Every week, Judy goes to the supermarket and buys the following: $5$ carrots at $\$1$ each, $4$ 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 promotion where buying one pineapple gets another free. Judy also has a coupon for $\$10$ off any order of $\$40$ or more. How much money does Judy spend on this shopping trip?


A hyperbola has its two foci at $(2, 3)$ and $(8, 6)$. Find the coordinates of its center.

Solution: \boxed{(5, 4.5)}


Let \(a\), \(b\), and \(c\) be the roots of the polynomial equation
\[x^3 - 5x^2 + 7x - 2 = 0.\]
Find the monic polynomial, in \(x\), whose roots are \(a - 3\), \(b - 3\), and \(c - 3\).

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


The measures of angles \( C \) and \( D \) are positive integer numbers of degrees where \( C \) is a multiple of \( D \). Angles \( C \) and \( D \) are supplementary angles. How many measures are possible for angle \( C \)?


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


Cylinder $C$'s height is equal to the radius of cylinder $D$ and cylinder $C$'s radius is equal to the height $h$ of cylinder $D$. If the volume of cylinder $D$ is three times the volume of cylinder $C$, the volume of cylinder $D$ can be written as $M \pi h^3$ cubic units. What is the value of $M$?


Compute
$$\sum_{k=1}^{500} k(\lceil \log_{\sqrt{3}}{k}\rceil - \lfloor\log_{\sqrt{3}}{k} \rfloor).$$


Calculate the value of $(10! \cdot 7! \cdot 3!) / (9! \cdot 8!)$.


Find the minimum value of
\[ \sqrt{x^2 + (1 + x)^2} + \sqrt{(1 + x)^2 + (1 - x)^2} \]
over all real numbers \( x \).


The chess club has 20 members: 12 boys and 8 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?


Define the function $g(x,y)$ as follows:
\[ g(x,y) = \left\{
\begin{array}{cl}
\dfrac{2xy - x + 3}{3x} & \text{if $x + y \leq 4$}, \\
\dfrac{xy - y - 3}{-3y} & \text{if $x + y > 4$}.
\end{array}
\right. \]
Calculate $g(3,1) + g(3,3)$.


To reach the center \( C \) of a circular fountain for maintenance, a crew uses a 20-foot plank from point \( A \) to point \( B \) and then a 12-foot plank from point \( D \) to \( C \), where \( D \) is the midpoint of \( \overline{AB} \). Determine the area of the circular base of the fountain in terms of \(\pi\).

Solution: \boxed{244 \pi \text{ square feet}}


The logo designed by Wei consists of a square and four identical circles. Each circle is tangent to two sides of the square and its two adjacent circles. For a version of the logo where the square has a side length of 30 inches, calculate:
1. The total shaded area in square inches.
2. The total circumference of all four circles.

Solution: \boxed{60\pi \text{ inches}}


What is the smallest possible positive integer value of $x$ if $2x^2 < 50$?


An ellipse is tangent to both the $x$-axis and the $y$-axis, and its foci are located at $(3, -4 + 2\sqrt{3})$ and $(3, -4 - 2\sqrt{3})$. Find the length of the major axis.


The six faces of a four-inch wooden cube are each painted blue. The cube is then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least two faces?


An equilateral triangle has two vertices at $(3,6)$ and $(11,6)$. If the third vertex is in the first quadrant, what is the y-coordinate of this third vertex?

Solution: \boxed{6 + 4\sqrt{3}}


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


In triangle $XYZ$, we have $\angle X = 90^\circ$, $YZ = 10$, and $\tan Z = 3\sin Z$. What is $XZ$?


Determine the number of angles between 0 and $2\pi$, excluding integer multiples of $\frac{\pi}{3}$, such that $\sin \theta$, $\cos \theta$, and $\tan \theta$ form an arithmetic sequence in some order.


A right-angled triangle has sides of lengths 8, 15, and 17. Determine the distance between the centers of the inscribed and circumscribed circles of this triangle.


Karlanna places 720 marbles into $m$ total boxes such that each box contains an equal number of marbles. There should be more than one box, and each box must contain more than one marble. Determine how many values of $m$ are possible.


The sequence $1, 4, 5, 16, 17, 20, 21 \cdots$ consists of all those positive integers which are powers of 4 or sums of distinct powers of 4. Find the $150^{\mbox{th}}$ term of this sequence.


At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. I have a sample of helium in a 3.00 liter container with a pressure of 8 kPa. If I transfer all of it to a 7.5 liter container at the same temperature, what will the new pressure be in kPa?

Solution: \boxed{3.2 \text{ kPa}}


Given the Cesaro sum of the 199-term sequence $(a_1,a_2,\dots,a_{199})$ is 500, find the Cesaro sum of the 200-term sequence $(2, a_1, \dots, a_{199})$.


On the game show $\text{\emph{Wheel of Fortune Plus}}$, the spinner is divided into six equal segments. The segment values are Bankrupt, $1000$, $500$, $300$, $400$, and $700$. If each segment is equally likely to be landed on, what is the probability of earning exactly $2200$ after three spins? Represent your answer as a common fraction.

Solution: \boxed{\frac{6}{216} = \frac{1}{36}}


Determine how many numbers less than 35 are abundant numbers.


Let's say that James and Sara each think of a polynomial. Each of their polynomials is monic, has degree 4, and has the same positive coefficient for the term $z^3$ and the same positive constant term. The product of their polynomials results in \[z^8 + 4z^7 + 5z^6 + 7z^5 + 9z^4 + 8z^3 + 6z^2 + 8z + 9.\] Determine the constant term of Sara's polynomial.


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


Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length?


Expand and find the sum of the coefficients of the terms containing a nonzero power of $y$ when $(5x + 3y + 2)(2x + 5y + 6)$ is multiplied.


What is the greatest divisor of 630 that is smaller than 60 and also a factor of 90?


If $\displaystyle\frac{p}{q} = 4$, $\displaystyle\frac{q}{r} = 3$, and $\displaystyle \frac{r}{s} = \frac{1}{5}$, then what is $\displaystyle\frac{s}{p}$?


In 2021, the sum of the year's digits is $5$. What is the first year after 2021 in which the sum of the digits is again 5?


Two standard decks of cards are combined, making a total of 104 cards (each deck contains 13 ranks and 4 suits, with all combinations unique within its own deck). The decks are randomly shuffled together. What is the probability that the top card is an Ace of $\heartsuit$?


Solve the inequality
\[9x^2 - 6x + 1 > 0.\]

Solution: \boxed{(-\infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty)}


A function \( f \) is defined on the complex numbers by \( f(z) = (a + bi)z^2 \), where \( a \) and \( b \) are real numbers. The function has the property that for each complex number \( z \), \( f(z) \) is equidistant from both \( z \) and the origin. Given that \( |a+bi| = 5 \), find \( b^2 \).


Solution: \boxed{2x^3 - x^2 - 11x + 27}


Given that $x$ is a multiple of $34560$, what is the greatest common divisor of $g(x) = (3x + 4)(8x + 5)(15x + 11)(x + 17)$ and $x$?


How many ways can 2210 be factored as a product of two two-digit numbers?


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.


What integer $n$ satisfies $0 \leq n < 25$ and $$-250 \equiv n \pmod{23}~?$$


On a Cartesian coordinate plane, points $(1,2)$ and $(4,6)$ are endpoints of a diagonal of a square. What is the area of the square?


Suppose $a$ and $b$ are integers where $a = b + c$ for some positive integer $c$. What is the smallest possible positive value of $\frac{a+b}{a-b} + \frac{a-b}{a+b}$?


Find the phase shift of the graph of $y = 3 \sin \left( 3x + \frac{\pi}{4} \right)$.


The Dunkers, coached by Coach Loudvoice, have 15 players, but three of them, Ben, Tom, and Dan, refuse to play together. Additionally, one player, Alex, is injured and cannot participate. How many starting lineups (of 5 players) can Coach Loudvoice make if the starting lineup cannot contain more than one among Ben, Tom, and Dan, and cannot include Alex at all?


How many positive four-digit integers are divisible by both $5$ and $8$?


A gardener plans to enclose a rectangular garden with 480 feet of fencing. However, one side of the garden will be twice as long as another side. What is the maximum area of this garden?


Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[
\begin{pmatrix} 2a & b \\ c & 2d \end{pmatrix}^{-1} = \begin{pmatrix} \frac{1}{2a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{2d} \end{pmatrix}.
\]


The square quilt block shown is made from sixteen unit squares, some of which have been divided in half to form triangles. Four of the unit squares in each corner are not divided and are shaded. Additionally, four half-triangles (formed from two squares) in the center are also shaded. What fraction of the square quilt is shaded? Express your answer as a common fraction.


How many positive integers, including $1,$ are divisors of both $60$ and $84?$ Furthermore, what is the least common multiple (LCM) of these common divisors?


The cost of three pencils and two pens is \$4.20, and the cost of four pencils and three pens is \$5.60. What is the cost of one pencil and one pen?


Find the equation of the directrix of the parabola \( x = -\frac{1}{4} y^2 + 1 \).

Solution: \boxed{x = 2}


A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its twelfth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$


Consider the set of all points $(x,y)$ in the coordinate plane for which one of the coordinates is exactly twice the other or equal to the other. If we were to plot all such points, into how many regions would the resulting graph split up the plane?


The foci of the ellipse \(\frac{x^2}{25} + \frac{y^2}{b^2} = 1\) and the foci of the hyperbola
\[\frac{x^2}{169} - \frac{y^2}{144} = \frac{1}{36}\] coincide. Find \(b^2.\)


John tells Mary his score in the Australian Senior Mathematics Competition (ASMC), which was exactly 100. From this, Mary was able to determine the number of problems John solved correctly. If John's score had been over 95 but less than 100, Mary could not have determined this. What was John's score? (Recall that the ASMC consists of 25 multiple choice questions and that one's score, $s$, is computed by the formula $s = 25 + 5c - 2w$, where $c$ is the number of correct answers and $w$ is the number of wrong answers. Students are not penalized for problems left unanswered.)


Determine how many positive perfect square integers are factors of the product $\left(2^8\right)\left(3^9\right)\left(5^{12}\right)\left(7^4\right)$.


Suppose $q(x)$ is a function such that $q(x) + (x^6 + 2x^4 + 5x^2 + 8x) = (3x^4 + 18x^3 + 20x^2 + 5x + 2)$. Express $q(x)$ as a polynomial with the degrees of the terms in decreasing order.

Solution: \boxed{-x^6 + x^4 + 18x^3 + 15x^2 - 3x + 2}


Determine the positive value of $x$ which satisfies:
\[\log_5 (x - 1) + \log_{5} (x^2 - 1) + \log_{\frac{1}{5}} (x - 1) = 3.\]


Calculate the product \(\left(\frac{4}{8}\right)\left(\frac{8}{12}\right)\left(\frac{12}{16}\right)\cdots\left(\frac{2008}{2012}\right)\). Express your answer as a common fraction.


Find the solutions to \(z^4 = 4i\) and sum the angles of the solutions in degrees. Each solution can be expressed using the polar form:
\[
z_k = r_k (\cos \theta_k + i \sin \theta_k)
\]
where \(r_k > 0\) and \(0^\circ \leq \theta_k < 360^\circ\) for \(k = 1, 2, 3, 4\).


Find the remainder when \(1^3 + 2^3 + 3^3 + \dots + 150^3\) is divided by 7.


Let $G'$ be a point inside triangle $ABC$ such that $G' = \frac{1}{4}A + \frac{1}{4}B + \frac{1}{2}C$. Let $Q$ be another arbitrary point. Determine the constant $k'$ such that
\[ QA^2 + QB^2 + QC^2 = k' \cdot QG'^2 + GA^2 + GB^2 + GC^2. \]


Suppose $a$ and $b$ are positive integers with $\gcd(a,b)$ divisible by exactly $5$ distinct primes and $\mathop{\text{lcm}}[a,b]$ divisible by exactly $18$ distinct primes.

If $a$ has fewer distinct prime factors than $b$, then determine the maximum number of distinct prime factors $a$ can have.


Find the largest even value of $k$ for which $3^{10}$ is expressible as the sum of $k$ consecutive positive integers.


Consider a standard set of dominoes, where each domino tile has two ends and each end is labeled with a number from 0 to 6. Each possible pair of numbers from this range appears exactly once in the set. A $\textit{double}$ in this context is a domino where both ends show the same number. What is the probability that a randomly selected domino from this standard set is a $\textit{double}$? Express your answer as a common fraction.


How many four-digit positive integers are there with thousands digit $7$ and an odd hundreds digit?


Given that the graphs of \( y=h(x) \) and \( y=j(x) \) intersect at \( (3,3) \), \( (6,9) \), \( (9,18) \), and \( (12,18) \), determine the sum of the coordinates of the point where the graphs of \( y=h(3x) \) and \( y=3j(x) \) intersect.


There are 10 sprinters in a championship 200-meter race finals. Four of the sprinters are Italians. The race also awards three medals: gold, silver, and bronze. Determine the number of ways the medals can be awarded if exactly one Italian must win a medal, and that medal must be the gold medal.


Find the smallest positive integer divisible by 15, 16, and 18.


In triangle $DEF$, side lengths are $DE = 25$, $EF = 29$, and $FD = 32$. Points $P$ and $Q$ are on $\overline{DE}$ so that $\overline{PQ}\parallel\overline{EF}$ and $PQ = \frac{1}{3}DE$. Points $R$ and $S$ are on $\overline{EF}$ such that $\overline{RS}\parallel\overline{FD}$ and $RS = \frac{1}{3}EF$. Points $T$ and $U$ are on $\overline{FD}$ with $\overline{TU}\parallel\overline{DE}$ and $TU = \frac{1}{3}FD$. Determine the length of segment $PQ$.


A particular coin is biased so that the probability of flipping heads is $\frac{2}{5}$ and the probability of flipping tails is $\frac{3}{5}$. If we get \$5 for flipping heads but lose \$4 for flipping tails, what is the expected value, in dollars, of our winnings after one flip?


You intend to distribute an assortment of gold coins evenly among 12 friends. After distributing the coins, you find that 3 friends would each receive an extra coin beyond an equal share. Given that you have fewer than 100 gold coins, what is the largest number of coins you could have?


Convert the point $(1, -2\sqrt{3}, 4)$ 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( \sqrt{29}, 2\pi - \tan^{-1}(2\sqrt{3}), \cos^{-1}\left(\frac{4}{\sqrt{29}}\right) \right)}


Let an integer be such that when divided by 20, the remainder is 11. What is the sum of the remainders when the same integer is divided by 4 and by 5? Additionally, find the smallest such integer greater than 50.


A collegiate sports league has 16 teams split into two divisions: Division A with 8 teams and Division B with 8 teams. Each team in Division A plays every other team in its division twice and each team in Division B once. Each team in Division B plays every other team in its division twice and each team in Division A once. How many total games are played in a complete season?


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


Place each of the digits 4, 5, 6, and 7 in exactly one square to make the smallest possible product. What is this product?


Find the units digit of $n$ given that $mn = 31^6$ and $m$ has a units digit of 3.


In triangle $DOG$, $\angle DGO = \angle DOG$ and $\angle DOG = 40^\circ$. If $\overline{OS}$ bisects $\angle DOG$, determine the measure of $\angle DSO$.


Determine the number of relatively prime dates in February (including considering leap years). Assume a relatively prime date means the day number and the month number (2 for February) are relatively prime.


How many three-digit integers leave a remainder of 3 when divided by 7?


Determine the smallest positive integer $N$ which, when divided by 5, leaves a remainder of 2, when divided by 6 leaves a remainder of 3, when divided by 7 leaves a remainder of 4, and when divided by 11 leaves a remainder of 9.


Thirty teams are participating in a knockout tournament where each match results in a win or lose outcome only, and the losing team is eliminated. Each team has a $50 \%$ chance of winning any game it plays. The probability that the final sequence of winners is in the exact increasing order of their initial arbitrary labels (team 1 to team 30) is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$


The interior of a right, circular cone is 12 inches tall with a 4-inch radius at the opening. The interior of the cone is filled with ice cream, and the cone has a hemisphere of ice cream exactly covering the opening of the cone. What is the volume of the total amount of ice cream? Provide your answer in terms of $\pi$.


What is the smallest 3-digit base 6 positive integer that is divisible by 7? (Express your answer in base 6.)


Parallelogram $EFGH$ with $E(3,7)$, $F(5,11)$, $G(7,7)$, and $H(5,3)$ is reflected across the $x$-axis to $E'F'G'H'$ and then $E'F'G'H'$ is reflected across the line $y=x-2$ to $E''F''G''H''$. This is done such that $H'$ is the image of $H$, and $H''$ is the image of $H'$. What is the ordered pair of $H''$ in the coordinate plane?


Find the ordered pair $(x,y)$ if

\begin{align*}
2x + 3y &= (6-x) + (6-3y),\\
x - 2y &= (x-2) - (y+2).
\end{align*}

Solution: \boxed{(-4, 4)}


Solution: \boxed{s = -573}


Let \(\mathbf{c}\) and \(\mathbf{d}\) be orthogonal vectors. If \(\operatorname{proj}_{\mathbf{c}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} -\frac{4}{5} \\ \frac{8}{5} \end{pmatrix},\) then find \(\operatorname{proj}_{\mathbf{d}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.\)

Solution: \boxed{\begin{pmatrix} \frac{24}{5} \\ -\frac{18}{5} \end{pmatrix}}


In $\triangle ABC$, $AB=AC=30$ and $BC=28$. Points $D, E, F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE} \parallel \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. Compute the perimeter of parallelogram $ADEF$.


Solution: \boxed{\frac{80}{3} \text{ miles per hour}}


Let $b$ and $c$ be the roots of the polynomial
\[x^2 - x - 1 = 0.\] Compute
\[\frac{1}{1 - b} + \frac{1}{1 - c}.\]


For some positive integer $k$, when 120 is divided by $k^2$, the remainder is 12. What is the remainder when 144 is divided by $k$?


Evaluate the following product of sequences: $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{2187} \cdot \frac{6561}{1}$.


In how many ways can 8 distinct beads be placed on a bracelet, considering that two arrangements are equivalent if one can be rotated or reflected to result in the other?


How many integers $n \neq 0$ satisfy the inequality $\frac{1}{|n|} \geq \frac{1}{5}$?


Determine the product of $0.\overline{356}$ and $12$, and express your result as a simplified fraction.


What is the largest 4-digit integer congruent to $7 \pmod{19}$?


Now, 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, dime, and half-dollar all come up showing the same side?


Define two sequences of vectors $(\mathbf{u}_n)$ and $(\mathbf{z}_n)$ as follows: Start with $\mathbf{u}_0 = \begin{pmatrix} 2 \\ 2 \end{pmatrix}$ and $\mathbf{z}_0 = \begin{pmatrix} 3 \\ -3 \end{pmatrix}$. For all $n \geq 1$, $\mathbf{u}_n$ is the projection of $\mathbf{z}_{n-1}$ onto $\mathbf{u}_0$, and $\mathbf{z}_n$ is the projection of $\mathbf{u}_n$ onto $\mathbf{z}_0$. Calculate the sum $\mathbf{u}_1 + \mathbf{u}_2 + \mathbf{u}_3 + \dots$.

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


Two circles, each with a radius of 12 cm, overlap such that the distance between their centers is 16 cm. Calculate the length of the common chord where the two circles intersect.

Solution: \boxed{8\sqrt{5} \text{ cm}}


A circular cylindrical post has a circumference of 6 feet and is 16 feet tall. A string is wrapped around it, spiraling from the bottom to the top of the post. The string makes exactly five complete loops around the post, starting from the bottom edge and finishing at the top edge. Calculate the length of the string.

Solution: \boxed{34 \text{ feet}}


A street has parallel curbs that are 50 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 20 feet, and each stripe is 65 feet long. Find the distance, in feet, between the stripes.

Solution: \boxed{\frac{200}{13} \; \text{feet}}


A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\sqrt{d} - b$ cm, where $d$ is a prime number. Find the value of $b + d$.


A stock investment increased by $30\%$ in 2006 and then by an additional $20\%$ in 2007. What percent decrease is needed in 2008 to bring the stock's value back to its original price at the beginning of 2006?


Let $g$ be a quadratic function such that $g(x+1)−g(x) = 2x + 3$ for all real numbers $x$. What is $g(2)−g(6)$?

Solution: \boxed{g(2) - g(6) = -40}


Consider three different lines given by the equations $2y - 3x = 4$, $x + 3y = 3$, and $3x - 4.5y = 7.5$. Determine how many points of intersection exist among these lines.


Find the remainder when $x^{60}$ is divided by $(x + 1)^3$.

Solution: \boxed{1770x^2 + 3480x + 1711}


My three-digit code is 734. Reckha cannot choose a code that matches mine in two of the three digit-positions or that is simply a transposition of two digits of my code (so 437 is forbidden, but 347 is allowed). Reckha can otherwise select any three-digit code where each digit is from the set $\{1, 2, ..., 9\}$. How many codes can Reckha choose?


Let $x$ be a real number. Consider these five statements:
1. $0 < x^2 < 1$
2. $x^2 > 1$
3. $-1 < x < 0$
4. $0 < x < 1$
5. $0 < x - x^3 < 1$

What is the maximum number of these statements that can be true for any value of $x$?


Let $f$ and $h$ be functions such that $f^{-1}(h(x)) = 3x - 2$. Find $h^{-1}(f(4))$.


Factor the expression $3x(x-5) + 7(x-5) - 2(x-5)$.


There are 6 blue chips, 5 red chips, and 4 yellow chips in a bag. One chip is drawn from the bag, then placed back into the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors?


Express $20.\overline{396}$ as a common fraction in lowest terms.


April now has five different basil plants and three 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?


Let \( a \) and \( b \) be positive real numbers. Find the maximum value of
\[ 2(a - x)(x - \sqrt{x^2 + b^2}) \] in terms of \( a \) and \( b \).


For how many integers $n=2,3,4,\ldots,99,100$ is the base-$n$ number $425426_n$ a multiple of $11$?


How many pairs of positive integers \( (m, n) \) satisfy \( m^2 \cdot n < 30 \)?


Each outcome on the spinner below has equal probability. The spinner has four sections labeled 2, 3, 5, and 0. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit and the third outcome is the units digit, what is the probability that you will end up with a three-digit number that is divisible by 5? Express your answer as a common fraction.


On the first two days of January, there is a $\frac{1}{4}$ chance that it will snow where Bob lives. On the next three days, there is a $\frac{1}{2}$ chance that it will snow. For the final two days of the week, there's a $\frac{1}{3}$ chance that it will snow. What is the probability that it snows at least once during the first week of January?


Given a prime $p=11$, determine the sum of all integers less than $11$ that are primitive roots modulo $11$ and are less than 8.


What is the number of square centimeters in the area of a circle with a diameter of 5 meters? Express your answer in terms of $\pi$.


Find all real $x$ such that \[\left\lfloor x \lfloor x \rfloor\right \rfloor = 49.\]

Solution: \boxed{\left[7, \frac{50}{7}\right)}


The function \( f(x) \) takes positive real numbers to real numbers, such that
\[ xf(y) - yf(x) = xyf \left( \frac{x}{y} \right) \]
for all positive real numbers \( x \) and \( y \). Find all possible values of \( f(50) \).


If 5 knicks = 3 knacks and 4 knacks = 6 knocks, how many knicks are equal to 36 knocks?


The sum of the areas of a square and a rectangle is 130, while the difference in their areas is 50. The rectangle has a length that is twice its width. Find the sum of the perimeters of both shapes.

Solution: \boxed{12\sqrt{10} + 12\sqrt{5}}


Peter has two numbers, $a$ and $b$. When he divides $a$ by 80, he gets a remainder of 75. When he divides $b$ by 90, he gets a remainder of 85. What remainder does he get when he divides $a+b$ by 40?


In how many ways can 8 distinct beads be placed on a necklace, considering that two arrangements are identical if one can be rotated or reflected to yield the other?


A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7, such that its base is parallel to and rests on the base of the hemisphere. What is the height of this cylinder?


Consider the graphs of the functions $r(x) = x^2 - 4$ and $s(x) = -|x+1|$. If $s(r(x))$ is evaluated at $x = -4, -3, -2, -1, 0, 1, 2, 3, 4$, what is the sum of the nine values obtained after squaring each of them?


A sphere is inscribed in a cube with edge length 16 inches. Then a smaller cube is inscribed in the sphere. What is the volume of this smaller inscribed cube? Express your answer in simplest radical form.


Let \( g(x) \) be defined as:
\[
g(x) = \begin{cases}
x^2-3 & \text{if } x \ge -3, \\
x + 4 & \text{if } -6 \le x < -3, \\
-2x & \text{otherwise},
\end{cases}
\]
Determine for how many distinct values of \( x \) the equation \( g(g(x)) = 4 \) holds.


Consider the geometric sequence $4$, $\dfrac{12}{3}$, $\dfrac{36}{9}$, $\dfrac{108}{27}$, $\ldots$. Find the tenth term of the sequence. Express your answer as a common fraction.


The polynomial $Q(x) = 3x^3 + dx^2 + ex + f$ has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = Q(x)$ is 9. Determine the value of $e$.


There are approximately 0.454 kilograms in a pound. To the nearest whole pound, how many pounds does a bag of rice that weighs 150 kg weigh?

Solution: \boxed{330\ \text{pounds}}


Jake wishes to organize his collection of marbles into $X$ groups, each containing the same number of marbles, $Y$. Each group must have more than one marble, but no single group can contain all the marbles. If there are 15 possible values for $Y$ given these conditions, what is the smallest number of marbles Jake could have?


The Benton Youth Soccer Team has 22 players on the team, including reserves. Of these, four are goalies. Today, the team is having a contest to see which goalie can block the most number of penalty kicks. For each penalty kick, a goalie stands in the net while the rest of the team (including other goalies) takes a shot on goal, one at a time, attempting to place the ball in the net.

How many penalty kicks must be taken to ensure that everyone has gone up against each of the goalies?


Let $a\star b = \dfrac{\sqrt{a+b}}{\sqrt{a-b}}$. If $x \star 36 = 12$, find $x$.


How many pairs of positive integers $(x, y)$ satisfy $x^2 - y^2 = 45$?


A cube with an edge length of 5 units has the same volume as a square-based pyramid with a base edge length of 10 units and a height of $h$ units. What is the value of $h$?

Solution: \boxed{h = 3.75}


Determine the smallest possible positive integer value of \( e \) for the polynomial
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
with integer coefficients, given that it has roots $-3$, $4$, $10$, and $-\frac{1}{4}$.


Tom has a red marble, a blue marble, four identical yellow marbles, and two identical green marbles. How many different groups of two marbles can Tom choose?


A rectangular prism has 6 faces, 12 edges, and 8 vertices. If a new pyramid is added using one of its rectangular faces as the base, calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting shape after the fusion of the prism and pyramid.


Given ten distinguishable rings, let $n$ be the number of possible seven-ring arrangements on the four fingers (not the thumb) of one hand, where no finger can have more than two rings. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.


The set of vectors \(\mathbf{v}\) such that
\[
\operatorname{proj}_{\begin{pmatrix} 6 \\ 3 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} -3 \\ -\frac{3}{2} \end{pmatrix}
\]
lie on a line. Enter the equation of this line in the form "$y = mx + b$".

Solution: \boxed{y = -2x - \frac{15}{2}}


There exists a constant $c$, so that among all chords $\overline{AB}$ of the circle $(x-1)^2 + (y-1)^2 = 4$ passing through $C = (0,c),$
\[ t = \sqrt{d_A^2 + d_B^2} \]
where $d_A$ and $d_B$ are the distances from $C$ to $A$ and $B$, respectively, is a fixed constant. Find the constant $t$.


Simplify the expression $3x - 5(2+x) + 6(2-x) - 7(2+3x)$.

Solution: \boxed{-29x - 12}


Given that $\triangle ABC$ has vertices $A=(0,0), B=(30,18)$, and the coordinates of vertex $C$ are integers, find the minimum area of $\triangle ABC$.


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


The area of a square equals the square of the length of one of its sides. The perimeter is four times the side length. If the sum of the areas of two squares is 85 and the difference of their areas is 41, what is the sum of their perimeters?

Solution: \boxed{4(\sqrt{63} + \sqrt{22})}


To calculate the squares of numbers near 50, Ethel uses mental math. She notes that $50^2 = 2500$. To compute $49^2$, she subtracts a number from $2500$. For $51^2$, she adds 101 to $2500$. What number does she subtract to determine $49^2$?


Bill can buy mags, migs, and mogs for $\$3$, $\$4$, and $\$8$ each, respectively. What is the largest number of mogs he can purchase if he must buy at least one of each item and will spend exactly $\$100$?


Determine the remainder when $(2x^5 - x^4 + x^2 - 1)(x^3 - 1)$ is divided by $x^2 - x + 1$.


Find the radius of the circle with equation $16x^2-32x+16y^2-48y+68=0.$


Suppose I have 8 shirts, 5 pants, 4 ties, and 2 jackets. If an outfit requires a shirt, pants, and optionally can have a tie and/or a jacket, how many different outfits can I create?


Determine where the lines given by the equations $8x - 5y = 10$ and $3x + 2y = 1$ intersect.

Solution: \boxed{\left(\frac{25}{31}, \frac{-22}{31}\right)}


The area of the base of a hemisphere is $144\pi$. The hemisphere is mounted on top of a cylinder that has the same radius as the hemisphere and a height of 10. What is the total surface area of the combined solid? Express your answer in terms of $\pi$.


Suppose \( z \) and \( w \) are complex numbers such that
\[
|z| = |w| = 2 \quad \text{and} \quad z \overline{w} + \overline{z} w = 2.
\]
Find the largest possible value of the real part of \( z + w \).


Given that $f(x) = x^3 - 3x^2 + 2x + 4$ and $g(x) = 2x - 1$, what is the value of $f(g(5)) - g(f(5))$?


Charlie and Dana play a game where they take turns rolling a standard die. If a player rolls $n$, they are awarded $g(n)$ points, where:
\[g(n) = \left\{
\begin{array}{cl}
8 & \text{if }n\text{ is a multiple of 3}, \\
3 & \text{if }n\text{ is a prime number}, \\
1 & \text{if }n\text{ is only even (and not a multiple of 3)},
\\ 0 & \text{otherwise}.
\end{array}
\right.\]
Charlie rolls the die five times and gets the numbers 6, 4, 5, 2, and 3. Dana rolls the die four times and gets the numbers 3, 3, 1, and 2. Calculate the product of Charlie's and Dana's total points.


The perimeter of a particular equilateral triangle and the circumference of a particular circle are equal. What is the ratio of the area of the triangle to the area of the circle? Express your answer as a common fraction in terms of $\pi$.


In triangle $ABC$, $BC = 40$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ at $D$ and extend to meet an extension of $AB$ at $E$. Find the length of $DE$.


How many distinct ordered pairs of positive integers $(x, y)$ are there such that the sum of the reciprocals of $x$ and $y$ equals $\frac{1}{6}$?


Let $\omega$ be a complex number such that $|\omega| = 3$. Determine $\lambda > 1$ such that $\omega$, $\omega^2$, and $\lambda \omega$ form an equilateral triangle in the complex plane.

Solution: \boxed{\frac{1 + \sqrt{33}}{2}}


Solution: \boxed{\frac{2}{\sqrt{3} + 6\sqrt{5} + 11}}


Alexia designed another version of a logo, originally 3 inches wide and 2 inches tall, to be used on her school's event posters. The school wants the logo on the poster to be 12 inches wide. How tall, in inches, will the logo be if it is enlarged proportionately? Additionally, the school also wants a smaller version of the same logo to be 1.5 inches wide for badges. How tall will this smaller logo be?


For how many values of the digit $A$ is it true that $70$ is divisible by $A$ and $546{,}2A4$ is divisible by 4?


For what base-6 digit $d$ is $2dd3_6$ divisible by the base 10 number 13? (Here $2dd3_6$ is a base-6 number whose first digit is 2, whose last digit is 3, and whose middle two digits are both equal to $d$).


How many three-digit perfect cubes are divisible by $16?$


Determine the complete factorization of the following expression:
\[(16x^6 + 36x^4 - 9) - (4x^6 - 12x^4 + 3).\]

Solution: \boxed{12(x^6 + 4x^4 - 1)}


Given the quadratic equation $x^2 + ax + b = 0$ with roots $r_1$ and $r_2$, find an equation where the roots are three times those of $x^2 + cx + a = 0$ and provide the value of $b/c$.


In rectangle \(ABCD\), \(AB = 80\). Let \(E\) be the point on \(AD\) such that \(AE = \frac{1}{3}AD\). Given that line \(AC\) and line \(BE\) are perpendicular, find the greatest integer less than \(AD\).


When $\sqrt[4]{5^9 \cdot 7^2}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$?


Fifty cards are placed into a box, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on five cards. Four cards are drawn from the box at random and without replacement. Let $p$ be the probability that all four cards bear the same number. Let $q$ be the probability that three of the cards bear a number $a$ and the other bears a number $b$ that is not equal to $a$. Calculate the value of $q/p$.


Both roots of the quadratic equation \(x^2 - 57x + k = 0\) are prime numbers. How many possible values of $k$ are there?


Let $s$ be a real number, $|s| < 3,$ and let $w$ be a complex number such that
\[w + \frac{2}{w} = s.\] Find $|w|$.


For all real numbers \(x\) except \(x=0\) and \(x=1\), the function \(f(x)\) is defined by
\[f\left(\frac{x}{x-1}\right) = \frac{1}{x}.\]
Suppose \(0 \leq t \leq \frac{\pi}{2}\). What is the value of \(f(\csc^2 t)\)?

Solution: \boxed{\cos^2 t}


Consider a modified sequence given by \( b_n = n^2! + n \) for \( n \geq 1 \). What is the maximum possible value of the greatest common divisor of two consecutive terms of this sequence?


A rectangular prism has dimensions with height twice the width and depth thrice the width. Similar to a cube's diagonals, a segment connecting non-edge vertices is also a diagonal in a rectangular prism. Compute the total number of such diagonals in the prism.


In the diagram, what is the perimeter of polygon $PQRST$? [asy]

import olympiad;

size(6cm); // ADJUST

pair p = (0, 8);
pair q = (4, 8);

pair r = (4, 4);
pair t = (0, 0);
pair s = (9, 0);

draw(p--q--r--s--t--cycle);
label("$P$", p, NW);
label("$Q$", q, NE);
label("$R$", r, E + NE);
label("$S$", s, SE);
label("$T$", t, SW);

label("$8$", p / 2, W);
label("$4$", p + (q - p) / 2, 2 * N);
label("$9$", s / 2, S);

draw(rightanglemark(p, t, s));
draw(rightanglemark(t, p, q));
draw(rightanglemark(p, q, r));
add(pathticks(p--q, s=6));
add(pathticks(q--r, s=6));
[/asy]


A circle is tangent to the lines $3x + 4y = 24$ and $3x + 4y = 0.$  The center of the circle lies on the line $x - 2y = 0.$  Find the center of the circle.

Solution: \boxed{\left(\frac{12}{5}, \frac{6}{5}\right)}


Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 13?


Compute the distance between the parallel lines given by:
\[\begin{pmatrix} 3 \\ -4 \end{pmatrix} + t \begin{pmatrix} 2 \\ -6 \end{pmatrix}\]
and
\[\begin{pmatrix} 0 \\ -1 \end{pmatrix} + s \begin{pmatrix} 2 \\ -6 \end{pmatrix}.\]


The diameter, in inches, of a sphere with three times the volume of a sphere of radius 7 inches can be expressed in the form $a\sqrt[3]{b}$ where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$.


Find a positive integer that is divisible by 14 and has a square root between 26 and 26.5.


The sum of five consecutive even numbers is 340. What is the smallest of these five numbers? Additionally, compute the sum of the squares of these five numbers.


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 980^\circ.$


Let $a_n$ be the integer obtained by writing all the integers from $1$ to $n$ sequentially from left to right. Compute the remainder when $a_{37}$ is divided by $45$.


A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits (\heartsuit and \diamondsuit) are red, and the other two (\spadesuit and \clubsuit) are black. Cards are shuffled, and four cards are drawn sequentially. What is the probability that the first two cards drawn are red and the next two cards drawn are black?


The Lucas-like sequence $M_n$ is defined by $M_0 = 3, M_1 = 2$, and the recursion $M_n = 2M_{n-1} + M_{n-2}$. Calculate the units digit of $M_{M_8}$.


Margo rides a bicycle to her friend’s house, taking 15 minutes. She then walks back home taking 25 minutes. The average rate for the entire trip is 6 miles per hour. How many total miles did she travel?

Solution: \boxed{4 \text{ miles}}


Let $z$ be a complex number such that $|z| = 2$. Find the largest possible distance between $(5 + 2i)z^3$ and $z^4$ when plotted in the complex plane.

Solution: \boxed{8(\sqrt{29} + 2)}


Let \( p, q, r, s, t, u, v, w \) be positive real numbers such that \( pqrs = 16 \) and \( tuvw = 25 \). Find the minimum value of
\[
(pt)^2 + (qu)^2 + (rv)^2 + (sw)^2
\]
with the additional condition that \( pt = qu = rv = sw \).


In $\triangle ABC$, $AB = 25$, $AC = 24$, and $BC = 23$. Points $D, E,$ and $F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?


The parabola with equation $y = ax^2 + bx + c$ has its vertex at $(3, -9)$ and passes through the point $(5, 15)$. Find the difference between the larger and smaller zeros of the quadratic $ax^2 + bx + c$.


John cycled continuously from 7:30 a.m. until 11:10 a.m. of the same day and covered a distance of 120 miles. What was his average speed in miles per hour?


What is the least positive integer that satisfies the following conditions?

a) When divided by 2, the remainder is 1.

b) When divided by 5, the remainder is 4.

c) When divided by 7, the remainder is 6.


A right pyramid has a square base with a perimeter of 32 inches. Its apex is 11 inches away from each of the other vertices. Calculate the height of the pyramid from its peak to the center of its square base, in inches.

Solution: \boxed{\sqrt{89} \text{ inches}}


A \(200\times 325\times 376\) rectangular solid is made by gluing together \(1\times 1\times 1\) cubes. An internal diagonal of this solid passes through the interiors of how many of the \(1\times 1\times 1\) cubes?


Calculate the sum of the first 7 terms of the geometric series given by the first term $\frac{1}{6}$ and a common ratio of $-\frac{1}{2}$. Express your answer as a common fraction.


How many positive multiples of $5$ that are less than $500$ end with the digit $5$?


There are four positive integers $a, b, c, d$ less than $10$ which are invertible modulo $10$. Find the remainder when $(a+b+c+d)(abcd)^{-1}$ is divided by $10$.


Let $f(x) : \mathbb{R} \to \mathbb{R}$ be a function such that
\[
\frac{f(x) f(y) - f(xy)}{2} = x + y + 1
\]
for all $x,$ $y \in \mathbb{R}.$ Find $f(x)$.

Solution: \boxed{x + 2}


There are $25n$ members in the Watertown marching band, and when they line up in rows of 29, there are 6 band members left over. If $n$ is an integer and there are fewer than 1200 band members, what is the maximum number of people that could be in the Watertown marching band?


Suppose that the graph of a certain function, $y = f(x)$, has the property that if it is shifted $20$ units to the right, then the resulting graph is identical to the original graph of $y = f(x)$.

What is the smallest positive $a$ such that if the graph of $y = f\left(\frac{x}{4}\right)$ is shifted $a$ units to the right, then the resulting graph is identical to the original graph of $y = f\left(\frac{x}{4}\right)$?


It is currently $7\!:\!45\!:\!00 \text{ a.m.}$ What time will it be in $9999$ seconds? (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)

Solution: \boxed{10\!:\!31\!:\!39 \text{ a.m.}}


The graphs of two functions, \( p(x) \) and \( q(x) \), are defined respectively as \( p(x) = 2|x| - 4 \) and \( q(x) = -|x| \). If \( q(p(x)) \) is evaluated at \( x = -4, -3, -2, -1, 0, 1, 2, 3, 4 \), what is the sum of the nine values obtained from this evaluation?


Determine the number of solutions of the equation $2^{x+2} - 3^{y+1} = 41$, where $x$ and $y$ are integers.


Express $0.\overline{1} + 0.\overline{02} + 0.\overline{0002}$ as a common fraction.


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 307 hours, 58 minutes, and 59 seconds the time will be $X:Y:Z$. What is the value of $X + Y + Z$?


Determine the integer $b$ where $0 \le b \le 16$, such that $352936524_{85}-b$ is a multiple of $17$.


Let \( n \) be a number composed of two different primes, such that it has exactly 22 positive divisors and \( n \) is still an odd number. Find the number of positive divisors of \( 8n^3 \).


If $a$, $b$, and $c$ are positive with $ab = 36$, $ac = 72$, and $bc = 108$, what is the value of $a+b+c$?


Find the volume of the region in space defined by
\[|x + y + 2z| + |x + y - 2z| \le 12\]
and $x, y, z \ge 0.$


Two positive integers differ by 8 and their product is 168. What is the larger integer?


What is the modulo $19$ residue of $348 + 8 \cdot 58 + 9 \cdot 195 + 6 \cdot 29$?


Tamika selects two different numbers at random from the set $\{7, 8, 9\}$ and adds them. Daniel takes two different numbers at random from the set $\{2, 4, 7\}$ and multiplies them. What is the probability that Tamika's result is greater than Daniel's result?  Express your answer as a common fraction.

Solution: \boxed{\frac{6}{9} = \frac{2}{3}}


Find the sum of the first seven terms in the geometric sequence $\frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \dots$. Express your answer as a common fraction.


Let \( b_1, b_2, \ldots \) be a sequence defined as follows:

(i) \( b_1 = 2 \),

(ii) \( b_{3n} = n \cdot b_n \) for any positive integer \( n \).

What is the value of \( b_{3^{100}} \)?

Solution: \boxed{2 \cdot 3^{4950}}


A deck of fifty-two cards consists of four $1$'s, four $2$'s,..., up to four $13$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards from the remaining deck also form a pair, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$


What is the value of $(10! \cdot 6! \cdot 3!)/(9! \cdot 7!)$?


Find the projection of the vector $\begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}$ onto the vector $\begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} \frac{-5}{21} \\ \frac{-20}{21} \\ \frac{10}{21} \end{pmatrix}}


Consider a graph on modulo $13$ graph paper. We have the following two congruences:
$$y \equiv 3x + 5 \pmod{13}$$
and
$$y \equiv 9x + 1 \pmod{13}.$$
What is the sum of the $x$-coordinates of points that are common to the graphs of both congruences?


The angles of a hexagon are in the ratio of 2:3:3:4:5:6. Calculate the measure in degrees of the largest angle.


What is the smallest base-10 integer that can be represented as $21_a$ in one base and $12_b$ in a different base, where $a$ and $b$ are any bases larger than 2?


Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $29x + y = 2035$. Find the number of such distinct triangles whose area is a positive integer.


The number of books read by each member of a book club over the summer is shown in the chart below. Calculate the average number of books read by each member, rounded to the nearest whole number. [asy]
size(150);
draw((0,7)--(0,0)--(10,0));
for(int i = 1; i <= 6; ++i){
    label((string)i,(2*i,0),S);
}
label("Books Read",(0,-1)--(12,-1),S);
void addDots(int pos, int num){
    for(int i = 1; i <= num; ++i){
        dot((2*pos,i));
    }
}
addDots(1,3); addDots(2,4); addDots(3,3); addDots(4,2); addDots(5,2); addDots(6,3);
picture perpLabel;
label(perpLabel,"Number of Members");
add(rotate(90)*perpLabel,(-1,3));
[/asy]


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


Suppose $215 \cdot 789 \equiv n \pmod{75}$, where $0 \le n < 75$.

What is the value of $n$?


In triangle $ABC$, $AB = 5$, $BC = 12$, $AC = 13$, and $BD$ is the angle bisector from vertex $B$. If $BD = k \sqrt{2}$, then find $k$.


Dots are spaced one unit apart, both horizontally and vertically. Find the area enclosed by the polygon formed by connecting these dots in sequence: (0,0), (20,0), (30,10), (20,20), (0,20), (10,10), (0,0).


Let $p, q, r, s$ be the roots of the polynomial $4x^4 - 8x^3 + 18x^2 - 14x + 7 = 0$. Find the sum of the products of the roots taken three at a time, namely $pqr + pqs + prs + qrs$.


How many non-empty subsets $S$ of $\{1,2,3,\ldots,20\}$ satisfy the following properties?
1. No two consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$.


A point has rectangular coordinates $(x, y, z)$ and spherical coordinates $\left(3, \frac{5 \pi}{6}, \frac{\pi}{3}\right)$. Find the spherical coordinates of the point with rectangular coordinates $(x, -y, z)$. 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(3, \frac{7\pi}{6}, \frac{\pi}{3}\right)}


Determine the volume of the region in space defined by
\[|x + y + z| + |x + y - z| \le 12\]
and \(x, y, z \ge 0.\)


Suppose $\cos R = \frac{3}{5}$ in the right triangle QRS shown below. What is $QS$?
[asy]
pair Q,R,S;
S = (0,0);
Q = (sqrt(91),0);
R = (sqrt(91),-6);
draw(S--Q--R--S);
draw(rightanglemark(S,Q,R,13));
label("$S$",S,NW);
label("$Q$",Q,NE);
label("$R$",R,SE);
label("$10$",(R+S)/2,SW);
[/asy]


In a certain sequence, the first term is \( a_1 = 2010 \) and the second term is \( a_2 = 2011 \). Furthermore, the values of the remaining terms are chosen so that
\[ a_n + a_{n + 1} + a_{n + 2} = 3n + 1 \] for all \( n \ge 1 \). Determine \( a_{1000} \).


When the digits in the number $3086$ are reversed, we obtain the number $6803$. Factorize $6803$ into $p \cdot q \cdot r$, such that $p$, $q$, and $r$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$, and $p_3$ such that $p_1 + p_2 + p_3 = p+q+r$?

Solution: \boxed{0 \text{ distinct integers}}


A gardener plants two pine trees, three cedar trees, and four fir trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{p}{q}$ in lowest terms be the probability that no two fir trees are next to one another. Find $p+q$.


In this diagram, the first polygon is a regular pentagon, and the second is a regular triangle. What is the value, in degrees, of the sum of the measures of angles $ABC$ and $ABD$?

[asy]
draw(10dir(18)--10dir(90)--10dir(162)--10dir(234)--10dir(306)--cycle,linewidth(2));  // Pentagon
draw(10dir(234)--10dir(306)--10dir(306)+(0,-10)--10dir(234)+(0,-10)--10dir(234)--cycle,linewidth(2));  // Triangle

label("A",10dir(234),W);
label("B",10dir(306),E);
label("C",10dir(18),E);  // A point on the pentagon
label("D",10dir(306)+(0,-10),E);  // A point on the triangle

draw(10dir(306)+2dir(-135)--10dir(306)+dir(-135),linewidth(2));
[/asy]


Determine all solutions to the inequality \[\frac{x^2}{(x-5)^2} > 0.\] (Provide your answer in interval notation.)

Solution: \boxed{(-\infty, 0) \cup (0, 5) \cup (5, \infty)}


Calculate the residue of $245 \cdot 15 - 18 \cdot 8 + 5 \pmod{17}$.


A $\frac{1}{1004}$-array is constructed similarly to a $\frac{1}{2008}$-array, where each succeeding term in a row is $\frac{1}{1004}$ times the previous term in the same row, and the first term of each row is $\frac{1}{2 \times 1004}$ times the first term of the previous row. If the sum of all the terms in this array is expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find the remainder when $a+b$ is divided by $1004$.


Let $x^2 + bx + c = 0$ be a quadratic whose roots are each three less than the roots of $2x^2 - 11x - 14 = 0$. What is $c$?


The expression $\cos 2x + \cos 4x + \cos 8x + \cos 10x$ can be written in an equivalent form \[a \cos bx \cos cx \cos dx\] for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$


Find the number of $x$-intercepts on the graph of $y = \sin \frac{2}{x}$ (evaluated in terms of radians) in the interval $(0.0002, 0.002).$


Determine the number of roots of unity that are also roots of the polynomial $z^3 + az^2 + bz + c = 0$ for integers $a$, $b$, and $c$.


How many positive integers $n$ satisfy \[\dfrac{n+1200}{80} = \lfloor \sqrt{n} \rfloor?\]


For the nonzero numbers $a$, $b$, and $c$, define $$
\text{{K}}(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a} + \frac{a \cdot c}{b^2}.
$$ Calculate $\text{{K}}(3, 8, 6)$.


Suppose $x$ and $y$ are positive integers such that $\gcd(x,y)$ is divisible by exactly $5$ distinct primes and $\mathop{\text{lcm}}[x,y]$ is divisible by exactly $20$ distinct primes.

If $x$ has fewer distinct prime factors than $y$, then $x$ has at most how many distinct prime factors?


Let $p$ and $q$ be real numbers. One of the roots of the polynomial
\[ x^3 + px^2 - 5x + q = 0 \]
is $2 - 3i$. Determine the ordered pair $(p,q)$.

Solution: \boxed{\left(\frac{1}{2}, \frac{117}{2}\right)}


A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 9 units has its center at $(20, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.


What is the base ten equivalent of $98765_8$?


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 both $a$ and $b$ are divisible by 4?


Calculate the area enclosed by the graph of the equation $|x| + |3y| = 12$.


A point $(x,y)$ is a distance of 15 units from the $x$-axis. It is a distance of 13 units from the point $(2,8)$. It is a distance $m$ from the origin. Given that $x>2$, what is $m$?

Solution: \boxed{m = \sqrt{349 + 8\sqrt{30}}}


A convex hexagon has interior angles with measures $2x+2$, $3x$, $4x+3$, $5x$, $6x-1$, and $7x$. What is the measure of the largest angle?

Solution: \boxed{\frac{5012}{27}\text{ degrees}}


Calculate the following two expressions for \(a=7\) and \(b=3\):
1. \((a^3+b^3)\div(a^2-ab+b^2)\).
2. \((a^2+b^2)\div(a+b)\).


Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 1024 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 1024 \right).\]


In triangle $PQR$, points $M$ and $N$ are on sides $\overline{PQ}$ and $\overline{PR}$, respectively. Point $M$ is the midpoint of $\overline{PQ}$, and point $N$ divides $\overline{PR}$ in the ratio $1:2$ (from $P$ to $R$). The area of triangle $PQR$ is 36 square units. What is the area of triangle $PMN$?


A rectangular box has a volume of $5000$ cubic inches, a surface area of $1800$ square inches, and the sum of the lengths of its $12$ edges is $210$ inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by two inches?

Solution: \boxed{7018 \text{ cubic inches}}


You have a whole cake in your pantry. On the first trip to the pantry, you consume one third of the cake. On each successive trip, you eat half of the remaining cake. After six trips to the pantry, what fractional part of the cake have you eaten?


A convex polyhedron \( Q \) has \( 30 \) vertices, \( 72 \) edges, and \( 44 \) faces, \( 34 \) of which are triangular and \( 10 \) of which are quadrilaterals. Find the number of space diagonals in polyhedron \( Q \).


Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 100.$ Find $\frac{n}{100}$.


Calculate $\sqrt{50x} \cdot \sqrt{18x} \cdot \sqrt{98x}$. Express your answer in simplest radical form in terms of $x$.

Solution: \boxed{210x \sqrt{2x}}


Sequence $C$ is a geometric sequence, and sequence $D$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $500.$ What is the least positive difference between a number selected from sequence $C$ and a number selected from sequence $D?$

$\bullet$ Sequence $C:$ $3,$ $9,$ $27,$ $81,$ $243,$ $\ldots$

$\bullet$ Sequence $D:$ $15,$ $45,$ $75,$ $105,$ $135,$ $\ldots$


A nine-digit integer is formed by repeating a three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form?


Let \(a\) and \(b\) be the roots of the equation \(x^2-mx+3=0.\) Suppose that \(a+(1/b)\) and \(b+(1/a)\) are the roots of another equation \(x^2 - px + r = 0.\) What is \(r?\)


For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{2100}$ terminate?


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


What is the digit in the tens place when $9^{1503}$ is expressed in decimal notation?


Given the complex number sum
\[\text{cis } 60^\circ + \text{cis } 70^\circ + \text{cis } 80^\circ + \dots + \text{cis } 140^\circ,\]
express the result in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees.


What is the smallest positive integer $n$ such that $827n \equiv 1369n \pmod{36}?$


Let $d$ be a positive number such that when $134$ is divided by $d$, the remainder is $7$. Compute the sum of all possible two-digit values of $d$.


In the diagram below, $WXYZ$ is a trapezoid such that $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. If $YZ = 15$, $\tan Z = \frac{4}{3}$, and $\tan X = \frac{3}{2}$, what is the length of $XY$?


What is the degree measure of the smaller angle formed by the hands of a clock at 2:30?


How many ways are there to distribute 5 distinguishable balls into 3 indistinguishable boxes, with at least 1 ball in each box?


How many integers 1-9 are divisors of the five-digit number 36,432?


Interior numbers begin in the third row of Pascal's Triangle. Find the sum of the interior numbers of the eighth row.


What is the units digit of the product of all odd multiples of 3 between 10 and 200?


Let \( f(x) = \frac{2ax + b}{3cx + 2d} \) where \( abcd \neq 0 \) and \( f(f(x)) = x \) for all \( x \) in the domain of \( f \). Determine the value of \( 2a + 2d \).


Jackie and Phil each have a fair coin and two other coins that come up heads with probabilities $\frac{3}{5}$ and $\frac{2}{3}$ respectively. Jackie flips all three coins, followed by Phil flipping the same three coins. Let $\frac{m}{n}$ be the probability that Jackie and Phil each get the same number of heads, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.


What is the quotient when \(6x^3 + 12x^2 - 5x + 3\) is divided by \(3x + 4\)?

Solution: \boxed{2x^2 + \frac{4}{3}x - \frac{31}{9}}


Solve the problem where the equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$ must be reduced to the equivalent equation $\cos ax \cos bx \cos cx = 0$ for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$


In the diagram below, $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{3}{5}$ and $\sin C = \frac{1}{4}$, what is the length of $DC$?


Calculate the sum of the coefficients in the polynomial \( 4(2x^8 + 3x^5 - 5) + 6(x^6 - 5x^3 + 4) \) once it is fully simplified.


Determine the period of the function \( y = \tan x - \cot x \).


The postal company charges an extra $\$0.15$ in postage if the length of an envelope, in inches, divided by its height, in inches, is less than $1.5$ or greater than $3.0.$ Determine for how many of these four envelopes the extra $\$0.15$ in postage must be paid:
\begin{tabular}[t]{ccc}
Envelope & Length in inches & Height in inches\\\hline
A &7 &5\\
B &10 &2\\
C &5 &5\\
D &12 &3
\end{tabular}


Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 31 + |x_1 + x_2 + \dots + x_n|.\] What is the smallest possible value of n?


Let $x$ and $y$ be positive real numbers such that $x^3 + y^3 = 3(x + y).$ Simplify
\[\frac{x}{y} + \frac{y}{x} - \frac{3}{xy}.\]


How many integers are between $(11.5)^3$ and $(11.7)^3$?


Suppose $a$, $b$, and $c$ are positive numbers such that $ab = 36$, $ac = 72$, and $bc = 108$. Find the value of $a + b + c$.


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha^2}{\beta}$ is a real number and $|\alpha - \beta| = 4$. Find $|\alpha|$.


Consider the following problem: A line parametrized by $\begin{pmatrix} a \\ -2 \end{pmatrix}$ passes through the points $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} -4 \\ 1 \end{pmatrix}.$ Find the value of $a.$


In triangle $ABC$, $\angle A = 90^\circ$, $BC = 12$, and $\tan C = 4\cos B$. What is $AB$?


An equilateral triangle and a rectangle have the same perimeter of 24 inches. The rectangle's length is twice its width. What is the ratio of the side length of the triangle to the width of the rectangle? Express your answer as a common fraction.


Find all real numbers \( x \) which satisfy \[\frac{x-2}{x-4} \ge 3.\]

Solution: \boxed{(4, 5]}


What is the sum of all of the positive even factors of $720$?


Simplify $5 \cdot \frac{21}{8} \cdot \frac{32}{-63}$.


Solution: \boxed{1 + i}


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 $\overline{OB}$?


Let $z$ and $w$ be complex numbers such that $|z| = 2$ and $|w| = 4$. If $|z+w| = 5$, what is $\left | \frac{1}{z} + \frac{1}{w} \right|$?


Find the largest value of $n$ such that $6x^2 + nx + 72$ can be factored as the product of two linear factors with integer coefficients.


What is the smallest odd number with five different prime factors?


Two numbers $120$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors?


When two fair 8-sided dice are tossed, the numbers $c$ and $d$ are obtained, where each die has numbers from 1 to 8. What is the probability that the two-digit number $cd$ (where $c$ and $d$ are digits) and $c$ and $d$ themselves are all divisible by 4?


An arithmetic sequence with first term $3$ has a common difference of $8$. A second sequence begins with $5$ and has a common difference of $9$. In the range of $1$ to $150$, what is the largest number common to both sequences?


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}}


Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ $d,$ and $e,$
\[\sqrt{\frac{a}{b + c + d + e}} + \sqrt{\frac{b}{a + c + d + e}} + \sqrt{\frac{c}{a + b + d + e}} + \sqrt{\frac{d}{a + b + c + e}} + \sqrt{\frac{e}{a + b + c + d}} > m.\]


Calculate the sum of the numbers in Row 10 of Pascal's Triangle.


Express $2.\overline{06}$ as a reduced fraction, given that $0.\overline{02}$ is $\frac{2}{99}$.


A list of seven positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element?


Suppose $z$ and $w$ are complex numbers such that:
\[|z| = |w| = 1\]
and
\[z \overline{w} + \overline{z} w = 2.\]
Find the largest possible value of the real part of $z + w.$


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


April has five different basil plants and three 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 half of the absolute value of the difference of the squares of 20 and 15?


What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,18]=90$?


Find the product of the greatest common divisor and the least common multiple of $90$ and $135$.


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 $x + y$?


In how many different ways can 4 men and 5 women be placed into two groups of three people and one group of three people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable.


If the eight solutions of $x^8 = -256$ are written in the form $a+bi$, where $a$ and $b$ are real, find the product of those solutions with $a>0.$


Find the largest integer $n$ for which $18^n$ evenly divides $30!$.


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

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


How many three-digit numbers are multiples of neither 6 nor 8?


What is the total area, in square units, of the four triangular faces of a right, square-based pyramid that has base edges measuring 8 units and lateral edges measuring 10 units?


Find the minimum value of
\[\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}\]over all real numbers $x.$


For how many positive integers $n$ does $1+2+\cdots+n$ evenly divide $8n$?


Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.


Find the value of $k$ so that the line $4x + 7y + k = 0$ is tangent to the parabola $y^2 = 16x.$


The sum of the first 1500 terms of a geometric sequence is 300. The sum of the first 3000 terms is 570. Find the sum of the first 4500 terms.


Find the number of positive integers $n$ that satisfy
\[(n - 1)(n - 3)(n - 5) \dotsm (n - 99) < 0.\]


Two distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7\}$. What is the probability that their product is an even number? Express your answer as a common fraction.


Find the sum of all possible positive integer values of \( b \) such that the quadratic equation \( 3x^2 + 7x + b = 0 \) has rational roots.


What is the sum of the digits of the decimal representation of $2^{2010} \times 5^{2008} \times 7$?


In trapezoid $EFGH$, sides $\overline{EF}$ and $\overline{GH}$ are parallel, $\angle E = 3\angle H$, and $\angle G = 2\angle F$. Find $\angle F$.

Solution: \boxed{\angle F = 60^\circ}


Find $100^{-1} \pmod{101}$, as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)


For each value of $x,$ $g(x)$ is defined to be the minimum value of the three numbers $3x + 3,$ $x + 2,$ and $-\frac{1}{2} x + 8.$ Find the maximum value of $g(x).$


In $\triangle ABC$, we have $AB = AC = 26$ and $BC = 24$. Point $D$ is the midpoint of $\overline{BC}$. Find $AD$.

Solution: \boxed{AD = 2\sqrt{133}}


Quadrilateral $ABCD$ has right angles at $B$ and $D$, and $AC=5$. If $ABCD$ has two sides with distinct integer lengths, then what is the area of $ABCD$? Express your answer in simplest radical form.


Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 73 \times 83 \times 93$ is divided by $5$.


What is the least three-digit whole number, the product of whose digits is 8?


Consider the fractional decomposition:
\[
\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2} + \frac{D}{x + 3} + \frac{E}{x + 4} + \frac{F}{x + 5}
\]
for some constants $A$, $B$, $C$, $D$, $E$, and $F$. Find $A + B + C + D + E + F$.


The product of three different positive integers is equal to $5^4$. What is the sum of these three integers?


Solution: \boxed{XY = 10 - 5\sqrt{2}}


A five-digit even integer will be chosen at random from all possible positive five-digit even integers. What is the probability that the number's units digit will be less than 3?


How many three-digit positive integers are congruent to 2 (mod 4)?


A circle of radius 1 is surrounded by 3 equal circles of radius \( r \) such that each smaller circle is tangent to two others and the central circle. Determine the radius \( r \).

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


What is the largest number, all of whose digits are either 4 or 3, and whose digits add up to $20$?


Find the absolute value of the difference of the solutions of $x^2 - 6x + 8 = 0$, and find the sum of the absolute values of these solutions.


Let $a$, $b$, and $c$ be positive integers such that $\gcd(a,b) = 130$ and $\gcd(a,c) = 770$. What is the smallest possible value of $\gcd(b,c)$?


The line joining $(2,3)$ and $(5,1)$ divides the square shown into two parts. Find the fraction of the area of the square that is above this line. The square vertices are at $(2,1)$, $(5,1)$, $(5,4)$, and $(2,4)$.


What is the coefficient of \(x^3y^5\) in the expansion of \(\left(\frac{2}{3}x - \frac{y}{3}\right)^8\)?


In right triangle $ABC$, with $AB=8$, $BC=6$, and hypotenuse $CA=10$. Point $D$ is on $\overline{AB}$, $E$ is on $\overline{BC}$, and $F$ is on $\overline{CA}$. Let $AD=p \cdot AB$, $BE=q \cdot BC$, and $CF=r \cdot CA$ where $p$, $q$, and $r$ are positive and satisfy $p+q+r=1$ and $p^2+q^2+r^2=1/2$. Find the ratio of the area of triangle $DEF$ to the area of triangle $ABC$ in its simplest form.

Solution: \boxed{\frac{[DEF]}{[ABC]} = \frac{1}{4}}


** What is the value of $0.\overline{567} + 0.\overline{345} - 0.\overline{234}$? Express your answer as a fraction in lowest terms.


Let \(\triangle PQR\) be a right triangle with \(Q\) as the right angle. A circle with diameter \(QR\) intersects side \(PR\) at point \(S\). If the area of \(\triangle PQR\) is \(120\) and \(PR = 24\), find the length of \(QS\).


Find a positive integer that is divisible by 18 and has a square root between 26 and 26.5.


What is the least number of colors needed to shade a tessellation consisting of alternating rows of squares and hexagons such that no two tiles sharing a side are the same color?


Find the intersection of the lines $8x - 5y = 40$ and $6x + 2y = 14.$ Express your answer as an ordered pair $(x, y)$.

Solution: \boxed{\left(\frac{75}{23}, \frac{-64}{23}\right)}


Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[a^3 + b^3 + c^3 + d^3 = 8.\]
Find the maximum value of $a^4 + b^4 + c^4 + d^4$.


Find the number of complex numbers $z$ satisfying $|z| = 1$ and
\[\left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} + 1 \right| = 2.\]


What is the smallest positive integer $n$ that satisfies $$77n \equiv 308 \pmod{385}~?$$


Mary is making peanut butter cookies for her school fair. If she divides the cookies equally among 9 of her classmates, she'll have 3 cookies left over. If she divides the cookies equally among 5 of her classmates, she'll have 2 cookies left over. Assuming Mary made fewer than 100 cookies, what is the sum of the possible numbers of cookies that she could have made?


Let $D$ be the determinant of the matrix whose column vectors are $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$. Find the determinant of the matrix whose column vectors are $2\mathbf{a} + \mathbf{b}, \mathbf{b} + 2\mathbf{c},$ and $\mathbf{c} + 2\mathbf{a},$ in terms of $D$.


In triangle $PQR$, the side-lengths are $PQ = 15$, $QR = 30$, and $PR = 25$. A line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y$. What is the perimeter of $\triangle PXY$?


What is the smallest four-digit number that is divisible by $35$?


What is the sum of all positive integers $n$ that satisfy $$\mathop{\text{lcm}}[n,120] = \gcd(n,120) + 300~?$$


Quadrilateral $PQRS$ is a square. A circle with center $S$ has arc $PMT$. A circle with center $Q$ has arc $PNT$. If $PQ = 4$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the nearest tenth.


In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are placed on top of the grid. The largest circle is placed centrally and touches midpoints of the grid sides, and three smaller circles of radius 1.5 cm each are placed at the top-left, top-right, and bottom-left corners of the grid. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value of $A+B$?


Find $x,$ given that $x$ is nonzero and the numbers $\{x\},$ $\lfloor x \rfloor + 1,$ and $x$ form an arithmetic sequence in that order.

Solution: \boxed{-2 \text{ and } -\frac{1}{2}}


The median of the set of numbers $\{15, 35, 50, x, 18\}$ is eight less than the mean. If $x$ is a positive integer, what is the value of $x$?


Two circles of radius 3 are centered at $(0,3)$ and at $(3,0)$. What is the area of the intersection of the interiors of these two circles? Express your answer in terms of $\pi$ in its simplest form.

Solution: \boxed{\frac{9\pi - 18}{2}}


A point is selected at random from the portion of the number line from 0 to 8. What is the probability that the point is closer to 4 than to 0 or 8?


In triangle \( PQR \), points \( M \) and \( N \) are on \(\overline{PQ}\) and \(\overline{PR}\), respectively, and angle bisector \(\overline{PS}\) intersects \(\overline{MN}\) at \( X \). If \( PM = 2, MQ = 6, PN = 3, \) and \( NR = 9, \) compute \(\frac{PX}{PS}\).

Solution: \boxed{\frac{5}{20} = \frac{1}{4}}


Five eight-sided dice (each sided from 1 to 8) are rolled. Calculate the probability that the sum of the numbers rolled is odd, provided that the product of the numbers is even.


Calculate the following sum:
\[\text{cis } 40^\circ + \text{cis } 50^\circ + \text{cis } 60^\circ + \dots + \text{cis } 140^\circ\]
and express in the form \(r \, \text{cis } \theta\), where \(r > 0\) and \(0^\circ \leq \theta < 360^\circ\). Find \(\theta\) in degrees.


Suppose $f(x) = ax + b$ and $g(x) = -4x + 6$. If $h(x) = f(g(x))$ and $h^{-1}(x) = x + 9$, find the value of $2a-b$.


How many different positive three-digit integers can be formed using only the digits in the set $\{1, 2, 2, 3, 4, 4, 4\}$ if no digit may be used more times than it appears in the given set of available digits?


$ABCD$ is a trapezoid where base $\overline{AB}$ is three times the length of base $\overline{CD}$. The diagonals of the trapezoid intersect at point $E$. Given that the length of diagonal $\overline{BD}$ is 15, determine the length of segment $\overline{ED}$.


Determine the maximum number of Tuesdays that can occur in the first 36 days of a year.


If $\frac{3}{4} = \frac{y}{215+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers.


Jack recorded times, in seconds, of 102, 107, 109, 103, and 105 for his first five 100-meter sprints. After his sixth sprint, he reduced his median time to 104 seconds. What was his time, in seconds, for his sixth attempt?


Let \( a, b, c, d \) be the roots of the polynomial \( x^4 + qx^2 + rx + s = 0 \). Compute the determinant
\[
\begin{vmatrix}
a & b & c & d \\
b & c & d & a \\
c & d & a & b \\
d & a & b & c
\end{vmatrix}.
\]


The sidewalk in front of Khali's house is now 30 feet long and 3 feet wide. Initially, the snow is $\frac{3}{4}$ foot deep. While he is shoveling, an additional snowfall of $\frac{1}{4}$ foot accumulates evenly across the sidewalk. How much snow does Khali have to shovel off in total?


You are given a sequence of $60$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times\ldots \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $1, 2, 3, \ldots, 60$. Let $N$ be the number of primes appearing in this sequence. What is $N$?

Solution: \boxed{N = 1 \text{ or } 0}


A Senate committee consists of 6 Democrats and 4 Republicans. In how many ways can they sit around a circular table if no two Republicans are allowed to sit next to each other?


The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.

\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular}


Given a quadrilateral $EFGH,$ side $\overline{EF}$ is extended past $F$ to $E'$ so that $E'F = EF.$ Points $F',$ $G',$ and $H'$ are similarly constructed.

After this construction, points $E,$ $F,$ $G,$ and $H$ are erased. You only know the locations of points $E',$ $F',$ $G'$ and $H',$ and want to reconstruct quadrilateral $EFGH.$

There exist real numbers $u,$ $v,$ $w,$ and $x$ such that
\[\overrightarrow{E} = u \overrightarrow{E'} + v \overrightarrow{F'} + w \overrightarrow{G'} + x \overrightarrow{H'}.\]Enter the ordered quadruple $(u,v,w,x).$

Solution: \boxed{\left( \frac{1}{15}, \frac{2}{15}, \frac{4}{15}, \frac{8}{15} \right)}


Let \(\mathbf{N} = \begin{pmatrix} 3 & 1 \\ 4 & -2 \end{pmatrix}\). Find constants \(c\) and \(d\) such that
\[\mathbf{N}^{-1} = c \mathbf{N} + d \mathbf{I}.\]
Enter the ordered pair \((c,d)\).

Solution: \boxed{\left(\frac{1}{10}, -\frac{1}{10}\right)}


Calculate $\left\lceil \left(\frac{15}{8}\right)^2 \cdot \frac{-34}{4}\right\rceil - \left\lfloor \frac{15}{8} \cdot \left\lfloor \frac{-34}{4}\right\rfloor \right\rfloor$.


What is the largest four-digit negative integer congruent to $1 \pmod{17}$?


Point $(x, y)$ is randomly picked from the rectangular region with vertices at $(0, 0), (4040, 0), (4040, 2020),$ and $(0, 2020)$. What is the probability that $x > 5y$? Express your answer as a common fraction.


Mrs. Carly awards extra credit in her class for scores exceeding the class mean. If 150 students take the quiz, what is the maximum number of students who can receive extra credit?


A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,3), (3,3), and (3,0). What is the probability that $x+y < 4$?


A spiral staircase turns $360^\circ$ as it rises 12 feet. The radius of the staircase is 3 feet. What is the length of the handrail? Express your answer as a decimal to the nearest tenth.


How many positive, three-digit integers contain at least one $6$ as a digit but do not contain a $4$ as a digit?


What is the least positive integer $n$ such that $7875$ is a factor of $n!$?


Let \( h(x) = x^3 + x^2 + x + 1 \). What is the remainder when the polynomial \( h(x^8) \) is divided by the polynomial \( h(x) \)?


Let $g(x) = \left\lceil\dfrac{2}{x+3}\right\rceil$ for $x > -3$, and $g(x) = \left\lfloor\dfrac{2}{x+3}\right\rfloor$ for $x < -3$. ($g(x)$ is not defined at $x = -3$.) Which integer is not in the range of $g(x)$?


Let $\mathbf{S}$ be the transformation matrix corresponding to scaling by a factor of 2 in the x-direction, followed by a rotation about the origin counter-clockwise by $45^\circ$. Find $\det \mathbf{S}$.


Two sides of a right triangle have the lengths 6 and 8. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth.


How many integers between 400 and 600 have the sum of their digits equal to 15?


How many positive multiples of 3 that are less than 150 have a units digit of 3 or 6?


My school's math club has 10 boys and 10 girls. I need to select a team of 8 people to send to the state math competition. The team must include at least 2 boys and at least 2 girls. In how many ways can I select the team under these restrictions?


Michael is writing a feature for the school newspaper about students who take language classes. The school offers German and Japanese language classes. Michael has a list of all 30 students in the school enrolled in at least one language class. He finds out that 20 students are taking German and 24 students are taking Japanese. If Michael randomly selects two students from his list to interview, what is the probability he can cover both the German and Japanese classes based on his interviews? Express your answer as a fraction in simplest form.


Consider the function $y = \frac{3-x}{4x+5}$. For which value of $x$, excluding any values making the denominator zero, is $y$ unattainable?


Two distinct positive integers from 1 to 100 inclusive are chosen.  Let $a$ and $b$ be these integers. Determine the probability that the sum and product of $a$ and $b$ are such that $ab + a + b + 1$ is a multiple of 7.


What is the number of square units in the area of trapezoid PQRS with vertices P(0,0), Q(-2,0), R(-2,5), and S(3,5)?


How many three-digit numbers are multiples of neither 3 nor 8?


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


Let \( A=(0,6) \) and \( B=(0,10) \). Points \( A' \) and \( B' \) are on the line \( y=x \), and \( \overline{AA'} \) and \( \overline{BB'} \) intersect at \( C=(3,7) \). Find the length of \( \overline{A'B'} \).


Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if $CX$ bisects $\angle ACB$. In the diagram, $AB = 34$, $BC = 35$, and $AC = 39$. Let the coordinates be:
- $A = (-17,0)$,
- $B = (17,0)$,
- $C = (0,30)$.
Express your answer as a common fraction.


**Mary can mow a lawn in three hours, and Tom can mow the lawn in five hours. If they start mowing the lawn together but Tom leaves after 1.5 hours, how much of the lawn remains to be mowed?**

Revised


The values of $a$, $b$, $c$, and $d$ are 2, 3, 4, and 5, but not necessarily in that order. What is the largest possible value of the sum of the four products $ab$, $ac$, $bd$, and $cd$?


I have a picture with dimensions $x$ and $y$ (in inches), with both $x$ and $y$ as integers greater than one. I want to set this picture in a frame of dimensions $(3x + 2)$ and $(y + 4)$. If the area of the frame (shaded region only) is measured to be $62$ square inches, what is the area of the picture in square inches?


Both roots of the quadratic equation \( x^2 - 67x + k = 0 \) are prime numbers. Find the number of possible values of \( k \).


Consider a function \( g(x) \) defined for all real numbers \( x \) that satisfies
\[ g(x - y) = g(x) + g(y) - 1 \]
for all real numbers \( x \) and \( y \), and \( g(x) \neq 0 \) for all \( x \). Find \( g(4) \).


Let
\[
\mathbf{C} = \begin{pmatrix} 3 & 1 \\ -4 & -1 \end{pmatrix}.
\]
Compute $\mathbf{C}^{50}$.

Solution: \boxed{\begin{pmatrix} 101 & 50 \\ -200 & -99 \end{pmatrix}}


Solution: \boxed{27 \sqrt[3]{3}}


Find the positive value of $x$ which satisfies
\[\log_3 (x - 1) + \log_{\sqrt{3}} (x^2 - 1) + \log_{\frac{1}{3}} (x - 1) = 3.\]

Solution: \boxed{\sqrt{1 + 3\sqrt{3}}}


Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2$, $\|\mathbf{b}\| = 6$, and $\|\mathbf{c}\| = \sqrt{32}$, and
\[\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}.\]Compute $\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}.$


The base of a triangular piece of paper $ABC$ is $15\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base of the paper. The area of the triangle that projects below the base is $25\%$ that of the area of triangle $ABC$. What is the length of $DE$, in cm?

Solution: \boxed{7.5\text{ cm}}


Let $p$ and $q$ be the roots of $m(x^2 - 2x) + 3x + 4 = 0.$  Let $m_1$ and $m_2$ be the values of $m$ for which $p$ and $q$ satisfy
\[\frac{p}{q} + \frac{q}{p} = 2.\]Find
\[\frac{m_1}{m_2} + \frac{m_2}{m_1}.\]


Find all real numbers $x$ such that \[4 \le \frac{x}{3x-7} < 9.\] (Give your answer in interval notation.)

Solution: \boxed{\left(\frac{63}{26}, \frac{28}{11}\right]}


Simplify
\[\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x}.\]

Solution: \boxed{\frac{2}{\sin x} = 2 \csc x}


What integer $n$ satisfies $0 \leq n < 25$ and $$-150 \equiv n \pmod{25}~?$$


Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He chooses a card at random. What is the probability that the number on the card he picks is a multiple of 4, 5 or 6? Express your answer as a reduced fraction.


Find the smallest positive integer solution for the sum, \(a + b + c\), such that the equation \(\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2.5\) can be reduced to the equation \(\cos ax \cos bx \cos cx = 0\) for some positive integers \(a\), \(b\), and \(c\).


How many integers from 1 through 8888, inclusive, do not contain any of the digits 2, 4, 5, or 9?


Find the arithmetic mean of the reciprocals of the first four prime numbers, including the number 7 instead of 5.


Given a cubic polynomial $ 60 x^3 - 80x^2 + 24x - 2 $, which has three roots $\alpha, \beta, \gamma$, all distinct and lying in the interval $(0, 1)$. Find the sum
\[
\frac{1}{1-\alpha} + \frac{1}{1-\beta} + \frac{1}{1-\gamma}.
\]


Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 6$ and $xy + xz + yz = 9.$  Find the largest possible value of $x.$


A segment with endpoints at $A(3, -3)$ and $B(15, 3)$ is extended through $B$ to point $C$. If $BC = \frac{1}{2} \cdot AB$, what are the coordinates for point $C$? Express your answer as an ordered pair.

Solution: \boxed{(21, 6)}


The value of $\log_{10}{25}$ is between which two consecutive integers $c$ and $d$? Find $c+d$.


The algae in a local lake expands to cover double its area each day. By the end of the 24th day of the month, the lake is entirely covered with algae. Determine on which day of the month was only 10% of the lake covered with algae.


In the diagram, if $\angle PQR = 60^\circ$, calculate the measure of $\angle PMN$, assuming $\triangle PMR$ is an isosceles triangle. Consider $\triangle RQN$ is also isosceles with $RQ = RN$. [asy]

size(6cm);

pair p = (0, 0);
pair m = dir(180 - 30);
pair n = dir(180 + 30);
pair r = 1.3 * dir(30);
pair q = 2 * 1.3 * cos(60) * dir(-30);

label("$M$", m, N);
label("$R$", r, N);
label("$P$", p, 1.5 * S);
label("$N$", n, S);
label("$Q$", q, SE);

draw(m--q--r--n--cycle);

add(pathticks(m--p, s=4));

add(pathticks(n--p, s=4));

add(pathticks(r--p, 2, spacing=0.9, s=4));

add(pathticks(r--q, 2, spacing=0.9, s=4));

[/asy]


Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(2)=0$, $30<f(5)<40$, $50<f(6)<60$, and $1000k<f(50)<1000(k+1)$ for some integer $k$. Find the value of $k$.


Find the common ratio of the infinite geometric series: $$\frac{7}{3} + \frac{49}{9} + \frac{343}{27} + \dots$$


In quadrilateral $EFGH$, sides $\overline{EF}$ and $\overline{FG}$ both have length 12, sides $\overline{GH}$ and $\overline{HE}$ both have length 20, and the measure of angle $EHG$ is $60^\circ$. What is the length of diagonal $\overline{EG}$?


Find the quotient when $x^5 - 22x^3 + 12x^2 - 16x + 8$ is divided by $x - 3$.

Solution: \boxed{x^4 + 3x^3 - 13x^2 - 27x - 97}


Four of the following test scores are Cyprian's and the other two are Margaret's: 78, 81, 85, 87, 90, 92. Cyprian's mean score is 84. What is Margaret's mean score?


A property value increased by 30% in 2018. Assuming the value started from this increased amount, what percent would it have to decrease in 2019 to return to its original price at the beginning of 2018?


I own 6 shirts, 6 pairs of pants, 6 hats, and 6 pairs of shoes, each available in 6 different colors. I refuse to wear an outfit where any two pieces have the same color. How many outfit combinations can I choose from?


Let \(x\) be a real number. Consider the following five statements:
1. \(0 < x^2 < 1\)
2. \(x^2 \geq 4\)
3. \(-1 < x < 0\)
4. \(0 < x < 1\)
5. \(0 < x - x^2 \leq \frac{1}{4}\)

What is the maximum number of these statements that can be true for any value of \(x\)?


For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{15400}$ terminate?


Find a positive integer that is divisible by 18 and has a square root between 30 and 30.2.


For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[15+15r+15r^2+15r^3+\cdots .\]Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=2430$. Find $T(b)+T(-b)$.


Find the 6-digit repetend in the decimal representation of $\frac{7}{29}$.


In a cube with side length 2 units, the centers of the six faces are the vertices of a smaller octahedron. Determine the ratio of the volume of the smaller octahedron to that of the cube. Provide the sum of the numerator and denominator of the simplified fraction.


The admission fee at a museum is $ \$30$ per adult and $ \$15$ per child. Last Friday, the museum collected a total of $ \$2700$ in admission fees, with the restriction that there were at least two adults and at least two children. Find the ratio of adults to children that is closest to $2$.


For constants $a$ and $b$, let
\[f(x) = \left\{
\begin{array}{cl}
ax + b & \text{if } x < 1, \\
7 - 2x & \text{if } x \ge 1.
\end{array}
\right.\]
The function $f$ has the property that $f(f(x)) = x$ for all $x$. What is the value of $a + b$?


Let \( y_1, y_2, y_3 \) be positive real numbers such that \( y_1 + 3y_2 + 4y_3 = 72 \). Find the smallest possible value of
\[
y_1^2 + y_2^2 + y_3^2.
\]


In Pascal's Triangle, each number is the sum of the number just above it and to the left and the number just above it and to the right. What is the sum of the numbers in Row 10 of Pascal's Triangle?


Determine the solution of the equation $9^{x+6} = 10^x$ in the form $x = \log_c 9^6$. Identify the value of $c$.


Compute the sum given by \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]


Let \( P \) be the point on line segment \( \overline{AB} \) such that \( AP:PB = 3:4 \). Then
\[ \overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B} \]
for some constants \( t \) and \( u \). Enter the ordered pair \( (t, u) \).

Solution: \boxed{\left( \frac{4}{7}, \frac{3}{7} \right)}


What is the base ten equivalent of $23456_{7}$?


Let $z$ be a complex number such that
\[ |z^2 + 9| = |z(z + 3i)|. \]
Find the smallest possible value of $|z + i|$.


For $0 \le p \le 2$ and $0 \le q \le 1$, define a new function $H(p, q)$ by
\[
H(p, q) = -3pq + 4p(1-q) + 2(1-p)q - 5(1-p)(1-q).
\]
Define $K(p)$ to be the maximum of $H(p, q)$ over all $q$ in the interval $0 \le q \le 1$. Determine the value of $p$ in the interval $0 \le p \le 2$ that minimizes $K(p)$.


Four points are each $4$ units away from the line $y = 10$ and $12$ units away from the point $(5,10)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?


The quadratic polynomial $Q(x),$ with real coefficients, satisfies
\[Q(x^3 - x) \ge Q(x^2 - 1)\] for all real numbers $x.$  Find the sum of the roots of $Q(x).$


Find the value of $y$ if $y$ is positive and $2y \cdot \lfloor y \rfloor = 162$. Express your answer as a decimal.


There are 5 companies at a convention, and each company sends 4 representatives. Each person shakes hands only with people from two designated companies, avoiding handshakes with their company members and two other companies. How many handshakes occur?


In rectangle $ABCD$, $AB = 8$ and $BC = 6$. Points $J$ and $K$ are on $\overline{CD}$ such that $DJ = 2$ and $KC = 3$. Lines $AJ$ and $BK$ intersect at $L$. Find the area of $\triangle ABL$. [asy]
pair A,B,C,D,J,K,L;
A=(0,0);
B=(8,0);
C=(8,6);
D=(0,6);
J=(2,6);
K=(5,6);
L=(2.2,8);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(A--J--L--cycle,linewidth(0.7));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$J$",J,SE);
label("$K$",K,SW);
label("$L$",L,N);
label("2",(1,6),N);
label("3",(6.5,6),N);
label("6",(0,3),W);
label("6",(8,3),E);
label("8",(4,0),S);
[/asy]


An 8 by 8 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard?


Humanity discovers 15 habitable planets; 8 are "Earth-like" and 7 are "Mars-like". If each Earth-like planet requires 3 units of colonization and each Mars-like planet requires 1 unit, and humanity can deploy 18 total units of colonization, how many different combinations of planets can be occupied assuming all planets are distinct?


Phoenix hiked the Serpent Ridge Trail last week. It took her five days to complete the trip. The first two days she hiked a total of 28 miles. The second and fourth days she averaged 15 miles per day. The last three days she hiked a total of 42 miles. The total hike for the first and third days was 30 miles. How many miles long was the trail?

Solution: \boxed{70 \text{ miles}}


Find the remainder when \(x^{150}\) is divided by \((x + 1)^3\).

Solution: \boxed{11175x^2 + 22200x + 11026}


Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \(a,\) \(b,\) and \(c\) are positive real numbers.


The Lucas sequence is given by the terms 1, 3, 4, 7, 11, $\ldots$ where each term is the sum of the previous two terms, starting from 1 and 3. What is the remainder when the $50^{\mathrm{th}}$ term of the sequence is divided by 5?


I have 7 books, three of which are identical copies of a mystery novel, and two others are identical copies of a science fiction novel, with the rest of the books being different. How many ways can I arrange these books on a shelf?


In an isosceles triangle, the two equal sides are 15 cm each, and the base is 24 cm. If the base of a similar triangle is 60 cm, find the perimeter of this larger triangle.

Solution: \boxed{135 \text{ cm}}


Roger collects the first 18 U.S. state quarters released in the order that the states joined the union. Five states joined the union during the decade 1790 through 1799. What fraction of Roger's 18 quarters represents states that joined the union during this decade? Express your answer as a common fraction.


Suppose that $a$, $b$, and $c$ are digits, not all nine and not all zero, and the repeating decimal $0.\overline{abc}$ is expressed as a fraction in lowest terms. How many different denominators are possible?


How many integers are solutions to the equation $$(x-3)^{(16-x^2)}=1?$$


In a two-day problem-solving tournament, Alpha and Gamma both attempted questions worth a total of 600 points. Alpha scored 210 points out of 350 points on the first day, and 150 points out of 250 points on the second day. Gamma, who did not attempt 350 points on the first day, had a positive integer score on each of the two days, and Gamma's daily success ratio (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $\frac{360}{600} = 3/5$.

Find the largest possible two-day success ratio that Gamma could have achieved.


In polar coordinates, what is the equivalent point in the standard polar coordinate representation for the point $\left(-3, \frac{5\pi}{6}\right)$?

Solution: \boxed{\left(3, \frac{11\pi}{6}\right)}


When the expression $3(x^2 - 3x + 3) - 8(x^3 - 2x^2 + 4x - 1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms?


A rectangle has vertices at \((-2, z), (6, z), (-2, 4)\), and \((6, 4)\). The area of the rectangle is 64 square units, and \( z \) is positive. Determine the value of \( z \).


The lengths of the sides of a triangle with positive area are $\log_{2}15$, $\log_{2}45$, and $\log_{2}m$, where $m$ is a positive integer. Determine the number of possible values for $m$.


Define function $g$ on the set of integers such that:
\[g(n)= \begin{cases}
n-3 & \mbox{if }n\ge 1200 \\
g(g(n+6)) & \mbox{if }n<1200.
\end{cases}\]
Find $g(84)$.

Solution: \boxed{g(84) = 1197}


The function $g(x)$ satisfies
\[ 4g(x) - 3g\left(\frac{1}{x}\right) = 2x^2 \]
for all $x \neq 0.$ Find $g(2)$.


The integers $a$ and $b$ are randomly selected, where $-4 < a < 7$ and $0 < b < 9$. What is the probability that the division $a \div b$ results in an integer? Express your answer as a common fraction.


Find the greatest common divisor of $50400$ and $37800$.


Let $x<y<z$ be three integers such that $x, y, z$ is an arithmetic progression and $x, z, y$ is a geometric progression. What is the smallest possible value of $z$?


What is the smallest whole number $b$ such that 124 can be expressed in base $b$ using only three digits?


For real numbers $x$, let \[f(x) = \left\{
\begin{array}{cl}
x^2 + x + 2 &\text{ if }x>3, \\
2x + a &\text{ if }x\le 3.
\end{array}
\right.\] What value must $a$ have to make the piecewise function continuous?


A circle is inscribed around a quadrilateral $WXYZ$ such that $\angle WYZ = 20^\circ$ and $\angle WXY = 70^\circ$. Determine the sum of degrees in $\angle YWX + \angle YZW$.


The sides of a triangle have lengths \( 13, 17, \) and \( k \), where \( k \) is a positive integer. For how many values of \( k \) is the triangle obtuse?


Compute \[ \left\lfloor \dfrac {102^3}{100 \cdot 101} - \dfrac {100^3}{101 \cdot 102} \right\rfloor,\]where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


A projectile is launched from the ground with an initial speed $u$ at an angle of $\phi$. The motion of the projectile can be modeled by the parametric equations:
\[
x = u t \cos \phi, \quad y = u t \sin \phi - \frac{1}{2} g t^2,
\]
where $t$ denotes time, $g$ is the acceleration due to gravity, and $\phi$ varies from $0^\circ$ to $180^\circ$. Determine the area enclosed as the highest points of the trajectories form a closed curve, expressed in the form $k \cdot \frac{u^4}{g^2}$, and find the value of $k$.


Let \( A \) be the greatest common factor and \( B \) be the least common multiple of 9, 18, and 36. What is the value of \( A + B \)?


If $\frac{1}{8}$ of $2^{36}$ equals $4^x$, then what is the value of $x$?


Each outcome on the spinner below has equal probability. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit and the third outcome is the units digit, what is the probability that you will end up with a three-digit number that is divisible by 5? The spinner has labels 5, 1, and 2.


Evaluate $\log_4 64\sqrt2$. Express your answer as an improper fraction.


For some positive integers $a$ and $b$, compute $a+b$ if the product \[\log_a(a+1) \cdot \log_{a+1} (a+2) \dotsm \log_{b-2} (b-1) \cdot \log_{b-1} b\] contains exactly $1560$ terms, and its value is $3.$


In a triangle ABC, angle A is $x$ degrees, angle B is $2x$ degrees, and angle C is $45^\circ$. What is the number of degrees in $x$?


In Ms. Johnson's class, each student averages two days absent out of thirty school days. What is the probability that out of any three students chosen at random, exactly two students will be absent and one will be present on a Monday, given that on Mondays the absence rate increases by 10%? Express your answer as a percent rounded to the nearest tenth.


Point \((x, y)\) is randomly selected from the rectangular region with vertices at \((0, 0), (3000, 0), (3000, 4000),\) and \((0, 4000)\). What is the probability that \(x > 4y\)? Express your answer as a common fraction.


A circle with radius 2 rolls around the inside of a right triangle with sides 9, 12, and 15. The circle remains tangent to at least one side of the triangle at all times. Determine the total distance traveled by the center of the circle when it returns to its starting position.


A sphere has a surface area of \(256\pi\text{ cm}^2\). Calculate its volume in cubic centimeters and express your answer in terms of \(\pi\).

Solution: \boxed{\frac{2048}{3}\pi \text{ cm}^3}


At a school cafeteria, Jennifer wants to buy a lunch consisting of one main course, one beverage, and one snack. The table below lists Jennifer’s options in the cafeteria. How many distinct possible lunches can she buy from these options?

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


For a polynomial $p(x),$ define its munificence as the maximum value of $|p(x)|$ on the interval $-2 \le x \le 2.$  For a monic quadratic polynomial, find the smallest possible munificence.


In base $b_4$, how many four-digit numbers exist where all four digits are distinct and there are exactly 168 such numbers? Find $b_4$.