Given that Sadie exchanged $d$ U.S. dollars with an exchange rate of $8$ Canadian dollars for every $5$ U.S. dollars, and after her shopping spent $80$ Canadian dollars and had exactly $d$ Canadian dollars left, calculate the sum of the digits of $d$.


In $\triangle ABC$, $D$ is on $AC$ and $E$ is on $BC$. Also, $AB \perp BC$, $AE \perp BC$, and $BD = DC = CE = x$. Find the length of $AC$.


Calculate the simplified value of the expression: $2 - \frac{1}{2 + \sqrt{5}} + \frac{1}{2 - \sqrt{5}}.$

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


Calculate the number of digits in the number $4^{25}5^{22}$ when written in decimal form.


A coffee shop offers a promotion stating, "25% off on coffee today: quarter-pound bags are available for $4.50 each." Determine the regular price for a half-pound of coffee, in dollars.


Given the sprinter's age is $30$ years, calculate the target heart rate, in beats per minute, which is $85\%$ of the theoretical maximum heart rate found by subtracting the sprinter's age from $225$.


Let $\angle PQR = 40^{\circ}$ and $\angle PQS = 15^{\circ}$. Find the smallest possible degree measure for $\angle SQR$.


Given a number twelve times as large as $x$ is increased by five, determine one third of the result in terms of $x$.

Solution: \boxed{4x + \frac{5}{3}}


The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements.


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


The point $Q(c,d)$ in the $xy$-plane is first rotated clockwise by $90^\circ$ around the point $(2,3)$ and then reflected about the line $y = x$. The image of $Q$ after these two transformations is at $(4, -1)$. Find the value of $d - c$.


To hit at least 90% of his targets in 60 rounds, John successfully hits 54 targets. After 40 rounds, he has 20 targets remaining. Calculate the maximum number of rounds he can afford to miss.


Given that Mina writes down one integer three times and another integer four times, and their sum is $135$, and one of the numbers is $15$, find the other number.


Two cards are dealt from a deck composed of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$. A winning pair consists of two cards of the same color or two cards with the same letter. Calculate the probability of drawing a winning pair.


Mary divides a circle into 15 sectors. The central angles of these sectors, also measured in degrees, are all integers and form an arithmetic sequence. Find the degree measure of the smallest possible sector angle.


Given that the least common multiple of $a$ and $b$ is $16$, and the least common multiple of $b$ and $c$ is $21$, determine the least possible value of the least common multiple of $a$ and $c$.


Let the number of $2 pairs of socks be x, the number of $3 pairs of socks be y, and the number of $5 pairs of socks be z. Given that x + y + z = 15 and 2x + 3y + 5z = 35, determine the value of x.


Given the set $A$ of positive integers that have no prime factors other than $2$ and $7$, express the infinite sum of the reciprocals of the elements of $A$ as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and calculate $m + n$.


Determine the number of significant digits in the measurement of the side of a square whose computed area is $2.3406$ square inches to the nearest ten-thousandth of a square inch.


Given that the area of triangle $XYZ$ is $100$ square units, and the geometric mean between sides $XY$ and $XZ$ is $15$ units, find $\sin X$.


Consider the operation "add the reciprocal of," defined by $a\diamond b=a+\frac{1}{b}$. Evaluate the expression $((3\diamond4)\diamond5)-(3\diamond(4\diamond5))$.


If Menkara has a $3 \times 7$ index card, and she shortens the length of one side by $2$ inches, resulting in an area of $15$ square inches, determine the area of the card if she shortens the length of the other side by $2$ inches.


Find the sum of the squares of all real numbers satisfying the equation $x^{64} - 64^{16} = 0$.


Eric sets a new goal to complete a modified triathlon in 3 hours. He plans to swim 0.5 miles at an average speed of 1.5 miles per hour, run 4 miles at an average speed of 5 miles per hour, and then ride a bicycle for 20 miles. What must his average speed in miles per hour be for the bicycle ride to achieve his goal?


Five different awards are to be given to four students. Each student will receive at least one award. Calculate the total number of ways the awards can be distributed.


Given that A can do a piece of work in $12$ days, and B is $66\frac{2}{3}\%$ more efficient than A, calculate the number of days it will take for A and B to complete the same piece of work when working together.


Given the area of a square is $1.2105$ square inches to the nearest ten-thousandth of a square inch, determine the number of significant digits in the measurement of the diagonal of the square.


Given a tetrahedron, calculate the sum of the number of edges, the number of vertices, and the number of faces, assuming Joe counted one vertex twice.


Triangle $DEF$ has a right angle at $D$. The sides of the triangle serve as diameters for semicircles. The area of the semicircle on $\overline{DE}$ equals $12.5\pi$, and the arc length of the semicircle on $\overline{DF}$ is $7\pi$. Determine the radius of the semicircle on $\overline{EF}$.


Each of a group of $60$ girls is either blonde or brunette and has either blue eyes or brown eyes. If $20$ are blue-eyed blondes, $36$ are brunettes, and $25$ are brown-eyed, then calculate the number of brown-eyed brunettes.


In John's first $6$ basketball games, he scored $10, 5, 8, 6, 11,$ and $4$ points. In his seventh game, he scored fewer than $15$ points and his points-per-game average for the seven games was an integer. For the eighth game, he also scored fewer than $15$ points, and the average for the $8$ games became an integer again. Determine the product of the number of points he scored in the seventh and eighth games.


A positive integer $n$ not exceeding $120$ is chosen such that if $n\le 60$, then the probability of choosing $n$ is $p$, and if $n > 60$, then the probability of choosing $n$ is $3p$. Determine the probability that a perfect square is chosen.


The bakery owner turns on his muffin machine at $\text{9:00}\ {\small\text{AM}}$ and completes one fourth of the day's job at $\text{12:15}\ {\small\text{PM}}$. Determine the time at which the muffin machine will complete the job.

Solution: \boxed{10:00 PM}


Given 4 mathematics courses and 8 available periods in a day, find the number of ways a student can schedule the courses so that no two mathematics courses are taken in consecutive periods.


A digital watch displays time in a 24-hour format showing only hours and minutes. Find the largest possible sum of the digits in the display.


Given that group A consisting of $a$ cows produces $b$ gallons of milk in $c$ days, and group B, which is $20\%$ more efficient in milk production than group A, has $d$ cows, calculate the number of gallons of milk that group B will produce in $e$ days.


Given that Peter won 5 games and lost 4 games, Emma won 4 games and lost 5 games, and Jordan lost 2 games, find the number of games Jordan won.


Given points $Q_1, Q_2, \ldots, Q_9$ on a straight line, in the order stated (not necessarily evenly spaced). Let $Q$ be a point on the line, and let $t$ be the sum of the undirected lengths $QQ_1, QQ_2, \ldots, QQ_9$. Determine the location of point $Q$ such that $t$ is minimized.


Given that 63 students wear red shirts and 81 students wear green shirts, and the 144 students are divided into 72 pairs, while 27 of these pairs consist of two students wearing red shirts, find the number of pairs with two students wearing green shirts.


Given that $a_n = \frac{(n+7)!}{(n-1)!}$, determine the rightmost digit of $a_k$ when it stops changing for the smallest positive integer $k$ such that the rightmost digit of $a_k$ stops changing after reaching $k+5$.


Given the track consists of 3 semicircular arcs with radii $R_1 = 120$ inches, $R_2 = 70$ inches, and $R_3 = 90$ inches, and a ball with a diameter of 6 inches rolls from point A to point B, calculate the distance the center of the ball travels.


Given that the activity lasts for 120 minutes and eight children participate, with only four children playing at any one time, calculate the total time each child plays.


Stephan has 60% more chocolates than Mary, and Clara has 40% more chocolates than Mary. Calculate the percentage difference in chocolates between Stephan and Clara.


Jasmine wishes to fill the decorative container holding 2650 milliliters entirely with craft bottles that hold 150 milliliters each. Calculate the number of craft bottles necessary for this purpose.


Right triangle DEF has leg lengths DE = 18 and EF = 24. If the foot of the altitude from vertex E to hypotenuse DF is F', then find the number of line segments with integer length that can be drawn from vertex E to a point on hypotenuse DF.


The base of an isosceles triangle is 20 inches. A line is drawn parallel to the base, which divides the triangle into two regions where the area of the smaller region is $\frac{1}{4}$ the area of the triangle. Determine the length of this line parallel to the base.


A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $3:1$ and the ratio of the rectangle's length to its width is $3:2$. Calculate the percentage of the rectangle’s area that is inside the square.


Given that Alex wins $\frac{1}{3}$ of the rounds, Mel's probability of winning is $\frac{p}{3}$ and Chelsea is three times as likely to win as Mel, determine the probability that Alex wins four rounds, Mel wins one round, and Chelsea wins one round.


Peter's family ordered a 16-slice pizza for dinner. Peter ate one slice of pizza alone and then shared another slice with his brother Paul and their sister Mary, each having an equal part of that slice. Calculate the amount of pizza that Peter ate in total.


Express the repeating decimal $7.171717\ldots$ as a fraction and compute the sum of its numerator and denominator in their lowest terms.


Two candles of the same length are made of different materials such that one burns out completely at a uniform rate in $5$ hours and the other in $3$ hours. At what time should the candles be lighted so that, at 5 P.M., one stub is three times the length of the other?

Solution: \boxed{2:30 \text{ PM}}


Consider a set of $n$ numbers where $n > 2$. Two of these numbers are $1 - \frac{1}{n}$ and $1 - \frac{2}{n}$, and the remaining $n-2$ numbers are all $1$. Calculate the arithmetic mean of these $n$ numbers.

Solution: \boxed{1 - \frac{3}{n^2}}


Given the expression $2 - (-3)^2 - 4 - (-5) - 6^2 - (-7)$, simplify the arithmetic expression.


If $7 = k \cdot 3^s$ and $126 = k \cdot 9^s$, determine the value of $s$.

Solution: \boxed{2 + \log_3 2}


John went to the bookstore and purchased 20 notebooks totaling $62. Some notebooks were priced at $2 each, some at $5 each, and some at $6 each. John bought at least one notebook of each type. Let x be the number of $2 notebooks, y be the number of $5 notebooks, and z be the number of $6 notebooks. Solve for x.


A $9 \times 9$ grid has a total of 81 squares, and the first square is dark. What is the difference in the number of dark squares and light squares on this grid?


Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE}$. Let $DA=20$, and let $FD=AE=12$. Calculate the area of rectangle $ABCD$.


In how many ways can $435$ be written as the sum of an increasing sequence of two or more consecutive positive integers?


Alex sent 200 text messages and talked for 51 hours, and the cell phone plan costs $25$ dollars each month, $5$ cents per text message sent, and $15$ cents for each minute used over $50$ hours. Calculate the total cost for Alex's cell phone plan in February.


A rectangle has a length of 12 meters and a width of 8 meters. The rabbit runs 8 meters along a diagonal toward the opposite corner, makes a 90-degree right turn, and runs 3 more meters. Calculate the average of the shortest distances from the rabbit to each side of the rectangle.


A line $x = k$ intersects the graphs of $y = \log_5 x$ and $y = \log_5 (x + 8)$. The distance between the points of intersection is $1$. Express $k$ in the form $a + \sqrt{b}$, where $a$ and $b$ are integers, and find $a + b$.


Given Michael walks at the rate of $4$ feet per second on a long straight path, and the garbage truck travels at $8$ feet per second in the same direction as Michael, calculate the number of times Michael and the truck will meet.


What is the smallest positive even integer $n$ such that the product $3^{1/6}3^{2/6}\cdots3^{n/6}$ is greater than $500$?


Find the total number of ordered pairs of integers \((x, y)\) that satisfy the equation \(x^{4} + y^2 = 4y\).


A standard 6-sided die and an 8-sided die are rolled once. The sum of the numbers rolled determines the side length of a square. What is the probability that the numerical value of the area of the square is less than the numerical value of the square's perimeter?


Given that a class with \( n > 15 \) students achieved an average quiz score of \( 12 \) and a subset of \( 15 \) students from this class has an average score of \( 20 \), determine the average score of the remaining students in terms of \( n \).


Given the sets of consecutive integers $\{1\}$,$\{2, 3\}$,$\{4,5,6\}$,$\{7,8,9,10\}$,$\; \cdots \;$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set, determine the sum $S_{15}$ of the elements in the 15th set.


Given that the Math Club has 15 students, the Science Club has 10 students, the Art Club has 12 students, and 5 students are enrolled in both the Science and Math Clubs but not in the Art Club, calculate the total number of students participating in these clubs.

Solution: \boxed{15 (Math) + 10 (Science) + 12 (Art) - 5 (overlap of Math and Science) = 15 + 10 + 12 - 5 = 32}


Given $\triangle ABC$, where $AB=6$, $BC=8$, $AC=10$, and $D$ is on $\overline{AC}$ with $BD=6$, find the ratio of $AD:DC$.


Given a rectangular box with dimensions $2 \times b \times c$, where $b$ and $c$ are integers with $b \leq c$, find the number of ordered pairs $(b, c)$ such that the sum of the volume and the surface area of the box equals 120.


Given the expression $2-(-3) \times 2 - 4 - (-5) \times 3 - 6$, evaluate the expression.


An insect lives on the surface of a regular octahedron with edges of length $\sqrt{2}$. It wishes to travel from the midpoint of one edge to the midpoint of an edge that shares no common faces with the starting edge. What is the length of the shortest such trip?
A) $\frac{\sqrt{2}}{2}$
B) $\frac{3\sqrt{2}}{2}$
C) $\sqrt{2}$
D) $2\sqrt{2}$
E) $2$


Given the sales from January through May are documented as $120$, $80$, $70$, $150$, and in May there was a 10% discount on the total sales of $50$, calculate the average sales per month in dollars after applying the discount for May.


Given that 300 students preferred spaghetti and 120 students preferred tortellini, and that these preferences were evenly distributed across the grade levels, find the ratio of the number of sophomores who preferred spaghetti to the number of sophomores who preferred tortellini.


Given the polynomial expansion of $(1+3x-2x^2)^5$, calculate the coefficient of $x^9$.


An inverted cone with a base radius of $8 \, \mathrm{cm}$ and a height of $24 \, \mathrm{cm}$ is full of water. This water is first poured into another cone with a base radius of $6 \, \mathrm{cm}$ to form a certain water level. Then all the water from this cone is completely poured into a cylinder with a radius of $6 \, \mathrm{cm}$. Determine the height of the water in the cylinder.


The parabolic arch has a height of $20$ inches and a span of $50$ inches. Find the height, in inches, of the arch at the point $10$ inches from the center.


The High School Ten basketball conference has 10 teams. Each team plays every other conference team twice and also plays 5 games against non-conference opponents. Calculate the total number of games in a season involving the High School Ten teams.


Find the polynomial whose roots satisfy the condition that the sum is double the product of the roots, with the sum of the roots being $2k$ and the product being $k$, and given that one of the roots is $1$.

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


Keiko walks around a rectangular track with semicircular ends each day. The straight lengths are constant, and the track width is 8 meters. It takes her 48 seconds longer to walk around the outside than around the inside. Given this, calculate Keiko's average speed in meters per second.


The number of lines that are simultaneously tangent to two circles with radii of 4 cm and 5 cm can be determined based on the different placements of the circles.

Solution: \boxed{0, 1, 2, 3, 4}


Jake and Ellie alternately toss their coins until someone gets heads. Jake's coin lands heads with probability $\frac{1}{4}$, and Ellie's coin lands heads with probability $\frac{1}{3}$. Jake goes first. Find the probability that Jake wins.


In a small reserve, a biologist counted a total of 300 heads comprising of two-legged birds, four-legged mammals, and six-legged insects. The total number of legs counted was 980. Calculate the number of two-legged birds.


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


Find the value of $x$ that satisfies the equation $25^{-3} = \frac{5^{60/x}}{5^{36/x} \cdot 25^{21/x}}.$


Given a positive number $y$, determine the condition that satisfies the inequality $\sqrt{y} < 3y$.

Solution: \boxed{y > \frac{1}{9}}


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

Solution: \boxed{7:16 \text{ and  7:56}}


Calculate the value of \(x[x\{x(3-x)-3\}+5]+1\).

Solution: \boxed{-x^4 + 3x^3 - 3x^2 + 5x + 1}


Given a quadrilateral with vertices $P(a,c)$, $Q(c,a)$, $R(-a, -c)$, and $S(-c, -a)$, where $a$, $c$ are integers with $a>c>0$, and with area 24, find the sum of $a$ and $c$.


Given that a fly is 3 meters from one wall, 7 meters from the other wall, and 10 meters from point P where two walls and the ceiling meet at a right angle, calculate the distance of the fly from the ceiling.


If the markings on a number line are equally spaced, what is the number y after taking four steps starting from 0, and reaching 25 after five steps?


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


Given that Beth gets the disease calcuvitis and must take one red pill and one blue pill each day for three weeks, and a red pill costs $3 more than a blue pill, and Beth's pills cost a total of $\textdollar 966$ for the three weeks, calculate the cost of one red pill.


Given square ABCD has side length s, calculate the ratio of the area of square EFGH to the area of square ABCD, where E, F, G, and H are the centers respectively of isosceles right-angled triangles with bases AB, BC, CD, and DA, each exterior to the square with the right angle at the base vertex.


Given that a randomly drawn positive factor of $84$ is selected, calculate the probability that it is less than $8$.


A circle is drawn through vertices $E$ and $H$ and tangent to side $FG$ of square $EFGH$ with side $12$ feet. Calculate the radius of the circle.


Suppose $m$ and $n$ are positive integers such that $45m = n^3$. Additionally, it is given that $n$ is a multiple of 5. Find the minimum possible value of $m + n$.


Evaluate the square of \(8 - \sqrt{x^2 + 64}\).

Solution: \boxed{x^2 + 128 - 16\sqrt{x^2 + 64}}


Initially, a square is completely painted black. Each time the square is changed, the middle half of each black square turns white. After three changes, what fraction of the original area of the black square remains black.


In an arithmetic progression, any term equals the average of the next two terms. The first term is 12. Find the common difference.


Given Max makes a larger batch of lemonade using 500 grams of lemon juice, 200 grams of sugar, and 1000 grams of water, and each 100 grams of lemon juice contains 30 calories, while each 100 grams of sugar has 400 calories, determine the total number of calories in 300 grams of this prepared lemonade.


Given that the sum of all the angles except one of a convex polygon is $2790^{\circ}$, calculate the number of sides of the polygon.


At a local beach festival, a vendor offers a special deal on flip-flops. If you buy one pair at the regular price of $45, you get the second pair at a 30% discount, and if you buy a third pair, it costs only 40% of the regular price. Maria opts to buy three pairs under this festival offer. Calculate the percentage of the total regular price that Maria saved.


One day a drink kiosk sold 360 cans of soda to 150 customers, with every customer buying at least one can of soda. Determine the maximum possible median number of cans of soda bought per customer on that day.


A cell phone plan costs $25$ dollars each month, plus $0.10$ dollars per text message sent, plus $0.15$ dollars for each minute used over $30$ hours, but not exceeding $31$ hours. Any usage beyond $31$ hours costs $0.20$ dollars per minute. Given that Lucas sent $150$ text messages and talked for $31.5$ hours, calculate the total cost of the plan.


Given the inequalities \(x - y + 3 > x\) and \(x + y - 2 < y\), find the relationship between \(x\) and \(y\).

Solution: \boxed{x < 2 \text{ and } y < 3}


Given that Ike and Mike enter a bakery with a total of $50.00 to spend, calculate the number of items, consisting of sandwiches at $5.00 each and cakes at $2.50 each, that they will buy altogether, with the condition that they do not want to buy more than 12 items in total.


Professor Lee has eleven different language books lined up on a bookshelf: three Arabic, four German, and four Spanish. Calculate the number of ways to arrange the eleven books on the shelf while keeping the Arabic books together.


A rectangular solid has side faces with areas of $18\text{ in}^2$, $50\text{ in}^2$, and $45\text{ in}^2$. Calculate the volume of this solid.

Solution: \boxed{90\sqrt{5}\text{ in}^3}


Evaluate the expression $\sqrt[3]{8 + 3 \sqrt{21}} + \sqrt[3]{8 - 3 \sqrt{21}}$ and square the result.


Triangle $OPQ$ is defined with $O=(0,0)$, $Q=(3,0)$, and $P$ in the first quadrant. Suppose $\angle PQO = 90^\circ$ and $\angle POQ = 45^\circ$. If $PO$ is rotated $-90^\circ$ (clockwise) around $O$, determine the new coordinates of $P$.

Solution: \boxed{(3, -3)}


The final answer is:
Let the original price of the jacket be $120.00. During a sale, this price is discounted by 25%, then the sales tax of 8% is applied. Two different methods of calculating the bill are presented. The first method involves adding 8% sales tax to the discounted price, whereas the second method involves applying a 25% discount to the original price, followed by adding 8% sales tax to this discounted price. What is the difference between the two totals.


The sum of the greatest integer less than or equal to \(x\) and the least integer greater than or equal to \(x\) is 7. What is the solution set for \(x\)?
A) $\left\{\frac{7}{2}\right\}$
B) $\{x | 3 \le x \le 4\}$
C) $\{x | 3 \le x < 4\}$
D) $\{x | 3 < x \le 4\}$
E) $\{x | 3 < x < 4\}$

Solution: \boxed{\{x | 3 < x < 4\}}


Maria bakes a $24$-inch by $30$-inch pan of brownies, and the brownies are cut into pieces that measure $3$ inches by $4$ inches. Calculate the total number of pieces of brownies the pan contains.


Given that $7 = k\cdot 3^r$ and $49 = k\cdot 9^r$, solve for $r$.

Solution: \boxed{\log_3 7}


A lemming starts at a corner of a square with side length $8$ meters. It moves $4.8$ meters along a diagonal towards the opposite corner, then makes a $90^{\circ}$ left turn and runs $2.5$ meters. Calculate the average of the shortest distances from the lemming to each side of the square.


A lemming sits at a corner of a rectangle with side lengths 12 meters and 16 meters. The lemming runs 9.6 meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ left turn, and runs 3 more meters. Calculate the average of the shortest distances from the lemming to each side of the rectangle.


Given a square chalkboard divided into squares of equal size, with one diagonal drawn in each square, find the smallest number of total squares that satisfies the condition that the total number of smaller squares along the perimeter of the chalkboard is twice the number of them along one of its diagonals.


Given that Jo needs to climb an 8-stair flight, find the total number of ways Jo can take 2 or 3 stairs at a time to reach the top.


Suppose $a$ and $b$ are nonzero real numbers such that $\frac{4a+b}{a-4b}=3$. Evaluate the value of $\frac{a+4b}{4a-b}$.


Given that a positive even multiple of $3$ less than 5000 is a perfect square, how many such numbers exist?


If $x > 0$ but not too large so that $\frac{x^2}{1+x}$ remains relevant, establish the correct relationship between $\log(1+x)$ and $\frac{x^2}{1+x}$.

Solution: \boxed{\log (1+x) > \frac{x^2}{1+x}}


If $\omega = \alpha t + \omega_0$ and $\theta = \frac{1}{2}\alpha t^2 + \omega_0t$, solve for $t$ in terms of $\theta$, $\omega$, and $\omega_0$.

Solution: \boxed{\frac{2\theta}{\omega + \omega_0}}


Let $S$ be the sum of all positive real numbers $x$ for which $x^{3^{\sqrt{3}}} = \sqrt{3}^{3^x}$. Determine the value of $S$.

Solution: \boxed{S = \sqrt{3}}


What is the product of all positive odd integers less than $20000$?

Solution: \boxed{\dfrac{20000!}{2^{10000} \cdot 10000!}}


What is the probability that a randomly drawn positive factor of $36$ is less than $6$?


Marie performs four equally time-consuming tasks consecutively without breaks. She begins the first task at 7:00 AM and completes the second task at 9:20 AM. Determine the time at which she finishes the fourth task.

Solution: \boxed{11:40 \text{ AM}}


Given that every 8-digit whole number is a potential telephone number except those that begin with $0$ or $\geq 6$, calculate the fraction of telephone numbers that begin with $5$ and end with $2$.


Given the sum of the interior angles of a convex polygon is $2240^\circ$, calculate the measure of the uncounted angle.


Given that the sum of the interior angles of a convex polygon is 1958°, determine the degree measure of the omitted angle.


Given that Ben spent some amount of money and David spent $0.5 less for each dollar Ben spent, and Ben paid $16.00 more than David, determine the total amount they spent together in the bagel store.


How many positive factors does $27,648$ have?


Triangle $OPQ$ has $O=(0,0)$, $Q=(6,0)$, and $P$ in the first quadrant. Additionally, $\angle PQO=90^\circ$ and $\angle POQ=45^\circ$. Suppose that $OP$ is rotated $90^\circ$ counterclockwise about $O$. Find the coordinates of the image of $P$.

Solution: \boxed{(-6, 6)}


A cuboid has its side lengths as 8 units, 6 units, and 4 units respectively. Calculate the number of edges that are not parallel to a given edge.


If the sum of the first 20 terms and the sum of the first 50 terms of a given arithmetic progression are 200 and 150, respectively, find the sum of the first 70 terms of the sequence.


Calculate the product $(1 + x^3)(1 - 2x + x^4)$.

Solution: \boxed{1 - 2x + x^3 - x^4 + x^7}


Given the taxi fare starts at $3.00 for the first $\frac{3}{4}$ mile, and each additional 0.1 mile is charged at $0.25, and a $3 tip is intended, determine the number of miles that can be ridden for a total cost of $15.


Given $2^9+1$ and $2^{17}+1$, inclusive, determine the number of perfect cubes between these two values.


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


Given the population of Newlandia in the year 2050 is 350,000,000, the area of Newlandia is 4,500,000 square miles, and there are $(5280)^2$ square feet in one square mile, calculate the average number of square feet per person in Newlandia.


Let $\frac{27x - 19}{x^2 - 5x + 6} = \frac{M_1}{x - 2} + \frac{M_2}{x - 3}$ be an identity in $x$. Find the value of $M_1M_2$.


The circle having $(2,2)$ and $(10,8)$ as the endpoints of a diameter intersects the $x$-axis at a second point. Calculate the $x$-coordinate of this point.


A box contains 3 red marbles, 3 green marbles, and 3 yellow marbles. Carol takes 3 marbles from the box at random; then Claudia takes another 3 marbles at random; lastly, Cheryl takes the remaining 3 marbles, calculate the probability that Cheryl gets 3 marbles of the same color.


Given the number $1036$ represented in factorial base numeration as $1036 = b_1 + b_2 \times 2! + b_3 \times 3! + \ldots + b_n \times n!$, where each $b_k$ satisfies $0 \leq b_k \leq k$, find the value of $b_4$.


Elmer's latest car offers $60\%$ better fuel efficiency, measured in kilometers per liter, than his old car. Additionally, the new car uses diesel fuel, which is $25\%$ more expensive per liter than the gasoline used by his old car. Calculate the percent by which Elmer will save money if he opts to use his new car instead of his old car for a long trip.


Given $221^8$, calculate the number of positive integer divisors that are perfect squares or perfect cubes or both.


Given the product of all even negative integers strictly greater than $-2020$, after adding 10 to the product, determine the sign and units digit of the result.


Let $g(t) = \frac{2t}{1+t}$, where $t \not= -1$. Express $w$ as a function of $z$, where $z = g(w)$.


Given that a baby is born every 6 hours and a death occurs every 2 days in the nation of North Southland, calculate the average annual increase in population to the nearest fifty.


Calculate the sum of the digits of the greatest prime number that is a divisor of $59,048$.


A sequence of squares is constructed with each square having an edge that is two tiles longer than the edge of the previous square. The sequence starts from a square with a side length of 1 tile. Calculate the difference in the number of tiles required for the eleventh square and the tenth square.


Find the second-next perfect square after a perfect square number $x$.

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


It takes Clea 75 seconds to walk down an escalator when it is not moving, and 30 seconds when it is moving. Calculate the time it would take Clea to ride the escalator down when she is not walking.


Jim traveled to the store at an average speed of 15 kilometers per hour. After traveling two-thirds of the total distance, the skateboard broke, and he walked the remaining distance at 4 kilometers per hour. It took him a total of 56 minutes to complete the trip. Calculate the total distance Jim traveled in kilometers, rounded to the nearest half kilometer.


Given that Rachel has $3030$ coins composed of pennies and nickels, and that she must have at least $10$ times as many pennies as nickels but no less than $3$ nickels, determine the difference in cents between the maximum and minimum monetary amounts Rachel can have.


A woman buys a property for $12,000 and plans to earn a $6\%$ return on her investment through renting it out. She sets aside $10\%$ of the monthly rent for repairs and maintenance and pays $400 annually in taxes. Calculate the necessary monthly rent.


Given that a reader is guessing at random, calculate the probability that a reader will match all four celebrity photos correctly.


$\log_5{(10x-3)}-\log_5{(3x+4)$ approaches what value as $x$ grows beyond all bounds.

Solution: \boxed{\log_5 \left(\frac{10}{3}\right)}


Consider a $5\times 5$ block of calendar dates as shown below. First, the order of the numbers in the second, third, and fifth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?


The yearly changes in the population census of a city for five consecutive years are, respectively, 20% increase, 10% increase, 30% decrease, 20% decrease, and 10% increase. Calculate the net change over these five years, to the nearest percent.


Point $E$ is the midpoint of side $\overline{BC}$ in square $ABCD$, and $\overline{AE}$ meets diagonal $\overline{BD}$ at $F$. The area of quadrilateral $AFED$ is $25$. Given this information, calculate the area of square $ABCD$.


In an All-District track meet, $320$ sprinters enter a $100-$meter dash competition. The track has 8 lanes, so only $8$ sprinters can compete at a time. At the end of each race, the seven non-winners are eliminated, and the winner will compete again in a later race. Calculate the number of races needed to determine the champion sprinter.


A rectangular board consists of an alternating pattern of light and dark squares similar to a chessboard. This board is $8$ rows high and $7$ columns wide. Determine how many more dark squares there are than light squares.


Consider those functions $f$ that satisfy $f(x+5)+f(x-5) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions.


Given the sequence $\{a_1, a_2, a_3, \ldots\}$ where each positive even integer $k$ appears exactly $k+1$ times, and the sequence starts with 2, determine the sum of the constants $b$, $\alpha$, $\beta$, and $d$ in the formula $a_n = b\lfloor \alpha\sqrt{n} + \beta \rfloor + d$ such that the sequence retains its noted pattern for all positive integers $n$.


Given that $720a$ is a square and $720b$ is a cube, where $a$ and $b$ are the two smallest positive integers, calculate the sum of $a$ and $b$.


Given the expression $\frac{b - 1}{b + \frac{b}{b-1}}$, simplify the given expression, where $b \neq 1$.


A rectangular piece of metal has dimensions 10 units by 6 units. A circular piece of maximum diameter is cut out from this rectangle (assuming the longest dimension is the diameter). Then, a square piece of maximum size is cut out from the circular piece. Calculate the area of the metal wasted.

Solution: \boxed{42 \text{ units}^2}


Let \( M = 42 \cdot 43 \cdot 75 \cdot 196 \). Find the ratio of the sum of the odd divisors of \( M \) to the sum of the even divisors of \( M \).


Given a Ferris wheel with a radius of 30 feet, revolving at the constant rate of one revolution every 2 minutes, determine the time in seconds it takes for a rider to travel from the bottom of the wheel to a point 15 vertical feet above the bottom.


Given that Ray's car averages 50 miles per gallon of gasoline, Tom's car averages 20 miles per gallon of gasoline, and Alice's car averages 25 miles per gallon, find the combined rate of miles per gallon for all three cars.


Solution: \boxed{\sqrt{x + \left(x + x^{\frac{1}{2}}\right)^{\frac{1}{2}}}}


In an All-Area track meet, 320 sprinters participate in a 100-meter dash competition on a track with 8 lanes, allowing 8 sprinters to compete simultaneously. At the end of each race, the seven non-winners are eliminated, and the winner advances to compete in subsequent races. How many races are necessary to determine the champion sprinter?


What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<0.02$?


Last year, 120 adult cats, 60 of whom were male, were brought into Rivertown Animal Shelter. 40% of the female cats, and 10% of the male cats were accompanied by a litter of kittens. The average number of kittens per litter was 5. Calculate the total number of cats and kittens received by the shelter last year.


Given that Ron reversed the digits of a two-digit number $a$ and added 5 to the product with $b$, and the mistaken result was 266, determine the correct value of the product of $a$ and $b$.


In Markville, the sales tax rate is 7%. A dress originally priced at $150.00 receives a 25% discount during a store sale. Two assistants, Ann and Ben calculate the final cost differently. Ann calculates by adding the tax to the original price before applying the 25% discount. Ben discounts the price first and then adds the 7% tax. Additionally, Ann applies a special 5% service charge after all other calculations due to premium packaging. Determine the difference between the amount paid by Ann's customer and Ben's customer.


A 4x4x4 cube is made of $64$ normal dice, where opposites sum to $7$. Calculate the smallest possible sum of all the values visible on the $6$ faces of the giant cube.


Determine the number of integer values of $n$ for which $8000 \cdot \left(\frac{2}{5}\right)^n$ is an integer.


Given that in $\triangle DEF$, $DE = 45$ and $DF = 75$, and a circle with center $D$ and radius $DE$ intersects $\overline{EF}$ at points $E$ and $Y$, with $\overline{EY}$ and $\overline{FY}$ having integer lengths, calculate the length of $EF$.


The maximum number of the eight integers that can be larger than $20$ if their sum is $-20$.


Given seven points on a circle numbered 1 through 7 in clockwise order, a bug jumps in a clockwise direction; if the point number is divisible by 3, it moves two points forward, otherwise, it moves three points forward. Determine the point where the bug will be after 2023 jumps if it starts on point 7.


In a botanical garden, three decorative plant beds are described, with Bed X containing 600 plants, Bed Y containing 500 plants, and Bed Z containing 400 plants. Beds X and Y share 80 plants, Beds X and Z share 120 plants, and Beds Y and Z share 70 plants. No plant is in all three beds. Calculate the total number of unique plants across all beds.


Given a $5 \times 5$ block of calendar dates as shown below, first reverse the order of the numbers in the second, third, and fifth rows, and then find the sum of the numbers on each diagonal. What is the absolute difference between the two diagonal sums?

$$
\begin{tabular}{|c|c|c|c|c|}
\hline 1 & 2 & 3 & 4 & 5 \\
\hline 6 & 7 & 8 & 9 & 10 \\
\hline 11 & 12 & 13 & 14 & 15 \\
\hline 16 & 17 & 18 & 19 & 20 \\
\hline 21 & 22 & 23 & 24 & 25 \\
\hline
\end{tabular}
$$


Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determine the average of the first and last numbers.


Find the number of four-digit passwords that can be formed from digits $0$ to $9$, with repeated digits allowable, and excluding passwords that begin with the sequence $1,2,3$.


When $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $715$ terms that include all four variables $a$, $b$, $c$, and $d$, each to some positive power. Find the value of $N$.

Solution: \boxed{N = 13}


Consider the set $\{1, 5, 9, 13, 17, 21, 25, 29\}$. Calculate the number of different integers that can be expressed as the sum of four distinct members of this set.


A lemming starts at a corner of a square with a side length of 15 meters. It runs 9.3 meters along a diagonal toward the opposite corner, stops, makes a 90-degree right turn, and runs 3 more meters. Determine the average of the shortest distances from the lemming to each side of the square after it stops.


Calculate the area of a rectangular garden which measures 2.5 meters in length and 0.48 meters in width.


Points $A$ and $B$ are on a circle of radius $7$ and $AB=8$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$?

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


Given that 132 is an even integer greater than 7, find the largest possible difference between two different prime numbers that sum to 132.


Find the number of positive factors of 48 that are also multiples of 6.


Given $x_{k+1} = x_k + \frac{1}{3}$ for $k=1, 2, \dots, n-1$ and $x_1=2$, calculate the sum of $x_1 + x_2 + \dots + x_n$.


What is the value of $\frac{3a^{-2} + \frac{a^{-3}}{3}}{a^2}$ when $a = 3$?


Numbers between 0 and 1 are chosen as follows: A fair coin is flipped twice. If the first flip is heads and the second is tails, the number is 1. If the first flip is heads and the second is heads, the number is 0. If the first flip is tails, regardless of the second flip, the number is 0.5. Two independent numbers x and y are chosen using this method. Compute the probability that the absolute difference between x and y, |x-y|, is greater than 0.5.


A group of students from Sequoia High School is volunteering for a local park cleaning event. The ratio of 9th-graders to 7th-graders is 3 : 2 while the ratio of 9th-graders to 6th-graders is 7 : 4. Determine the smallest number of students that could be volunteering in this event.


Given that a circle is divided into 10 sectors, the central angles of these sectors, measured in degrees, increase arithmetically. Find the degree measure of the smallest possible sector angle.


Determine the number of ordered pairs of positive integers $(A, B)$ that satisfy the equation $\frac{A}{8} = \frac{8}{B}$, given that $A$ and $B$ are even numbers.


Given that there are $12$ students in Mrs. Germain's class, $10$ students in Mr. Newton's class, and $11$ students in Mrs. Young's class taking the AMC 8, and $5$ students are in an advanced study group and are counted in both Mrs. Germain's and Mr. Newton's totals, calculate the number of unique mathematics students at Euclid Middle School taking the contest.


A digital watch is set to a 24-hour format displaying both hours and minutes. Calculate the largest possible sum of the digits in the display when the watch displays a time between 12:00 and 23:59.


Dana goes first and Carl's coin lands heads with probability $\frac{2}{7}$, and Dana's coin lands heads with probability $\frac{3}{8}$. Find the probability that Carl wins the game.


A half-sector of a circle of radius 6 inches is rolled up to form the lateral surface area of a right circular cone by taping together along the two radii. Calculate the volume of the cone.

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


A rectangular piece of metal has dimensions of 8 units by 10 units. Determine the total amount of metal wasted after a circular piece of maximum size is cut from this rectangle, and then a square piece of maximum size is cut from the circular piece.

Solution: \boxed{48 \text{ units}^2}


In how many ways can $420$ be written as the sum of an increasing sequence of two or more consecutive positive integers?


A nine-digit number is formed by repeating a three-digit number three times; for example, $256256256$. Determine the common factor that divides any number of this form exactly.


Determine the number of ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $10$, inclusive, that satisfy the equation $(\log_b a)^3 = 2 \log_b(a^{3})$.


Given that the lighthouse is 60 meters high and the tank has a capacity of 150,000 liters, determine the height of a scale model of the lighthouse with a tank capacity of 0.15 liters.


The number of possible license plates under the old scheme is given by $26\cdot 10^5$, and the number of possible license plates under the new scheme is given by $26^2\cdot 10^4$. Calculate the ratio of the number of license plates under the new scheme to the number of license plates under the old scheme.


Given that Annie runs 50% faster than Bonnie on a 500-meter circular track, determine the number of laps Annie will have run when she first laps Bonnie.


Given that our number system has a base of eight, determine the fifteenth number in the sequence.


Determine $x$ in terms of $m$ and $n$ given that the equation $(x+m)^2 - 3(x+n)^2 = m^2 - 3n^2$.

Solution: \boxed{x = 0 \text{ or } x = m - 3n}


How many pairs of positive integers \((a, b)\) with \(a+b \leq 150\) satisfy the equation \(\frac{a+b^{-1}}{a^{-1}+b} = 17?\)


Let \( c_1, c_2, \ldots \) and \( d_1, d_2, \ldots \) be arithmetic progressions such that \( c_1 = 30, d_1 = 90 \), and \( c_{50} + d_{50} = 120 \). Find the sum of the first fifty terms of the progression \( c_1 + d_1, c_2 + d_2, \ldots \).


Alice starts her new job with a schedule of 4 work-days followed by 2 rest-days, and Bob's schedule is 5 work-days followed by 1 rest-day. Determine how many days in the first 800 days do both have rest-days on the same day.


How many whole numbers between $200$ and $500$ contain the digit $3$?


$\triangle ABC$ has a right angle at $C$ and $\angle A = 15^\circ$. $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$. Compute $\angle BDC$.


A convex polyhedron $S$ has vertices $W_1,W_2,\ldots,W_m$, and $150$ edges. Each vertex $W_j$ is intersected by a plane $Q_j$ that cuts all edges connected to $W_j$. These cuts produce $m$ pyramids and a new polyhedron $T$. Assuming no two planes intersect inside or on $S$, determine the number of edges on polyhedron $T$.


Given that Rani can cycle 20 miles in 2 hours and 45 minutes when she was a girl and 12 miles in 3 hours as an older woman, calculate the difference in time it takes for her to cycle a mile now compared to when she was a girl.


Calculate the area of each triangle with sides of length 15, 15, and 20, and with sides of length 15, 15, and 30, and determine the relationship between their areas.


Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments.


Samantha lives 3 blocks north and 2 blocks east of the northeast corner of City Park. Her school is 3 blocks south and 3 blocks west of the southwest corner of City Park. On school days she bikes on streets to the northeast corner of City Park, then takes one of two diagonal paths through the park to the southwest corner, and then bikes on streets to school. Assuming she takes the shortest on-street routes, determine the total number of different routes she can take.


Given rectangle ACDE, where AC = 48 and AE = 36, point B is located one-fourth the length of AC from A, and F is located one-third the length of AE from A, calculate the area of quadrilateral ABDF.


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


Given that the total amount of money originally owned by Moe, Loki, and Nick was $72, and each of Loki, Moe, and Nick gave Ott$\, 4, determine the fractional part of the group's money that Ott now has.


What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 100?


Determine the radius $r$ of the circle with center $O$ tangent to one side of the equilateral triangle $ABC$ and to the coordinate axes.


Jacqueline has 2 liters of soda. Liliane has 60% more soda than Jacqueline, and Alice has 40% more soda than Jacqueline. Calculate the percentage difference between the amount of soda Liliane has compared to Alice.


Paul owes Paula 60 cents and has a pocket full of 5-cent coins, 10-cent coins, 25-cent coins, and a new 50-cent coin that he can use to pay her. Find the difference between the largest and smallest number of coins he can use to pay her exactly.


Given the expression $2-(-3)-4-(-5) \times 2 -6-(-7)$, calculate its value.


Consider each positive integer $n$, let $g_1(n)$ be thrice the number of positive integer divisors of $n$ raised to the power of 2, and for $j \ge 2$, let $g_j(n) = g_1(g_{j-1}(n))$. Determine for how many values of $n \le 30$ is $g_{50}(n) = 243$.


Given $\dfrac{15! - 14! - 13!}{11!}$, calculate the value.


Find the smallest positive integer $x$ such that $1260x = N^2$, where $N$ is an integer.


Let $n$ be the smallest positive integer such that $n$ is divisible by $30$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. Find the number of digits of $n$.


Let \( M = 57^{4} + 4\cdot57^{3} + 6\cdot57^{2} + 4\cdot57 + 1 \). Determine the number of positive integer factors of \( M \).


Let $\angle ABC = 40^\circ$, $\angle ABD = 30^\circ$, and $\angle ABE = 15^\circ$. Given that point $E$ is on line segment $BD$, calculate the smallest possible degree measure for $\angle EBC$.


Given the two numbers have their difference, their sum, and their product in the ratio $1:8:30$, find the product of these two numbers.


Determine the minimum number of fence posts required to fence a rectangular garden plot measuring 30 m by 50 m, where the fourth side is an existing 80 m wall.


Bricklayer Brenda would take $8$ hours to build a chimney alone, and bricklayer Brandon would take $12$ hours to build it alone. When they work together, their efficiency is diminished, resulting in a decreased output of $15$ bricks per hour due to their chatting. Given that they complete the chimney in $6$ hours when working together, determine the total number of bricks in the chimney.


Given that a store prices an item so that when 6% sales tax is added, the total cost is exactly $m$ dollars, where $m$ is a positive integer, determine the smallest value of $m$.


Given a bouquet containing purple tulips, yellow tulips, purple lilies, and yellow lilies, where half of the purple flowers are tulips, two-thirds of the yellow flowers are lilies, and seven-tenths of the flowers are purple, calculate the percentage of flowers that are lilies.


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


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


The Tigers beat the Sharks 3 out of the 4 times they played. They then played N more times, and the Sharks ended up winning at least 90% of all the games played. Determine the minimum possible value for N.


What is the smallest prime number dividing the expression $2^{12} + 3^{10} + 7^{15}$?


Consider a circle with center \(O\) and a given radius. There are two tangents to the circle; one is at a distance \(r\) (radius of the circle) above the center, and the other is at distance \(2r\) below the center. Determine the number of points which are equidistant from the circle and both tangents.


In an isosceles trapezoid $ABCD$, with $AB \parallel CD$, the sides $AB$ and $CD$ are equal, and the legs $BC$ and $DA$ are also equal. The sum of the lengths of the bases $AB$ and $CD$ is 24 units. If the length of each leg is 13 units, calculate the perimeter of trapezoid $ABCD$.


What is the hundreds digit of $(17! - 12!)$?


Evaluate the expression $3 - (-3)^{-\frac{2}{3}}$.

Solution: \boxed{3 - \frac{1}{\sqrt[3]{9}}}


The conference has 12 teams. Each team plays every other team twice and an additional 6 games against non-conference opponents. Calculate the total number of games in a season involving the conference teams.


You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits.


Given that Lakeview Academy has 1500 students, each student takes 6 classes per day, and each class has 25 students, each class is taught by a main teacher and a co-teacher, and each teacher teaches 5 classes, determine the total number of teachers at Lakeview Academy.


Let U be the set of the 3000 smallest positive multiples of 5, and let V be the set of the 3000 smallest positive multiples of 7. Determine the number of elements common to U and V.


It takes Clea 120 seconds to walk up an escalator when it is not operating, and only 48 seconds to walk up the escalator when it is operating. Calculate the time it takes Clea to ride up the operating escalator when she just stands on it.


Given the equation $a \cdot b + 125 = 30 \cdot \text{lcm}(a, b) + 24 \cdot \text{gcd}(a, b) + a \mod b$, where $\text{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a, b)$ denotes their least common multiple, assuming $a \geq b$, calculate the number of ordered pairs $(a, b)$ of positive integers that satisfy this equation.


If $7 = k \cdot 3^r$ and $49 = k \cdot 9^r$, solve for $r$.

Solution: \boxed{\log_3 7}


For how many positive integers $n$ is $n^3 - 9n^2 + 23n - 15$ a prime number?


Seven points on a circle are given. Four of the chords joining pairs of the seven points are selected at random. What is the probability that the four chords form a convex quadrilateral?


How many perfect cubes lie between $3^6+1$ and $3^{12}+1$, inclusive?


If $\frac{1}{4}$ of all ninth graders are paired with $\frac{1}{3}$ of all sixth graders, what fraction of the total number of sixth and ninth graders are paired?


In a collection of red, blue, and yellow balls, all but 9 are red, all but 5 are blue, and all but 6 are yellow, calculate the total number of balls in the collection.


A triangle and trapezoid share the same height. The triangle's base is expanded to 36 inches and their areas remain equal. What is the length of the median of the trapezoid?


A square floor is covered with congruent square tiles, such that if the sum of the tiles on the two diagonals is equal to 25, determine the total number of tiles needed to cover the entire floor.


Let $f$ be a function defined on the set of positive rational numbers such that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Assume additionally that $f(p^2)=2p$ for every prime number $p$. Determine $f(x)$ for $x = \frac{121}{36}$.


Determine the number of solution-pairs in the non-negative integers of the equation $4x + 7y = 600$.


A clock takes $7$ seconds to strike $9$ o'clock starting precisely from $9:00$ o'clock. If the interval between each strike increases by $0.2$ seconds as time progresses, calculate the time it takes to strike $12$ o'clock.


Given that triangle ABC is right-angled at B, and AD=5, DC=2, determine the area of triangle ABC.


In a physics test, 20% of the students scored 60 points, 40% scored 75 points, 25% scored 85 points, and the remaining students scored 95 points. Calculate the difference between the mean and median score of the students' scores on this test.


The pressure of wind on a sail varies jointly as the area of the sail and the square of the velocity of the wind. When the velocity of the wind is 20 miles per hour, the pressure on a 2 square foot area is 4 pounds. Find the velocity of the wind when the pressure on 4 square feet is 64 pounds.

Solution: \boxed{40 \sqrt{2} \text{ mph}}


Given that the average cost of a long-distance call in the USA in 1995 was $35$ cents per minute, and the average cost of a long-distance call in the USA in 2015 was $15$ cents per minute, find the approximate percent decrease in the cost per minute of a long-distance call.


Given the expression $(1296^{\log_6 4096})^{\frac{1}{4}}$, calculate its value.


The sum of the digits of the greatest five-digit number M, whose digits have a product of 90.


Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus ride.


Jenny has two identical 12-ounce cups. She pours six ounces of milk into the first cup and six ounces of juice into the second cup. She then transfers one-third of the milk from the first cup to the second cup. After mixing thoroughly, she transfers one-fourth of the mixture from the second cup back to the first cup. What fraction of the liquid in the first cup is now juice?


A hamster is placed at a corner of a square garden with side length $12$ meters. It scurries $7.2$ meters along the diagonal towards the opposite corner, then takes a $90^{\circ}$ left turn and darts $3$ meters further. Calculate the average of the shortest distances from the hamster to each side of the square garden.


Given that p, q, r, and s are integers in the set {0, 1, 2, 3, 4}, calculate the number of ordered quadruples (p, q, r, s) such that p·s + q·r is odd.


What is the ratio of the least common multiple of 280 and 476 to the greatest common factor of 280 and 476.


Given the triangle with side lengths $12$, $30$, and $x$, determine how many integers $x$ satisfy the condition that the triangle has one right angle.


Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 30 miles per hour, he will be late by 5 minutes. If he drives at an average speed of 50 miles per hour, he will be early by 5 minutes. Calculate the speed in miles per hour that Mr. Bird needs to drive to get to work exactly on time.


Susie buys $6$ muffins and $4$ bananas. Calvin spends thrice as much as Susie, paying for $3$ muffins and $24$ bananas. Calculate how many times as expensive as a banana is a muffin.


At Mountain Valley School, the ratio of $9^\text{th}$-graders to $7^\text{th}$-graders is $7:4$, and the ratio of $9^\text{th}$-graders to $8^\text{th}$-graders is $9:5$. Determine the smallest number of students that could be participating in this project.


A woman born in the first half of the twentieth century was $x$ years old in the year $x^2 + x$. Find the year in which she was born.


Find the units digit of the decimal expansion of $\left(17 + \sqrt{210}\right)^{20} + \left(17 + \sqrt{210}\right)^{83}$.


Given triangle XYZ has a total area of 180 square units, point M is the midpoint of XY, and point N is the midpoint of YZ. A line segment is drawn from X to N, dividing triangle XMN into two regions. Calculate the area of triangle XPN when point P is the midpoint of segment XM and line segment PN is drawn.


A professional soccer league consists of teams with 23 players each. Each player must be paid at least $20,000, and the total of all players' salaries for each team cannot exceed $800,000. Additionally, no single player can earn more than $450,000. Find the maximum possible salary, in dollars, for a single player.


A point Q(c,d) in the xy-plane is first rotated counterclockwise by 90° about the point (2,3) and then reflected about the line y = x. If the image of Q after these transformations is at (7, -4), calculate d - c.


Given the town's original population, it increases by $15\%$ and then decreases by $13\%$, and the resulting population is $50$ fewer people than the original population, find the original population.


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


What is the value of $\dfrac{13! - 12!}{10!}$?


How many primes less than $150$ have $3$ as the ones digit?


How many whole numbers between $200$ and $500$ contain the digit $3$?


If $2x, 4x+4, 6x+6, \dots$ are in geometric progression, determine the fourth term.


Given that Abe holds 2 blue and 1 green jelly bean, Bob holds 1 blue, 2 green, and 1 yellow jelly bean, and Cara holds 3 blue, 2 green, and 1 red jelly bean, calculate the probability that all three pick jelly beans of the same color.


An integer $N$ is selected at random in the range $1 \leq N \leq 2030$. Calculate the probability that the remainder when $N^{12}$ is divided by $7$ is $1$.


Elmer's new car provides 60% better fuel efficiency measured in kilometers per liter compared to his old car, but uses diesel fuel that is 25% more expensive per liter than the gasoline used by his old car. Calculate the percentage by which Elmer will save money when using his new car for a 1000 km trip.


In $\bigtriangleup ABC$, $AB = 75$, and $AC = 100$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. It is known that $\overline{BX}$ and $\overline{CX}$ have integer lengths. Calculate the length of segment $BC$.


In a similar mistake during multiplication, Tom reversed the digits of the two-digit number c and mistakenly multiplied the reversed number with d, resulting in the erroneous product of 143. Determine the correct product of c and d.


The largest possible difference between the two primes which add up to the even number 138.


Let $ABCD$ be a rectangle in the $xy$-plane where $AB=2$ and $BC=1$ with coordinates $A(0,0,0)$, $B(2,0,0)$, $C(2,1,0)$, and $D(0,1,0)$. Let $\overrightarrow{AA'}$, $\overrightarrow{BB'}$, $\overrightarrow{CC'}$, and $\overrightarrow{DD'}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA' = 12$, $BB' = 10$, $CC' = 16$, and $DD' = 20$, and $M$ and $N$ are the midpoints of $A'C'$ and $B'D'$ respectively, find the length of $MN$.


Given a regular pyramid with a square base area of $144 \mathrm{cm}^2$ and a height of $27 \mathrm{cm}$, determine the height in centimeters of the water in a cylindrical container with a base radius of $9 \mathrm{cm}$ after being filled with this water.


Compute the difference between Mark's total and Mary's total bill for a coat originally priced at $120.00, which is being sold at a 25% discount, when the sales tax rate is 7%.


Let $n$ be the smallest positive integer such that $n$ is divisible by 30, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$?


Given the containers A, B, and C, where A is $\tfrac{4}{5}$ full, B is initially empty and becomes $\tfrac{3}{5}$ full after pouring from A, and C is initially empty and becomes $\tfrac{3}{4}$ full after pouring from B, determine the ratio of the volume of A to the volume of C.


Given the expression $(64)^{-2^{-3}}$, calculate its value.


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


Triangle OCD has O = (0, 0) and D = (7, 0), with ∠CDO = 90° and ∠COD = 45°. Find the coordinates of the image of point C after OC is rotated 120° counterclockwise about O.

Solution: \boxed{\left(-\frac{7(1+\sqrt{3})}{2}, \frac{7(\sqrt{3}-1)}{2}\right)}


Suppose in February a city recorded a total rainfall of 280 inches. Calculate the average rainfall in inches per hour during that month.


Given in $\bigtriangleup ABC$, $AB = 75$, and $AC = 120$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. Find the length of $BC$.


Given that Emma's age is $E$ years and this is equal to the sum of the ages of her four children, and her age $M$ years ago was three times the sum of their ages then, calculate the value of $E/M$.


Given rectangle ABCD where E is the midpoint of diagonal BD, point E is connected to point F on segment DA such that DF = 1/4 DA. Find the ratio of the area of triangle DFE to the area of quadrilateral ABEF.


A regular hexagon $ABCDEF$ with side length 2 is given. Two circles are positioned outside the hexagon. The first circle is tangent to line $\overline{AB}$ and the second circle is tangent to line $\overline{CD}$. Both circles are tangent to line $\overline{EF}$. Determine the ratio of the area of the second circle to the first circle.


Solution: \boxed{2 \cdot 3000^{1501}}


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


Given a string of length 2, cut at a point chosen uniformly at random, calculate the probability that the longer piece is at least 3 times as large as the shorter piece, and the length of the smaller piece is exactly $\frac{1}{2}$.


The number of significant digits in the measurement of the side of a square whose computed area is $2.401$ square inches to the nearest thousandth of a square inch is $4$.


Given hexadecimal (base-16) numbers only use digits $0$ through $9$ and letters $A$ through $F$ for values $10$ through $15$, determine the number of positive integers less than $500$ with a hexadecimal representation that contains only numeric digits.


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


How many 3-digit whole numbers, whose digits sum up to 27 and are even, and end specifically in 4?


If an examination consisting of 30 questions awards 4 points for each correct answer, -1 point for each incorrect answer, and 0 points for unanswered questions, and Amy's total score is 85, calculate the maximum number of questions she could have answered correctly.


Given that a circle is divided into 15 sectors with all sector angles forming an arithmetic sequence and are all integers, determine the degree measure of the smallest possible sector angle.


What is the number of terms with rational coefficients among the 726 terms in the expansion of \((x\sqrt[4]{3} + y\sqrt[5]{5})^{725}\)?


If 105 customers purchased a total of 300 cans of soda, with every customer purchasing at least one can of soda, determine the maximum possible median number of cans of soda bought per customer that day.


Let (b_1, b_2, ... b_7) be a list of the first 7 odd positive integers such that for each 2 ≤ i ≤ 7, either b_i + 2 or b_i - 2 (or both) must appear before b_i in the list. How many such lists are there?


An iterative average of the numbers 1, 3, 5, 7, 9, and 11 is computed by arranging the six numbers in some order. First, find the mean of the first two numbers, then find the mean of that result with the third number, then continue this process with the fourth, fifth, and sixth numbers. What is the difference between the largest and smallest possible values that can be obtained using this procedure?


Given a set $\{8, -4, 3, 27, 10\}$, determine the smallest possible sum of three different numbers from this set such that their sum must be even.


What is the smallest positive integer \( n \) such that \( \sqrt{n} - \sqrt{n-1} < 0.05 \)?


Given that $20\%$ of the students scored $60$ points, $40\%$ scored $75$ points, and $25\%$ scored $85$ points, with the remaining students scoring $95$ points, calculate the difference between the mean and median score of the students' scores on this test.
.


In a plane, points $A$ and $B$ are $12$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $60$ units and the area of $\triangle ABC$ is $180$ square units?


Given the graphs of $y = -|x-a| + b$ and $y = |x-c| -d$ intersect at points $(1,4)$ and $(7,2)$. Find the value of $a+c$.


Given an archery contest with 120 shots, Chelsea is ahead by 70 points after 60 shots, with scoring as 10, 8, 5, 3, or 0 points, and Chelsea scoring at least 5 points per shot. Chelsea will achieve victory by scoring the next n shots as bullseyes (10 points) regardless of her opponent’s scores in the remaining shots. Find the smallest n needed to guarantee her win.


Calculate the area of a polygon with vertices at points (1, -1), (4, 2), (6, 1), (3, 4), and (2, 0).


Consider a scenario where $89$ out of the first $90$ balls are red. Afterwards, $8$ out of $9$ balls counted are also red. If at least $92\%$ of the total balls are red, find the maximum value of $n$, the total number of balls.

Solution: \boxed{n = 90 + 9y = 90 + 9 \times 22 = 210 + 78 = 288}


Given circle $O$, points $E$ and $F$ are on the same side of diameter $\overline{AB}$, $\angle AOE = 60^\circ$, and $\angle FOB = 90^\circ$. Calculate the ratio of the area of the smaller sector $EOF$ to the area of the circle.


When Neva was young, she could cycle 20 miles in 2 hours and 45 minutes. Now, as an older adult, she walks 8 miles in 3 hours. Calculate the difference in time it takes her to walk a mile now compared to when she was young.


Eight spheres, each with a radius of 2 and located in each octant, are tangent to the coordinate planes. Find the radius of the smallest sphere, centered at the origin, that contains these eight spheres.

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


Given x be a real number selected uniformly at random between 200 and 400, and \(\lfloor \sqrt{x} \rfloor = 18\), find the probability that \(\lfloor \sqrt{100x} \rfloor = 180\).


Given $(n+i)^5$ is a real number, calculate the number of integer values of n.


Given \( x \) is real and positive and grows beyond all bounds, find the limit as \( x \) approaches infinity of \( \log_4{(8x-7)}-\log_4{(3x+2)} \).

Solution: \boxed{\log_4 \left(\frac{8}{3}\right)}


A box contains triangular and pentagonal tiles totaling 30 tiles. There are 100 edges in total. Find the number of pentagonal tiles in the box.


Points $P$ and $Q$ are on a circle of radius $10$ and $PQ = 12$. Point $R$ is the midpoint of the minor arc $PQ$. Determine the length of the line segment $PR$.


An ordered pair (b, c) of integers is chosen such that b is non-negative and less than or equal to 5, and c is between -10 and 10, inclusive. Find the probability that the equation x^2 + bx + c = 0 will not have distinct positive real roots.


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


Calculate the probability that all four of them flip their coins the same number of times.


A rectangular box measuring $8$ cm by $4$ cm by $15$ cm is filled with liquid $Y$. This liquid forms a circular film of thickness $0.2$ cm when poured into a large body of water. What is the radius, in centimeters, of the resulting circular film?


Given that a square-shaped area is covered entirely by congruent square tiles, and the total number of tiles that lie along the two diagonals of the square is 25, calculate the number of tiles used to cover the entire square area.


Medians \( AF \) and \( BE \) of triangle \( ABC \) are perpendicular, \( AF = 10 \), and \( BE = 15 \). Find the area of triangle \( ABC \).


Given a positive integer n not exceeding 120, the probability of choosing n when n ≤ 60 is q, and the probability of choosing n when n > 60 is 4q. Calculate the probability that an odd number is chosen.


Given $d$ is a digit, find the number of values of $d$ for which $2.0d05 > 2.015$.


A painting measuring 20 inches by 30 inches is to be placed in a wooden frame, with the longer dimension horizontal this time. The wood at the top and bottom is three times as wide as the wood on the sides. If the area of the frame equals that of the painting itself, determine the ratio of the smaller to the larger dimension of the framed painting.
A) $1:3$
B) $1:2$
C) $2:3$
D) $3:4$
E) $1:1$

Solution: \boxed{1:1 \text{ (E)}}


In a classroom test, $20\%$ of the students scored $65$ points, $25\%$ scored $75$ points, another $25\%$ scored $85$ points, and the rest scored $95$ points. Find the difference between the mean and median score of the students' scores in this test.


Two poles, one 30 feet high and the other 90 feet high, are 150 feet apart. Halfway between the poles, there is an additional smaller pole of 10 feet high. The lines go from the top of each main pole to the foot of the opposite main pole, intersecting somewhere above the smaller pole. Determine the height of this intersection above the ground.


Given that we have the digits 4, 4, 6, 6, 7, 7, 8, determine the number of 7-digit palindromes that can be formed.


How many different integers can be expressed as the sum of three distinct members of the set $\{3, 5, 9, 13, 17, 21, 27\}$?


Given that half of Marcy's marbles are red, three-eighths of them are blue, and seven of them are green, determine the smallest number of yellow marbles she could have.


In rectangle $JKLM$, $JL=2$, and $LM=4$. Points $N$, $O$, and $P$ are midpoints of $\overline{LM}$, $\overline{MJ}$, and $\overline{JK}$, respectively. Point $Q$ is the midpoint of $\overline{NP}$. Find the area of the triangle formed by points $Q$, $O$, and $P$.


Given that the sequence starts with 2 and alternates by adding 2 between consecutive terms, find the 30th term of this arithmetic sequence.


Given a set $\{4,6,8,12,14,18\}$, select three different numbers, add two of these numbers, multiply their sum by the third number, and finally subtract the smallest number you initially selected. Find the smallest result that can be obtained from this process.


The diameter of a circle is divided into n equal parts. On each part a quarter-circle is constructed. As n becomes very large, calculate the sum of the lengths of the arcs of the quarter-circles.

Solution: \boxed{\frac{\pi D}{2}}


Given a three-digit number with digits that sum exactly to $14$ and have the first digit equal to the last digit, find the total count of such numbers that are not divisible by $5$.


Given that $(xy-2)^2 + (x+y-1)^2$ represents a sum of squares of real numbers, calculate the least possible value.


Star lists the whole numbers $1$ through $50$ once. Emilio copies Star's numbers, but he replaces each occurrence of the digit $2$ by the digit $1$ and each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum and Emilio's sum.


In a quadrilateral ABCD, if $\angle A = 70^\circ$, $\angle B = 60^\circ$, and $\angle C = 40^\circ$, calculate the measurement of $\angle D$.


In ∆ABC, points D and E are interior points such that lines AD, BE, and CE intersect at point F. Let x, y, z, w, v represent the angles at F in degrees. Solve for x in terms of y, z, w, v.

Solution: \boxed{360^\circ - y - z - w - v}


Queen High School has 1500 students, each taking 6 classes a day, and there are 25 students in each class. Each teacher teaches 5 classes per day. Calculate the number of teachers needed at Queen High School.


A circle with a radius of 6 is inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Calculate the area of the rectangle.


Eight spheres, each with a radius of 2, are placed one in each octant. Each sphere is tangent to the coordinate planes. Calculate the radius of the smallest sphere, centered at the origin, that can enclose all eight of these spheres.

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


Ten friends ate at a restaurant and agreed to share the bill equally. However, because Sam forgot his money, each of his nine friends paid an extra $3 to cover his share of the total bill. What is the total bill?


Determine the number of integer solutions $(x, y)$ to the equation $x^3 + 4x^2 - 11x + 30 = 8y^3 + 24y^2 + 18y + 7$.


A fair 8-sided die is rolled twice. Find the probability that the sum of the numbers that come up is a prime number.


A circle is inscribed in a triangle with side lengths 12, 16, and 20. Let the segments of the side of length 12, made by a point of tangency, be $r$ and $s$, with $r<s$. Determine the ratio $r:s$.


Andrew and Bella are jogging on a circular track at different speeds, with Andrew completing a lap every 75 seconds and Bella finishing a lap every 120 seconds. Both start from the same line simultaneously. A photographer, standing at a point one-third of the way from the starting line around the track, will take a photo between 9 and 10 minutes after they start, capturing one-fourth of the track around his standing point. Determine the probability that both Andrew and Bella will be captured in this photo.


How many ordered pairs (m, n) of positive integers, with m ≥ n, have the property that their squares differ by 150?


A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $3:1$. Calculate the percent of the rectangle's area that is contained within the square.


Given $x$ such that $0 \leq x < 4$, calculate the minimum value of the expression $\frac{x^2 + 2x + 6}{2x + 2}.$


Given the set $B$ of all positive integers exclusively having prime factors from the set $\{2, 3, 5, 7\}$, Calculate the infinite sum:

\[
\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{14} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \frac{1}{21} + \dots
\]

and express this sum, $\Sigma$, as a fraction in simplest form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers.


Given a list of the first 12 positive integers such that for each $2\le i\le 12$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list, calculate the number of such lists.


Emily is riding her bicycle at 15 miles per hour and Emerson is jogging at 9 miles per hour in the same direction. If Emily sees Emerson 1 mile ahead and then sees him 1 mile behind her after passing, how long can she see him in minutes.


Let $\overline{AB}$ be a diameter in a circle of radius $7$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=3$ and $\angle AEC = 45^{\circ}.$ Calculate $CE^2+DE^2$.


In a square $EFGH$ with side length 2 units, a circle $\Omega$ is inscribed. The circle intersects the line segment $\overline{HG}$ at $N$. The line $\overline{EN}$ intersects $\Omega$ again at a point $Q$ (different from $N$). Calculate the length of $EQ$.


Given the product $2! \cdot 4! \cdot 6! \cdot 8! \cdot 10!$, find the number of perfect squares that are divisors of this product.


Given that Maria scored $5, 2, 4, 3, 6, 2, 7, 4, 1,$ and $3$ goals in her first $10$ soccer matches, her total score after $11$ matches is divisible by $11$, and her total score after $12$ matches is divisible by $12$. Find the product of the number of goals she scored in the eleventh and twelfth matches.

Solution: \boxed{7 \times 4 = 28}


Let $M = 39 \cdot 48 \cdot 77 \cdot 150$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$.


Given that x and y are distinct nonzero real numbers such that x + 3/x = y + 3/y, find the value of xy.


The polynomial $(x+y)^{11}$ is expanded in decreasing powers of $x$. Evaluate the third and fourth terms of the expansion when $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is two. Determine the value of $p$.


A cell phone plan costs $25 each month, plus $8 cents per text message sent, plus $12 cents for each minute used over 40 hours. Calculate the total cost for Jenny in February, given that she sent 150 text messages and talked for 41 hours.


Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE}$. Let $DA=20,$ and let $FD=AE=7.$ Calculate the area of $ABCD$.


Determine the minimum number of small droppers required to fill a medicine container completely with a total volume of $265$ milliliters, using only small droppers that hold $19$ milliliters.


Given that Jenny has 30 coins consisting of nickels and dimes, and if her nickels were dimes and her dimes were nickels, she would have $\$1.20 more, determine the total value of her coins.


Given Elmer upgraded to a new car with $60\%$ better fuel efficiency in kilometers per liter and uses a type of diesel that is $30\%$ more expensive per liter, calculate the percentage that Elmer saves on fuel costs using his new car over the same distance compared to his previous car.


Compute the sum $g \left(\frac{1}{2021} \right) - g \left(\frac{2}{2021} \right) + g \left(\frac{3}{2021} \right) - g \left(\frac{4}{2021} \right) + \cdots - g \left(\frac{2020}{2021} \right)$ where $g(x) = x^2(1-x)^2$.


The sum of two numbers is \( S \). Suppose \( x \) is added to each number and then each of the resulting numbers is multiplied by \( k \). What is the sum of the final two numbers?

Solution: \boxed{kS + 2kx}


The parabolas $y = ax^2 - 4$ and $y = 6 - bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $24$. Determine the value of $a + b$.


Given the number $15!$, calculate the probability that a randomly chosen positive integer divisor of this number is odd.


Given a cube with 4-inch edges is to be constructed from 64 smaller cubes with 1-inch edges, where 48 are colored red and 16 are colored white, determine the fraction of the surface area that is white when constructed to have the smallest possible white surface area showing.


Three primes $p, q$, and $r$ satisfy $p + q + r = s$ where $s$ is the smallest square number greater than 15. If $1 < p < q < r$, find $p$.


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


Given quadrilateral $EFGH$, with side lengths $EF = 6$, $FG = 19$, $GH = 6$, and $HE = 10$, and where $EG$ is an integer, calculate the value of $EG$.


Given a quadrilateral $Q$ with vertices $A$, $B$, $C$, and $D$, determine how many circles in the plane of $Q$ can be drawn such that a diameter of each circle is defined by a pair of vertices from $Q$.


Given the operation $x \otimes y = x^3 + 3 - y$, calculate the value of $k \otimes (k \otimes (k\otimes k))$.

Solution: \boxed{k^3 + 3 - k}


Two cyclists start from two points that are $50$ miles apart. If they travel towards each other, they meet in $2$ hours. However, if one cyclist starts from a point $10$ miles behind his original position and the other remains at his starting position, and they would be $70$ miles apart in $3$ hours. Determine the ratio of the speed of the faster cyclist to the speed of the slower cyclist.


Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white.


Determine how many pairs of positive integers (a, b) with $a+b \leq 90$ satisfy the equation $\frac{a+b^{-1}}{a^{-1}+b} = 17.$


In a chess tournament, there are $n$ junior players and $3n$ senior players, with each participant playing exactly one game against every other player. If there are no draws and the ratio of the number of games won by junior players to the number of games won by senior players is $3/7$, find the value of $n$.


When $10^{100} - 94$ is expressed as a single whole number, calculate the sum of its digits.


A rectangular solid has its side, front, and bottom faces with areas $15\text{ in}^2$, $20\text{ in}^2$, and $12\text{ in}^2$ respectively. Determine the volume of this solid.


Using a calculator with only two keys [+1] and [x2], starting with the display "1," calculate the fewest number of keystrokes needed to reach "256".


Three primes $p, q$, and $s$ satisfy $p + q = s + 4$ and $1 < p < q$. Solve for $p$.


In a triangle, if one angle remains the same, one of its enclosing sides is tripled, and the other side is doubled, calculate the factor by which the area of the triangle is multiplied.


Given that in a kingdom $\frac{1}{3}$ of the knights are green, and $\frac{1}{5}$ of the knights are magical, and the fraction of green knights who are magical is $3$ times the fraction of yellow knights who are magical, find the fraction of green knights who are magical.


Ahn chooses a two-digit integer, subtracts it from 300, and triples the result. Find the smallest number Ahn can get.


Given Alice has 26 apples and each person must receive at least 3 apples, calculate the total number of ways of distributing the apples among Alice, Becky, and Chris.


Given the function $g$ defined by the table and given that $v_0=3$ and $v_{n+1} = g(v_n + 1)$ for $n \ge 0$, find $v_{2004}$.


Determine the number of distinct triangles that can be formed using three of the dots below as vertices, where the dots form a $3 \times 3$ grid.


Given that 30% of the objects in the urn are beads, and 50% of the coins in the urn are silver, calculate the percentage of objects in the urn that are gold coins.


Kiana has two older twin siblings, and their ages, together with her own, multiply together to equal 162. Calculate the sum of their ages.


Tom's Hat Shoppe increased all original prices by $30\%$. Then, the shoppe is having a sale where all prices are $10\%$ off these increased prices. What is the sale price of an item relative to its original price?


Orvin went to the store with enough money to buy 40 balloons at full price. He discovered a revised store promotion: after buying 4 balloons at full price, the next balloon could be purchased at half off. Determine the greatest number of balloons Orvin could buy if he takes full advantage of this offer.


Given that Fluffy the rabbit ate twice as many carrots each day from April 1st to April 3rd, and a total of 84 carrots over these three days, calculate the number of carrots Fluffy ate on April 3rd.


Peter's family ordered a 16-slice pizza for dinner. Peter ate two slices and shared another slice equally with his two brothers, Paul and John. What fraction of the pizza did Peter eat.


Determine the maximum number of quarters that could be in a coin box containing 120 coins, consisting of only nickels, dimes, and quarters, and totaling $10.00.


Given that each licence candy costs $24$ cents, calculate the smallest possible value of $n$ when Casper has exactly enough money to buy $10$ pieces of jelly candies, $16$ pieces of chocolate candies, or $18$ pieces of gummies.


Given two angles form a linear pair and are both prime numbers, and the larger angle $a^{\circ}$ is more than the smaller angle $b^{\circ}$, find the least possible value of $b$.


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


Given a test with 30 questions, a scoring system of +4 points for each correct answer, -1 point for each incorrect answer, and 0 points for each unanswered question, and Mary scores 70 points, calculate the maximum number of questions she could have answered correctly.


Given that $P$ is a point inside rectangle $ABCD$, the distances from $P$ to the vertices of the rectangle are $PA = 5$ inches, $PD = 12$ inches, and $PC = 13$ inches. Find $PB$, which is $x$ inches.


Given that Penelope takes 50 equal waddles to walk between consecutive telephone poles, and Hector takes 15 equal jumps to cover the same distance, and the 51st pole is 6336 feet from the first pole, calculate how much longer Hector's jump is than Penelope's waddle.


A positive integer \( n \) has 48 divisors and 5n has 72 divisors. Find the greatest integer k such that 5^k divides n.


Two circles with radii $r$ and $s$ (where $r > s$) are placed such that the distance between their centers is $d$. Determine the possible number of different values of $k$ representing the number of lines that are simultaneously tangent to both circles.


The expression $\frac{2022}{2021} - \frac{2021}{2022}$ is equal to the fraction $\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is $1$. Find the value of $p$.


Calculate the sum of the digits of the number \(10^{100} - 57\).


A car moves such that its speed for the third and subsequent kilometers varies inversely as the square of the integral number of kilometers already traveled. If the third kilometer is traversed in $3$ hours, determine the time needed, in hours, to traverse the $n$th kilometer.


Given Emma's telephone number has the form $555-ab-cde-fgh$, where $a$, $b$, $c$, $d$, $e$, $f$, $g$, and $h$ are distinct digits in increasing order, and none is either $0$, $1$, $4$, or $5$, determine the total number of different telephone numbers Emma could have.


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


The number $24!$ has many positive integer divisors. What is the probability that a divisor randomly chosen from these is odd?


Three primes p, q, and r satisfy p + q = r + 2 and 1 < p < q. Find the value of p.


If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 12$, find the sum of $a$ and $b$.


Carl encounters three types of rocks in a different section of the cave: $6$-pound rocks worth $\$$20 each, $3$-pound rocks worth $\$$9 each, and $2$-pound rocks worth $\$$4 each. Given that Carl can carry up to $24$ pounds and there are at least $10$ of each type available, find the maximum value, in dollars, of the rocks he can carry out of the cave.


Find the number of ordered triples of integers (x, y, z) that satisfy |x+y| + z = 23 and xy + |z| = 119.


Suppose the estimated cost to build an interstellar telescope is $30$ billion dollars and it is shared equally by the $300$ million people in the country. Calculate the amount each person needs to pay.


Evaluate the expression $\dfrac{13!-12!}{10!}$.


The maximum number of the five integers that can be less than -5.


Given a rectangle with dimensions $a$ and $b$ ($a < b$), determine the number of rectangles with dimensions $x$ and $y$ such that $x < \frac{a}{2}$ and $y < \frac{a}{2}$, and the perimeter of the new rectangle is half the perimeter of the original rectangle and its area is half the area of the original rectangle.


Given that there are eleven books, including three Italian, three German, and five French books, determine the number of ways to arrange the books on the shelf, keeping the Italian books together, the German books together, and the French books together.


A list of $2520$ positive integers has a unique mode, which occurs exactly $12$ times, calculate the least number of distinct values that can occur in the list.


Determine how many solution-pairs in the positive integers exist for the equation $4x + 7y = 548$.


Determine how many numbers between 3000 and 4000 have the property that their units digit equals the product of the other three digits modulo 10.


Given that a semipro basketball league has teams with 15 players each and each player must receive a minimum salary of $20,000, and the total of all players' salaries for each team must not exceed $500,000, determine the maximum possible salary for the highest-paid player on the team.


Given that spinner P has numbers 2, 4, 5, spinner Q has numbers 1, 3, 5, and spinner R has numbers 2, 5, 7, determine the probability that the sum of numbers obtained from rotating spinners P, Q, and R is an odd number.


Given the mean score of the students in the morning class is 90, the mean score of the students in the afternoon class is 75, and the ratio of the number of students in the morning class to the afternoon class is $\frac{5}{6}$, calculate the mean score of all the students in both classes combined.


Brenda's rate of building bricks per hour is 1/8, Brandon's rate of building bricks per hour is 1/12, but when they work together their combined rate is reduced by 15 bricks per hour, and they complete the chimney in 6 hours, calculate the number of bricks in the chimney.


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


A store owner bought 2000 markers at $0.20 each. To make a minimum profit of $200, if he sells the markers for $0.50 each, calculate the number of markers he must sell at least to achieve or exceed this profit.


Given that $q_1 (x)$ and $r_1$ are the quotient and the remainder, respectively, when the polynomial $x^9$ is divided by $x - 1$, and if $q_2 (x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x - 1$, find $r_2$.


Positive integers $a$ and $b$ are each less than $8$. Find the smallest possible value for $3a-2ab$.


Given the fractions \(\frac{5}{7}\) and \(\frac{3}{4}\), find the value of \(\text{rem} \left(\frac{5}{7}, \frac{3}{4}\right)\).


The area of the ring between two concentric circles is $50\pi$ square centimeters. Find the length of a chord of the larger circle which is tangent to the smaller circle.


Given that each of $9$ standard dice, labeled from $1$ to $6$, are rolled, determine the sum that has the same probability as the sum of $15$ when all dice show their top faces.


How many integer values of $x$ satisfy $|x| < 4.5\pi$?


A garden pond contains 300 gallons of water. Each day, 1 gallon of water evaporates, and on every third day, it rains adding 2 gallons of water back into the pond. Determine the amount of water in the pond after 45 days.


Soda is now sold in packs of 8, 18, and 30 cans. Find the minimum number of packs needed to buy exactly 144 cans of soda.


The three-digit number $3a7$ is added to the number $414$ to give the three-digit number $7c1$. If $7c1$ is divisible by 11, then calculate the value of $a+c$.


A square diagram shows a uniform $11 \times 11$ grid of points including the points on the edges. Point $P$ is located at the center of the square. If point $Q$ is randomly picked from the remaining $120$ points, what is the probability that the line $PQ$ forms a line of symmetry for the square?


Calculate the value of the expression: \[ \frac{2 \times 6}{12 \times 14} \times \frac{3 \times 12 \times 14}{2 \times 6 \times 3} \times 2 \]


Given a circular park 20 feet in diameter with a straight walking path 5 feet wide passing through its center, calculate the area of the park covered by grass after the path is laid.

Solution: \boxed{100\pi - 100}


Given positive integers \(a\) and \(b\) are each less than 10, find the smallest possible value for \(2 \cdot a - a \cdot b\).


Given points $P(-1, -3)$ and $Q(5, 3)$ are in the $xy$-plane, and point $R(2, n)$ lies on the line $y = 2x - 4$, find the value of $n$ such that the total distance $PR + RQ$ is minimized.


One third of Marcy's marbles are blue, one third are red, and ten of them are green. What is the smallest number of yellow marbles that Marcy could have?


Three vertices of parallelogram $ABCD$ are $A(-1, 3), B(2, -1), D(7, 6)$ with $A$ and $D$ diagonally opposite. Calculate the product of the coordinates of vertex $C$.


Before the district play, the Zebras had won $40\%$ of their soccer games. During district play, they won eight more games and lost three, to finish the season having won $55\%$ of their games. Calculate the total number of games the Zebras played.


Given the product of the digits of a 3-digit positive integer is 30, find how many such integers exist.


In a math contest, 65 students are wearing blue shirts, and 67 students are wearing yellow shirts, making a total of 132 contestants. These students are grouped into 66 pairs. If in 28 of these pairs, both students are wearing blue shirts, determine the number of pairs that consist of students both wearing yellow shirts.


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

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


If a worker receives a 30% cut in wages, calculate the percentage of the raise he must obtain in order to regain his original pay.


Given that $\sqrt{18}$ and $\sqrt{120}$ are the lower and upper limits, respectively, calculate the number of whole numbers between them.


If $x$ and $y$ are non-zero numbers such that $x = 2 + \frac{1}{y}$ and $y = 2 + \frac{1}{x}$, find the value of $y$.

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


Given an urn contains only coins and beads, and 30% of the objects in the urn are beads. Also, 30% of the coins in the urn are silver, find the percentage of the objects in the urn that are gold coins.


The two spinners shown are spun once. Spinner 1 has sectors numbered 1, 4, 5, and Spinner 2 has sectors numbered 2, 3, 6. Determine the probability that the sum of the numbers in the two sectors falls between 6 and 10, inclusive.


In the Metropolitan County, there are $25$ cities. From a given bar chart, the average population per city is indicated midway between $5,200$ and $5,800$. If two of these cities, due to a recent demographic survey, were found to exceed the average by double, calculate the closest total population of all these cities.


Determine the ratio of the shorter side to the longer side of the rectangular park, given that by taking a shortcut along the diagonal, a boy saved a distance equal to $\frac{1}{3}$ of the longer side of the park.


Given that Paul owes Paula $45$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, $20$-cent coins, and $25$-cent coins, find the difference between the largest and smallest number of coins he can use to pay her.


Given that a triangle with integral sides is isosceles and has a perimeter of 12, find the area of the triangle.


Given $a$, $b$, and $a-b\cdot\frac{1}{a}$ are non-zero values, evaluate $\frac{a^2-\frac{1}{b^2}}{b^2-\frac{1}{a^2}}$.


Given a \(5 \times 5\) block of calendar dates, the numbers are reversed in the second, third, and fourth rows. Then, find the positive difference between the two diagonal sums.


Given that real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$, calculate the probability that $\lfloor\log_3x\rfloor=\lfloor\log_3y\rfloor$.


Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. Find the probability that $\lfloor\log_3x\rfloor = \lfloor\log_3y\rfloor$.


Given the condition that a positive integer less than $1000$ is $7$ times the sum of its digits, determine the number of such integers.


Given that Marla has a large white cube with an edge of 12 feet and enough green paint to cover 600 square feet, find the total area of the cube's faces that remain white after painting complete faces green and leaving exactly two adjacent faces untouched.


What is the minimum possible product of three different numbers from the set $\{-10, -7, -5, -3, 0, 2, 4, 6, 8\}$?


Given that Elmer's new car gives 60% better fuel efficiency, measured in kilometers per liter, than his old car, and the diesel fuel for his new car is 25% more expensive per liter than the gasoline for his old car, calculate the percentage by which Elmer will save money using his new car instead of his old car for a 100-kilometer trip.


Six semicircles are evenly arranged along the inside of a regular hexagon with a side length of 3 units. A circle is positioned in the center such that it is tangent to each of these semicircles. Find the radius of this central circle.

Solution: \boxed{\frac{3 (\sqrt{3} - 1)}{2}}


Two pitchers, one with a capacity of 800 mL and the other 700 mL, are initially filled with apple juice: the first pitcher is 1/4 full and the second pitcher is 3/7 full. Water is then added to each pitcher to reach its full capacity, and both pitchers are then poured into a larger container. What fraction of the liquid in the larger container is apple juice?


Calculate the value of $\text{rem} \left(\frac{5}{7}, \frac{3}{4}\right)$ and then multiply the result by $-2$.