Bella and Ella start moving towards each other from a distance of 3 miles apart. Bella is walking while Ella is riding a scooter at a speed 4 times as fast as Bella's walking speed. If Bella covers 3 feet with each step, determine the number of steps she has taken by the time they meet.


Let $\triangle ABC$ be a triangle with angles $\alpha = 59^\circ$, $\beta = 60^\circ$, and $\gamma = 61^\circ$. For each positive integer $n$, define $\alpha_n = 59^\circ + n\cdot 0.02^\circ$, $\beta_n = 60^\circ$, and $\gamma_n = 61^\circ - n\cdot 0.02^\circ$. What is the least positive integer $n$ for which $\triangle ABC$ with angles $\alpha_n$, $\beta_n$, and $\gamma_n$ is obtuse?


In a square grid composed of $121$ uniformly spaced points, calculate the probability that a line connecting point $P$, at the center of the square, to a randomly chosen point $Q$ from the other $120$ points is a line of symmetry for the square.


A triangle with vertices at $A=(2,4)$, $B=(7,2)$, and $C=(6,5)$ is plotted on an $8 \times 6$ grid. Calculate the fraction of the grid covered by the triangle.


Find the simplest form of $1 - \frac{2}{1 + \frac{b}{1 - 2b}}$ where $b \neq \frac{1}{2}$.

Solution: \boxed{\frac{3b - 1}{1 - b}}


Suppose that $\frac{3}{4}$ of $12$ bananas are worth as much as $6$ oranges. Determine the number of oranges that are worth as much as $\frac{2}{3}$ of $9$ bananas.


For how many three-digit whole numbers does the sum of the digits equal $26$?


Given a basketball player made 7 baskets during a game, with each basket worth either 1, 2, or 3 points, determine the number of different total scores the player could achieve.


Find the area of the region bounded by the graphs of \((x+2ay)^2 = 16a^2\) and \((2ax-y)^2 = 4a^2\), where \(a\) is a positive real number.

Solution: \boxed{\frac{32a^2}{1 + 4a^2}}


Sally is 20% younger than Danielle, Adam is 30% older than Mary, and Mary is 25% younger than Sally. Given that the sum of their ages is 60 years, determine Sally's next birthday age.


Let $x = .123456789101112....109810991100$, where the digits are obtained by writing the integers $1$ through $1099$ in order. Determine the $2500$th digit to the right of the decimal point.


Given the rug display featuring sections in three distinct colors, where the areas of these sections form an arithmetic progression, and the width of the center section is 2 feet, and each of the two colored rings around the center section extends outward by 2 feet on all sides, determine the length in feet of the center section.


A woman buys an apartment for $20,000 and rents it out. She sets aside 10% of each month's rent for maintenance and pays $460 a year in taxes. Determine the monthly rent she needs to charge to realize a 6% return on her investment annually.


Calculate the number of positive integers less than 1200 that are divisible by neither 6 nor 8.


Moe, Loki, and Nick are companions with distinct amounts of money. Moe gives Ott one-fifth of his money, Loki gives Ott one-third of his money, and Nick gives Ott one-half of his money. If each gave Ott the same amount of money, find the fraction of the group's total money that Ott now possesses.


A 10x10 arrangement of alternating black and white squares has a black square $R$ in the second-bottom row and a white square $S$ in the top-most row. Given that a marker is initially placed at $R$ and can move to an immediately adjoining white square on the row above either to the left or right, and the path must consist of exactly 8 steps, calculate the number of valid paths from $R$ to $S$.


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


Given a circle of radius \(3\), find the area of the region consisting of all line segments of length \(6\) that are tangent to the circle at their midpoints.


Given Jo and Blair take turns counting numbers in a sequence where each number is two more than the last number said by the other person, and Jo starts by saying $2$, determine the $30^{\text{th}}$ number said.


Given a warehouse stores 110 containers of oranges, where each container has at least 98 oranges and at most 119 oranges, determine the largest integer n such that there must be at least n containers having the same number of oranges.


Let \( f \) be a function defined by \( f\left(\frac{x}{3}\right) = x^2 + x + 1 \). Calculate the sum of all values of \( z \) for which \( f(3z) = 4 \).


Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $3$, $2$, and $4$, respectively, and constant terms $2$, $3$, and $6$, respectively. If $P(z)$ and $Q(z)$ each have $z=-1$ as a root, determine the minimum possible value of the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) = R(z)$.


The set of all real numbers \(y\) for which the expression \(\log_{10}(\log_{9}(\log_{8}(\log_{7}(y^{2}))))\) is defined. Represent this set as \(\{y \mid y > d\}\). Determine the value of \(d\).


Given 300 swimmers compete in a 50-meter freestyle event with 8 lanes, each allowing 8 swimmers to compete simultaneously, and the top two swimmers in each race advance to the next round, determine the number of races required to find the champion swimmer.


The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class.


If one side of a triangle is $20$ inches and the opposite angle is $45^{\circ}$, calculate the diameter of the circumscribed circle.

Solution: \boxed{20 \sqrt{2}\text{ inches}}


There are 30 tiles in the box, containing 100 edges in total. Find the number of square tiles in the box.


The average (arithmetic mean) age of a group consisting of doctors and lawyers is 47. If the doctors average 45 years old and the lawyers 55 years old, calculate the ratio of the number of doctors to the number of lawyers.


Given the sets of fractions $S_1$ and $S_2$ defined as $S_1$ includes fractions from $\dfrac{1}{10}$ up to $\dfrac{9}{10}$ incrementing by $1$ in the numerators, and $S_2$ is created by the constant $\dfrac{20}{10}$ repeated four times, calculate the sum of all fractions from $S_1$ and $S_2$.


Given that one root of $ax^2 + bx + c = 0$ is triple the other, determine the relationship between the coefficients $a, b, c$.

Solution: \boxed{3b^2 = 16ac}


Calculate the sum of the sequence $3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + 5\left(1-\dfrac{1}{5}\right) + \cdots + 11\left(1-\dfrac{1}{11}\right)$.


Given a square with a side length of 10 cut into two unequal rectangles, one of which is twice the area of the other, find the dimensions of the smaller rectangle.

Solution: \boxed{\left(\frac{10}{3}, 10\right)}


A positive integer divisor of $10!$ is chosen at random. Calculate the probability that the divisor chosen is a perfect square, expressed as a simplified fraction $\frac{m}{n}$, and find the sum of the numerator and denominator.


Find the smallest integer value of $N$ so that when $(a+b+c+d+e+1)^N$ is expanded and like terms are combined, the expression contains exactly $3003$ terms that include all five variables $a, b, c, d,$ and $e$, each to some positive power.


Consider the sequence $2, -4, 6, -8, 10, -12, \ldots,$ where each term $a_n$ in the sequence is calculated by $(-1)^n \cdot 2n$. Find the average of the first $300$ terms of this sequence.


Consider the set $\{45, 52, 87, 90, 112, 143, 154\}$. How many subsets containing three different numbers can be selected from this set so that the sum of the three numbers is odd?


Suppose the estimated $30$ billion dollar cost to send a person to Mars and another $10$ billion dollars to establish a base on the Moon is shared equally by $200$ million people in a country. Calculate each person's share.


Define a two-digit positive integer as snuggly if it is equal to the sum of its nonzero tens digit, the cube of its units digit, and 5. How many two-digit positive integers are snuggly?


Determine the least possible value of \((x+2)(x+3)(x+4)(x+5) + 2024\) where \(x\) is a real number.


Let's say Cagney can frost a cupcake every 25 seconds, and Lacey can frost a cupcake every 35 seconds. Calculate the number of cupcakes they can frost in 10 minutes if they work together.


Calculate the square of $7 - \sqrt[3]{x^3 - 49}$.

Solution: \boxed{49 - 14\sqrt[3]{x^3 - 49} + (\sqrt[3]{x^3 - 49})^2}


Determine the total cost David had to pay for his cell phone plan, given a basic monthly fee of $30, a cost of $0.10 per text message, and a cost of $0.15 for each minute over a standard usage of 20 hours, after sending 150 text messages and using the phone for 21 hours.


What is the sum of the mean, median, and mode of the numbers $1,2,1,4,3,1,2,4,1,5$?


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


Successive discounts of $15\%$ and $25\%$ are equivalent to a single discount.


Given that $a \oplus b$ means $4a + 2b$, find the value of $y$ if $3 \oplus (4 \oplus y) = -14$.


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


Leah places six ounces of tea into a twelve-ounce cup and six ounces of milk into a second cup of the same size. She then pours one-third of the tea from the first cup to the second and, after stirring thoroughly, pours one-fourth of the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now milk?


Given the expression $n = \frac{2kcd}{c+d}$, solve for $d$.

Solution: \boxed{\frac{nc}{2kc - n}}


Given $a$ and $b$ are even single-digit positive integers chosen independently and at random, find the probability that the point $(a,b)$ lies above the parabola $y = ax^2 + bx$.


Given that the sum of two or more consecutive positive integers is 20, find the number of sets.


Find the value of $y$ that satisfies the equation $16^{-3} = \frac{4^{72/y}}{4^{42/y} \cdot 16^{25/y}}$.


For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius $12$ centered at $(3, 3)$?


Calculate the product $6 \times 0.5 \times \frac{3}{4} \times 0.2$.


The parabolas \( y = ax^2 - 3 \) and \( y = 5 - bx^2 \) intersect the coordinate axes in exactly four points, and these four points form the vertices of a kite with an area of 18. Determine \( a + b \).


Given a bag of popcorn kernels containing $\frac{3}{4}$ white kernels and $\frac{1}{4}$ yellow kernels, and with $\frac{2}{5}$ of the white kernels and $\frac{3}{4}$ of the yellow kernels popping when placed in a popper, determine the probability that a kernel was white if it was selected at random and successfully popped.


Given the expression \(125\sqrt[3]{16}\sqrt[6]{64}\), evaluate the logarithm to the base \(5\).

Solution: \boxed{3 + \frac{7}{3}\log_5(2)}


In a bucket, there are $34$ red balls, $25$ green balls, $23$ yellow balls, $18$ blue balls, $14$ white balls, and $10$ black balls. Find the minimum number of balls that must be drawn from the bucket without replacement to guarantee that at least $20$ balls of a single color are drawn.


In trapezoid EFGH, the bases EF and GH are not equal, where EF = 10 units and GH = 14 units. The lengths of the non-parallel sides are EG = 7 units and FH = 7 units. Calculate the perimeter of trapezoid EFGH.


Two concentric circles have radii in the ratio $2:5$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 8$, calculate the radius of the larger circle.


An equilateral triangle is entirely painted blue. Each iteration, the middle ninth of each blue region turns green. After three iterations, what fractional part of the original area of the blue triangle remains blue.


A set $T$ of points in the $xy$-plane is symmetric about both coordinate axes and the line $y=-x$. If $(1,4)$ is in $T$, calculate the smallest number of points in $T$.


A circle is inscribed in a triangle with side lengths $9, 12$, and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $u$ and $v$, with $u<v$. Find the ratio $u:v$.


Given that $\boxed{n}$ is defined to be the sum of the positive factors of $n$, find $\boxed{\boxed{13}}$.


Find the largest power of 2 that divides $17^4 - 13^4$.


Given Gabriel's weight of 96 pounds, along with the weights of his quadruplet brothers, 10, 10, 12, and 18 pounds, calculate the difference between the average and median weights of the five children.


Given the list of the first 12 positive integers such that for each \(2 \le i \le 12\), either \(b_i+1\) or \(b_i-1\) or both appear somewhere before \(b_i\) in the list, find the number of such lists.


A square with area $16$ is inscribed in a square with area $25$, where each vertex of the smaller square touches a side of the larger square. One vertex of the smaller square divides a side of the larger square into two segments, and one length is three times the other length. Find the product of the lengths of the two segments.


Given that Star lists the whole numbers from $1$ to $50$ once, and Emilio copies Star's numbers replacing each occurrence of the digit '3' by the digit '2', calculate the difference between Star's sum and Emilio's sum of the copied numbers.


A rectangle with dimensions $10$ by $4$ shares the same center with an ellipse having semi-major axis lengths of $3$ and semi-minor axis lengths of $2$. Calculate the area of the region common to both the rectangle and the ellipse.


Given 1 brown tile, 2 purple tiles, 2 red tiles, and 3 yellow tiles, determine the number of distinguishable arrangements of these tiles in a row from left to right.


A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Calculate the value of $d$ to the nearest tenth.


A school has $100$ students divided among $5$ classes taught by $5$ teachers, with each teacher teaching one class. The student distribution across the classes is $40, 40, 10, 5, 5$. Compute the value of $t-s$, where $t$ is the expected number of students per class when choosing a teacher at random and $s$ is the expected number of students per class when choosing a student at random.


Given a palindrome between $1000$ and $10,000$ is chosen at random, find the probability that it is divisible by both 3 and 11.


Given that $\frac{4}{7}$ of the marbles are blue, if the number of red marbles is tripled while the number of blue marbles remains the same, calculate the resulting fraction of marbles that will be red.


Solution: \boxed{5^{128} - 4^{128}}


In a room where $3/7$ of the people are wearing gloves, and $5/6$ of the people are wearing hats, determine the minimum number of people in the room wearing both a hat and a glove.


A lemming starts at a corner of a rectangular area measuring 8 meters by 15 meters. It dashes diagonally across the rectangle towards the opposite corner for 11.3 meters. Then the lemming makes a $90^{\circ}$ right turn and sprints upwards for 3 meters. Calculate the average of the shortest distances to each side of the rectangle.


A ticket to a school concert costs $x$ dollars, where $x$ is a whole number. A group of 8th graders buys tickets costing a total of $36$, and a group of 9th graders buys tickets costing a total of $63$. Determine the number of values for $x$ that are possible.


Given the function $g$ defined by the table
\[
\begin{tabular}{|c||c|c|c|c|c|}  \hline   x & 1 & 2 & 3 & 4 & 5 \\   \hline  g(x) & 5 & 3 & 1 & 2 & 4 \\  \hline \end{tabular}
\]
calculate the value of $u_{2023}$ given that $u_0=5$ and $u_{n+1} = g(u_n)$ for $n \ge 0$.


How many even positive 4-digit integers are divisible by 5 but do not contain the digit 5?


Given the product $\frac{4}{3}\times\frac{5}{4}\times\frac{6}{5}\times\cdots\times\frac{2010}{2009}$, find the value of this product.


Given a triangle with vertices at $A = (2, 2)$, $B = (6, 3)$, and $C = (5, 6)$ plotted on a $7 \times 6$ grid, calculate the fraction of the grid that is covered by the triangle.


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


Calculate the value of $3000 \cdot (3000^{2500}) \cdot 2$.

Solution: \boxed{2 \cdot 3000^{2501}}


Given a cylindrical water tank that stands 50 meters high and can hold 200,000 liters of water, and a miniature water tank that holds only 0.2 liters, determine the height of the miniature water tank.


If the sum of the first $20$ terms and the sum of the first $50$ terms of a given arithmetic progression are $150$ and $20$, respectively, then find the sum of the first $70$ terms.


A belt is fitted without slack around two circular pulleys with radii of $10$ inches and $6$ inches. The belt runs externally and does not cross over itself. The distance between the points where the belt contacts with the pulleys is $30$ inches. Find the distance between the centers of the two pulleys.


Given quadrilateral $ABCD$ is divided into four triangles by point $P$, solve for $\alpha$ in terms of $\beta$, $\gamma$, and $\delta$.

Solution: \boxed{360^\circ - \beta - \gamma - \delta}


Jasmine has a large bottle that can hold 450 milliliters and a small bottle that holds 45 milliliters. She obtained an extra large bottle with a capacity of 900 milliliters. Determine the minimum total number of bottles required to completely fill the extra large bottle.


Given positive integers $a$ and $b$ that are each less than 10, calculate the smallest possible value for $3 \cdot a - a \cdot b$.


The taxi fare in Gotham City is $3.00 for the first $\frac12$ mile and additional mileage charged at the rate $0.25 for each additional 0.1 mile, and a $3 tip is planned. Calculate the number of miles that can be ridden for $15.


Points $A, B, C,$ and $D$ lie on a line in that order, with $AB = CD$ and $BC = 8$. Point $E$ is not on the line, and $BE = CE = 8$. Find $AB$ given that the perimeter of $\triangle AED$ is three times the perimeter of $\triangle BEC$.


In a triangle with sides 50, 110, and 120 units, an altitude is dropped upon the side of length 120. Determine the length of the larger segment cut off on this side by the altitude.


Sharon's car uses a gallon of gas every 40 miles, and her gas tank holds 16 gallons when it is full. One day, Sharon started with a full tank of gas, drove 480 miles, bought 6 gallons of gas, and continued driving to her destination. When she arrived, her gas tank was a quarter full. Calculate the total number of miles Sharon drove that day.


The student locker numbers at Liberty High are numbered consecutively starting from locker number $1$. Each digit costs three cents. If it costs $206.91 to label all the lockers, how many lockers are there?


The sum of seven test scores has a mean of 84, a median of 85, and a mode of 88. Calculate the sum of the three highest test scores.


Determine how many non-prime numbers exist where the digits sum to 7, and each digit in the number is greater than 1.


A cylindrical water tank has a radius $R$ of $10$ inches and a height $H$ of $5$ inches. Determine the value of $x$ such that the increase in volume, when the radius is increased by $x$ inches, is twice the increase in volume when the height is increased by $x$ inches.
A) $x = 0$
B) $x = 10$
C) $x = 20$
D) $x = 40$

Solution: \boxed{C) \ x = 20}


Let $Q$ equal the product of 1,234,567,890,123,456,789 and 987,654,321,098,765 multiplied by 123. Calculate the number of digits in $Q$.


Quadrilateral $EFGH$ is a rhombus with a perimeter of $80$ meters. The length of diagonal $\overline{EG}$ is $36$ meters. Calculate the area of rhombus $EFGH$.


Given a regular decagon, a triangle is formed by connecting three randomly chosen vertices of the decagon. Calculate the probability that exactly one of the sides of the triangle is also a side of the decagon.


A paper triangle with sides of lengths $4, 4, 6$ inches is folded so that point $A$ falls on point $C$ (where $AC = 6$ inches). What is the length, in inches, of the crease?


Three congruent circles are centered at points $A$, $B$, and $C$ such that each circle passes through the center of the other two circles. The line containing both $A$ and $B$ is extended to intersect the circles at points $D$ and $E$ respectively. The circles centered at $A$ and $C$ intersect at a point $F$ distinct from where they intersect with the circle centered at $B$. Calculate the degree measure of $\angle DFE$.


Given a 7-period day, and the condition that no two mathematics courses can be taken in consecutive periods, and the first and last periods cannot be math courses, calculate the number of ways a student can schedule 3 mathematics courses -- algebra, trigonometry, and calculus.


Given a town's population increased by $1,500$ people, and then this new population decreased by $15\%$, and after these changes the town had $45$ fewer people than it did before the increase, determine the original population of the town.


Define a sequence $\{u_n\}$ by $u_1=7$ and the recurrence relation $u_{n+1}-u_n=5+2(n-1)$ for $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, find the algebraic sum of its coefficients.


In a regular hexagon with side length 4, there are six alternating semicircles along its sides such that the diameters of three of them equal the side of the hexagon, and the other three have diamimeters equal to half the side length. Calculate the area of the region inside the hexagon but outside all the semicircles.

Solution: \boxed{24\sqrt{3} - \frac{15\pi}{2}}


Given the sequence $1, 4, 9, -16, 25, -36,\ldots$, whose $n$th term is $(-1)^{n+1}\cdot n^2$, find the average of the first $100$ terms of the sequence.


A rectangular area is to be fenced off on three sides using part of a 150 meter rock wall as the fourth side, with fence posts placed every 10 meters along the fence, including the two posts where the fence meets the rock wall. Given the area dimensions are 40 m by 100 m, determine the fewest number of posts required to fence this area.


How many 4-digit numbers greater than 1000 can be formed using the digits of 2025?


The length of the paper used to make a $3$ cm diameter cylindrical cardboard tube into an $11$ cm diameter cylindrical cardboard tube by wrapping it $400$ times is what length in meters?


Given a right circular cone with a base that is a circle with radius twice that of a given sphere, and the volume of the cone is one-third that of the sphere, determine the ratio of the altitude of the cone to the radius of its base.


Given that soda is sold in packs of 6, 12, 24, and 48 cans, find the minimum number of packs needed to buy exactly 126 cans of soda.


Given $A$ can finish a task in $10$ days, and $B$ is $75\%$ more efficient than $A$, calculate the time it takes for $B$ to complete the same task.


Given Alice took 80% of a whole pie. Bob took one fourth of the remainder. Cindy then took half of what remained after Alice and Bob. What portion of the whole pie was left?


Given the equation $\sin(3x) = \sin(x)$, determine the number of solutions on the interval $[0,2\pi]$.


A regular decagon is given. A triangle is formed by connecting three randomly chosen vertices of the decagon. Calculate the probability that none of the sides of the triangle is a side of the decagon.


If the perimeter of rectangle ABCD is 30 inches, determine the least value of diagonal AC in inches.


Points $M$ and $N$ are on a circle of radius $10$ and $MN = 12$. Point $P$ is the midpoint of the minor arc $MN$. What is the length of the line segment $MP$?


Given that Bob spends a total of $36.00 for lunch, including a 12% sales tax and an 18% tip on the pre-tax amount, calculate the cost of his lunch without tax or tip in dollars.


Given a minor league soccer team with $25$ players, where each player must be paid at least $20,000$ dollars, and the total of all players' salaries cannot exceed $900,000$ dollars, determine the maximum possible salary for a single player.


Given that a grocer stacks oranges in a pyramid-like stack whose rectangular base is $6$ oranges by $7$ oranges, and each orange above the first level rests in a pocket formed by four oranges below, determine the total number of oranges in the stack.


Suppose $m$ and $n$ are positive integers such that $108m = n^3$. Find the minimum possible value of $m + n$.


Given the original price of a jacket is $120.00, an 8% sales tax is applied, and then a 25% discount is applied, find the difference between the total bill calculated by Bob and the total bill calculated by Alice.


The median of the numbers 3, 7, x, 14, 20 is equal to the mean of those five numbers. Calculate the sum of all real numbers \( x \) for which this is true.


Macy wants to buy the minimum number of standard jars necessary to fill at least a large pack that contains 2000 grams of jam, without exceeding it by more than the weight of one standard jar that can hold 140 grams of jam. What is the number of jars she must buy?


Chips are drawn randomly, one at a time without replacement, from a magician's hat containing 4 red chips and 3 green chips until all 4 red chips are drawn or until at least 2 green chips are drawn. Calculate the probability that all 4 red chips are drawn.


A blue ball and a yellow ball are randomly and independently tossed into bins numbered with the non-negative integers so that for each ball, the probability that it is tossed into bin $0$ is $\frac{1}{4}$ and into bin $k$ (where $k \geq 1$) is $\left(\frac{1}{2}\right)^k \cdot \frac{3}{4}$. Find the probability that the blue ball is tossed into a higher-numbered bin than the yellow ball.


The complex number $z$ satisfies $z + |z| = 5 - 3i$. What is $|z|^{2}$.


Given the quadratic equation $x^2 + px + q = 0$, where $p$ and $q$ are positive numbers, and the roots of this equation differ by $2$, find the value of $p$.


Given that a recipe to make $8$ servings of a smoothie requires $3$ bananas, $2$ cups of strawberries, $1$ cup of yogurt, and $4$ cups of milk, and if Sarah has $10$ bananas, $5$ cups of strawberries, $3$ cups of yogurt, and $10$ cups of milk, determine the greatest number of servings of smoothie she can make maintaining the same ratio of ingredients.


Two tangents are drawn to a circle from an exterior point A; they touch the circle at points B and C respectively. A third tangent intersects segment AB in P and AC in R, and touches the circle at Q. If AB = 24, and the lengths BP = PQ = x and QR = CR = y with x + y = 12, find the perimeter of triangle APR.


Given Professor Miller has nine different language books on a bookshelf: two Arabic, three German, and four Spanish, calculate the number of ways to arrange the nine books on the shelf keeping all German books together and all Spanish books also together.

Solution: \boxed{4! \times 3! \times 4! = 24 \times 6 \times 24 = 3456}


Given the modified sequence where the first number is $1$, and for all $n\ge 2$, the product of the first $n$ numbers in the sequence is $n^3$, find the sum of the third and the fifth numbers in this sequence.


Given that $2^{10} \cdot 3^6 \cdot 5^4 = d^e$, find the smallest possible value for $d+e$.


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


Sara lists the whole numbers from 1 to 50. Lucas copies Sara's numbers, replacing each occurrence of the digit '3' with the digit '2'. Calculate the difference between Sara's sum and Lucas's sum.


Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions.


Given that $3x^3 - tx + q$ has $x - 3$ and $x + 1$ as its factors, determine the value of $|3t - 2q|$.


Two poles, one 30 feet high and the other 90 feet high, are 150 feet apart. Find the height at which the lines joining the top of each pole to the foot of the opposite pole intersect.


Given a triangle with side lengths $15, 20,$ and $x$, determine the number of integers $x$ for which the triangle has all its angles acute.


Given that Mary thought of a positive two-digit number and performed the following operations: multiplied it by $4$ and subtracted $7$, then switched the digits of the result, and obtained a number between $91$ and $95$, inclusive, determine the original two-digit number.


Given Alice's weekly allowance was $B$ dollars, the cost of the book was $25\%$ of the difference between $B$ and the magazine cost, and the cost of the magazine was $10\%$ of the difference between $B$ and the book's cost, calculate the fraction of $B$ that Alice spent on the book and magazine combined.


Josanna's current test average is $\dfrac{85 + 75 + 80 + 70 + 90}{5}$. If she wants to raise her test average by $5$ points, then her new overall test average will be $\dfrac{85 + 75 + 80 + 70 + 90}{5} + 5$. Find the minimum test score Josanna would need to achieve this new average.


Suppose $d$ is a digit. For how many values of $d$ is $3.1d3 > 3.123$?


Given a circle with center O and radius r, alongside two horizontal parallel tangents to the circle that are each at a distance of d from the center where d > r, calculate the number of points equidistant from the circle and both tangents.


If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 18$, calculate the sum of $a$ and $b$.


Given the radii ratio of two concentric circles is $1:4$, if $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=8$, determine the radius of the larger circle.


For how many positive integers $n$ is $\frac{n}{40-n}$ also a positive integer?


Consider a map with points labeled as P, Q, R, S, and T. The allowed paths are P to Q, Q to R, Q to S, S to T, Q directly to T, and R to T. Determine how many different routes are there from P to T.


Evaluate the expression $\sqrt{7+4\sqrt{3}} - \sqrt{7-4\sqrt{3}}$.


Given the water tower in real life is 60 meters high and the spherical top holds 200,000 liters of water, and Logan wants his model's water tower to hold 0.2 liters of water, determine the height of Logan's model tower.


Given a class of $60$ students with $20$ blue-eyed blondes, $36$ brunettes, and $23$ brown-eyed students, calculate the number of blue-eyed brunettes in the class.


Given the product \(6^3 \cdot 15^4\), calculate the number of digits in the result.


A shopkeeper purchases 2000 pencils at $0.20 each. If he plans to sell them for $0.50 each, determine the number of pencils he must sell to make a profit of exactly $200.


Let $ABCD$ be a square. Let $E, F, G, H$ be the centers, respectively, of isosceles right-angled triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. Find the ratio of the area of square $EFGH$ to the area of square $ABCD$.


Given that the fourth degree polynomial function y=p(x) and the fifth degree polynomial function y=q(x) have leading coefficients of 1, determine the maximum number of points of intersection of their graphs.


Given the expression $G = \frac{5x^2 + 20x + 4n}{5}$, which is the square of an expression linear in $x$, determine the range between which $n$ must fall.


The sums of three whole numbers taken in pair are 18, 23, and 27. Express the middle number in terms of the other two numbers and solve for its value.


A telephone number has the form $\text{ABC-DEF-GHIJ}$, where each letter represents a different digit. In this number, $A > B > C$, $D > E > F$, and $G > H > I > J$. Moreover, $D$, $E$, and $F$ are consecutive odd digits; $G$, $H$, $I$, and $J$ are consecutive even digits. Additionally, $A + B + C = 12$. Determine the value of $A$.


Given that Alice visits every 4th day, Bianca visits every 6th day, and Carmen visits every 8th day, all three friends visited Daphne yesterday, and the next 365-day period is considered, calculate the number of days when exactly two friends will visit.


What is the greatest possible sum of the digits in the base-eight representation of a positive integer less than $1728$?


In a trapezoid, the line segment joining the midpoints of the diagonals has length 5.  The height of the trapezoid is 10 and the longer base is 63.  Find the area of the trapezoid.


A trapezoidal field with one pair of parallel sides of lengths 15 meters and 9 meters respectively is part of a rectangular plot. The non-parallel sides of the trapezoid are equal and perpendicular to the parallel sides, and the trapezoid fills one end of the rectangle. Given that the longer parallel side of the trapezoid is also the length of the short side of the rectangle, and the total perimeter of the rectangle is 52 meters, calculate the perimeter of the trapezoidal field.


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 at least half of the other 7 balls.


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


The difference between the largest and smallest roots of the cubic equation \(x^3 - px^2 + \frac{p^2 - 1}{4}x = 0\), calculate the difference.


The population of Cyros doubles every 30 years and is currently 500, with each inhabitant requiring 2 acres. Given the island can support 32000 acres, calculate the number of years from 2020 until the population meets or exceeds the island's capacity.


Three different numbers are randomly selected from the set $\{ -3, -2, -1, 0, 1, 3, 4\}$ and multiplied together. What is the probability that the product is positive?


Given a triangle with vertices $A=(2,2)$, $B=(6,2)$, and $C=(5,5)$ plotted on an $8\times6$ grid, and a smaller rectangle within the grid defined by its diagonal with endpoints $(1,1)$ and $(6,5)$. Determine the fraction of the grid area covered by the portion of the triangle that lies within the smaller rectangle.


Given a positive integer $N$, with four digits in its base four representation, is chosen at random with each such number having an equal chance of being chosen, calculate the probability that $\log_2 N$ is an integer and $N$ is even.


Determine the twenty-fifth number in a sequence obtained by counting in base 5.


A woman has part of $6000 invested at 3% and the rest at 5%. If her annual return on each investment is the same, calculate the average rate of interest she realizes on the $6000.


Given that the faces of each of $9$ standard dice are labeled with the integers from $1$ to $6$, let \( p \) be the probability that when all $9$ dice are rolled, the sum of the numbers on the top faces is $15$. What other sum occurs with the same probability as \( p \)?


Given the enclosure dimensions are 15 feet long, 8 feet wide, and 7 feet tall, with each wall and floor being 1 foot thick, determine the total number of one-foot cubical blocks used to create the enclosure.


How many positive even multiples of $3$ less than $3000$ are perfect squares?


Given that the scores of the test are 65, 70, 75, 85, and 95, and the class average is always an integer after each score is entered, determine the last score Mrs. Johnson entered.


Fifteen points are selected on the positive $x$-axis, \( X^+ \), and six points are selected on the positive $y$-axis, \( Y^+ \). All possible segments connecting points from \( X^+ \) to \( Y^+ \) are drawn. What is the maximum possible number of points of intersection of these segments that could lie in the interior of the first quadrant if none of the points on \( Y^+ \) have an integer coordinate?


Given the function $g(x) = 2x^3 + 5x^2 - 2x - 1$, find $g(x+h) - g(x)$.

Solution: \boxed{h(6x^2 + 6xh + 2h^2 + 10x + 5h - 2)}


How many four-digit numbers are divisible by 17?


Given three different numbers are randomly selected from the set $\{-2, -1, 0, 0, 3, 4, 5\}$ and multiplied together, calculate the probability that the product is $0$.


A box contains 30 red balls, 25 green balls, 23 yellow balls, 14 blue balls, 13 white balls, and 10 black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn.


Given that a product is discounted by 60% and the buyer also has a coupon for an additional 30% off the sale price, determine the percentage off the original price that the buyer effectively gets.


Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187} - \cdots$. Determine the sum $S$ of this series.


Five friends did yardwork for their neighbors, and their earnings were \$18, \$22, \$30, \$36, and \$50, respectively. Calculate the amount the friend who earned \$50 will give to the others.


Given a list of 21 names with altered lengths recorded as follows: 6 names of 4 letters each, 4 names of 5 letters each, 1 name of 6 letters, 5 names of 7 letters, and 5 names of 8 letters, calculate the median length of these names.


Given that point P is the center of the regular hexagon ABCDEF, and Z is the midpoint of the side AB, determine the fraction of the area of the hexagon that is in the smaller region created by segment ZP.


Five friends earn $18, $22, $30, $35, and $45 respectively. Determine the amount the friend who earned $45 needs to give to the others.


Given Oscar buys $15$ pencils and $5$ erasers for $1.25$, and a pencil costs more than an eraser, and both items cost a whole number of cents, determine the total cost, in cents, for one pencil and one eraser.


If Michael was 15 years old when he participated in the 10th IMO, determine the year Michael was born.


A circle is inscribed in a triangle with side lengths $9, 12$, and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $r$ and $s$, with $r<s$. Find the ratio $r:s$.


A class contains \( k > 20 \) students and their quiz scores have a mean of \( 10 \). A group of \( 20 \) students from the class has a mean score of \( 16 \). Determine the mean of the quiz scores of the remaining students in terms of \( k \).


Circles of diameter $2$ inches and $6$ inches have the same center. Find the ratio of the blue-painted area to the red-painted area.


When the radius of a circle is incremented by \( k \) units, find the ratio of the new circumference to the increase in area compared to the original.

Solution: \boxed{\frac{2(r+k)}{2rk + k^2}}


Given Tamara has $3030$ coins, consisting of pennies ($1$-cent coins) and dimes ($10$-cent coins), and she has at least one penny and one dime, calculate the difference in cents between the greatest possible and least amounts of money Tamara can have.


Given Linda drove the first quarter of her trip on a gravel road, the next 30 miles on pavement, and the remaining one-sixth on a dirt road, determine the total length of Linda's trip in miles.


A novel is recorded onto compact discs, taking a total of 505 minutes to read aloud. Each disc can hold up to 53 minutes of reading. Assuming the smallest possible number of discs is used and each disc contains the same length of reading, calculate the number of minutes of reading each disc will contain.


Given that you and five friends and four of their family members are raising $1800 in donations for a charity event, determine the amount each person will need to raise for equal distribution.


As the number of sides of a polygon increases from $3$ to $n$, determine how the sum of the interior angles of the polygon changes.

Solution: \boxed{(n-2) \times 180^\circ}


Given two boxes, each containing the chips numbered $1$, $2$, $4$, $5$, a chip is drawn randomly from each box. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$.


Consider a list where each integer $n$ from 1 to 300 appears exactly $n$ times. What is the median of this number list?


Given that Country X has $a\%$ of the world's population and $b\%$ of the world's wealth, and Country Y has $b\%$ of the world's population and $a\%$ of its wealth, find the ratio of the wealth of a citizen of X to the wealth of a citizen of Y.


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


Determine the number of solution-pairs in the positive integers for the equation $4x + 7y = 975$.


What is the probability that a randomly drawn positive factor of $90$ is less than $10$?


Given that the length of $\overline{AB}$ is triple the length of $\overline{BD}$, and the length of $\overline{AC}$ is $7$ times the length of $\overline{CD}$, calculate the fraction of the length of $\overline{AD}$ that is the length of $\overline{BC}$.


Given the four-digit number 5005, calculate the total number of different numbers that can be formed by rearranging its four digits.


Tom, Dick, and Harry each flip a fair coin repeatedly until they get their first tail. Calculate the probability that all three flip their coins an even number of times and they all get their first tail on the same flip.


Given that $\triangle XYZ$ exists, and $P$ divides side $XZ$ in the ratio $2:3$, while $K$ is the midpoint of $YP$, and $Q$ is the point of intersection of $XK$ and $YZ$, determine the ratio in which $Q$ divides side $YZ$.


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


Given the number of terms in an arithmetic progression (A.P.) is even, the sum of the odd-numbered terms is 36, the sum of the even-numbered terms is 45, and the last term exceeds the first term by 15, and the common difference is a whole number, determine the number of terms in the A.P.


Given a triangle with side lengths $15, 30$, and $x$, determine the number of integers $x$ for which all its angles are acute.


Given there are 121 grid points in a square, arranged in an 11 x 11 grid, and point $P$ is at the center of the square, compute the probability that the line $PQ$ is a line of symmetry for the square, given that point $Q$ is randomly selected from the other 120 points.


Given that $A > B > 0$ and $C = A + B$, if $C$ is $y\%$ greater than $B$, calculate the value of $y$.

Solution: \boxed{100 \cdot \frac{A}{B}}


Given the fraction $\frac{987654321}{2^{27}\cdot 5^3}$, determine the minimum number of digits to the right of the decimal point needed to express this fraction as a decimal.


In the $xy$-plane, find the number of lines whose $x$-intercept is a positive prime number greater than 5 and whose $y$-intercept is a positive integer that pass through the point $(6,5)$.


Given John thought of a positive two-digit number, he multiplied it by $5$ and added $13$, then switched the digits of the result, obtaining a number between $82$ and $86$, inclusive. Determine John's original number.


Determine how many pairs consist of both students wearing red shirts at a study group event where $65$ students are wearing green shirts and $85$ students are wearing red shirts, with a total of $150$ students divided into $75$ pairs, and exactly $30$ of these pairs consisting of both students wearing green shirts.


Point $P$ moves along a vertical line that intersects side $AB$ of $\triangle PAB$. Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ respectively. As $P$ moves vertically, calculate how many of the four quantities listed change: (i) the length of the segment $MN$, (ii) the perimeter of $\triangle PAB$, (iii) the area of $\triangle PAB$, (iv) the area of trapezoid $ABNM$.


Let $S$ be a square one of whose diagonals has endpoints $(1/4,3/4)$ and $(-1/4,-3/4)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\leq x\leq 100$ and $0\leq y\leq 100$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly three points with integer coordinates in its interior?
A) $\frac{3}{50}$
B) $\frac{7}{100}$
C) $\frac{1}{50}$
D) $\frac{1}{25}$
E) $\frac{3}{100}$


Sam drove 150 miles in 120 minutes. His average speed during the first 30 minutes was 75 mph, and his average speed during the second 30 minutes was 70 mph. Calculate Sam's average speed in mph during the last 60 minutes.


Given that there exist positive integers $A$, $B$, and $C$, with no common factor greater than $1$, such that $A \log_{50} 5 + B \log_{50} 2 = C$, calculate $A + B + C$.


A digital clock is set to display time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits in this display.


Given right triangle $ABC$ where angle $B$ is a right angle, point $D$ is the foot of the altitude from $B$ onto hypotenuse $AC$. If $AD = 5$ cm and $DC = 3$ cm, calculate the area of triangle $ABC$.


Given $y = x^2 + px + q$, and the least possible value of $y$ is 1, determine the value of $q$.

Solution: \boxed{1 + \frac{p^2}{4}}


Given the equation \(\frac{9}{x^2} - \frac{6}{x} + 1 = 0\), find the value of \(\frac{2}{x}\).


The bakery owner turns on his doughnut machine at $\text{8:00}\ {\small\text{AM}}$. By $\text{11:40}\ {\small\text{AM}}$, the machine has completed one-fourth of the day's job. At what time will the doughnut machine complete the entire job?

Solution: \boxed{10:40\ \text{PM}}


A number $m$ is added to the set $\{2, 5, 8, 11\}$ to make the mean of the set of five numbers equal to its median. Determine the number of possible values of $m$ that satisfy this condition.


Given Peter's family ordered a 16-slice pizza, Peter ate 2 slices alone, shared 1 slice equally with his brother Paul, and shared 1 slice equally with their sister Sarah, and brother Paul. What fraction of the pizza did Peter eat in total?


The sum of the circumferences of the small circles constructed on the circumference of a given circle behaves as $n$ becomes very large.

Solution: \boxed{2\pi^2 R}


Mia chooses a real number uniformly at random from the interval $[0, 3000]$. Independently, Jake chooses a real number uniformly at random from the interval $[0, 6000]$. Calculate the probability that Jake's number is greater than Mia's number.


Let $R_k$ denote an integer whose base-ten representation consists of $k$ ones. For instance, $R_3=111$ and $R_5=11111$. Consider the quotient $Q = R_{30}/R_5$. Calculate the sum of the digits of $Q$, which is an integer whose base-ten representation comprises of only ones and zeros.


Four years ago, Jim was twice as old as his sister Eliza, and six years before that, Jim was three times as old as Eliza. Let x be the number of years until the ratio of their ages is 3 : 2. Determine the value of x.


A circle is inscribed in a triangle with side lengths $9, 12$, and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $r$ and $s$, with $r<s$. Find the ratio $r:s$.


There are $12$ horses, each taking unique prime minutes to complete a lap: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37$ minutes respectively. Determine the least time $T > 0$, in minutes, where at least $6$ horses are back at the starting point again together, and calculate the sum of the digits of $T$.


Given the expression $12 - (3 \times 4)$, calculate Harry's answer, and then subtract Terry's answer, where Terry's answer is obtained by ignoring the parentheses and calculating $12 - 3 \times 4$.


Given that Jack purchased 15 pairs of socks for a total of $36, some of which cost $2 a pair, some $3 a pair, and some $4 a pair, and he bought at least one pair of each type, determine the number of pairs of $2 socks Jack bought.


Determine the remainder when the polynomial $x^4 - 3x^2 + 1$ is divided by the polynomial $x^3 - x - 1$.

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


Find the difference in the total value of Liam's and Mia's fifty-cent coins, where Liam has $3p + 2$ fifty-cent coins and Mia has $2p + 7$ fifty-cent coins, expressed in pennies.

Solution: \boxed{50p - 250}


In a right trapezoid, one of the non-rectangular vertex angles is divided into angles $x^\circ$ and $y^\circ$, where both angles are prime numbers and the total angle is $90^\circ$, find the least possible value of $y^\circ$.


Given Josie makes lemonade by using 150 grams of lemon juice, 200 grams of sugar, and 300 grams of honey, and there are 30 calories in 100 grams of lemon juice, 386 calories in 100 grams of sugar, and 304 calories in 100 grams of honey, determine the total number of calories in 250 grams of her lemonade.


In a $3 \times 3$ grid, numbers $1$ to $9$ are arranged such that if two numbers are consecutive, they must share an edge. The numbers in the four corners add up to $20$. The sum of the numbers along one of the diagonals also equals $15$. Determine the number in the center square.


Given that $f(x+5) + f(x-5) = f(x)$ for all real $x$, find the least common positive period $p$ for all such functions $f$.


Given that Lin can climb a flight of stairs 1, 2, 3, or 4 steps at a time, find the total number of different ways Lin can climb 8 stairs.


Ralph walked down a street and passed five houses in a row, each painted a different color: green, blue, orange, red, and yellow. He passed the orange house before the red house and after the green house. He also passed the blue house before the yellow house, which was not next to the blue house. Additionally, the blue house was not next to the orange house. Determine the total number of orderings of the colored houses.


Karl's car uses a gallon of gas every 30 miles, and his gas tank holds 16 gallons when it is full. One day, Karl started with a full tank of gas, drove 360 miles, bought 10 gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. Determine the total distance Karl drove that day.


What is the maximum value of $\frac{(3^t - 4t)t}{9^t}$ for real values of $t$?


Given the complex numbers $2+i$, $-1+2i$, and $-2-i$ are vertices of a square in the complex plane, find the fourth complex number which completes the square.


Given that the sum of a polygon's interior angles is $2843^\circ$ and one angle was accidentally left out, determine the degree measure of the missed angle.


Find the integer $d$ such that the remainder $r$ is the same when each of the numbers $1210, 1690$, and $2670$ is divided by $d$, and calculate the value of $d-4r$.


Given the numbers $1, 2, 2, 3, 4, 4, 4, 5$, find the sum of the mean, median, and mode of these numbers.


An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 16, find the area of the hexagon.


If 2 pints of paint are needed to paint a statue 8 ft. high, calculate the number of pints it will take to paint 800 statues similar to the original but each only 2 ft. high.


For how many integers x does a triangle with side lengths 7, 15, and x have all its angles acute?


Given that there are 10 children and a total of 26 wheels, determine the number of tricycles.


Given that point $E$ is the midpoint of side $\overline{BC}$ in square $ABCD$, and $\overline{DE}$ meets diagonal $\overline{AC}$ at $F$, determine the area of square $ABCD$ if the area of quadrilateral $DFEC$ is $36$.


Given that Shea is now 75 inches tall, having grown 25% from a common initial height with Mika, and Mika has grown 10% of her original height, determine Mika's current height in inches.


Given that $\frac{3}{8}$ of the knights are red, and the rest are blue, and $\frac{1}{5}$ of all the knights are magical, the fraction of magical red knights is $3$ times the fraction of magical blue knights, calculate the fraction of red knights that are magical.


Determine the number of factors the polynomial $x^{15} - x$ has when factored completely using polynomials and monomials with integral coefficients.


Julia leaves Green Bay at 7:00 AM heading for Appleton on her scooter, traveling at a uniform rate of 15 miles per hour, and Mark leaves Appleton at 7:45 AM heading for Green Bay on his scooter, traveling at a uniform rate of 20 miles per hour. They travel on the same 85-mile route between Green Bay and Appleton. Determine the time at which they meet.


A store announces a "40% off everything" sale. Furthermore, a coupon offers an additional 30% discount on the discounted sale price. Calculate the percentage off the original price that the final price represents.


Let's consider a number $300^7$. Calculate the number of positive integer divisors of $300^7$ that are perfect squares or perfect cubes (or both).


Let x, y, and z be the three numbers such that their sum is 120. The ratio of the first to the second is 3/4 and the ratio of the second to the third is 3/5. Find the value of y.


Given that the erroneous product of two positive integers $a$ and $b$, where the digits of the two-digit number $a$ are reversed, is $189$, calculate the correct value of the product of $a$ and $b$.


Given Paula initially had enough paint to paint 36 rooms, but lost four cans leaving her enough paint for only 28 rooms, determine the number of cans of paint she used for the 28 rooms.


Given a rectangular box-shaped fort with dimensions 15 feet in length, 12 feet in width, and 6 feet in height, and walls and floor uniformly 1.5 feet thick, calculate the total number of one-foot cubical blocks used to build the fort.


Given Tom can make 6 pies in one batch, and each pie is a circle with a radius of 8 cm, determine the number of pies Bob can make in one batch, if Bob's pies are right-angled triangles with legs of 6 cm and 8 cm, using the same amount of dough as Tom.


Consider a $3 \times 5$ rectangular grid, determine the maximum number of X's that can be placed such that no four X's align in a row vertically, horizontally, or diagonally.


Simplify the expression $(x - y)^{-2}(x^{-1} - y^{-1})$ and express it with negative exponents.

Solution: \boxed{-\frac{1}{xy(x - y)}}


A circle centered at O is divided into 16 equal arcs labeled from $A$ through $P$. Points X and Y are chosen so that angle X at the circumference spans 1 arc and angle Y at the circumference spans 6 arcs. Calculate the sum of the degrees in the angles X and Y.


Positive integers $a$ and $b$ are each less than $11$. Calculate the smallest possible value for $2 \cdot a - a \cdot b$.


A digital watch in a 24-hour format displays hours and minutes. What is the largest possible sum of the digits in this display?


Given that Sarah and Jill start a swimming race from opposite ends of a 50-meter pool, and they cross paths two minutes after they start, determine the time it takes for them to cross paths for the second time.


Eight semicircles are inscribed in a regular octagon such that their diameters coincide with the sides of the octagon. Each side of the octagon has a length of 3. Calculate the area of the region inside the octagon but outside all of the semicircles.

Solution: \boxed{18(1+\sqrt{2}) - 9\pi}


What is the value of $\sqrt{\log_4{8} + \log_8{4}}$?


Given the graphs of $y = -2|x-a| + b$ and $y = 2|x-c| + d$ intersect at points $(1,7)$ and $(7, 3)$, determine $a+c$.


The maximum possible median number of cans of soda bought per customer on that day can be calculated if each of 70 customers purchased 5 cans and each of the other 50 customers purchased only 1 can.


Given that point D is a point on side BC such that AD = DC and angle DAC measures 50 degrees, calculate the degree measure of angle ADB.


For each positive integer n > 1, let \( P(n) \) denote the greatest prime factor of \( n \). Determine how many positive integers n satisfy both \( P(n) = \sqrt{n} \) and \( P(n+50) = \sqrt{n+50} \).


The probability that when all 10 dice are rolled, the sum of the numbers on the top faces is 15, equals the probability that the sum of the numbers on the top faces is what value.


A month with 30 days has the same number of Sundays and Tuesdays. How many of the seven days of the week could be the first day of this month?


How many even positive 3-digit integers are divisible by 5 but do not contain the digit 5?


If the ratio of the legs of a right triangle is 2:3, determine the ratio of the corresponding segments of the hypotenuse made by a perpendicular from the vertex.


Given quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at $O$, $BO=4$, $OD=5$, $AO=9$, $OC=2$, and $AB=7$, find the length of $AD$.


Given the function $g(x) = bx^3 - 1$, determine the value of $b$ if $g(g(1)) = -1$.


A triangle has an area of $50$ square units, one side of length $12$, and the median to that side of length $13$. Calculate the sine of the acute angle $\theta$ formed by that side and the median.


The probability that the sum of the numbers on the top faces of the 8 dice is 12 is equal to the probability that the sum of the numbers on the top faces of the 8 dice is another value.


Three different numbers are randomly selected from the set $\{-3, -2, -1, 1, 2, 3, 4\}$ and multiplied together. Calculate the probability that the product is negative.


Given $2^{3x} = 128$, calculate the value of $2^{-x}$.


Given the set $\{-10, -7, -3, 0, 4, 6, 9\}$, find the minimum possible product of three different numbers from this set.


Given that Larry, Julius, and Nina are playing a game where they take turns throwing a ball at a bottle with probabilities of $\frac{1}{3}$ of knocking it off the ledge at each turn, calculate the probability that Larry wins the game.


George walks $2$ miles to school at a normal speed of $4$ miles per hour. He walked the first $\frac{3}{4}$ mile at $3$ miles per hour and the next $\frac{3}{4}$ mile at $4$ miles per hour. Find the speed at which George must run the last $\frac{1}{2}$ mile to arrive just as school begins.


Given the set ${2, 5, 8, 11, 14, 17, 20}$, determine the number of different integers that can be expressed as the sum of three distinct members of this set.


Handy Fiona helped her neighbor, spending 1 hour 30 minutes on Monday, 1 hour 15 minutes on Tuesday, from 9:10 AM to 12:20 PM on Wednesday, and 45 minutes on Thursday. She is paid $\textdollar4$ per hour. Calculate the total amount Handy Fiona earned for the week.


In $\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ bisect the right angle. If the area of $\triangle CHA$ is $K$, calculate the area of $\triangle ABC$.


Determine for how many positive integers \( n \) the expression \( n^3 - 7n^2 + 17n - 11 \) is a prime number.


Given $2^w \cdot 3^x \cdot 5^y \cdot 7^z = 13230$, calculate the value of $3w + 2x + 6y + 4z$.


Given the quadratic equation $3x^2 - mx + 2x + 6 = 0$, find the values of $m$ for which the quadratic equation will have real and equal roots.

Solution: \boxed{2 - 6\sqrt{2} \text{ and } 2 + 6\sqrt{2}}


Find the hundreds digit of $(30! - 25!)$


If $5(2y + 3\sqrt{3}) = S$, calculate the value of $10(4y + 6\sqrt{3})$.


Given that $n$ is the smallest positive integer such that $n$ is divisible by $12$, $n^2$ is a perfect cube, $n^3$ is a perfect square, and $n^4$ is a perfect fifth power, find the number of digits of $n$.


What is the maximum value of $\frac{(2^t - 5t)t}{4^t}$ for real values of $t$?


Four red beads, two white beads, and two blue beads are placed in line in random order. Calculate the probability that no two beads of the same color are neighbors.


A biased coin has a probability of $\frac{3}{4}$ of landing heads and $\frac{1}{4}$ of landing tails on each toss. The outcomes of the tosses are independent. The probability of winning Game C, where the player tosses the coin four times and wins if either all four outcomes are heads or all four are tails, can be compared to the probability of winning Game D, where the player tosses the coin five times and wins if the first two tosses are the same, the third toss is different from the first two, and the last two tosses are the same as the first two. Determine the difference in the probabilities of winning Game C and Game D.


Given a two-digit number, determine how many numbers satisfy the condition that the unit digit of the number formed by subtracting the sum of its digits is divisible by 7.


A pentagon is inscribed in a circle. Find the sum of the angles inscribed in the five arcs cut off by the sides of the pentagon.


Given that a medium bottle can hold $60$ milliliters and a large bottle can hold $720$ milliliters, and there are small bottles that hold $40$ or $45$ milliliters, determine the total number of small bottles that Jasmine must buy to completely fill the large bottle.


Given that Linda has 30 coins consisting of nickels and dimes, and that if her nickels were dimes and her dimes were nickels, she would have 90 cents more, determine the total value of her coins in dollars.


A 4x4x4 cube is made of $64$ normal dice, each with opposite faces summing to $7$. Find the smallest possible sum of all the values visible on the $6$ faces of the large cube.


Given that 20% of the students scored 60 points, 25% scored 75 points, 40% scored 85 points, and the remainder scored 95 points, find the difference between the mean and median score of the students' scores in this competition.


Given rectangle $EFGH$ with $EF=10$ and $EG=15$, and point $N$ on $\overline{EH}$ such that $EN=6$, calculate the area of $\triangle ENG$.


For the even number 144, determine the largest possible difference between two prime numbers whose sum is 144.


Given the four-digit number $2552$, determine the total number of different arrangements of its digits.


Given two poles, one 30 ft high and the other 70 ft high, are 150 ft apart, calculate the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole.


Given $S = i^{n+k} + i^{-(n+k)}$ where $i = \sqrt{-1}$, $n$ is any integer and $k \in \{1, 2\}$, determine the total number of distinct values for $S$.


If three plus the reciprocal of $(2-y)$ equals twice the reciprocal of $(2-y)$, find $y$.


Let $N = 36 \cdot 42 \cdot 49 \cdot 280$. Find the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$.


Calculate Glenda's average speed in miles per hour given that the odometer reading on her scooter was a palindrome of $1221$ and after riding for $5$ hours and $7$ hours, the odometer showed another palindrome of $1881$.


Given four primes $p, q, s$, and $r$ satisfy $p + q + s = r$ and $2 < p < q < s$, determine the smallest value of $p$.


Positive integers $a$ and $b$ are such that $a < 6$ and $b \leq 7$. Calculate the smallest possible value for $2 \cdot a - a \cdot b$.


Evaluate the expression: $\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{4}}}}$


Given that the ratio of $2x - 5$ to $y + 20$ is constant, and $y = 6$ when $x = 7$, find the value of $x$ when $y = 21$.


A point is chosen at random within a square in the coordinate plane whose vertices are $(0, 0), (2500, 0), (2500, 2500),$ and $(0, 2500)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{7}{16}$. Find the value of $d$ to the nearest tenth.


Given the expressions \(\frac{x + a}{y}\) and \(\frac{y - b}{x}\), where \(x \neq 0\) and \(y \neq 0\), evaluate the arithmetic mean of these expressions.

Solution: \boxed{\frac{x^2 + ax + y^2 - by}{2xy}}


A grocer stacks apples in a cubic-like stack where the base cube consists of $4$ apples by $4$ apples by $4$ apples. Each apple above the first level rests in a pocket formed by four apples below, with the next layer reducing by one apple in each dimension. The stack is completed by a single apple. Find the total number of apples in the stack.


Determine the number of real values of \( x \) that satisfy the equation \((2^{5x+2})(4^{2x+4}) = 8^{3x+7}\).


Evaluate the expression $3 + 2\sqrt{3} + \frac{1}{3 + 2\sqrt{3}} + \frac{1}{2\sqrt{3} - 3}$.

Solution: \boxed{3 + \frac{10\sqrt{3}}{3}}


What is the greatest possible sum of the digits in the base-eight representation of a positive integer less than $1728$?


If $\alpha$, $\beta$, $p$, and $q$ are positive numbers, and $q<50$, then the number obtained by increasing $\alpha \times \beta$ by $p\%$ followed by a decrease by $q\%$ exceeds $\alpha \times \beta$ if and only if...
A) $p > q$
B) $p > \dfrac{q}{100-q}$
C) $p > \dfrac{q}{1-q}$
D) $p > \dfrac{100q}{100+q}$
E) $p > \dfrac{100q}{100-q}$

Solution: \boxed{p > \dfrac{100q}{100-q}}


Yesterday, Han drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Jan drove 3 hours longer than Ian at an average speed 15 miles per hour faster than Ian. Han drove 100 miles more than Ian. Given this information, calculate the difference in the distances driven by Jan and Ian.


How many whole numbers are between $\sqrt{18}$ and $\sqrt{200}$?


Consider an isosceles triangle $ABC$ where $CA = CB = 15$ inches and $AB = 24$ inches. A circle with radius $15$ inches has its center at $C$ and passes through vertices $A$ and $B$. Extend $AB$ through $B$ to intersect the circle at point $E$. Determine the number of degrees in angle $AEC$.


Consider an arithmetic progression where the sum of the first 15 terms is three times the sum of the first 8 terms, find the ratio of the first term to the common difference.


How many 4-digit numbers greater than 1000 are there that use the four digits 2, 0, 3, and 3?


What is the hundreds digit of $(25! - 20!)$?


A triangle is inscribed in a circle. Calculate the sum of three angles, each inscribed in one of the three segments outside the triangle.


Given Jane reads $\frac{1}{4}$ of the pages plus $10$ additional pages on the first day, $\frac{1}{5}$ of the remaining pages plus $20$ pages on the second day, and $\frac{1}{2}$ of what's left plus $25$ pages on the third day, calculate the total number of pages in Jane's book if there are $75$ pages remaining.


A box contains $35$ red balls, $25$ green balls, $22$ yellow balls, $15$ blue balls, $12$ white balls, and $10$ black balls. Calculate 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.


Let \( x = 1729 \). Determine the value of \(\Bigg\vert\Big\vert |x| + x\Big\vert + |x|\Bigg\vert + x\).


In a botanical garden, there are three overlapping flower beds. Bed X contains 600 plants, bed Y contains 500 plants, and bed Z contains 400 plants. Beds X and Y share 70 plants, beds X and Z share 80 plants, and beds Y and Z share 60 plants. Additionally, 30 plants are shared among all three beds. Calculate the total number of distinct plants in the garden.


Kiana has two older twin brothers. The product of their three ages is 162. What is the sum of their three ages?


A rectangle $ABCD$ has a length of 8 inches and a width of 6 inches. Diagonal $AC$ is divided into four equal segments by points $P$, $Q$, and $R$. Calculate the area of the triangle $BPQ$.


Evaluate the sum of the sequence: $1342 + 2431 + 3124 + 4213$, then calculate the product of the sum and $3$.


The sum of the first n positive odd integers from 1 to 2021, minus the sum of the first n positive even integers from 2 to 2020.


What is the maximum value of $\frac{(3^t - 5t)t}{9^t}$ for real values of $t$?


A circle of radius 3 is centered at point $A$. An equilateral triangle with side 6 has a vertex at $A$. Calculate 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})}


The circumference of the Earth at this latitude is 3000 \times 2 \pi, and the jet flies at 600 miles per hour, so calculate the number of hours the flight will take.


Calculate the sum of the numbers $1357 + 7531 + 3175 + 5713$.


Ms. Linda teaches mathematics to 22 students. Before she graded Eric's test, the average score for the class was 84. After grading Eric's test, the class average rose to 85. Determine Eric's score on the test.


Given that ten points are chosen on the surface of a sphere, and lines are drawn connecting every pair of points, such that no three lines intersect in a single point inside the sphere, calculate the number of tetrahedrons with all four vertices in the interior of the sphere.


Using the new operation \( x \otimes y = x^2 + y^2 \), simplify the expression \( x \otimes (x \otimes x) \).

Solution: \boxed{x^2 + 4x^4}


In $\bigtriangleup ABC$, $E$ is a point on side $\overline{AB}$, and $D$ is a point on side $\overline{BC}$ such that $BD=DE=EC$. Let $\angle BDE$ be $90^{\circ}$. Determine the degree measure of $\angle AED$.


Given the permutation $(a_1, a_2, a_3, a_4, a_5, a_6)$ of $(1, 2, 3, 4, 5, 6)$, calculate the number of such permutations that satisfy the condition $a_1 + a_2 + a_3 < a_4 + a_5 + a_6$.


How many 3-digit whole numbers, whose digit-sum is 27, are divisible by 3 and even?


The dimensions of a rectangular box are all positive integers and the volume of the box is $3003$ in$^3$. Find the minimum possible sum of the three dimensions.


Given Laura is adding two distinct three-digit positive integers, where all six digits in these numbers are different, and the sum, $S$, is a four-digit number, calculate the smallest possible value for the sum of the digits of $S$.


Alice needs to take one blue pill and one red pill each day for three weeks. A blue pill costs $2 more than a red pill, and Alice's pills cost a total of $945 for the three weeks. Find the cost of one blue pill.


A point is chosen at random within the square in the coordinate plane whose vertices are (0, 0), (4040, 0), (4040, 4040), and (0, 4040). The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{4}$. What is $d$ to the nearest tenth?


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


Sophia's age is $S$ years, which is thrice the combined ages of her two children. Her age $M$ years ago was four times the sum of their ages at that time. Find the value of $S/M$.


Given a set of 36 square blocks arranged into a 6 × 6 square, determine the number of different combinations of 4 blocks that can be selected from that set so that no two are in the same row or column.


Given triangle $PQR$, where $P = (0,0)$, $R = (8,0)$, and $Q$ is in the first quadrant with $\angle QRP = 90^\circ$ and $\angle QPR = 45^\circ$. Suppose that $PQ$ is rotated $90^\circ$ counterclockwise about $P$. Find the coordinates of the image of $Q$.

Solution: \boxed{(-8, 8)}


Point $O$ is the center of the regular octagon $ABCDEFGH$, $X$ is the midpoint of side $\overline{AB}$, and $Y$ is the midpoint of side $\overline{CD}$. Calculate the fraction of the area of the octagon that is shaded if the shaded region includes the full area of triangles $\triangle BCO$, $\triangle CDO$, $\triangle DEO$, $\triangle EFO$, and half the area each of $\triangle ABO$ and $\triangle CDO$.
A) $\frac{3}{8}$
B) $\frac{7}{16}$
C) $\frac{13}{32}$
D) $\frac{1}{2}$
E) $\frac{5}{8}$


Angie, Bridget, Carlos, Diego, and Edwin are seated at random around a circular table, one person to each chair. Calculate the probability that Angie and Carlos are seated in positions that are exactly two seats apart.


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


Calculate the sum of the series given by the expression:
\[3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + 5\left(1-\dfrac{1}{5}\right) + \cdots + 12\left(1-\dfrac{1}{12}\right)\]


A box contains $30$ red balls, $25$ green balls, $22$ yellow balls, $15$ blue balls, $12$ white balls, and $6$ black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $18$ balls of a single color will be drawn.


Given that teams $X$ and $Y$ are participating in a hockey league, and team $X$ has won $\frac{3}{4}$ of its games and team $Y$ has won $\frac{2}{3}$ of its games, while team $Y$ has won $4$ more games, lost $5$ more games, and drawn $3$ more games than team $X$, determine the number of games played by team $X$.


Given that the product of the digits of a 3-digit positive integer equals 36, calculate the number of such integers.


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


Gabriela has a younger brother and an older sister. The product of their three ages is 72. Find the sum of their three ages.


Ana's monthly salary was $2500 in May. In June, she received a 15% raise and an additional $300 bonus. In July, she received a 25% pay cut. Calculate Ana's monthly salary after these adjustments in June and July.


Given that the distance light travels in one year is approximately $6,000,000,000,000$ miles, calculate the distance light travels in $50$ years.

Solution: \boxed{3 \times 10^{14} \text{ miles}}


Calculate the smallest product obtainable by multiplying any two numbers from the set $\{-9, -5, -1, 1, 4\}$.


In a trapezoid, the line segment joining the midpoints of the diagonals has length $5$, and the longer base has a length of $115$. Calculate the shorter base.


If Yen has a 5 × 7 index card and reduces the length of the shorter side by 1 inch, the area becomes 24 square inches. Determine the area of the card if instead she reduces the length of the longer side by 2 inches.


Given that Chelsea is leading by 60 points halfway through a 120-shot archery competition, each shot can score 10, 7, 3, or 0 points, and Chelsea always scores at least 3 points. If Chelsea's next \(n\) shots are all for 10 points, she will secure her victory regardless of her opponent's scoring in the remaining shots. Find the minimum value for \(n\).


If $a > 2$, then find the sum of the real solutions of the equation:
\[ \sqrt{a - \sqrt{a - x}} = x \]
A) $\frac{\sqrt{4a} - 1}{2}$
B) $\frac{\sqrt{4a-1} - 1}{2}$
C) $\frac{\sqrt{4a-2} - 1}{2}$
D) $\frac{\sqrt{4a-3} - 2}{2}$
E) $\frac{\sqrt{4a-3} - 1}{2}$

Solution: \boxed{\frac{\sqrt{4a-3} - 1}{2}}


Given the product $\frac{5}{4}\cdot \frac{6}{5}\cdot \frac{7}{6}\cdot \ldots \cdot \frac{a}{b} = 42$, calculate the sum of $a$ and $b$.


$\triangle ABC$ has a right angle at $C$ and $\angle A = 30^{\circ}$. If $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, then find the measure of $\angle BDC$.


Let $M$ be the greatest five-digit number whose digits have a product of $72$. Calculate the sum of the digits of $M$.


For how many positive integers $x$ is $\log_{10}(x-50) + \log_{10}(70-x) < 2$?


Given $g(x) = 2 - 3x^2$ and $f(g(x)) = \frac{2-3x^2}{2x^2}$ when $x \neq 0$, determine $f\left(\frac{2}{3}\right)$.


For all real numbers $x$, calculate the value of the expression $x[x\{x(x-3)-5\}+11]+2$.

Solution: \boxed{x^4 - 3x^3 - 5x^2 + 11x + 2}


If Billy Bob counts a total of 29 wheels and observes 10 children riding past his house, using bicycles, tricycles, and scooters, determine the number of tricycles among the children.


Given that Connor takes four tests, each worth a maximum of 100 points, and his scores on the first two tests are 82 and 75, calculate the lowest score he could earn on one of the remaining two tests in order to have a mean score of 85 for all four tests.


If the product $\dfrac{5}{3}\cdot \dfrac{6}{5}\cdot \dfrac{7}{6}\cdot \dfrac{8}{7}\cdot \ldots\cdot \dfrac{a}{b} = 16$, calculate the sum of $a$ and $b$.


Given a basketball player made 7 shots during a game, each shot was worth either 1, 2, or 3 points, determine how many different numbers could represent the total points scored by the player.


If Shauna takes five tests, each worth a maximum of 100 points, and her scores on the first three tests are 76, 94, and 87, while aiming for an average of 85 across all five tests, given that she scores 92 on the fourth test, determine the lowest score she could earn on the fifth test to meet her goal.


Let $p$ and $q$ be the roots of the quadratic equation $x^2 - 2px + q = 0$, where $p \neq 0$ and $q \neq 0$. Calculate the sum of the roots $p$ and $q$.


Given David's previous math contest scores of 88, 92, 75, 83, and 90, what is the minimum score he must achieve on his next contest to increase his average score by at least 4 points?


The region consisting of all points in three-dimensional space within $4$ units of line segment $\overline{CD}$ has volume $544\pi$. Calculate the length of $CD$.


For how many three-digit whole numbers does the sum of the digits equal 27?


Given a box containing five pieces of paper, each labeled with one of the digits $1$, $2$, $4$, $5$, or $6$, calculate the probability that a three-digit number formed by drawing three of these digits (one at a time without replacement) has a sum of digits that is a multiple of $5$.


What is the hundreds digit of $(25! - 20!)$?


What is the total cost of three purchases for $\textdollar 2.99$, $\textdollar 6.51$, and $\textdollar 10.49$, rounded to the nearest dollar using the rounding rule where exact half amounts ($x.50$) are rounded down instead of up?


Consider the 16 integers from $-8$ to $7$, inclusive, arranged to form a 4-by-4 square. What is the value of the common sum of the numbers in each row, the numbers in each column, and the numbers along each of the main diagonals?


Given the polynomial $x^3+2x^2-3$ divided by the polynomial $x^2+2$, determine the remainder.

Solution: \boxed{-2x - 7}


Find the units digit of the decimal expansion of $\left(12 + \sqrt{245}\right)^{17} + \left(12 + \sqrt{245}\right)^{76}$.


An equilateral triangle has a side length of 2. Inside it, three congruent isosceles triangles are constructed such that their bases collectively cover the entire perimeter of the equilateral triangle equally. The sum of the areas of these three isosceles triangles equals the area of the equilateral triangle. Find the length of one of the congruent sides of one of the isosceles triangles.


Emily traveled the first one-fifth of her journey on a country road, the next 30 miles through a city, and then one-third on a highway before completing the final one-sixth on a rural path. Calculate the total length of Emily's journey in miles.


Given that $800$ students were surveyed and $320$ students preferred spaghetti and $160$ students preferred manicotti, calculate the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti.


If \( a \) and \( b \) are positive numbers such that \( a^b = b^a \) and \( b = 4a \), find the value of \( a \).


While Steve and LeRoy are fishing 2 miles from shore, their boat springs a leak, and water comes in at a constant rate of 12 gallons per minute. The boat will sink if it takes in more than 40 gallons of water. Steve starts rowing towards the shore at a constant rate of 3 miles per hour while LeRoy bails water out of the boat. Determine the slowest rate, in gallons per minute, at which LeRoy must bail if they are to reach the shore without sinking.


Points $P$ and $Q$ are on a circle of radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of the line segment $PR$.

Solution: \boxed{\sqrt{98 - 14\sqrt{33}}}


A box contains 34 red balls, 25 green balls, 18 yellow balls, 21 blue balls, and 13 purple balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 18 balls of a single color will be drawn.


Maria buys computer disks at a price of 5 for $7 and sells them at a price of 4 for $7. Find the number of computer disks Maria must sell in order to make a profit of $125.


Two carts are racing down two parallel hills. The first cart travels $6$ inches in the first second and accelerates so that each successive $1$-second time interval, it travels $8$ inches more than during the previous $1$-second interval. It takes $35$ seconds to reach the bottom of the hill. The second cart starts $2$ seconds after the first and travels $7$ inches in the first second, then each successive $1$-second time interval, it travels $9$ inches more than the previous interval. Find the distance, in inches, the second cart travels by the time the first cart reaches the bottom of the hill.


A circle with area $A_1$ is inside a larger circle with area $A_1 + A_2$, where the larger circle has a radius of 5. If $A_1, A_2, A_1 + A_2$ are in an arithmetic progression, find the radius of the smaller circle.


A pair of standard 10-sided dice is rolled once, each die numbered from 1 to 10. The sum of the numbers rolled determines the diameter of a circle. Calculate the probability that the numerical value of the circle's area is less than the numerical value of its circumference.


The average age of 8 people in a room is 28 years. A 20-year-old person leaves the room. Then a 25-year-old person enters the room. Calculate the new average age of the people now in the room.


Find the number of positive factors of 90 that are also multiples of 6.


Given a team with 25 players, where each player must receive a minimum salary of $18,000, and the total salary for all players cannot exceed $900,000, calculate the maximum possible salary, in dollars, for any single player on the team.


Starting from "1," determine the fewest number of keystrokes needed to reach "480" using only the keys [+1] and [x2].


Given strides, leaps, and bounds are specific units of length. If $3$ strides equals $4$ leaps, $5$ bounds equals $7$ strides, and $2$ bounds equals $9$ meters, then determine the number of leaps in one meter.


Ed and Rose start by taking turns counting aloud, beginning with Rose saying the number $4$. Each subsequent number is obtained by adding $3$ to the last number spoken by the other person. Find the $26^{\text{th}}$ number spoken in this sequence.


Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $12$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks, she helps around the house for $10$, $13$, $9$, $14$, and $11$ hours. Find the number of hours that Theresa must work in the final week to earn the tickets.


Suppose $d$ is a digit. For how many values of $d$ is $2.0d05 > 2.010$?


Given externally tangent circles with centers at points $A$ and $B$ and radii of lengths $6$ and $4$, respectively, a line externally tangent to both circles intersects ray $AB$ at point $C$. Calculate $BC$.


Josh has $40$ thin rods, one each of every integer length from $1$ cm through $40$ cm. Given that he places the rods with lengths $5$ cm, $12$ cm, and $20$ cm on a table, how many of the remaining rods can he choose as the fourth rod to form a quadrilateral with positive area?


A 7 × 7 grid of square blocks is arranged. Determine how many different combinations of 4 blocks can be selected from this set such that no two blocks are in the same row or column.


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 exactly half of the other 7 balls.


Carlos took $80\%$ of a whole pie, and Maria took one fourth of the remainder. Calculate the portion of the whole pie that was left.


Given that 50% of the students at Central College like dancing, and those who like dancing have a 30% false negative rate, with the remaining 70% accurately expressing their liking, and those who dislike dancing have an 20% false positive rate, with the remaining 80% accurately expressing their dislike, find the fraction of students who say they dislike dancing who actually like dancing.


A cylindrical tank with radius $6$ feet and height $12$ feet is lying on its side. The tank is filled with water to a depth of $3$ feet. Calculate the volume of the water in the tank in cubic feet.

Solution: \boxed{144\pi - 108\sqrt{3}}


Chelsea is 60 points ahead halfway through a 120-shot archery competition, where each bullseye scores 10 points and other possible scores are 7, 3, 1, and 0. Chelsea consistently scores at least 3 points per shot. Calculate the minimum number of consecutive bullseyes Chelsea needs over her next $n$ shots to secure victory, assuming her competitor can score a maximum in each shot hereafter.


Given that $\angle XYZ = 40^\circ$ and $\angle XYW = 15^\circ$, determine the smallest possible degree measure for $\angle WYZ$.


Twenty-five percent less than 80 is twenty-five percent more than what number?


Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated.


Olivia drove the first quarter of her trip on a mountain road, the next $30$ miles on a paved road, and the remaining one-sixth on a desert road. What is the total length of Olivia's entire trip?


If $2(5x + 3\pi)=Q$, evaluate the expression $4(10x + 6\pi + 2)$.

Solution: \boxed{4Q + 8}


The Eagles won 3 out of the 5 games they initially played against the Hawks. If the Hawks won all subsequent games that they played against the Eagles, determine the minimum number of additional games they must win to reach a winning percentage of at least 90%.


Let $\omega = -\frac{1}{2} + \frac{i\sqrt{3}}{2}$ and $\omega^2 = -\frac{1}{2} - \frac{i\sqrt{3}}{2}$. Define $T$ as the set of all points in the complex plane of the form $a + b\omega + c\omega^2 + d$, where $0 \leq a, b, c \leq 1$ and $d \in \{0, 1\}$. Find the area of $T$.


Given the expression $\sqrt{36\sqrt{12\sqrt{9}}}$, evaluate the value of this expression.


Given that $\operatorname{log}_{8}(p) = \operatorname{log}_{12}(q) = \operatorname{log}_{18}(p-q)$, calculate the value of $\frac{q}{p}$.

Solution: \boxed{\frac{\sqrt{5} - 1}{2}}


Segments AD=12, BE=8, CF=20 are drawn from vertices of triangle ABC, each perpendicular to a straight line RS, not intersecting the triangle. Points D, E, F are the intersection points of RS with the perpendiculars. If x is the length of the perpendicular segment GH drawn to RS from the centroid G of the triangle, calculate the value of x.


Calculate the probability of a contestant winning the quiz given that they answer at least 3 out of 4 questions correctly, assuming random guesses for each question with 3 possible answer options.


A student must schedule 3 mathematics courses — algebra, geometry, and number theory — in a 7-period day, such that at most one pair of mathematics courses can be taken in consecutive periods. Calculate the total number of ways this can be done.


Three fair tetrahedral dice (four sides each) are tossed at random. Calculate the probability that the three numbers turned up can be arranged to form an arithmetic progression with a common difference of one.


Ray's car averages 50 miles per gallon of gasoline, and Tom's car averages 15 miles per gallon of gasoline. If they both drive the same number of miles, calculate their cars' combined rate of miles per gallon of gasoline.


Let $n = \frac{3xy}{x-y}$. Solve for $y$ in terms of $x$ and $n$.

Solution: \boxed{\frac{nx}{3x + n}}


What is the minimum test score Josanna would need on her next test to increase her test average by $10$ points given that she received scores of $75, 85, 65, 95, 70$ on her previous tests?


A painting $4$ feet wide is hung in the center of a wall that is $26$ feet wide. Calculate the distance from the end of the wall to the nearest edge of the painting.


Given $T' = \{(0,0),(5,0),(0,4)\}$ be a triangle in the coordinate plane, determine the number of sequences of three transformations—comprising rotations by $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, and reflections across the lines $y = x$ and $y = -x$—that will map $T'$ onto itself.


Given a wooden cube $n$ units on a side is painted blue 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 blue. What is $n$?


Given two positive numbers $p$ and $q$ that are in the ratio $3:5$, and the sum of the squares of $p$ and $q$ equals $2k$, calculate the smaller of $p$ and $q$.


Given that the number $3103$ has four digits, determine the number of different four-digit numbers that can be formed by rearranging these digits.


Mia drove the first one-fourth of her journey on a highway, the next 25 miles through a town, and the final one-sixth on a country road. Calculate the total length of Mia's trip in miles.


Given distinct points $G$, $H$, $I$, and $J$ lie on a line, with $GH = HI = \frac{1}{2}$, and $IJ = 1$, and points $K$ and $L$ lie on a second line, parallel to the first, with $KL = 2$, determine the number of possible values for the area of a triangle formed by selecting any three of the six points.


In an arithmetic sequence, the first term is $3$, the last term is $34$, and the sum of all the terms is $222$. Find the common difference.


Let $x = 0.\overline{527}$, and express $x$ in its simplest fractional form, and then calculate the sum of the numerator and denominator.


Points $A(13, 11)$ and $B(5, -1)$ are vertices of $\triangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(2, 7)$. Determine the coordinates of point $C$.

Solution: \boxed{(-1, 15)}


The numbers $-3, 5, 7, 10, 15$ are rearranged according to the following rules:
1. The largest number is not in the last place, but it is within the last three places.
2. The smallest number is not in the first place, but it is within the first three places.
3. The median number is neither in the first nor in the last place. Determine the sum of the second and fourth numbers.


A digital watch displays hours and minutes in a 24-hour format. What is the largest possible sum of the digits in the display?


Given a circle with radius 10 units, and two semicircles within it, one with radius 8 units and another with radius 6 units, both passing through the center of the circle, determine the ratio of the combined areas of these two semicircles to the area of the larger circle.


An expanded grid of a city has intersections labeled A, C, and B. Rectangle blocks are surrounded by streets. The student goes from Home (A) situated at the top left of the grid to School (B) located at the bottom right of a 4x4 block grid. The coffee shop (C) is located such that the student needs to make exactly 3 eastward moves and 2 southward moves from A to C. Calculate the probability that the student, choosing randomly at intersections, will walk through point C on his way to school.


Given the progression \(8^{\frac{2}{11}}, 8^{\frac{3}{11}}, 8^{\frac{4}{11}}, \dots, 8^{\frac{(n+1)}{11}}\), calculate the least positive integer \(n\) such that the product of the first \(n\) terms of the progression exceeds 1,000,000.


For how many positive integer values of $N$ does the expression $\dfrac{48}{N+3}$ result in an integer?


Given the inequality $(x-1)^2 < 12 - x$, determine the range of values for $x$ that satisfy this inequality.

Solution: \boxed{\left(\frac{1 - 3\sqrt{5}}{2}, \frac{1 + 3\sqrt{5}}{2}\right)}


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


Given that the probability of a boy being chosen is $\frac{3}{4}$ of the probability that a girl is chosen, find the ratio of the number of boys to the total number of boys and girls in the class.


On hypotenuse $AB$ of a right triangle $ABC$, a second right triangle $ABD$ is constructed with hypotenuse $AB$. If $\overline{BC} = 3$, $\overline{AC} = a$, and $\overline{AD} = 1$, then calculate $\overline{BD}$.

Solution: \boxed{$\sqrt{a^2 + 8}$}


Given that there are three times as many third graders as fourth graders, twice as many fourth graders as fifth graders, and the average daily running times for third, fourth, and fifth graders are 14, 18, and 8 minutes, respectively, calculate the average number of minutes run per day by these students.


Given that Thomas the painter initially had enough paint for 50 identically sized rooms, and 4 cans of paint were lost, determining how many cans of paint he used for 42 rooms.


Given $S = i^n + i^{-2n}$, where $i = \sqrt{-1}$ and $n$ is an integer, determine the total number of possible distinct values for $S$.


Determine the times between $7$ and $8$ o'clock, to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$.


A retailer purchases a product at $50 less 20%. He then intends to sell the product at a profit of 25% relative to his cost after applying a 10% discount on his marked price. What should the marked price be, in dollars?


Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. Determine the number of two-digit values of $x$ for which $\clubsuit(\clubsuit(x))=4$.


Consider the year 2020, which is not a palindrome. What is the product of the digits of the next year after 2020 that reads the same forwards and backwards?


Given the sum of the first 15 terms of an arithmetic progression is four times the sum of the first 8 terms, find the ratio of the first term to the common difference.


Given that point $E$ lies on $\overline{AB}$ and line $\overline{ED}$ bisects $\angle ADC$, determine the ratio of the area of $\triangle AED$ to the area of square $ABCD$.


Given that $a$ cows give $b$ gallons of milk in $c$ days with a standard efficiency factor of 100%, calculate the number of gallons of milk $d$ cows will give in $e$ days when the efficiency declines by 10%.


The maximum number of possible points of intersection of a circle and a rectangle.


Given six test scores have a mean of $85$, a median of $86$, and a mode of $88$. Determine the sum of the two lowest test scores.


In a triangle, one side measures 16 inches and the angle opposite this side is 45°. Calculate the diameter of the circumscribed circle surrounding this triangle.

Solution: \boxed{16\sqrt{2}\text{ inches}}


Determine the sum of the prime factors of the number $2550$.


Given the sequence $1, -2, 3, -4, 5, -6, \ldots, $ whose nth term is $(-1)^{n+1}\cdot n$, determine the average of the first $201$ terms of this sequence.


Let $S=\{(x,y) : x\in \{0,1,2,3\}, y\in \{0,1,2,3\},\text{ and } (x,y)\ne (2,2)\}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $B$, find the product of the values of $\tan(\angle{ACB})$ for every $t$ in $T$.


A quadrilateral has vertices $P(a+1,b-1)$, $Q(b+1,a-1)$, $R(-a-1, -b+1)$, and $S(-b-1, -a+1)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $24$. Calculate the value of $a+b$.


Given Paula initially planned to paint 50 rooms with a total amount of paint that is just enough to cover 40 rooms after losing 5 cans, calculate the number of cans of paint she used for these 40 rooms.


A palindrome between $1000$ and $10,000$ is chosen at random. Calculate the probability that it is divisible by $11$.


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


Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is 7:3, and the ratio of blue to green marbles in Jar 2 is 9:1. There are 80 green marbles in all. Calculate the difference in the number of blue marbles in Jar 1 and Jar 2.


Five friends earned $18, $23, $28, $35, and $45. If they split their earnings equally among themselves, how much will the friend who earned $45 need to give to the others?


Given an unfair die that has a probability 4 times as likely to result in an odd number as an even number, three such dice are rolled. Find the probability that the sum of the numbers rolled is odd.


Given that the roots of the polynomial $Q(x) = x^3 + px^2 + qx + r$ are $\cos \frac{\pi}{7}, \cos \frac{3\pi}{7},$ and $\cos \frac{5\pi}{7}$, where angles are in radians, determine the value of $pqr$.

Solution:
\boxed{1. Let $z = e^{\frac{\pi i}{7}}$. The roots of $Q(x) = x^3 + px^2 + qx + r$ are $\cos \frac{\pi}{7}, \cos \frac{3\pi}{7},$ and $\cos \frac{5\pi}{7}$. Using Euler's formula, this can be expressed as:
   \[
   \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}.
   \]

2. The sum of all the real parts (cosines) of $z^k$, where $k=1$ to $6$, noting that $\cos \theta = \cos (2\pi - \theta)$ and $\sum_{k=1}^{6} z^k = -1$, simplifies to:
   \[
   \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} = -\frac{1}{2}.
   \]

3. Using trigonometric identities for sum of cosines and polynomial coefficients, the polynomial becomes:
   \[
   x^3 + \frac{1}{2}x^2 - \frac{1}{2}x - \frac{1}{8} = 0.
   \]
   Here, $p = \frac{1}{2}$, $q = -\frac{1}{2}$, and $r = -\frac{1}{8}$.

4. The product of the coefficients $p$, $q$, and $r$ is given by $pqr = \left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)\left(-\frac{1}{8}\right) = \frac{1}{32}.$}


Suppose that $\tfrac{3}{4}$ of $14$ apples are worth as much as $9$ lemons. How many lemons are worth as much as $\tfrac{5}{7}$ of $7$ apples?


In a pentagon $ABCDE$, $\angle A = 100^\circ$, $\angle D = 120^\circ$, and $\angle E = 80^\circ$. If $\angle ABC = 140^\circ$, find the measure of $\angle BCD$.


Let $F=\frac{8x^2+20x+5m}{8}$ be the square of an expression which is linear in $x$. Find the range in which $m$ lies for this condition to hold.


Given that Ava's odometer displayed $14941$ miles, and four hours later another palindrome appeared on the odometer, calculate Ava's average speed, in miles per hour, during these 4 hours.


If two congruent circles centered at points $P$ and $Q$ each pass through the other circle's center, a line containing both $P$ and $Q$ is extended to intersect the circles at points $F$ and $G$, and the circles intersect at point $H$, where $\triangle PFQ$ is such that $PF = FQ$ and $PQ = QH = PH$, find the degree measure of $\angle FHG$.


Given Tara bought some notebooks for a total of $\textdollar 5.20$ at the school supplies store and Lea bought some of the same notebooks for $\textdollar 7.80$, assuming the notebooks cost more than a dollar each, determine the difference in the number of notebooks bought by Lea and Tara.


Given the quadratic equation \( Ax^2 + Bx + C = 0 \) with roots \( r \) and \( s \), formulate a new quadratic equation \( x^2 + px + q = 0 \) such that its roots are \( r^3 \) and \( s^3 \). Determine the value for \( p \).

Solution: \boxed{\frac{B^3 - 3ABC}{A^3}}


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


Two concentric circles have radii of 15 meters and 30 meters. An aardvark starts at point $A$ on the smaller circle and runs along the path that includes half the circumference of each circle and each of the two straight segments that connect the circumferences directly (radial segments). Calculate the total distance the aardvark runs.

Solution: \boxed{45\pi + 30}