaneously tangent to both circles.


Determine the number of distinct terms in the expansion of the expression $[(a+2b)^3(a-2b)^3]^2$ when fully simplified.


Let a sequence $\{u_n\}$ be defined by $u_1=7$ and the relationship $u_{n+1} - u_n = 5 + 3(n-1), n=1,2,3,\dots$. Express $u_n$ as a polynomial in $n$ and find the algebraic sum of its coefficients.


A cylindrical tank with radius $6$ feet and height $10$ feet is lying on its side. The tank is filled with water to a depth of $3$ feet. Calculate the volume of water in the tank.

Solution: \boxed{120\pi - 90\sqrt{3}}


Given 500 members voting, a city council initially defeated a proposal. In a subsequent vote, with the same members participating, the proposal was passed by one and a half times the margin by which it was originally defeated. The number voting for the proposal in the re-vote was $\frac{13}{12}$ of the number voting against it originally. Determine how many more members voted for the proposal in the re-vote compared to the initial vote.


Samantha lives 3 blocks west and 3 blocks south of the southwest corner of City Park. Her school is 3 blocks east and 3 blocks north of the northeast corner of City Park. On school days, she bikes on streets to the southwest corner of City Park, then can choose between 2 different diagonal paths through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, calculate the number of different routes she can take.


Given A is $x\%$ greater than B and also $y\%$ greater than C, and A, B, and C are positive numbers such that A > B > C > 0, determine the expressions for x and y.

Solution: \boxed{100 \left(\frac{A-C}{C}\right)}


Given that a piece of purple candy costs $30$ cents, and Casper has enough money to buy $16$ pieces of red candy, $18$ pieces of yellow candy, or $20$ pieces of black candy, determine the smallest possible value of $m$ if he can also buy $m$ pieces of purple candy.


The number $15!$ has several positive integer divisors. What is the probability that a randomly chosen divisor is odd?


Given quadrilateral $ABCD$ is a rhombus with a perimeter of 80 meters and the length of diagonal $\overline{AC}$ is 30 meters, calculate the area of rhombus $ABCD$ in square meters.


Given that in right trapezoid ABCD, AD is perpendicular to DC, AD = AB = x, DC = 2x, and E is the midpoint of DC such that DE = EC = x. Find the area of triangle BEC when x = 5.


Given that there are 3-pounds stones worth $9 dollars each, 6-pounds stones worth $15 dollars each, and 1-pounds stones worth $1 dollar each, and that Tanya can carry at most 24 pounds, determine the maximum value, in dollars, of the stones Tanya can carry.


Given the series $1 - 2 + 3 - 4 + \cdots - 100 + 101$, calculate the sum of the series.


In a 150-shot archery competition, Nora leads by 80 points after 75 shots. Each shot can score 10, 8, 5, 3, or 0 points. Nora always scores at least 5 points per shot. To ensure her victory, if Nora's next $n$ shots are all scoring 10 points each, calculate the minimum number of such shots required for her guaranteed victory.


Suppose three whole numbers in ascending order have pairwise sums of 18, 23, and 27, respectively. Find the middle number.


Laura has an 8-hour workday in which she attends three meetings. The first meeting lasts 40 minutes, the second meeting lasts twice as long as the first, and overlaps the last 10 minutes of the first meeting. The third meeting is 30 minutes long and does not overlap with the others. Calculate the percentage of Laura's workday that was spent in meetings.


Given the number of minutes studied by Clara and Mira over a period of one week, calculate the average difference in minutes per day in the number of minutes studied by Mira and Clara.


Each of two wheels contains numbers from 1 to 8. When the wheels are spun, a number is selected from each wheel. Find the probability that the sum of the two selected numbers is divisible by 4.


If four $\Delta$'s and two $\diamondsuit$'s balance twelve $\bullet$'s, and one $\Delta$ balances a $\diamondsuit$ and two $\bullet$'s, determine the number of $\bullet$'s that balance three $\diamondsuit$'s in this balance.


Orvin went to the store with just enough money to buy 40 balloons. During his visit, he found out that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at 50% off the regular price. What is the greatest number of balloons Orvin could buy?


Determine the minimum number of digits to the right of the decimal point required to express the fraction $\frac{987654321}{2^{30} \cdot 5^5}$ as a decimal.


Given that a light bulb is located $15$ centimeters below the ceiling in Bob's living room, the ceiling is $2.8$ meters above the floor, Bob is $1.65$ meters tall and can reach $55$ centimeters above his head, and Bob standing on a chair can just reach the light bulb, calculate the height of the chair, in centimeters.


Given the ceiling is 300 centimeters above the floor, Alice is 160 centimeters tall and can reach 50 centimeters above her head, and standing on a box she can just reach a light bulb located 15 centimeters below the ceiling and 10 centimeters above a decorative shelf, calculate the height of the box in centimeters.


Let ($a_1$, $a_2$, ... $a_{20}$) be a list of the first 20 positive integers such that for each $2\le$ $i$ $\le20$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. Determine the number of such lists.


How many ordered pairs of integers \((x, y)\) satisfy the equation \(x^4 + y^2 = 2y + 3\)?


An uncrossed belt is fitted without slack around two circular pulleys with radii of $12$ inches and $3$ inches. If the distance between the points of contact of the belt with the pulleys is $20$ inches, calculate the distance between the centers of the pulleys in inches.


Let H represent the number of holes dug by the hamster and G represent the number of holes dug by the guinea pig. The hamster stores 5 seeds per hole, and the guinea pig stores 6 seeds per hole. If both animals stored the same total number of seeds, and the guinea pig needed 3 fewer holes, calculate the total number of seeds stored by the hamster.


Let $\overline{AB}$ be a diameter in a circle with a radius of $10$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at point $E$ such that $BE=3$ and $\angle AEC = 45^{\circ}$. Find $CE^2+DE^2$.


Given that Alice, Bob, Cindy, Dave, and Emma have internet accounts, some but not all of whom are internet friends with each other, and none have friends outside this group, and each has the same number of internet friends, determine the number of ways the configuration of friendships can occur.


A right triangle with legs of length a and b is drawn. A new right triangle is formed by joining the midpoints of the hypotenuse of the first one. Then a third right triangle is formed by joining the midpoints of the hypotenuse of the second; and so on forever. Find the limit of the sum of the perimeters of all the triangles thus drawn.

Solution: \boxed{2(a+b+\sqrt{a^2 + b^2})}


The sum of the first 15 terms and the sum of the first 90 terms of a given arithmetic progression are 150 and 15, respectively. Find the sum of the first 105 terms.


Given that the price of each piece of yellow candy is 24 cents, and that Lena has exactly enough money to buy 18 pieces of red candy, 16 pieces of green candy, 20 pieces of blue candy, or n pieces of yellow candy, calculate the smallest possible value of n.


Suppose that $12 \cdot 3^x = 7^{y+5}$. Determine $x$ when $y = -4$.

Solution: \boxed{\log_3 \left(\frac{7}{12}\right)}


Three A's, three B's, and three C's are placed in the nine spaces so that each row and column contains one of each letter. If B is placed in the upper left corner, calculate the number of arrangements possible.


In a right triangle XYZ, where XZ=15, YZ=8, and angle Z is a right angle, a circle is inscribed completely inside the triangle. Determine the radius of this circle.


Julia has a large white cube with an edge of 15 feet. She also possesses enough blue paint to cover 500 square feet. Julia decides to paint a blue square centered on each face of the cube, leaving a white border around it. Find the area of one of the white squares in square feet.


Lara and Darla are racing on a 500-meter circular track. Starting simultaneously, Lara runs at a speed 20% faster than Darla. Calculate the number of laps Lara has completed when she first overtakes Darla.


A half-sector of a circle of radius $6$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. Calculate the volume of the cone in cubic inches.

Solution: \boxed{9\pi \sqrt{3}}


A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-third of the total number of faces of the unit cubes are red. What is $n$?


A small bottle of lotion can hold 60 milliliters, while a large bottle can hold 750 milliliters. Additionally, pad material in the large bottle absorbs 5% of the lotion from the small bottles during transfer. Determine the minimum number of small bottles Kyla must buy to fill the large bottle completely.


How many primes less than $100$ have $3$ as the ones digit?


In a city of 400 adults, 370 adults own scooters and 80 adults own bikes, calculate the number of the scooter owners who do not own a bike.


How many whole numbers are between $\sqrt{50}$ and $\sqrt{200}$?


In a class, $20\%$ of the students are juniors and $80\%$ are seniors. The overall average score of the class on a recent test was $85$. Juniors all received the same score, and the average score of the seniors was $84$. What score did each of the juniors receive on this test?


Given an 81-number grid ranging from 1 to 81 is written on a 9x9 checkerboard, calculate the sum of the numbers in the four corners of the checkerboard.


Mr. $X$ owns a property worth $15,000. He sells it to Mr. $Y$ with a 15% profit, then Mr. $Y$ sells it back to Mr. $X$ at a 5% loss. Determine the net outcome of these transactions for Mr. $X$.


How many whole numbers between $200$ and $500$ contain the digit $1$?


Given positive integers $a$ and $b$ are each less than $8$, calculate the smallest possible value for $2 \cdot a - a \cdot b$.


Given a pentagon inscribed in a circle, determine the sum of the angles inscribed in the five arcs cut off by the sides of the pentagon.


Given the number 2550, calculate the sum of its prime factors.


What is the least possible value of (x+1)(x+3)(x+5)(x+7) + 2050, where x is a real number?


Given that the monogram consists of three initials in alphabetical order with a last initial of 'X', and the first and middle initials must be distinct, determine the total number of possible monograms.


In an isosceles triangle $\triangle ABC$, base angles $A$ and $B$ are such that $A = B$. Let's say $A = 30^\circ$. The altitude from vertex $C$ to the base $AB$ divides the angle $C$ into two parts $C_1$ and $C_2$, with $C_2$ adjacent to side $a$. Calculate $C_1 - C_2$.