Find the sum of the coefficients in the polynomial $3(x^8 - x^5 + 2x^3 - 6) - 5(x^4 + 3x^2) + 2(x^6 - 5)$.


The matrix $\mathbf{B} = \begin{pmatrix} 4 & 5 \\ 7 & e \end{pmatrix}$ satisfies
\[\mathbf{B}^{-1} = m \mathbf{B}\]for some constant $m.$ Find the ordered pair $(e,m).$

Solution: \boxed{(-4, \frac{1}{51})}


Factor the following expression: $294b^3 + 63b^2 - 21b$.

Solution: \boxed{21b(14b^2 + 3b - 1)}


Define \( h(x) = 7x - 6 \). If \( h(x) = f^{-1}(x) - 5 \) and \( f^{-1}(x) \) is the inverse of the function \( f(x) = cx + d \), find \( 7c + 7d \).


Determine the area enclosed by the graph of $|2x| + |5y| = 20$.


In a bag, there are 7 blue chips, 5 yellow chips, and 4 red chips. One chip is randomly drawn from the bag and then placed back into the bag before a second chip is drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 9:35?


Solution: \boxed{14x^{18} - 28x^{10} - 63x^3 + 21}


The arithmetic mean of these five expressions is 26. What is the value of $x$? $$x + 10 \hspace{.5cm} 17 \hspace{.5cm} 2x \hspace{.5cm} 15 \hspace{.5cm} 2x + 6$$

Solution: \boxed{x = 16.4}


Calculate the sum of the squares of the coefficients when the expression $3(x^3 - 4x^2 + x) - 5(x^3 + 2x^2 - 5x + 3)$ is fully simplified.


Determine the value of `p` for the quadratic equation $5x^2 + 7x + p$ when the roots are given as $\frac{-7 \pm i\sqrt{231}}{10}$.


Two circles, one of radius 7 inches, the other of radius 3 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one crawling along the larger circle at $4\pi$ inches per minute, the other crawling along the smaller circle at $3\pi$ inches per minute. How many minutes is it before their next meeting at point P?


Given the table below, calculate the median salary among the 67 employees of a different company:

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO&1&$\$150{,}000$\\\hline
Senior Vice-President&4&$\$105{,}000$\\\hline
Manager&15&$\$80{,}000$\\\hline
Team Leader&8&$\$60{,}000$\\\hline
Office Assistant&39&$\$28{,}000$\\\hline
\end{tabular}


Solution: \boxed{27x^6 - 64y^9}


I run at a constant speed, and it takes me 30 minutes to run to the park from my house. The park is 4 miles away. There is a library exactly midway between my house and the park. How many minutes will it take me to run from my house to the library and then to the park?

Solution: \boxed{30 \text{ minutes}}


If the parabola defined by $y = bx^2 + 4$ is tangent to the line $y = 2x + 2$, then calculate the constant $b$.


A person has $440.55$ in their wallet. They purchase goods costing $122.25$. Calculate the remaining money in the wallet. After this, calculate the amount this person would have if they received interest annually at a rate of 3% on their remaining money over a period of 1 year.


In parallelogram $EFGH$, the measure of angle $EFG$ is twice the measure of angle $FGH$. Determine the measure of angle $EHG$.


If \(a + 3b = 27\) and \(5a + 2b = 40\), what is the value of \(a + b\)?


A running track is now composed of three concentric circles. The circumferences of the inner and middle circles differ by $20\pi$ feet, and the circumferences of the middle and outer circles differ by $30\pi$ feet. What is the total width of the track from the innermost to the outermost circle?

Solution: \boxed{25 \text{ feet}}


The hyperbolas
\[\frac{y^2}{49} - \frac{x^2}{25} = 1\]
and
\[\frac{x^2}{T} - \frac{y^2}{18} = 1\]
have the same asymptotes. Find \(T.\)


Fifty slips are placed into a hat, each bearing a number from 1 to 10, with each number entered on five slips. Five slips are drawn from the hat at random and without replacement. Let \( p \) be the probability that all five slips bear the same number. Let \( q \) be the probability that three of the slips bear a number \( a \) and the other two bear a number \( b \neq a \). What is the value of \( q/p \)?


Given that $f(x) = x^2 - 3x + 7$ and $g(x) = 2x + 4$, find the value of $f(g(5)) - g(f(5))$.


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0<y\le10$, then what is the largest possible value of $a+b+c$?


A tailor needs to make a pair of pants for Jessica, who provides her waist size in inches. Given that there are $12$ inches in a foot and $25.4$ centimeters in a foot, determine Jessica's waist size in centimeters if she states her waist is $28$ inches. Round your answer to the nearest tenth.

Solution: \boxed{59.3\ \text{cm}}


Let $g(x) = 3x^4 - 22x^3 + 47x^2 - 44x + 24$. Find the value of $g(5)$ using the Remainder Theorem.


Determine the count of positive integers less than 150 that satisfy the congruence $x + 20 \equiv 70 \pmod{45}$.


How many four-character license plates can be formed if the plate must start with an even digit, followed by a consonant, then a vowel, and end with a consonant? Assume Y is considered a vowel.


Find the smallest, positive five-digit multiple of $18.$


The numbers $x$ and $y$ are inversely proportional. If the sum of $x$ and $y$ is now 60, and $x$ is three times $y$, find the value of $y$ when $x=-12$.


What is the value of $x + y$ if the sequence $3, 9, 15, \ldots, x, y, 45$ is an arithmetic sequence?


What is the probability that the same odd number will be facing up on each of four standard eight-sided dice that are tossed simultaneously? Express your answer as a common fraction.


Determine the greatest common divisor (GCD) of the numbers 4410 and 10800.


Bryan now has some 5 cent stamps and some 6 cent stamps. What is the least number of stamps he can combine to make a total of 60 cents?


In the right triangle $XYZ$, we know $\angle X = \angle Z$ and $XZ = 12$. What is the perimeter of $\triangle XYZ$?

Solution: \boxed{12\sqrt{2} + 12}


At a school cafeteria, Sam wants to buy a lunch consisting of one main course, one beverage, and one snack. The table below lists Sam's options available in the cafeteria. How many different lunch combinations can Sam choose from?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages} & \textbf{Snacks} \\ \hline
Burger & Water & Apple \\ \hline
Pasta & Soda & Banana \\ \hline
Salad & Juice & \\ \hline
Tacos & & \\ \hline
\end{tabular}


Ten people decide to attend a basketball game, but five of them are only 1/3 sure that they will stay for the entire time (the other five are certain they'll stay the whole time). What is the probability that at the end, at least 9 people stayed the entire time?


A triangle in a Cartesian coordinate plane has vertices at (3, -3), (8, 4), and (3, 4). Calculate the area of this triangle.

Solution: \boxed{17.5 \text{ square units}}


Climbing the first flight of stairs takes Jimmy 25 seconds, and each following flight takes 10 seconds more than the preceding one. How many total seconds does it take to climb the first seven flights of stairs?


Find the integer $n$, $0 \le n \le 11$, that satisfies \[n \equiv -5033 \pmod{12}.\]


If the two roots of the quadratic $5x^2 - 2x + m$ are $\frac{2 \pm i \sqrt{78}}{10}$, what is $m$?


Find the number of integers $n$ that satisfy
\[25 < n^2 < 144.\]


What is the sum of all integer solutions to $|n| < |n-5| < 5$?


Calculate $(-6)^5 \div 6^2 + 4^3 - 7^2$ and express it as an integer.


What is the 150th digit to the right of the decimal point in the decimal representation of $\frac{16}{81}$?


In a pet shelter housing 100 dogs, 20 dogs enjoy apples, 70 dogs enjoy chicken, and 10 dogs enjoy cheese. 7 dogs enjoy both apples and chicken, 3 dogs enjoy both apples and cheese, and 5 dogs enjoy both chicken and cheese. 2 dogs enjoy all three. How many dogs enjoy none of these foods?


What is the smallest positive integer that leaves a remainder of 6 when divided by 8 and a remainder of 5 when divided by 9?


What is the coefficient of $x^2$ when $3x^3 - 4x^2 - 2x - 3$ is multiplied by $2x^2 + 3x - 4$ and the like terms are combined?


Kadin makes a larger snowman by stacking four snowballs with radii of 2 inches, 4 inches, 5 inches, and 6 inches. Assuming all snowballs are perfect spheres, calculate the total volume of snow used only for the snowballs with radii greater than 3 inches. Express your answer in terms of $\pi$.


How many two-digit numbers are there in which the tens digit is greater than the ones digit and the ones digit is odd?


A bin contains 10 black balls and 10 white balls. Four balls are drawn at random. What is the probability of drawing exactly 2 black balls and 2 white balls?


If $n = 2^{12} \cdot 3^{15} \cdot 5^{9}$, how many of the natural-number factors of $n$ are multiples of 360?


What is the fourth power of the cube of the third smallest prime number?


What is the first year after 2010 for which the sum of the digits is 8?


The sum of the first 1000 terms of a geometric sequence is 500. The sum of the first 2000 terms is 950. Find the sum of the first 3000 terms.


Semicircles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in an 18-inch length of this pattern? Express your answer in terms of \(\pi\).


If $\frac{7}{12}$ is expressed in decimal form, what digit is in the 101st place to the right of the decimal point?


Four concentric circles are drawn with radii of 2, 4, 6, and 8. The inner circle is painted white, the ring around it is black, the next ring is white, and the outer ring is black. What is the ratio of the black area to the white area? Express your answer as a common fraction.


An infinite geometric series has a first term of $18$ and a second term of $6.$ A second infinite geometric series has the same first term of $18,$ a second term of $6+n,$ and a sum of five times that of the first series. Find the value of $n.$


Let $p$ and $q$ be the solutions to the equation $3x^2 - 5x - 8 = 0$. Compute the value of $(5p^3 - 5q^3)(p - q)^{-1}$.


Marvin has forgotten his bike lock code. The code is a sequence of four numbers: the first number is between 1 and 25 and is a prime number, the second number is a multiple of 4 between 1 and 20, the third number is a multiple of 5 between 1 and 30, and the fourth number is an even number between 1 and 12. How many different codes could Marvin's combination be?


Calculate the value of \( n \) for which the sum $$2 + 33 + 444 + 5555 + 66666 + 777777 + 8888888 + 99999999$$ is congruent to \( n \) modulo 9, where \( 0 \leq n < 9 \). Additionally, find out if \( n \) is even or odd.


Find the distance from the point $(0,3,-1)$ to the line passing through $(1,-2,0)$ and $(2,-5,3)$.


The greatest common divisor of 24 and some number between 70 and 90 is 6. What is the smallest such number?


The ratio of irises to roses in Carter's garden is 3:7. He currently has 42 roses. He plans to add 35 more roses and needs to add enough irises to maintain the same ratio. How many irises will he need in total after this addition?


Calculate $\sqrt{50} \cdot \sqrt{18} \cdot \sqrt{8}$.


What is the base-10 integer 624 when expressed in base 7?


Calculate the area, in square units, of a triangle with vertices at $(1,1)$, $(1,8)$, and $(10,15)$. Use the determinant method for a solution.


What is the smallest integer value of $x$ for which $3x^2 - 4 < 20$?


Find the number of zeros between the decimal point and the first non-zero digit when $\frac{7}{5000}$ is written as a decimal.


Calculate the product:
\[
\begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 7 \\ 0 & 1 \end{pmatrix} \dotsm
\begin{pmatrix} 1 & 101 \\ 0 & 1 \end{pmatrix}.
\]

Solution: \boxed{\begin{pmatrix} 1 & 2600 \\ 0 & 1 \end{pmatrix}}


Given that $f(x) = x^{k/n}$ where $k < 0$ and $n > 0$, determine the range of $f(x)$ on the interval $[2, \infty)$.

Solution: \boxed{(0, 2^{k/n}]}


Find the ones digit of $13^{13(7^{7})}$.


Alice rolls six fair 6-sided dice, each numbered with numbers from 1 to 6. What is the probability that exactly three of the dice show a prime number?


According to Moore's law, the number of transistors on a chip doubles every 18 months. In 1995, a typical CPU contained about $2,\!500,\!000$ transistors. How many transistors did a typical CPU contain in the year 2010?

Solution: \boxed{2,\!560,\!000,\!000 \text{ transistors}}


Multiply the base-10 numbers 354 and 78. Write the product in base-7. What is the units digit of the base-7 representation?


Determine the number of ways to arrange the letters of the word "MATHEMATICS".


What is the remainder when the sum $3 + 11 + 19 + \cdots + 291 + 299$ is divided by $8$?


Solve
\[\arcsin x + \arcsin 3x = \frac{\pi}{2}.\]


How many integers from 100 through 999, inclusive, do not contain any of the digits 0, 1, 8, or 9?


A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and that each number rolled is a prime number? Express your answer as a common fraction.


A valid license plate in Yonderville consists of three letters followed by three digits. The first letter must be a vowel (A, E, I, O, U), which cannot repeat in the license plate, while the other two letters can be any letter except the first chosen vowel. How many valid license plates are possible?


A very large number $y$ is equal to $2^33^44^55^66^77^88^99^{10}$. What is the smallest positive integer that, when multiplied with $y$, produces a product that is a perfect square?


What is the greatest integer less than 200 for which the greatest common divisor of that integer and 18 is 9?


Find \( d \) given that \( \lfloor d \rfloor \) satisfies
\[ 3x^2 + 11x - 20 = 0 \]
and \( \{ d \} = d - \lfloor d \rfloor \) satisfies
\[ 4x^2 - 12x + 5 = 0. \]


I have two 12-sided dice that each have 3 red sides, 4 green sides, 2 blue sides, and 3 yellow sides. If I roll both dice, what is the probability that they come up the same?


What common fraction (i.e., a fraction reduced to its lowest terms) is equivalent to $.4\overline{37}$?


How many positive integers smaller than $2,000,000$ are powers of $2$, but are not powers of $4$?


What is the smallest four-digit integer $n$ that satisfies $$75n \equiv 225 \pmod{450}~?$$


Two numbers \(180\) and \(n\) share exactly three positive divisors. What is the greatest of these three common divisors?


Consider the quadratic function \( g(x) = 3x^2 - 9x + 4 \). Determine the largest interval that includes the point \( x=-1 \) on which \( g \) can be made invertible.

Solution: \boxed{(-\infty, \frac{3}{2}]}


In 1970, there were 700,000 cases of a hypothetical disease reported in a particular country. By 2010, the cases decreased to 1,000. Calculate how many cases would have been reported in 1995 if the number of cases had decreased linearly over this period.


The chart below provides the air distance in miles between selected world cities. If two different cities from the chart are chosen at random, what is the probability that the distance between them is less than $8000$ miles? Express your answer as a common fraction.

\begin{tabular}{|c|c|c|c|c|}
\hline
& Bangkok & Cape Town & Honolulu & London \\ \hline
Bangkok & & 6300 & 6609 & 5944 \\ \hline
Cape Town & 6300 & & 11,535 & 5989 \\ \hline
Honolulu & 6609 & 11,535 & & 7240 \\ \hline
London & 5944 & 5989 & 7240 & \\ \hline
\end{tabular}


Given positive integers $x$ and $y$ such that $x \neq y$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$, what is the smallest possible value for $x + y$?


What is the remainder when $2002 \cdot 1493$ is divided by $300$?


Factor the expression $5x(x-2) + 9(x-2) - 4(x-2)$.


Find the greatest common divisor of $7384$ and $12873$.


The graph of $y = ax^2 + bx + c$ has a maximum value of 36, and passes through the points $(-3,0)$ and $(3,0).$ Find $a + b + c.$


If an integer is divisible by both $4$ and $3$, and the sum of its last two digits is $12$, then what is the product of its last two digits?


If the six digits 1, 2, 3, 5, 7, and 8 are randomly arranged into a six-digit positive integer, what is the probability that the integer is divisible by both 15 and 2? Express your answer as a common fraction.


A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?


A rectangle has a perimeter of 40 units and one of its sides must be an even number. What is the maximum possible area of the rectangle if its dimensions are whole numbers?


Compute the distance between the vertices of the parabolas formed by the graph of the equation \[\sqrt{x^2+y^2} + |y-1| = 5.\]


The average of four different positive whole numbers is $5$. If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers?


If $a, b, c$ are integers from the set of positive integers less than $5$ such that
\begin{align*}
abc &\equiv 1\pmod{5}, \\
3c &\equiv 2\pmod{5}, \\
4b &\equiv 3 + b\pmod{5},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $5$?


Find the sum of the coefficients in the polynomial \[ -3(x^8 - 2x^5 + 4x^3 - 6) + 5(x^4 + 3x^2) - 2(x^6 - 5) \].


When the base-12 integer $2615_{12}$ is divided by $9$, what is the remainder?


In Pascal's Triangle, each number is the result of the sum of the number above it to the left and the number above it to the right. Consider Row 10, calculate the sum of the numbers in this row, and also determine the sum for Row 11. Verify the pattern mentioned in the previous problem.


What is the period and phase shift for the function \( y = \sin(3x + \frac{\pi}{4}) \)?


Express $8.\overline{9}$ as a common fraction.


Calculate the expected value of rolling an 8-sided die where the numbers 1 through 4 have a probability of $\frac{1}{10}$ each, and the numbers 5 through 8 have a probability of $\frac{3}{20}$ each.


A student must choose a program of five courses from a list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Biology. This program must contain English and at least two mathematics courses. In how many ways can this program be chosen?


Add $518_{12} + 276_{12}$. Express your answer in base 12, using $A$ for $10$ and $B$ for $11$ if necessary.


Diane has borrowed $200$ clams from Harold at a $4\%$ simple daily interest. Ian has borrowed $100$ clams from Jackie at a $8\%$ simple daily interest. How many days will it take until the total amount they both owe is $400$ clams, assuming that they will not make any repayments during this time?

Solution: \boxed{6.25 \text{ days}}


Calculate the domain of the function \( f(x) = \log_5(\log_3(\log_2(x^2))) \).

Solution: \boxed{(-\infty, -\sqrt{2}) \cup (\sqrt{2}, \infty)}


In a new laboratory experiment, a colony of bacteria triples in number every day. The colony begins with 4 bacteria. Determine the first day when the number of bacteria exceeds 200.


Suppose that $a,$ $b,$ and $c$ are three positive numbers that satisfy the equations $abc = 1,$ $a + \frac {1}{c} = 8,$ and $b + \frac {1}{a} = 20.$ Find $c + \frac {1}{b}.$


Given that the point \((6,10)\) is on the graph of \(y=f(x)\), there is one point that must be on the graph of \(2y=5f(3x)+7\). What is the sum of the coordinates of that point?


Given the function \( h(x) = 3x^3 + 3x^2 - x - 1 \), find the value of \( h(h(3)) \).


In a panel discussion at a sports conference, five athletes from different teams are participating: two are from the Lakers, two are from the Celtics, and one is from the Warriors. If athletes from the same team insist on sitting together, how many ways can the five athletes be seated in a row?


For how many integers $n$ is it true that $\sqrt{3n} \le \sqrt{5n - 8} < \sqrt{3n + 7}$?


The average age of 40 fifth-graders is 10 years. The average age of 60 of their parents is 35 years. Additionally, there are 10 teachers with an average age of 45 years. What is the average age of all these fifth-graders, parents, and teachers combined?


How many even perfect square factors does $2^6 \cdot 5^4 \cdot 7^3$ have?


Solution: \boxed{\frac{20y^7 + 7y^5}{8}}


What is the probability that the square root of a randomly selected two-digit whole number is less than nine? Express your answer as a common fraction.


Math City is now designed with ten streets, all of which are straight. No two streets are parallel to one another. However, two of the ten streets have been built with tunnels that cross other streets without creating intersections. Assuming these tunnels are uniquely positioned such that each only bypasses one potential intersection with any other street, find the greatest number of police officers needed for the intersections.


Fifteen prime numbers are randomly selected without replacement, from the first fifteen prime numbers. What is the probability that the sum of the four selected numbers is odd? Express your answer as a common fraction.


A sphere is perfectly inscribed in a cube. If the edge of the cube measures 10 inches, determine the volume of the sphere in cubic inches. Express your answer in terms of \(\pi\).


In a triangle $PQR$, where $P=(0,8)$, $Q=(0,0)$, and $R=(10,0)$, point $S$ is the midpoint of $\overline{PQ}$, and point $T$ is the midpoint of $\overline{QR}$. Determine the sum of the $x$ and $y$ coordinates of $U$, the point of intersection of $\overline{PT}$ and $\overline{RS}$.

Solution: \boxed{\frac{18}{3} = 6}


Factor completely: $x^6 + 2x^4 - x^2 - 2$.


A mineralogist is hosting a competition to guess the age of an ancient mineral sample. The age is provided by the digits 2, 2, 3, 3, 5, and 9, with the condition that the age must start with an odd number.


The operation $\star$ is redefined as $a \star b = a^2 + \frac{a}{b}$. What is the value of $5 \star 2$?


Cecilia is solving two quadratic equations simultaneously. The first equation is $9x^2 - 36x - 81 = 0$. She wants to solve it by completing the square. In addition, she is also considering the equation $y^2 + 6y + 9 = 0$. Find the values of $x + y$.

Solution: \boxed{-1 - \sqrt{13}}


What is the radius of the circle inscribed in triangle $ABC$ if $AB = 6, AC = 8, BC = 10$? Express your answer in simplest radical form.


What is the coefficient of $a^3b^2$ in $(a+b)^5\left(c+\dfrac{1}{c}\right)^6$?


Find \(n\) such that \(2^6 \cdot 3^3 \cdot n = 10!\).


What is the slope of the line defined by any two solutions to the equation $\frac{3}{x} + \frac{4}{y} = 0$? Express your answer as a common fraction.


How many three-digit numbers are there in which the second digit is greater than the third digit?


Find the sum of the distinct prime factors of $7^7 - 7^4$.


James rode 40 miles at 8 miles per hour and 20 miles at 40 miles per hour. Additionally, James took a 30-minute break during his ride. What was his average speed, in miles per hour, for the entire ride excluding the break?


What is the fifteenth term in the geometric sequence $12, 4, \frac{4}{3}, \ldots$?


When the base-15 integer $2643_{15}$ is divided by 9, what is the remainder?


Consider a new sequence of numbers arranged in rows where the number in each row starts with $3n$, where $n$ is the row number, and each row $n$ contains $n^3$ times this number. Given this pattern, what is the $40^{\mathrm{th}}$ number in the sequence?


Given $ab + ac + bd + cd = 40$ and $a + d = 6$, find the value of $b + c$ assuming $a \neq d$.


For how many values of $c$ in the interval $[0, 2000]$ does the equation \[5 \lfloor x \rfloor + 4 \lceil x \rceil = c\] have a solution for $x$?


Express \( 5.\overline{317} \) as a common fraction in lowest terms.


What is the base ten equivalent of $54123_{6}$?


The polynomial $g(x)=x^4+px^3+qx^2+rx+s$ has real coefficients, and $g(3i)=g(3+2i)=0$. What is $p+q+r+s$?


What is the largest divisor of 540 that is less than 80 and also a factor of 180?


Mandy Monkey starts hopping on a number line at 0. She aims to reach 1, but can only hop 1/4 of the distance towards the goal at each jump. Each subsequent hop proceeds as 1/4 of the remaining distance to 1. Calculate the total distance Mandy has traveled after six hops. Express your answer as a common fraction.


Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}.$ Given $\mathbf{m} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10,$ find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$


Suppose $a$ and $b$ are positive integers such that $\gcd(a,b)$ is divisible by exactly $5$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $20$ distinct primes. Additionally, let the sum of the distinct prime factors of $a$ be greater than $50$.

If $a$ has fewer distinct prime factors than $b$, then what is the maximum number of distinct prime factors that $a$ can have?


What is the sum of the digits of the base $8$ representation of $888_{10}$?


Convert $2 e^{15 \pi i/4}$ to rectangular form.

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


Every week, Judy goes to the supermarket and buys the following: $5$ carrots at $\$1$ each, $3$ bottles of milk at $\$3$ each, $2$ pineapples at $\$4$ each, $2$ bags of flour at $\$5$ each, and a giant $\$7$ container of ice cream. This week the store has a sale, offering 25% off on flour. Judy also has a coupon for $\$10$ off any order of $\$30$ or over. How much money does Judy spend on this shopping trip?


The line \( x = k \) intersects the graph of the parabola \( x = -3y^2 + 2y + 7 \) at exactly one point. What is \( k \)?


Find all $c$ which satisfy $$\frac{c}{4} \le 3+c < -3(1+c).$$ Express your answer in interval notation, simplifying any fractions which occur in your answer.

Solution: \boxed{[-4, -\frac{3}{2})}


Consider the sum $$2+33+444+5555+66666+777777+8888888+99999999.$$ Calculate its congruence modulo 9, denoted as $m$, where $0 \le m < 9$.


If 18 bahs are equal to 30 rahs, and 6 rahs are equivalent to 10 yahs, how many bahs are equal to 1200 yahs?


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 8 and an altitude of 10, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.


Triangle $ABC$ has vertices $A(1, 10)$, $B(3, 0)$, and $C(9, 0)$. A horizontal line with equation $y=t$ intersects line segment $\overline{AB}$ at $J$ and line segment $\overline{AC}$ at $K$, forming $\triangle AJK$ with area 7.5. Compute $t$.


Fido's leash is tied to a stake at the center of his yard, which is now in the shape of a square. His leash is long enough to reach exactly the midpoint of each side of the yard. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest form as \(\frac{a}{b}\pi\), what is the value of \(a+b\)?


A certain integer has $5$ digits when written in base $3$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$?


Three distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6\}$. What is the probability that the smallest number divides the other two numbers? Express your answer as a common fraction.


Let the operation $\&$ be defined as $\&(a, b, c) = b^3 - 3abc - 4ac^2$, for all real numbers $a, b$ and $c$. What is the value of $\&(2, -1, 4)$?


Convert $\rm{BFACE}_{16}$ to a base 10 integer, where the 'digits' A through F represent the values 10, 11, 12, 13, 14, and 15 respectively.


Convert the point $(0, 3, -3 \sqrt{3})$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$

Solution: \boxed{\left( 6, \frac{\pi}{2}, \frac{5\pi}{6} \right)}


**
Consider a set of numbers $\{1, 2, 3, 4, 5, 6, 7\}$. Two different natural numbers are selected at random from this set. What is the probability that the greatest common divisor (gcd) of these two numbers is one? Express your answer as a common fraction.

**


The foci of the ellipse \[\frac{x^2}{25} + \frac{y^2}{b^2} = 1\] and the foci of the hyperbola
\[\frac{x^2}{64} - \frac{y^2}{36} = \frac{1}{16}\] coincide. Find $b^2$.

Solution: \boxed{b^2 = \frac{75}{4}}


Suppose the point $(2,3)$ is on the graph of $y = 2f(x)$. Determine the point that must be on the graph of $y = \frac{f^{-1}(x)}{3}$ and find the sum of that point's coordinates.


Simplify $8 \cdot \frac{15}{4} \cdot \frac{-40}{45}$.


What is the coefficient of \( x^5 \) when $$x^5 - 4x^4 + 6x^3 - 5x^2 + 2x + 1$$ is multiplied by $$3x^4 - 2x^3 + x^2 + 4x - 8$$ and the like terms are combined?


Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? Express your answer as a common fraction.


What is the sum of the positive even divisors of 180?


At the beginning of my bike ride I feel good, so I can travel 15 miles per hour. Later, I get tired and travel only 10 miles per hour. If I travel a total of 100 miles in a total time of 8 hours, for how many hours did I feel good? Express your answer as a common fraction.


A 12-foot by 15-foot floor is tiled with square tiles of size 2 feet by 2 feet. Each tile has a pattern consisting of four white quarter circles of radius 1 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

Solution: \boxed{180 - 45\pi}


The positive integers \( A, B \) and \( C \) form an arithmetic sequence, while the integers \( B, C \) and \( D \) form a geometric sequence. If \( \frac{C}{B} = \frac{7}{4}, \) what is the smallest possible value of \( A + B + C + D \)?


How many positive integers smaller than $10{,}000$ are powers of $2$, but are not powers of $4$?


Determine the number of pairs $(x, y)$ such that $12, x, y, xy$ form an arithmetic progression.


Let points $A = (4, \phi_1)$ and $B = (10, \phi_2)$ in polar coordinates, where $\phi_1 - \phi_2 = \frac{\pi}{4}$. Determine the distance $AB$.

Solution: \boxed{\sqrt{116 - 40\sqrt{2}}}


Find the area of a triangle with side lengths 10, 11, and 11.


What is the sum of every third odd integer between $200$ and $500$?


Starting with the number 200, Declan repeatedly divides his number by 3 and then takes the greatest integer less than or equal to that result. How many times must he do this before he reaches a number less than 2?


For the quadrilateral given, how many different whole numbers could be the length of the diagonal represented by the dashed line?

[asy]
draw((0,0)--(5,5)--(12,1)--(7,-8)--cycle,linewidth(0.7));
draw((0,0)--(12,1),dashed);
label("9",(2.5,2.5),NW);
label("11",(8.5,3),NE);
label("17",(9.5, -3.5),SE);
label("13",(3.5,-4),SW);
[/asy]


How many positive integers less than $1000$ are either multiples of $5$ or multiples of $10$?


April has five different basil plants and four different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other?


What is the area enclosed by the graph of $|x| + |3y| = 12$?


How many integers $n$ satisfy the double inequality $-12\sqrt{\pi} \le n^2 \le 15\pi$?


What is the sum of the $x$-values that satisfy the equation $7=\frac{x^3 - 3x^2 - 10x}{x+2}$?


Given that $\sec x + \tan x = \frac{5}{4},$ find all possible values of $\sin x.$


Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is an ace?


Solution: \boxed{\log_{13} 339}


For some real numbers $c$ and $d$, the equation $9x^3 + 5cx^2 + 6dx + c = 0$ has three distinct positive roots. If the sum of the base-2 logarithms of the roots is 6, what is the value of $c$?


In isosceles triangle $ABC$, the base $BC$ is $10$ units long and the height from $A$ perpendicular to $BC$ is $6$ units. What is the perimeter of triangle $ABC$?

Solution: \boxed{2\sqrt{61} + 10}


Using the same rules for assigning values to letters as described (with a repeating pattern of $1, 2, 1, 0, -1, -2, -1, 0$):
Calculate the sum of the numeric values of the letters in the word "algebra".


Determine the smallest possible perimeter of a scalene triangle where:
1.  All sides are distinct prime numbers greater than 3.
2.  The perimeter is a prime number.


When a polynomial is divided by $3x^2 - 5x + 12,$ what are the possible degrees of the remainder?

Solution: \boxed{0, 1}


Four dwarf planets have been added to the solar system tally, each with distinct moon counts. Now, what is the median number of moons per celestial body in the expanded list? The counts are as follows:
\begin{tabular}{c|c}
Celestial Body & $\#$ of Moons \\
\hline
Mercury & 0 \\
Venus & 0 \\
Earth & 1 \\
Mars & 2 \\
Jupiter & 20 \\
Saturn & 22 \\
Uranus & 14 \\
Neptune & 2 \\
Pluto & 5 \\
Ceres & 0 \\
Eris & 1 \\
Haumea & 2 \\
Makemake & 3 \\
\end{tabular}


In a classroom arranged in a row of 10 chairs, Alex and Jamie choose their seats at random. What is the probability that they do not sit next to each other?


A pyramid has a rectangular base measuring $10 \times 12$. Each of the four edges joining the apex to the corners of the rectangular base has length $15$. Calculate the volume of the pyramid.


Let $\mathbf{B} = \begin{pmatrix} x & 2 \\ -3 & y \end{pmatrix}$ for some real numbers $x$ and $y.$ If
\[\mathbf{B} + 2\mathbf{B}^{-1} = \mathbf{0},\] then find $\det \mathbf{B}.$


If \( h(x) = 3x^2 + x - 4 \), what is the value of \( h(h(3)) \)?


Find the integer \( n \), \( -90 \le n \le 90 \), such that \( \sin n^\circ = \sin 721^\circ \).


How many distinct, positive factors does $1320$ have?


What is the sum of all integer solutions to the inequality \(1 < (x-3)^2 < 36\)?


What is the least integer greater than $\sqrt{500}$?


Given the following data, determine how much cheaper, in cents, is the cheapest store's price for Camera $Y$ compared to the most expensive one?
\begin{tabular}{|l|l|}
\hline
\textbf{Store} & \textbf{Sale Price for Camera $Y$} \\ \hline
Mega Deals & $\$12$~off the list price~$\$52.50$ \\ \hline
Budget Buys & $30\%$~off the list price~$\$52.50$ \\ \hline
Frugal Finds & $20\%$~off the list price~$\$52.50$ plus an additional $\$5$~off \\ \hline
\end{tabular}


What is the shortest distance between the circles defined by $x^2 - 8x + y^2 + 6y + 9 = 0$ and $x^2 + 10x + y^2 - 2y + 25 = 0$?

Solution: \boxed{\sqrt{97} - 5}


The least common multiple of $x$, $15$, and $21$ is $105$. What is the greatest possible value of $x$?


A bizarre weighted coin comes up heads with probability $\frac{1}{4}$, tails with probability $\frac{1}{2}$, and rests on its edge with probability $\frac{1}{4}$. If it comes up heads, you win 1 dollar. If it comes up tails, you win 3 dollars. But if it lands on its edge, you lose 8 dollars. What are your expected winnings from flipping this coin? Express your answer as a dollar value.


Find the value of $x$ if the fourth power of the square root of $x$ is 256.


What is the smallest positive integer $n$ such that the product $n(n+1)$ is divisible by some but not all integer values of $k$ within the range $1\leq k \leq n$?


A cylinder has a radius of 5 cm and a height of 12 cm. Calculate both the longest segment that would fit inside this cylinder and the volume of the cylinder.

Solution: \boxed{300\pi \text{ cm}^3}


The first $20$ numbers of an arrangement are shown below. What would be the value of the $50^{\mathrm{th}}$ number if the arrangement were continued?

$\bullet$ Row 1: $2,$ $2$

$\bullet$ Row 2: $4,$ $4,$ $4,$ $4$

$\bullet$ Row 3: $6,$ $6,$ $6,$ $6,$ $6,$ $6$

$\bullet$ Row 4: $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8$


Find the sum of the $x$-coordinates of the solutions to the system of equations $y = |x^2 - 4x + 3|$ and $y = 7 - 2x$.


A sports conference is organized into three divisions. Division A contains 6 teams, Division B contains 7 teams, and Division C contains 5 teams. Each team must play every other team in its own division twice and every team in the other divisions exactly twice. How many games are in a complete season for the conference?


How many positive integers less than $300$ are multiples of $5$, but not multiples of $15$?


Similarly, by restricting the domain of the function $f(x) = 3x^2 + 6x - 8$ to an interval, we can make it invertible. What is the largest such interval that includes the point $x=2$?

Solution: \boxed{[-1, \infty)}


In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 48. What is the perimeter of the shaded region formed similarly as before with the triangular region where each circle touches the other two?


The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 4 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} -\frac{3}{2} \\ -2 \end{pmatrix}\] lie on a line. Find the equation of this line in the form "$y = mx + b$".

Solution: \boxed{y = -\frac{3}{4}x - \frac{25}{8}}


What is the area, in square units, of a triangle with vertices at $A(1, 3), B(6, 3), C(4, 9)$?

Solution: \boxed{15 \text{ square units}}


If $5x + 3 \equiv 1 \pmod{18}$, then $3x + 8$ is congruent $\pmod{18}$ to what integer between $0$ and $17$, inclusive?


In a mixed-doubles tennis exhibition, there were four teams, each consisting of one man and one woman. After the exhibition, each player shook hands exactly once with every other player except with their team partner. How many handshakes occurred?


Find the sum of the coefficients in the polynomial $-3(x^8 - 2x^5 + 4x^3 - 6) + 5(x^4 + 3x^2) - 4(x^6 - 5)$.


Suppose $179\cdot 933 / 7 \equiv n \pmod{50}$, where $0 \le n < 50$.


$ABCD$ is a trapezoid where $\overline{AB}$ is three times the length of $\overline{CD}$, and the height from $D$ to line $AB$ is 5 units. Point $E$ is the intersection of the diagonals. If the length of diagonal $\overline{AC}$ is 15, find the length of segment $\overline{EC}$. Express your answer as a common fraction.


**Sarah has misplaced her friend Layla's phone number. Sarah recalls that the first four digits are either 3086, 3089, or 3098. The remaining digits are 0, 1, 2, and 5, but she is not sure about their order. If Sarah randomly calls a seven-digit number based on these criteria, what is the probability that she correctly dials Layla's phone number? Express the answer as a common fraction.**


Let $f(x) = 5x^2 - 4$ and $g(f(x)) = x^2 + x + x/3 + 1$. Find the sum of all possible values of $g(49)$.


How many ways can 2023 be factored as a product of two two-digit numbers? (Two factorizations of the form $a\cdot b$ and $b\cdot a$ are considered the same).


In a diagram, there are two concentric circles where the visible gray area between the larger and smaller circles is equal to four times the area of the smaller white circular region enclosed by the smaller circle. Determine the ratio of the radius of the small circle to the radius of the large circle. Express your answer as a common fraction.


What is the remainder when \( 4x^8 - 3x^7 + 2x^6 - 8x^4 + 5x^3 - 9 \) is divided by \( 3x - 6 \)?


Rationalize the denominator for the expression $\sqrt[3]{\frac{4}{9}}.$


Simplify the product \[\frac{10}{5}\cdot\frac{15}{10}\cdot\frac{20}{15} \dotsm \frac{5n+5}{5n} \dotsm \frac{1030}{1025}.\]


Let $a$ and $b$ be integers such that $ab = 144$. Find the minimum value of $a + b$.


I randomly pick an integer $p$ between $1$ and $15$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 5p - 3q = 3$? Express your answer as a common fraction.


Each letter represents a non-zero digit. What is the value of $s?$ \begin{align*}
a + b &= x \\
x + c &= s \\
s + a &= z \\
b + c + z &= 16
\end{align*}


Find the distance between the centers of the inscribed and the circumscribed circles of a right triangle with sides 6, 8, and 10 units.


A number $x$ is equal to $6 \cdot 18 \cdot 42$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?


Warren wishes to solve the equation \(25x^2+30x-45=0\) by completing the square. He reformulates the equation as \((ax+b)^2=c\), where \(a\), \(b\), and \(c\) are integers and \(a>0\). Determine the value of \(a + b + c\).


A circle centered at $O$ is circumscribed about $\triangle ABC$ as shown in the diagram. Assume angle measurements $\angle AOC = 140^\circ$ and $\angle BOC = 90^\circ$. What is the measure of $\angle BAC$, in degrees?


Find the remainder when \( x^5 + 2x^3 + x + 3 \) is divided by \( x-4 \).


Add $91.234$ to $42.7689$ and round your answer to the nearest hundredth.


A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? Additionally, calculate the total surface area of this cylinder.


Let $B,$ $S,$ $N,$ and $K$ be positive real numbers such that
\begin{align*}
\log_{10} (BK) + \log_{10} (BN) &= 3, \\
\log_{10} (NK) + \log_{10} (NS) &= 4, \\
\log_{10} (SB) + \log_{10} (SK) &= 5.
\end{align*}
Compute the value of the product $BSNK.$


In a row of 10 chairs, two chairs (numbered 5 and 6) are broken and cannot be used. Mary and James each choose their remaining seats at random. What is the probability that they do not sit next to each other?


The line $y = c-x$ where $0 < c < 6$ intersects the $y$-axis at $P$ and the line $x=6$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 4:16, what is the value of $c$?


Find $n$ such that $2^6 \cdot 3^3 \cdot n = 9!$.


Let $Q$ be a cubic polynomial such that $Q(0) = m$, $Q(1) = 3m$, and $Q(-1) = 4m$. Find the value of $Q(2) + Q(-2)$.


Consider license plates consisting of a sequence of four digits followed by two letters. Assume each arrangement is equally likely for these plates. What is the probability that such a license plate contains at least one palindrome sequence (either the four-digit sequence or the two-letter sequence)? Express your result as a simplified fraction.


Chris and Dana each arrive at a cafe at a random time between 3:00 PM and 4:00 PM. Each stays for 20 minutes. What is the probability that Chris and Dana meet at the cafe?


Two spinners are divided into fifths and sixths, respectively. Spinner A (in fifths) has numbers 2, 4, 5, 7, and 9, while Spinner B (in sixths) displays 3, 6, 7, 8, 10, and 12. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number?


A geologist has discovered a peculiar mineral that has an age represented by the digits 1, 1, 2, 3, 7, and 9. He claims that the age of the mineral begins with a prime number. How many different ages can be formed under these conditions?


Evaluate $\log_3 81\sqrt{9}$. Express your answer as an improper fraction.


Line segment $\overline{AB}$ is extended past $B$ to point $Q$ such that $AQ:QB = 5:2.$ Then
\[\overrightarrow{Q} = s \overrightarrow{A} + v \overrightarrow{B}\] for some constants $s$ and $v.$ Enter the ordered pair $(s,v).$

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


Rationalize the denominator of $\frac{7}{3+\sqrt{15}}$ and then subtract $\frac{1}{2}$ from the result.

Solution: \boxed{-4 + \frac{7\sqrt{15}}{6}}


Suppose we have two numbers, $29_{10}$ and $45_{10}$. If $29_{10}$ is first converted to base 4 and $45_{10}$ to base 5, what is the sum of these two numbers in base 5?


For how many integer values of \(b\) does the equation \(x^2 + bx + 9b = 0\) have integer solutions for \(x\)?


For how many positive integers $n$ less than or equal to 500 is $$(\sin (t+\frac{\pi}{4})+i\cos (t+\frac{\pi}{4}))^n=\sin (nt+\frac{n\pi}{4})+i\cos (nt+\frac{n\pi}{4})$$ true for all real $t$?


Consider rows 1003, 1004, and 1005 of Pascal's triangle.

Let \((a_i)\), \((b_i)\), \((c_i)\) be the sequences, from left to right, of elements in the 1003rd, 1004th, and 1005th rows, respectively, with the leftmost element occurring at \(i = 0\). Compute
\[
\sum_{i = 0}^{1004} \frac{b_i}{c_i} - \sum_{i = 0}^{1003} \frac{a_i}{b_i}.
\]


Let \(x\) and \(y\) be positive real numbers such that \(2x + 3y = 4.\) Find the minimum value of
\[
\frac{2}{x} + \frac{3}{y}.
\]


Henry's little brother now has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to put all of the stickers on the sheets of paper, if only the number of stickers on each sheet matters and each sheet must have at least one sticker?


Find the second smallest positive integer $x$ which is greater than $1$ and relatively prime to $210$.


The interior of a right, circular cone is 12 inches tall with a 3-inch radius at the opening. The interior of the cone is filled with ice cream, and the cone has a right circular cylinder of ice cream, 2 inches tall with the same radius as the opening, exactly covering the opening of the cone. What is the volume of ice cream? Express your answer in terms of $\pi$.

Solution: \boxed{54\pi \text{ cubic inches}}


Find the number of lattice points that satisfy both $x^2 - y^2 = 75$ and $x - y = 5$ on the $xy$-plane.


Find the minimum value of
\[ x^2 + 4xy + 5y^2 - 8x - 6y, \]
over all real numbers $x$ and $y$.


What is the remainder when 3,452,179 is divided by 7, after adding 50 to it?


What is the sum of the tens digit and the ones digit of the integer form of $(3+4)^{15}$?


120 people were surveyed and asked the question: "Is teal kinda blue, or greenish?" Of them, 70 believe that teal is "kinda blue" and 35 believe it is both "kinda blue," and also "greenish." Another 20 think that teal is neither "kinda blue" nor "greenish."

How many of those 120 people believe that teal is "greenish"?


Jacob has run fifteen half-marathons in his life. Each half-marathon is $13$ miles and $193$ yards. One mile equals $1760$ yards. If the total distance Jacob covered in these half-marathons is $m$ miles and $y$ yards, where $0 \le y < 1760$, what is the value of $y$?


Twelve points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the twelve points as vertices?


In rhombus $ABCD$, angle $A$ measures $120^\circ$. What is the number of degrees in the measure of angle $C$?


How many positive three-digit integers less than 600 have at least two digits that are the same?


Suppose $a$, $b$, and $c$ are three positive numbers that satisfy the equations $abc = 1$, $a + \frac{1}{c} = 7$, and $b + \frac{1}{a} = 16$. Find $c + \frac{1}{b}$.


A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?

Solution: \boxed{10\sqrt{2} \text{ cm}}


In rectangle $ABCD$, $AB = 10$ cm, $BC = 12$ cm, and $DE = DF$. The area of triangle $DEF$ is one-third the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.


What is the smallest positive integer \( n \) such that all the roots of \( z^6 - z^3 + 1 = 0 \) are \( n^{\text{th}} \) roots of unity?


What is the base ten equivalent of $23456_{8}$?


The sides of a triangle have lengths $5, 12,$ and $k,$ where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse?


Let $x = (3 + \sqrt{8})^{1000}$, let $n = \lfloor x \rfloor$, and let $f = x - n$. Find
\[x(1 - f).\]


Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors (i.e., the smallest prime), and $n$ is the second largest integer less than 150 with exactly three positive divisors. Calculate $m+n$.


Our school's girls volleyball team has 16 players, including a set of triplets: Alicia, Amanda, and Anna, and a set of twins: Beth and Brenda. In how many ways can we choose 6 starters if at most one of the triplets and at most one of the twins can be in the starting lineup?


Find the sum of the values of $x$ which satisfy $x^2 + 2010x = 2011 + 18x$.


The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 110^{\circ}$ and $\angle AOB = 150^{\circ}$. Determine the degree measure of $\angle ABC$.


A point is randomly selected from the portion of the number line from 0 to 10. What is the probability that the point is closer to 7 than to 0? Express your answer as a decimal to the nearest tenth.


Let \( g(x) = x^2 - 2x + 2022 \). What is the greatest common divisor of \( g(50) \) and \( g(52) \)?


If the value of $x$ is doubled and then this increased value is divided by 4, the result is 12. What is the value of $x$?


There are seven unmarked envelopes on a table, each with a letter for a different person. If the mail is randomly distributed to these seven people, with each person getting one letter, what is the probability that exactly two people get the correct letter?


What is the smallest prime divisor of $3^{25} + 11^{19}$?


A bag contains 4 red, 7 green, 9 yellow, and 10 blue jelly beans. A jelly bean is selected at random. What is the probability that it is not yellow?


Two distinct numbers are selected simultaneously and at random from the set $\{2, 3, 4, 5, 6\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction.


Sue now owns 16 pairs of shoes: seven identical black pairs, four identical brown pairs, three identical gray pairs, and two identical red pairs. If she picks two shoes at random, what is the probability that they are a matching color and one is a left shoe and the other is a right shoe? Express your answer as a common fraction.


Simplify $\displaystyle \frac{2-2i}{3+4i}$, where $i^2 = -1$.

Solution: \boxed{-\frac{2}{25} - \frac{14}{25}i}


The pages of a book are numbered from 1 to n. Due to an error, one of the page numbers was added twice, resulting in a total sum of 2076. Determine which page number was incorrectly added twice.


A triangle is formed from wood sticks of lengths 10, 24, and 26 inches joined end-to-end. Pieces, with length $x$ from each of the sticks, are to be cut so that the remaining pieces can no longer form a triangle. Determine the smallest value of $x$ that meets this condition.

Solution: \boxed{8\text{ inches}}


Let \( A \) and \( B \) be two points on the parabola \( y = x^2 + 1 \), such that the tangents at \( A \) and \( B \) are perpendicular. Find the \( y \)-coordinate of their point of intersection \( P \).


Find the smallest possible perimeter of a scalene triangle where each side length is a different prime number, and the triangle's perimeter is also a prime number, given that the smallest side length is at least 5.


Let $p$ and $q$ be the solutions of the equation $3x^2 - 9x - 15 = 0$. Calculate the value of $(3p-5)(6q-10)$.


What is the greatest four-digit number that is one more than a multiple of 7 and five more than a multiple of 8?


Given $\begin{vmatrix} x & y \\ z & w \end{vmatrix} = 5$, find the value of:
\[\begin{vmatrix} x & 2x + 4y \\ z & 2z + 4w \end{vmatrix}\]


If the product $(4x^2 - 6x + 5)(8 - 3x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are real numbers, then find $8a + 4b + 2c + d$.


I randomly pick an integer $p$ between $1$ and $15$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 6p - 3q = 3$? Express your answer as a common fraction.


A rectangular tank holds 216 cubic feet of water. Determine the volume of this tank in cubic meters, knowing that 1 cubic meter is approximately 35.315 cubic feet.


A circle with center $A$ and radius four feet is tangent at $C$ to a circle with center $B$, as shown. If point $B$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\pi$.


For how many values of the digit $A$ is it true that $75$ is divisible by $A$ and $536{,}1A4$ is divisible by $4$?


A three-inch cube ($3\times3\times3$) of titanium weighs 5 pounds and is worth $\$300$. How much is a five-inch cube of titanium worth? Assume that, due to material characteristics, the price weight ratio increases by 20% with every inch increase in dimension.


The values of \(x\) and \(y\) are always positive, and \(x^3\) and \(y\) vary inversely. If \(y = 5\) when \(x = 2\), find \(x\) when \(y = 2000\).

Solution: \boxed{x = \frac{1}{\sqrt[3]{50}}}


Find the smallest composite number that has no prime factors less than 20.


A regular decagon \(Q_1 Q_2 \dotsb Q_{10}\) is drawn in the coordinate plane with \(Q_1\) at \((2,0)\) and \(Q_6\) at \((-2,0)\). If \(Q_n\) is the point \((x_n,y_n)\), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{10} + y_{10} i).
\]


Given the following stem and leaf plot, calculate the positive difference between the median and the mode. In this plot, $6|7$ represents $67$.

\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
2 & $1 \hspace{2mm} 3 \hspace{2mm} 3 \hspace{2mm} 5 \hspace{2mm} 5$ \\ \hline
3 & $2 \hspace{2mm} 2 \hspace{2mm} 2 \hspace{2mm} \hspace{2mm} \hspace{2mm} \hspace{1.5mm}$ \\ \hline
4 & $0 \hspace{2mm} 0 \hspace{2mm} 7 \hspace{2mm} 8 \hspace{2mm} \hspace{1.9mm}$ \\ \hline
5 & $1 \hspace{2mm} 2 \hspace{2mm} 3 \hspace{2mm} 4 \hspace{2mm} \hspace{1.9mm}$ \\ \hline
6 & $3 \hspace{2mm} 7 \hspace{2mm} 8 \hspace{2mm} \hspace{2mm} \hspace{2mm} \hspace{1.5mm}$ \\\hline
\end{tabular}


The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane is $15$, and the sum of the $y$-coordinates of the vertices is also $15$. Find the sum of the $x$-coordinates and the sum of the $y$-coordinates of the midpoints of the sides of the triangle.


Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that the penny and dime both come up the same, and the nickel and quarter also both come up the same?


A line with slope $-3$ intersects the positive $x$-axis at a point $A$ and the positive $y$-axis at $B$. Another line passes through the points $C(6,0)$ on the $x$-axis and $D$ on the $y$-axis. The lines intersect at $E(3,3)$. What is the area of the shaded quadrilateral formed by $OBEC$?


Let $D$ be the circle with equation $x^2 + 4y - 16 = -y^2 + 12x + 16$. Find the values of $(c,d)$, the center of $D$, and $s$, the radius of $D$, and calculate $c+d+s$.

Solution: \boxed{4 + 6\sqrt{2}}


I received \$40 in pocket money and spent it as indicated in the pie graph below. How many dollars did I spend on video games?

[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(.707,.707);
B=(-.707,.707);
C=(-.707,-.707);
D=(.707,-.707);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(0,.6);
label("Books", W, N);
label("$\frac{2}{5}$", W, S);
X=(-.6, 0);
label("Video Games", X, S);
Y=(0,-.6);
label("Snacks", Y, N);
label("$\frac{1}{4}$", Y, S);
Z=(.6, 0);
label("Toys", Z, N);
label("$\frac{1}{5}$", Z, S);
[/asy]


A line is described by the equation $y - 3 = 6(x - 5)$. What is the sum of its $x$-intercept and $y$-intercept?


Find the total number of odd integers \( n \) that satisfy
\[ 25 < n^2 < 144. \]


Convert the point $(2, -2\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2\pi.$

Solution: \boxed{\left( 4, \frac{5\pi}{3} \right)}


Given that $b$ is a multiple of $2700$, find the greatest common divisor of $b^2 + 27b + 75$ and $b + 25$.


If $m$, $n$, and $p$ are positive integers such that $\gcd(m,n) = 180$ and $\gcd(m,p) = 240$, then what is the smallest possible value of $\gcd(n,p)$?


If $|x| + x + y = 12$ and $x + |y| - y = 10$, find $x + y + z$ given that $x - y + z = 5$.


How many distinct sequences of five letters can be made from the letters in PROBLEM if each letter can be used only once, each sequence must begin with L, and does not end with P?


Calculate $\frac{4 \cdot 6! + 24\cdot 5!}{7!}$


If $q(x) = x^4 - 4x + 5$, then find the coefficient of the $x^3$ term in the polynomial $(q(x))^3$.


Each day, Jenny eats $25\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of the third day, 27 remained. How many jellybeans were in the jar originally?


The function $g(x)$ satisfies $g(1) = 2$ and
\[g(x + y) = 5^y g(x) + 3^x g(y)\]
for all real numbers $x$ and $y.$ Find the function $g(x).$

Solution: \boxed{5^x - 3^x}


What is the arithmetic mean of the integers from -6 through 7, inclusive? Express your answer as a decimal.


Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations:
\[
  17x +  by +  cz = 0, \\
   ax + 31y +  cz = 0, \\
   ax +  by + 53z = 0.
\]
Assuming that $ a \ne 17$ and $ x \ne 0$, what is the value of
\[
 \frac{a}{a - 17} + \frac{b}{b - 31} + \frac{c}{c - 53} \,?
\]


Consider the function $f(x) = x^2 + 4\sqrt{x}$. Evaluate $2f(3) - f(9)$.

Solution: \boxed{-75 + 8\sqrt{3}}


Suppose that \(x,\) \(y,\) and \(z\) are three positive numbers such that \(xyz = 1,\) \(x + \frac{1}{z} = 8,\) and \(y + \frac{1}{x} = 20.\) Find \(z + \frac{1}{y}.\)


What is the smallest four-digit palindrome that is divisible by 3 and has an odd first digit?


The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $60^{\text{th}}$ integer in the list?


A rectangle with even integer length and width has a perimeter of 120 units. What is the number of square units in the least possible area?


Suppose \(x^2\) varies inversely with \(y^3\). If \(x = 10\) when \(y = 2\), find the value of \(x^2\) when \(y = 4\).


Consider the sequence where $x_1+1=x_2+2=x_3+3=\cdots=x_{100}+100=x_1+x_2+x_3+\cdots+x_{100}+101$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{100}x_n$.


Evaluate the ratio $\frac{10^{2010}+10^{2013}}{10^{2011}+10^{2014}}$ and determine which whole number it is closest to.


What is the distance between points (-3, 7) and (4, -9), and what is the midpoint of the segment connecting these points?

Solution: \boxed{\left(\frac{1}{2}, -1\right)}


How many divisors of $9!$ are multiples of $10$?


Find the smallest solution to \[\lfloor x \rfloor = 3 + 50 \{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$


The graph of the rational function $\frac{p(x)}{q(x)}$ has a horizontal asymptote of $y = 0$ and a vertical asymptote at $x = -2$. If $q(x)$ is quadratic, $p(2)=4$, and $q(2) = 8$, and there is a hole at $x = 3$, find $p(x) + q(x)$.

Solution: \boxed{-2x^2 - 2x + 24}


Define a new operation $ \diamond $ for non-zero integers where $a \diamond b = \frac{1}{a} + \frac{1}{b}$. Given that $a * b = 10$ and $ab = 24$, what is $a \diamond b$? Express your answer as a common fraction.


For what value of $k$ does $6 \times 9 \times 2 \times k = 8!$?


What is the greatest integer less than 200 for which the greatest common factor of that integer and 24 is 4?


Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{2} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]Let $\phi$ be the angle between $\mathbf{v}$ and $\mathbf{w}.$  Find $\sin \phi.$


In a regular nonagon (a 9-sided polygon), two diagonals are chosen randomly. What is the probability that these diagonals intersect inside the nonagon?


Find the area of the parallelogram generated by the vectors $3\mathbf{a} + 4\mathbf{b}$ and $2\mathbf{a} - 6\mathbf{b}$ if the area of the parallelogram generated by $\mathbf{a}$ and $\mathbf{b}$ is 15 units\(^2\).


Let $x, y,$ and $z$ be three positive real numbers such that $x + y + z = 60.$ Find the ordered triple $(x, y, z)$ for which $x^3 y^2 z^4$ is maximized.

Solution: \boxed{(20, \frac{40}{3}, \frac{80}{3})}


Find the function $g(x)$ if $g(2) = 2$ and for all real numbers $x$ and $y$,
\[ g(x + y) = 5^y g(x) + 3^x g(y) \]

Solution: \boxed{g(x) = \frac{5^x - 3^x}{8}}


Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k$ such that $Q$ is divisible by $3^k.$


Let the line \( p \) be the perpendicular bisector of points \( A = (20, 12) \) and \( B = (-4, 3) \). Determine the point \( C = (x, y) \) where line \( p \) meets segment \( AB \), and calculate \( 3x - 5y \).


What is the greatest positive integer that must divide the sum of the first twelve terms of any arithmetic sequence whose terms are positive integers?


A sequence consists of $2000$ terms. Each term after the first is $1$ larger than the previous term. The sum of the $2000$ terms is $5010$. When every second term is added up, starting with the second term and ending with the last term, what is the sum?


Let \( A \), \( B \), and \( C \) be nonnegative integers such that \( A + B + C = 15 \). What is the maximum value of \[A\cdot B\cdot C + A\cdot B + B\cdot C + C\cdot A?\]


Consider a sequence $(v_n)$ where each term follows the relation:
\[ v_{n+2} = 3v_{n+1} - 2v_n \]
Given that $v_3 = 5$ and $v_6 = -76$, find the value of $v_5$.


The diagonal lengths of a rhombus are 18 units and 26 units. Calculate both the area and the perimeter of the rhombus.


A triangle has three different integer side lengths and a perimeter of 30 units. What is the maximum length of any one side?


When the decimal point of a certain positive decimal number is moved two places to the right, the new number is nine times the reciprocal of the original number. What is the original number?


A certain club consists of seven leaders and some number of regular members. Each year, the current leaders leave the club. Every regular member then recruits three new people to join as regular members. Following this, seven new leaders are elected from outside the club to serve for the next year. Initially, there are twenty-one people in total in the club. How many people in total will be in the club five years from now?


The Jackson High School Football Club has 24 players, including 4 goalkeepers. The club organizes a competition to discover which goalkeeper can save the most goals from penalty kicks. For each penalty kick, one goalkeeper will defend the goal while the remaining players (including the other goalkeepers) attempt to score.

How many penalty kicks are required so that every player has the opportunity to shoot against each goalkeeper?


A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row, and each column contains the same number of chairs as every other column. With the condition that there must be at least two chairs in every row and column, and all the chairs in the room must be included, how many arrays are possible in a classroom containing $48$ chairs?

Solution: \boxed{4 \times 2 = 8}


What is the positive difference between the $1002^{\mathrm{th}}$ term and the $1008^{\mathrm{th}}$ term of the arithmetic sequence $-11,$ $-4,$ $3,$ $10,$ $\ldots$?


If $5 \cot \theta = 4 \sin \theta$ and $0 < \theta < \pi$, find the value of $\cos \theta$.

Solution: \boxed{\cos \theta = \frac{-5 + \sqrt{89}}{8}}


Triangle $ABC$ has vertices $A(-2, 10)$, $B(3, 0)$, $C(10, 0)$. A line through $B$ cuts the area of $\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line.


How many three-digit numbers remain if we exclude all three-digit numbers where exactly two digits are the same and these two digits are adjacent?


If \( f(x) = 3x^2 + x - 4 \), what is the value of \( f(f(3)) \)?


An 8-sided die is rolled. If \( n \) is the number rolled, I win \( n^3 \) dollars. What is the expected value of my winnings? Express your answer as a dollar value rounded to the nearest cent.


Let \( a \), \( b \), and \( c \) be the roots of the polynomial equation \( x^3 - 2x^2 + x - 1 = 0 \). Calculate \( \frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2} \).


If $2x - 3y = 18$ and $x + 2y = 8$, what is the value of $x$?


A circle with a radius of 5 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Find the length of the longer side of the rectangle, in centimeters. Express your answer in terms of $\pi$.


When the binary number $110110111010_2$ is divided by 8, what is the remainder (give your answer in base 10)?


A mini-domino set is made in a similar way to a standard domino set, but only uses integers from 0 to 6. Each integer within this range is paired with every other and itself exactly once to form a complete mini-set. A $\textit{double}$ in this set is a domino where the two numbers are identical. What is the probability that a domino randomly selected from this mini-set will be a $\textit{double}$? Express your answer as a common fraction.


The expression \(512y^3 - 27\) is a difference of cubes. Express \(512y^3 - 27\) in the form \((ay + b)(cy^2 + dy + e)\) and find \(a+b+c+d+e\).


Factor the following expression: $58a^2 + 174a$.


Starting with the number 250, Tim repeatedly divides his number by 3 and then takes the greatest integer less than or equal to that number. How many times must he perform this operation before the result is less than or equal to 2?


A rectangle has a perimeter of 40 inches and each side must have an integer length. Determine how many non-congruent rectangles can be formed under these conditions.


In right triangle $PQR$ with $\angle Q = 90^\circ$, we have $$5\sin R = 4\cos R.$$ What is $\sin R$?


In triangle $ABC,$ $AB = 4,$ $AC = 7,$ $BC = 9,$ and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$  Find $\sin \angle BAD.$


Find the positive solution to
\[\sqrt{x + \sqrt{x + \sqrt{x + \dotsb}}} = \sqrt{x \sqrt{x \sqrt{x \dotsm}}}.\]


Suppose $a$ and $b$ are positive integers such that $(a+bi)^3 = 2 + 11i$. What is $a+bi$?

Solution: \boxed{2 + i}


Find the sum of all solutions to the equation $2^{|x|^2} + 2|x|^2 = 34.$


Dr. Math's four-digit house number $WXYZ$ contains no zeroes and can be split into two different two-digit primes ``$WX$'' and ``$YZ$'' where the digits $W$, $X$, $Y$, and $Z$ are not necessarily distinct. If each of the two-digit primes is less than 60, how many such house numbers are possible?


What is the greatest divisor of 540 that is smaller than 100 and also a factor of 180?


When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers all of which are nonprime?


Let $S$ be the set of 12-tuples $(a_0, a_1, \dots, a_{11})$, where each entry is 0 or 1, so $S$ contains $2^{12}$ 12-tuples. For each 12-tuple $s = (a_0, a_1, \dots, a_{11})$ in $S$, let $p_s(x)$ be the polynomial of degree at most 11 such that
\[p_s(n) = a_n\] for $0 \le n \le 11.$ Find
\[\sum_{s \in S} p_s(12).\]


Find the sum of all integral values of \( c \) with \( c \leq 30 \) for which the equation \( y = x^2 - 9x - c \) has two rational roots.


Let $a$ and $b$ be integers such that $ab = -72.$ Find the maximum value of $a + b.$


What is the smallest positive number that is a multiple of $45$ and also exceeds $100$ by a multiple of $7$?


A clock has a second hand that is 10 cm long. Calculate the distance in centimeters that the tip of the second hand travels in 15 minutes. Express your answer in terms of $\pi$.


A gardener plants four maple trees, five oak trees, and three birch trees in a row. He plants them randomly, each arrangement being equally likely. Calculate the probability, in lowest terms, that no two birch trees are adjacent. Let this probability be $\frac{m}{n}$ and find $m+n$.


What is the value of $a + b$ if the sequence $3, ~9, ~15, \ldots, ~a, ~b, ~33$ is an arithmetic sequence?


Twenty teams play a tournament where each team plays against every other team exactly once. No ties occur, and each team has a \(50\%\) chance of winning any game it plays. Calculate the probability that no two teams win the same number of games, expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Determine \(\log_2 n\).


When two fair 8-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two-digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 4?


Determine the values of $p$ and $q$ if $18^3 = \frac{27^2}{3} \cdot 2^{9p} \cdot 3^{3q}$.


Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.


Solution: \boxed{-5 + 9i}


Triangles $PQR$ and $STU$ are similar. The lengths of $\overline{PQ}$ and $\overline{PR}$ are 7 cm and 9 cm, respectively, while $\overline{ST}$ is 4.2 cm. Additionally, the area of triangle $PQR$ is 18 cm². Calculate the length of $\overline{SU}$. Express your answer in cm to the nearest tenth.

Solution: \boxed{5.4 \text{ cm}}


In triangle $XYZ$, $\angle X = 90^\circ$, $XZ = 4$, and $YZ = \sqrt{17}$. What is $\tan Y$?


How many positive integers smaller than $1{,}000{,}000{,}000$ are powers of $2$, but are not powers of $8$?


Larry now starts with the number 4 on his pinky finger. He applies function $f$ to 4 and writes the output on his ring finger. Then, instead of applying $f$ again, he uses a new function $g$ defined on the same set of points. If Larry alternates between applying $f$ and $g$, what number will Larry write on his tenth finger? The functions $f$ and $g$ at different points are defined as follows based on the previous graph data:
- $f(4) = 3$
- $g(3) = 1$
- $f(1) = 8$
- $g(8) = 7$
- $f(7) = 2$
- $g(2) = 1$
- $f(1) = 8$ (repetition starts here)


The surface area of a sphere is \(256\pi\text{ cm}^2\). What is the volume, in cubic centimeters, of the sphere? Express your answer in terms of \(\pi\).


The least common multiple of $x$, $15$, and $21$ is $105$. What is the greatest possible value of $x$?


What is the total number of digits used when the first 1500 positive even integers are written?


A new survey was conducted where 150 men and 750 women were questioned about their support for increased healthcare funding. It was found that 55% of the men and 85% of the women surveyed are in favor of it. What percentage of the total surveyed population supports the proposed funding increase?


Determine the radius of the circle inscribed in triangle $XYZ$ where $XY = 8, XZ = 13, YZ = 15$.


As $x$ ranges over all real numbers, find the range of the function
\[ f(x) = \sin^4 x + \cos^4 x. \]
Enter your answer using interval notation.

Solution: \boxed{\left[ \frac{1}{2}, 1 \right]}


Let $a$ and $b$ be the roots of $x^2 - 5x + 6 = 0.$ Compute
\[
a^3 + a^4b^2 + a^2b^4 + b^3 + ab(a+b).
\]


The weather forecast predicts a 75 percent chance of rain each day for the upcoming five-day holiday weekend. If it doesn't rain, the weather will be sunny. Sam and Alex want exactly two of those days to be sunny for their outdoor activities. What is the probability they get the weather they want?


Find the real root of the equation \[\sqrt{x+4} + \sqrt{x+6} = 12.\]


Triangle $PQR$ has side lengths $PQ=7$, $QR=8$, and $RP=9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$?

Solution: \boxed{QS = 5}


What is the units digit of the product of all the odd positive integers between 20 and 50 that are not multiples of 5?


A standard deck of 52 cards consists of 13 ranks and 4 suits (Spades, Hearts, Diamonds, Clubs), each suit represented equally. If two cards are drawn simultaneously from the deck, what is the probability that both cards are Diamonds?


How many different 8-digit positive integers are there such that the last digit is not zero?


If $\lceil{\sqrt{x}}\rceil=17$, how many possible integer values of $x$ are there?


Suppose that a real number \( y \) satisfies \[\sqrt{64-y^2}-\sqrt{36-y^2}=4.\] What is the value of \(\sqrt{64-y^2}+\sqrt{36-y^2}\)?


Two concentric circles have the radius of the inner circle as \( r \) feet and the outer circle as \( 3r \) feet. If the width of the gray region between these circles is \( 4 \) feet, what is the area of the gray region, expressed in terms of \( \pi \) and \( r \)?

Solution: \boxed{8\pi r^2}


What is the greatest common divisor of 256, 162, and 450?


When $\frac{7}{8000}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit?


Simplify $$(2x^5 - 3x^4 + 5x^3 - 9x^2 + 8x - 15) + (5x^4 - 2x^3 + 3x^2 - 4x + 9).$$ After simplification, evaluate the polynomial at \(x = 2\).


For a nonnegative integer $n$, let $r_8(n)$ stand for the remainder left when $n$ is divided by $8$. Define a sequence where each term $n$ satisfies $$r_8(7n) \leq 3.$$ What is the $15^{\text{th}}$ entry in this list, starting with the first entry as $0$?


Evaluate the complex number \[\frac{\tan \frac{\pi}{6} + i}{\tan \frac{\pi}{6} - i}\] and determine if it is a twelfth root of unity. If so, find the specific root it represents (i.e., find $m$ such that the number equals $\cos \frac{2m \pi}{12} + i \sin \frac{2m \pi}{12}$).


When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$.


Let $(x,y)$ be an ordered pair of real numbers that satisfies the equation $x^2+y^2=18x+40y$. Find the minimum value of $y$ and the maximum value of $x$.

Solution: \boxed{9 + \sqrt{481}}


Find the remainder when $7 \times 17 \times 27 \times \ldots \times 87 \times 97$ is divided by $3$.


Circles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in a 1.5-foot length of this pattern?


Find the matrix $\mathbf{M}$ that triples every element of a given matrix $\mathbf{A}$.  In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 3a & 3b \\ 3c & 3d \end{pmatrix}.\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}}


Let \(x,\) \(y,\) and \(z\) be real numbers such that
\[ x^3 + y^3 + z^3 - 3xyz = 8. \]
Find the minimum value of \( x^2 + y^2 + z^2 \).


What integer $n$ satisfies $0 \leq n < 251$ and $$250n \equiv 123 \pmod{251}~?$$


The bottoms of two vertical poles are 18 feet apart and are placed on flat ground. The height of the first pole is 9 feet and the second pole is 24 feet tall. Find the length of the wire stretched from the top of the first pole to the top of the second pole.


What is the remainder when $3x^2 - 22x + 63$ is divided by $x - 4 + 1$?


In trapezoid \(ABCD\), the lengths of the bases \(AB\) and \(CD\) are 10 and 25 respectively. The legs of the trapezoid are extended beyond \(A\) and \(B\) to meet at point \(E\). What is the ratio of the area of triangle \(EAB\) to the area of trapezoid \(ABCD\)? Express your answer as a common fraction.


Let $X$, $Y$, and $Z$ be points on a circle of radius $12$. If $\angle XZY = 90^\circ$, what is the circumference of the minor arc $XY$? Express your answer in terms of $\pi$.


Let $a,$ $b,$ $c$ be real numbers such that $a + 3b + c = 5.$ Find the maximum value of
\[ab + ac + bc.\]


Determine the value of $r$ if $8 = 3^{4r-2}$.

Solution: \boxed{\frac{3\log_3(2) + 2}{4}}


Find the value of $x$ that satisfies $\frac{\sqrt{2x+7}}{\sqrt{8x+10}}=\frac{2}{\sqrt{5}}$. Express your answer as a common fraction.


Point $Q$ lies on the line $y = 4$ and is 15 units from the point $(-2,-3)$. Find the product of all possible $x$-coordinates that satisfy the given conditions.


In $\triangle ABC$, we have $AC=BC=10$ and $AB=4$. Suppose $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$, and $CD=12$. What is $BD$?

Solution: \boxed{4\sqrt{3} - 2}


If $4x + 5 \equiv 3 \pmod{17}$, what is $2x + 8$ congruent to $\pmod{17}$?


How many positive integers less than $1000$ are either a perfect cube or a perfect square?


What is the modulo $17$ residue of $255 + 7 \cdot 51 + 9 \cdot 187 + 5 \cdot 34$?


Given the graph of the rational function $\frac{1}{q(x)}$, where $q(x)$ is a quadratic polynomial, and it's known that $q(2) = 2$. Further, there are vertical asymptotes at $x = -1$ and $x = 1$. Additionally, $q(1) = 0$. Determine the quadratic polynomial $q(x)$.

Solution: \boxed{\frac{2}{3}x^2 - \frac{2}{3}}


A bookstore has a sale on days of the month that are multiples of 3 (such as June 3, June 6...). A shoe store has a sale every 7 days, starting from June 1. How many times in the months of June, July, and August do the two stores have sales on the same date?


In how many ways can 10 people sit around a round table if two specific individuals must sit next to each other? (Two seatings are considered the same if one is a rotation of the other.)


Four congruent equilateral triangles, each of a different color, are used to construct a regular tetrahedron. How many distinguishable ways are there to construct the tetrahedron? (Two colored tetrahedrons are distinguishable if neither can be rotated to look just like the other.)
[asy] import three; import math; unitsize(1.5cm); currentprojection=orthographic(2,0.2,1); triple A=(1,1,1); triple B=(-1,-1,1); triple C=(-1,1,-1); triple D=(1,-1,-1); draw(A--B--C--cycle); draw(A--C--D--cycle); draw(A--D--B--cycle); draw(B--C--D--cycle); [/asy]


A ball is dropped from a height of 20 meters above the ground. On each bounce, it rises to $\frac{2}{3}$ of the height it fell from previously. The ball is caught when it reaches the high point after hitting the ground for the fourth time. To the nearest meter, how far has it travelled?


In $\triangle ABC$, lines $CE$ and $AD$ are drawn such that $\dfrac{CD}{DB}=\dfrac{4}{1}$ and $\dfrac{AE}{EB}=\dfrac{4}{3}$. Let $r=\dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Compute $r$.


A "super ball" is dropped from a window 20 meters above the ground. On each bounce, it rises to $\frac{2}{3}$ of the height it fell from. The ball is caught when it reaches the high point after hitting the ground for the fourth time. To the nearest meter, how far has it travelled?


Let $x$, $y$, and $z$ be positive numbers such that:
\begin{align*}
x^2/y &= 3, \\
y^2/z &= 4, \\
z^2/x &= 5.
\end{align*}
Find $x$.


Determine the value of $m$ such that $100^m = 100^{-3} \times \sqrt{\frac{100^{55}}{0.0001}}$.


When the base-10 integers 500 and 1800 are expressed in base 2, how many more digits does 1800 have than 500 (after being converted)?


The deli has now increased its variety and offers five kinds of bread, seven kinds of meat, and six kinds of cheese. A sandwich still consists of one type of bread, one type of meat, and one type of cheese. Turkey, roast beef, Swiss cheese, and rye bread are each available at the deli. If Al never orders a sandwich with a turkey/Swiss cheese combination nor a sandwich with a rye bread/roast beef combination, how many different sandwiches could Al order?


Without using a calculator, find the largest prime factor of \(16^4 + 2 \times 16^2 + 1 - 15^4\).


Ryan has 4 red lava lamps and 2 blue lava lamps. He arranges all 6 lamps in a row on a shelf randomly, then randomly turns 3 of them on. What is the probability that the leftmost lamp is blue and off, the third from the left is red and off, and the rightmost lamp is red and on?


There are $30$ different complex numbers $z$ such that $z^{30}=1$. For how many of these is $z^5$ a real number?


Let \(a\), \(b\), and \(c\) be the roots of the polynomial \(x^3 - 5x + 7 = 0\). Find the monic polynomial, in \(x\), whose roots are \(a - 3\), \(b - 3\), and \(c - 3\).

Solution: \boxed{x^3 + 9x^2 + 22x + 19}


The science club now has 23 members: 13 boys and 10 girls. A 4-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?


For what values of $z$ is $z^2-40z+400 \leq 36$? Express your answer in interval notation.

Solution: \boxed{[14, 26]}


A sports league has 16 teams in two divisions of 8 each. How many games are in a complete season if each team plays every other team in its own division thrice and each team in the other division once?


Jason is trying to remember the five-digit combination to his safe. He knows that he only used digits 1 through 6 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible combinations does Jason need to try?


Solution: \boxed{65 \frac{4}{5}}


How many sequences of 8 digits $x_1, x_2, \ldots, x_8$ can we form such that no two adjacent digits have the same parity? Leading zeroes are allowed.


The force needed to push a car varies inversely with the number of people pushing it. If 4 people can push a car with an effort of 120 newtons each, how much effort would each person need to exert if 6 people are pushing the car?


The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\text{th}}$ integer in the list?


When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 120$.


Compute the exact value of the expression $\left|\pi - | \pi - \frac{11}{2} | \right|$. Write your answer using only integers, π, and fractions, without any absolute value signs.

Solution: \boxed{2\pi - \frac{11}{2}}


The operation $\odot$ is redefined as $a \odot b = a + \frac{5a}{2b}$. What is the value of $10 \odot 4$?


Tiffany is constructing a fence around a rectangular garden next to her tennis court. She must use exactly 400 feet of fencing. The fence must enclose all four sides of the garden. Regulations state that the length of the fence enclosure must be at least 100 feet and the width must be at least 50 feet. Tiffany wants the area enclosed by the fence to be as large as possible to include various plants and decorations. What is the optimal area, in square feet?


If $\mathbf{c}$ and $\mathbf{d}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, then compute the volume of the parallelepiped generated by $\mathbf{c},$ $\mathbf{d} + 2\mathbf{d} \times \mathbf{c},$ and $\mathbf{d}.$


How many distinct arrangements of the letters in the word "balloon" are there?


Find the greatest common factor of \(7!\) and \(8!\).


The lattice shown is continued for $15$ rows. What will be the fourth number in the $15$th row if the number of elements per row increases to $7$ and starts from 3? \begin{tabular}{rccccccc}
Row 1: & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
Row 2: & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
Row 3: & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\
Row 4: & 24 & 25 & 26 & 27 & 28 & 29 & 30
\end{tabular}


A ball is thrown downward from a height with its height (in feet) after $t$ seconds given by the expression $-20t^2 - 40t + 50$. Determine the time at which it hits the ground.

Solution: \boxed{-1 + \frac{\sqrt{14}}{2}}


Evaluate the sum
\[ \text{cis } 80^\circ + \text{cis } 90^\circ + \text{cis } 100^\circ + \dots + \text{cis } 130^\circ \]
and express it in the form \( r \, \text{cis } \theta \), where \( r > 0 \) and \( 0^\circ \le \theta < 360^\circ \). Find \( \theta \) in degrees.


In triangle $DEF$, $\cos(3D-E) + \sin(D+E) = 2$ and $DE = 6$. What is $EF$?

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


How many positive four-digit integers of the form $\_\_25$ are divisible by 25?


Suppose that $d, e,$ and $f$ are positive integers satisfying $(d+e+f)^3 - d^3 - e^3 - f^3 = 300$. Find $d+e+f$.


In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively, where $A(0,8)$, $B(0,0)$, and $C(10,0)$. Determine the area of $\triangle DBC$.


Determine $\cos B$ in the right triangle shown below where side AC is 7 units and side BC (hypotenuse) is 25 units.

[asy]
pair A,B,C;
A = (0,0);
B = (24,0);
C = (0,7);
draw(A--B--C--A);
draw(rightanglemark(B,A,C,10));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$25$",(B+C)/2,NE);
label("$7$",C/2,W);
[/asy]


Two circles are drawn in a 15-inch by 20-inch rectangle. Each circle has a diameter of 4 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?

Solution: \boxed{\sqrt{377}\text{ inches}}


Let $p(x) = x^2 + px + q$ and $r(x) = x^2 + sx + t$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $p$ is a root of $r,$ and the $x$-coordinate of the vertex of $r$ is a root of $p,$ and both $p$ and $r$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(50, -50),$ what is the value of $p + s$?


Calculate:
\[\cos \left( 6 \arccos \frac{1}{4} \right).\]


John arranges his playing cards in a suit in the order $$K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2, A, K, Q, J, \cdots.$$ What is the $45^{th}$ card?


Consider a rectangle with dimensions $x + 3$ and $3x - 4$ where the area of the rectangle is $5x + 14$. Find the value of x.


A 4x4x4 cube is formed by assembling 64 unit cubes. Four unit squares are painted on the center of each of the six faces of the cube. How many of the 64 unit cubes have no paint on them?


A right triangle has sides of 8, 15, and 17. Calculate the radius of the circumscribed circle and the distance between the circumcenter and the incenter of the triangle.


The first term of the given sequence is 2, and each subsequent term is the sum of the squares of all previous terms. What is the value of the first term which exceeds 10000?


A triangle has three different integer side lengths and a perimeter of 30 units. What is the maximum length of any one side?


Two calculations are given: $\left(6^2-3^2\right)^4$ and $\left(7^2-2^2\right)^4$. What is the sum of the results of these two calculations?


The second hand on a clock is 10 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\pi$.


Determine the function $g(x)$ if $g(1) = 2$ and
\[ g(x + y) = 5^y g(x) + 4^x g(y) \]
for all real numbers $x$ and $y$.

Solution: \boxed{2(5^x - 4^x)}


The first stage of a geometric shape made entirely of matchsticks is a square and each subsequent stage adds an additional square on one side, each requiring 5 new matchsticks to complete without breaking any matches. How many matchsticks are needed to complete the 100th stage?


Compute without using a calculator: $10! - 9! + 8! - 7!$.


Solve the inequality
\[\frac{x^2 - 49}{x + 7} < 0.\]

Solution: \boxed{(-\infty, -7) \cup (-7, 7)}


Solution: \boxed{\frac{6125 \sqrt{10}}{32768}}


Solution: \boxed{\frac{1}{3} \log_7 6}


How many degrees are there in the measure of angle $P$?

[asy]
size (6cm);
pair A,B,C,D,E,F;

A=(0,0);
B=(1,-1);
C=(3,-1);
D=(4,0);
E=(3,2);
F=(1,2);

draw (A--B--C--D--E--F--A, linewidth(1));

label("$P$", D, NE);
label("$135^\circ$", shift(0,-0.6) * A);
label("$120^\circ$", B, S);
label("$105^\circ$", C, S);
label("$150^\circ$", E, N);
label("$110^\circ$", F, N);
draw(anglemark(E,D,C), blue);
[/asy]


What is the product of the digits in the base 8 representation of $8654_{10}$?


The solution of the equation $6^{x+6} = 9^x$ can be expressed in the form $x = \log_b 6^6$. What is $b$?


In right triangle $ABC$ with $\angle C = 90^\circ$, we have $AB = 15$ and $AC = 7$. Find $\sin A$ and $\cos A$.

Solution: \boxed{\sin A = \frac{4\sqrt{11}}{15}, \quad \cos A = \frac{7}{15}}


The points $(2, 9), (14, 18)$ and $(6, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum?


Find $n$ such that $2^8 \cdot 3^4 \cdot 5^1 \cdot n = 10!$.


Calculate the dot product of the vectors $\begin{pmatrix} 4 \\ 5 \\ -6 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} -3 \\ 7 \\ 0 \\ -4 \end{pmatrix}$.


Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be three mutually orthogonal unit vectors, such that
\[\mathbf{u} = s (\mathbf{u} \times \mathbf{v}) + t (\mathbf{v} \times \mathbf{w}) + u (\mathbf{w} \times \mathbf{u})\]
for some scalars \(s\), \(t\), and \(u\), and \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = 1.\) Find \(s + t + u.\)


Isosceles triangle $ABF$ has an area of 180 square inches and is divided by line $\overline{CD}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 135 square inches. If the altitude of triangle $ABF$ from $A$ is 30 inches, determine the length of $\overline{CD}$.

Solution: \boxed{6 \text{ inches}}


Find the integer $m$, $-90 \le m \le 90$, such that $\sin m^\circ = \sin 710^\circ$.


Fourteen stones are arranged in a circle. They are counted clockwise from 1 to 14, and then continue to count counter-clockwise starting from stone 14 again as 15, onward until reaching stone 2 again as 27. This pattern continues indefinitely. Determine which original stone is counted as 99.


Express $\sqrt{a} \div \sqrt{b}$ as a common fraction, given:
$$\frac{{\left(\frac{1}{3}\right)}^2 + {\left(\frac{1}{4}\right)}^2}{{\left(\frac{1}{5}\right)}^2 + {\left(\frac{1}{6}\right)}^2} = \frac{25a}{61b}$$


What is the product of the positive odd divisors of $180$?


A triangle has three different integer side lengths and a perimeter of 24 units. What is the maximum length of any one side?


Jenna distributes her stamp collection evenly across pages and then stores these pages into three different stamp books. One book contains 924 stamps, another has 1386 stamps, and the third contains 1848 stamps. Determine the greatest number of stamps Jenna could be putting on each page.


Two dice each with 8 sides are rolled. What is the probability that the sum rolled is a perfect square?


Add $24_8 + 157_8.$ Express your answer in base 8.


Let $T = 2 - 3 + 4 - 5 + \cdots + 2010 - 2011$. What is the residue of $T$, modulo 2011?


If $x^{2y} = 4$ and $x = 2$, what is the value of $y$? Express your answer as a common fraction.


Let $x$, $y$, $z$, and $w$ be real numbers with $|x-y|=1$, $|y-z|=2$, and $|z-w|=3$. What is the sum of all possible values of $|x-w|$?


Cara is sitting at a circular table with her seven friends. Two of her friends, Alice and Bob, insist on sitting together but not next to Cara. How many different possible pairs of people could Cara be sitting between?


In triangle $ABC$, we have $\angle B = 60^\circ$ and $\cos A = \frac{3}{5}$. Find $\sin C$.

Solution: \boxed{\sin C = \frac{3\sqrt{3} + 4}{10}}


An infinite geometric series has a common ratio of $\frac{1}{4}$ and sum $10$. What is the second term of the sequence?


What is the remainder when $5x^7 - 3x^6 - 8x^5 + 3x^3 + 5x^2 - 20$ is divided by $3x - 9$?


Find the quadratic polynomial \( q(x) \) such that \( q(-2) = 7 \), \( q(1) = 4 \), and \( q(3) = 10 \).

Solution: \boxed{q(x) = \frac{4}{5}x^2 - \frac{1}{5}x + \frac{17}{5}}


What is the probability that the square root of a randomly selected two-digit whole number is less than nine? Express your answer as a common fraction.


Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \cap E| = |E \cap F| = |F \cap D| = 1$ and $D \cap E \cap F = \emptyset$. Let $M$ be the number of such ordered triples where each set is a subset of $\{1,2,3,4,5,6,7,8\}$. Find $M$ modulo $1000$.


The sum of 36 consecutive integers is $6^5$. What is their median?


A polynomial of degree $15$ is divided by $d(x)$ to give a quotient of degree $9$ and a remainder of $2x^4 + 3x^3 - 5x + 7$. What is $\deg d$?


Let $a,$ $b,$ and $c$ be the roots of
\[x^3 - 4x^2 + 6x - 3 = 0.\]Find the monic polynomial, in $x,$ whose roots are $a + 3,$ $b + 3,$ and $c + 3.$

Solution: \boxed{x^3 - 13x^2 + 57x - 84}


Let \(P\), \(Q\), and \(R\) be points on a circle of radius \(12\). If \(\angle PRQ = 90^\circ\), what is the circumference of the minor arc \(PQ\)? Express your answer in terms of \(\pi\).


Let $x = (1 + \sqrt{2})^{500},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$  Find
\[x(1 - f).\]


Find the largest value less than 1000 that is common to the arithmetic progressions $\{4, 9, 14, \ldots \}$ and $\{5, 13, 21, \ldots \}$.


The quadratic \( x^2 - 20x + 49 \) can be written in the form \( (x+b)^2 + c \), where \( b \) and \( c \) are constants. Find the value of \( b+c \).


Robert decides to visit the milk bottling plant daily over a week-long period. Each day, the plant has a 3/4 chance of bottling chocolate milk. What is the probability that the plant bottles chocolate milk exactly 5 out of the 7 days he visits?


Two rays with common endpoint $O$ form a $45^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=2$. What is the maximum possible length of $\overline{OB}$?


How many even four-digit integers have the property that their digits, read left to right, are in strictly increasing order?


Points $P$, $Q$, $R$, and $S$ lie on a line, in that order. If $PQ=4$ units, $QR=10$ units, and $PS=28$ units, calculate the ratio of $PR$ to $QS$. Express your answer as a common fraction.


Piravena needs to travel from city $X$ to city $Y$, then from $Y$ to city $Z$, and finally from $Z$ back to $X$. Each segment of her journey is completed entirely by bus or by airplane. The cities form a right-angled triangle with city $Z$ as the right angle. The distance from $X$ to $Z$ is 4000 km and from $X$ to $Y$ is 5000 km. Bus fare is $\$0.20$ per kilometer, and airplane travel costs a $\$120$ booking fee plus $\$0.12$ per kilometer. Piravena chooses the least expensive travel option for each segment. Determine the total cost of her trip.


Consider the lines:
\[
y = 4x + 6, \quad 2y = 6x + 8, \quad 3y = 12x - 3, \quad 2y = 3x - 4, \quad 4y = 2x - 7.
\]
Determine how many pairs of lines are either parallel or perpendicular.


What is the least positive integer value of $x$ such that $(2x)^2 + 2\cdot 43\cdot 2x + 43^2$ is a multiple of 53?


Given the number $A485B6$, where $A$ and $B$ are single digits, find the sum of all possible values of $A+B$ such that the number is divisible by 9.


A rhombus has an area of 192 square units. The lengths of its diagonals have a ratio of 4 to 3. Determine the length of the longest diagonal, in units.


Calculate the number of different rectangles with sides parallel to the grid that can be formed by connecting four of the dots in a $5\times 5$ square array of dots. (Two rectangles are different if they do not share all four vertices.)


Let $f(x) = 4x^2 - 3$ and $g(f(x)) = x^2 - x + 2$. Find the sum of all possible values of $g(47)$.


In the diagram below, we have $\sin \angle RPQ = \frac{3}{5}$. What is $\sin \angle RPT$?

[asy]
pair R,P,Q,T;
T = (4,0);
P = (0,0);
Q = (2,0);
R = rotate(aSin(3/5))*(1.5,0);
dot("$T$",T,E);
dot("$Q$",Q,S);
dot("$R$",R,N);
dot("$P$",P,S);
draw(Q--T);
draw(P--R);
[/asy]


What is $\left(\frac{8}{9}\right)^2 \cdot \left(\frac{1}{3}\right)^2 \cdot \left(\frac{1}{4}\right)^2$?


Cory needs to buy 18 identical notebooks and has a budget of $160. He must pay a $5 entrance fee at the store. Each notebook costs the same whole-dollar amount and a 5% sales tax is added to the price at checkout. What is the greatest possible price per notebook that allows Cory to stay within his budget?


In a regular nonagon (a 9-sided polygon), two diagonals are chosen. What is the probability that they intersect inside the nonagon?


A regular hexagon $ABCDEF$ serves as the base of right pyramid $\allowbreak PABCDEF$. Assume triangle $PAD$ is an isosceles right triangle with the equal sides each of length 10 units. Determine the volume of the pyramid.


What is the median number of moons per celestial body? Consider this updated data about several planets and the dwarf planet Pluto:
\begin{tabular}{c|c}
Celestial Body & $\#$ of Moons\\
\hline
Mercury & 0 \\
Venus   & 0 \\
Earth   & 1 \\
Mars    & 2 \\
Jupiter & 67 \\
Saturn  & 82 \\
Uranus  & 27 \\
Neptune & 14 \\
Pluto   & 5 \\
Ceres   & 0 \\
\end{tabular}


A maintenance team needs to replace tiles at the center $C$ of a round patio. They lay a 20-foot plank from $A$ to $B$ and then a 15-foot plank from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$. Calculate the area of the circular patio. Express your answer in terms of $\pi$.

Solution: \boxed{325 \pi \text{ square feet}}


Each vertex of this parallelogram has integer coordinates. The perimeter of this parallelogram is $p$ units, while the area is $a$ square units. Determine the value of the sum $p + a$ given the parallelogram with vertices at $(1,1)$, $(6, 3)$, $(9, 3)$, and $(4,1)$.

Solution: \boxed{2\sqrt{29} + 12}


A function $f$ has domain $[0,2]$ and range $[0,1]$. Define another function by
\[ g(x) = 1 - f(x^2 + 1) \]
Determine the domain and range of $g(x)$ and provide them as an ordered quadruple $(a, b, c, d)$.

Solution: \boxed{(-1, 1, 0, 1)}


In $\triangle ABC$, $AC = BC$, $AB = BC$, $m\angle ACB = 60^{\circ}$, and $CD$ is the altitude from point $C$ to line $AB$. Point $E$ is the midpoint of $CD$. What is the number of degrees in $m\angle ECB$?

[asy]
pair A,B,C,D,E;
A = (-1,0);
B = (1,0);
C = (0, 1.732/2);
D = (0, 0);
E = (0, 1.732/4);
draw(A--B--C--cycle);
draw(C--D);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, N);
dot("$D$", D, S);
dot("$E$", E, E);
[/asy]


Let \( z \) and \( w \) be complex numbers such that \( |z| = 2 \) and \( |w| = 4 \). If \( |z+w| = 3 \), what is \( \left | \frac{1}{z} + \frac{1}{w} \right| \)?


Let $\mathbf{N}$ be a matrix such that
\[
\mathbf{N} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{N} \begin{pmatrix} -4 \\ 6 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}.
\]
Compute $\mathbf{N} \begin{pmatrix} 7 \\ 2 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 24 \\ 11 \end{pmatrix}}


If $x$ and $y$ are positive real numbers such that $8x^2 + 16xy = 2x^3 + 4x^2y,$ find the value of $x$.


Define $A \diamond B$ as $A \diamond B = \frac{A^2 + B^2}{5}$. What is the value of $(3 \diamond 7) \diamond 4$?


The sum of Alice's weight and Clara's weight is 240 pounds. If you subtract Alice's weight from Clara's weight, you get one third of Alice's weight. How many pounds does Clara weigh?


Six scores are distributed between Alice and Bob. These scores are 68, 72, 76, 80, 85, and 90. Alice's scores are in ascending order. If the mean score of Alice is 75, what is the mean score of Bob?


If \(a\), \(b\), and \(c\) are positive numbers with \(ab = 24\sqrt[3]{3}\), \(ac = 40\sqrt[3]{3}\), and \(bc = 15\sqrt[3]{3}\), find the value of \(abc\).


Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ in the interval $(0.0001, 0.0005).$


Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$?


Calculate $\frac{4 \cdot 7! + 28 \cdot 6!}{8!}$


How many real numbers \(x\) are solutions to the equation \[ |x-2| = |x-3| + |x-4| \] ?


The sequence described in rows is as follows:
- Row 1: $2, 2$
- Row 2: $4, 4, 4, 4$
- Row 3: $6, 6, 6, 6, 6, 6$
- Row 4: $8, 8, 8, 8, 8, 8, 8, 8$

Continuing this pattern, what would be the value of the $60^{\mathrm{th}}$ number if the arrangement were continued?