Rationalize the denominator of $\frac{2+\sqrt{5}}{3-\sqrt{5}}$. Express your answer in the form $A+B\sqrt{C}$, where $A$, $B$, and $C$ are integers, and compute the value of $ABC$.


Let $A$ equal the number of six-digit odd numbers. Let $B$ equal the number of six-digit multiples of 3. Find $A+B$.


What is the sum of the tens digit and the units digit in the decimal representation of \(8^{100}\)?


What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 360 = 0$ has integral solutions with one root being a multiple of the other?


Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the ninth row?


Chloe wants to buy a pair of sneakers that cost $\$47.50$. She has two $20$ bills, six $1$ bills, ten quarters, and a number of nickels. What is the minimum number of nickels that must be in her pile so she can pay for the sneakers?


Find the distance between the foci of the ellipse and its eccentricity when defined by the equation:
\[\frac{x^2}{16} + \frac{y^2}{9} = 8.\]


The following line is parameterized so that its direction vector is of the form $\begin{pmatrix} a \\ 2 \end{pmatrix}$. Find $a$.

[asy]
unitsize(0.4 cm);

pair A, B, L, R;
int i, n;

for (i = -8; i <= 8; ++i) {
  draw((i,-8)--(i,8),gray(0.7));
  draw((-8,i)--(8,i),gray(0.7));
}

draw((-8,0)--(8,0),Arrows(6));
draw((0,-8)--(0,8),Arrows(6));

A = (3, -1);
B = (-1, 4);
L = extension(A, B, (0,8), (1,8));
R = extension(A, B, (0,-8), (1,-8));

draw(L--R, blue);

label("$x$", (8,0), E);
label("$y$", (0,8), N);
[/asy]


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 1:2$. Point $F$ is on $AD$ such that $AF:FD = 1:3$. If the area of $\triangle DEF$ is 30, determine the area of $\triangle ABC$.


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


In an isosceles right-angled triangle, the sum of the squares of the three side lengths is 1452. What is the length of the hypotenuse?


The polynomial \( g(x)=x^4+ax^3+bx^2+cx+d \) has real coefficients, and \( g(-3i)=g(1+i)=0 \). What is \( a+b+c+d \)?


The grid has nine equally spaced points forming a 3x3 grid with a distance of 1 unit between each neighboring point. What is the area, in square units, of the region where a triangle with vertices at the top left, middle right, and bottom left of the grid overlaps with another triangle having vertices at the top right, middle left, and bottom right?


If $x$, $y$, and $z$ are positive integers satisfying $xy+z = yz+x = zx+y = 47$, what is the value of $x+y+z$?


Sue owns 12 pairs of shoes: seven identical black pairs, three identical brown pairs, and two identical gray pairs. If she picks two shoes at random, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? Express your answer as a common fraction.


If the ratio of $b$ to $a$ is 4, then what is the value of $a$ when $b=16-6a$?


Find the remainder when \(x^4 + 5\) is divided by \(x^2 - 4x + 7.\)

Solution: \boxed{8x - 58}


Given the vertices $(0, 3)$ and $(4, 0)$ of one square, find the total area when an identical square with its side adjacent to the first is added. Assume that both squares are aligned along one side.


Let $g(x)$ be a function defined for all positive integers, satisfying $g(x) > 0$ for all $x > 0$, and
\[g(x - y) = \sqrt{g(x/y) + 3}\] for all $x > 2y > 0$. Determine $g(2023)$.

Solution: \boxed{\frac{1 + \sqrt{13}}{2}}


How many multiples of 12 are between 35 and 247?


In trapezoid $PQRS$, the parallel sides $PQ$ and $RS$ have lengths of 10 and 30 units, respectively, and the altitude is 18 units. Points $T$ and $U$ are the midpoints of sides $PR$ and $QS$, respectively. What is the area of quadrilateral $TURS$?


A weather forecast predicts that from Sunday to Saturday in a particular week, there is a $30\%$ chance each day will be sunny and dry, a $20\%$ chance of 5 inches of rain, a $25\%$ chance of 7 inches of rain, and a $25\%$ chance of cloudiness with no rain. What is the expected value of the total number of inches of rain that will fall over these seven days? Provide the answer rounded to the nearest tenth.

Solution: \boxed{19.3 \text{ inches}}


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


The time right now is exactly midnight. What time will it be in 2405 minutes?

Solution: \boxed{4\!:\!05 \text{ p.m.}}


Given that \(7^{-1} \equiv 55 \pmod{102}\), find \(49^{-1} \pmod{102}\), as a residue modulo 102. (Give an answer between 0 and 101, inclusive.)


The set of vectors $\mathbf{u}$ such that
\[\mathbf{u} \cdot \mathbf{u} = \mathbf{u} \cdot \begin{pmatrix} 6 \\ -24 \\ 12 \end{pmatrix}\] forms a solid in space. Find the volume of this solid.

Solution: \boxed{756\sqrt{21} \pi}


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


What is the base ten equivalent of the base three number $102012_3$?


An infinite geometric series has a first term of $15$ and a second term of $9.$ A second infinite geometric series has the same first term of $15,$ a second term of $9+m,$ and a sum of three times that of the first series. Find the value of $m.$


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


Find the number of real solutions to the equation:
\[\frac{3}{x - 1} + \frac{4}{x - 2} + \frac{5}{x - 3} + \dots + \frac{102}{x - 100} = x.\]


My friend Kira has a preference for numbers divisible by both 3 and 4. How many different last digits are possible in numbers that Kira likes?


The line \(y = \frac{5}{3}x - 15\) crosses the \(x\)-axis at point \(P\) and the \(y\)-axis at point \(Q\). Point \(T(r, s)\) is on segment \(PQ\). If the area of \(\triangle POQ\) is twice the area of \(\triangle TOQ\), what is the value of \(r+s\)?


In the provided diagram, there are two overlapping triangles with three labeled angles each. In the first triangle, angles are labeled 1, 3, 5 and sum to $180^\circ$. In the second triangle, angles are labeled 2, 4, 6 and also sum to $180^\circ$. Another smaller triangle is formed by the intersection of these two triangles, with its angles labeled as A, B, C. How many degrees are in the sum of the measures of angles 1, 2, 3, 4, 5, 6, A, B, and C?


Determine the integer nearest to $500 \sum_{n=2}^{5000} \frac{1}{n^2 - 1}$.


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.6\overline{37}$?


How many integers between 1 and 300 are multiples of both 6 and 5 but not of either 8 or 9?


There are 25 people in a book club. 15 of them prefer reading fiction. 12 of them are left-handed. 3 of them are right-handed and do not prefer reading fiction. How many club members are left-handed and prefer reading fiction?


Solution: \boxed{65536 \cdot \sqrt[3]{2}}


How many rows of Pascal's Triangle contain the number $101$?


Given that $x$ is a multiple of $2520$, what is the greatest common divisor of $g(x) = (4x+5)(5x+2)(11x+8)(3x+7)$ and $x$?


Mary wants to compute the area of her rectangular garden plot, and she measures the length as 15 meters and the width as 10 meters. Since her measuring tools aren't perfect, each dimension could be off by up to $10\%$. What is the largest possible percent error, in percent, in Mary's computed area of the plot?


Ten positive integers from a list include $5, 9, 3, 6, 10, 8$. What is the largest possible value of the median of this list of ten positive integers if all newly added integers must be consecutive starting from 11?


There are four points that are $3$ units from the line $y=10$ and $10$ units from the point $(5,10)$.  What is the sum of the $x$- and $y$-coordinates of all four of these points?


What is the greatest common factor of all three-digit palindromes? (Note: A palindrome is a number that reads the same forwards as backwards.)


Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between?


If $x^2 + 3x + 4$ is a factor of $x^4 + Px^2 + Q,$ find $P + Q.$


In writing the integers from 100 through 999 inclusive, how many times is the digit 6 written?


Phil has 8 green marbles and 4 purple marbles in a bag. He removes a marble at random, records the color, puts it back, and then repeats this process until he has withdrawn 8 marbles. What is the probability that exactly five of the marbles that he removes are green? Express your answer as a decimal rounded to the nearest thousandth.


Given the equation
\[\frac{\sin^4 \beta}{\sin^2 \alpha} + \frac{\cos^4 \beta}{\cos^2 \alpha} = 1,\]
find the sum of all possible values of
\[\frac{\cos^4 \alpha}{\cos^2 \beta} + \frac{\sin^4 \alpha}{\sin^2 \beta}.\]


A school has between 150 and 200 students enrolled. Each afternoon, all the students gather to participate in a sports event and are divided into eight distinct sections. If one student is absent, the sections can still each have the same number of students. Determine the sum of all possible numbers of students enrolled at the school where this condition holds.


What non-zero, real value of $x$ satisfies $(3x)^3 = (9x)^2$? Express your answer as a common fraction.


All positive integers whose digits add up to 13 are listed in increasing order. What is the eleventh number in that list?


Expand and simplify the product $$(x^2 - 3x + 3)(x^2 + 3x + 3).$$

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


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


In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 4$, and $XZ = \sqrt{20}$. What is $\tan X$?


Find the integer $n$, $-180 < n < 180$, such that $\tan n^\circ = \tan 1500^\circ$.


Let
\[\mathbf{P} = \begin{pmatrix} 3 & 2 \\ -4 & -3 \end{pmatrix}.\]
Compute $\mathbf{P}^{50}$.

Solution: \boxed{\mathbf{P}^{50} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}}


Consider a quadratic equation $ax^2 + 10x + c = 0$ that has exactly one solution. If $a+2c = 20$, and $a < c$, find the ordered pair $(a, c)$.

Solution: \boxed{\left(10 - 5\sqrt{2}, 5 + \frac{5\sqrt{2}}{2}\right)}


In right triangle $DEF$ where $\angle D = 90^\circ$, the lengths of $DE = 8$ and $EF = 17$. Calculate $\sin F$.


Solution: \boxed{140 \sqrt[3]{20}}


If $\mathbf{v} \times \mathbf{w} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix},$ then find $(2\mathbf{v} - 3\mathbf{w}) \times (4\mathbf{v} + 5\mathbf{w}).$

Solution: \boxed{\begin{pmatrix} 66 \\ -22 \\ 44 \end{pmatrix}}


What is the domain of the function
$$k(x) = \frac{1}{x+9} + \frac{1}{x^2+9x+20} + \frac{1}{x^3+9}?$$
Express your answer as a union of intervals.

Solution: \boxed{(-\infty, -9) \cup (-9, -5) \cup (-5, -4) \cup (-4, -\sqrt[3]{9}) \cup (-\sqrt[3]{9}, \infty)}


Given $k > 0$, let $J_k = 10\ldots 01600$, where there are $k$ zeros between the $1$ and the $1600$.  Find the maximum value of $M(k)$, where $M(k)$ is the number of factors of $2$ in the prime factorization of $J_k$.


A workshop has fewer than $60$ employees. When these employees are grouped in teams of $8$, $5$ employees remain without a team. When arranged in teams of $6$, $3$ are left without a team. How many employees are there in the workshop?


If $x = 51$ and $x^3y - 2x^2y + xy = 51000$, what is the value of $y$?


Three of the five numbers 1875, 2001, 2027, 2033, and 2117 have a mean (average) of 2010. What is the mean (average) of the other two numbers?


Chewbacca has 25 pieces of cherry gum and 40 pieces of grape gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly \(y\) pieces of gum. If Chewbacca loses two packs of cherry gum, then the ratio of the number of pieces of cherry gum he has to the number of pieces of grape gum will be exactly the same as if he instead finds 4 packs of grape gum. Find \(y\).


How many nonnegative integers can be written in the form \[ b_8\cdot2^8 + b_7\cdot2^7 + b_6\cdot2^6 + b_5\cdot2^5 + b_4\cdot2^4 + b_3\cdot2^3 + b_2\cdot2^2 + b_1\cdot2^1 + b_0\cdot2^0, \] where \( b_i \in \{-1, 0, 1\} \) for \( 0 \leq i \leq 8 \)?


What is the smallest even abundant number greater than 12?


Find the value of $x$ such that \[|x-24| + |x-30| = |3x-72|.\]


I have 13 distinguishable socks in my drawer: 5 white, 5 brown, 2 blue, and 1 red. In how many ways can I choose a pair of socks, provided that I get two socks of different colors?


How many different 8-digit positive integers are there?


For what real value of $u$ is $\frac{-15-\sqrt{145}}{8}$ a root of $4x^2+15x+u$?


There are two concentric spheres with radii 5 units and 10 units, respectively. Calculate the volume of the space contained within the larger sphere but outside the smaller sphere. Express your answer in terms of $\pi$.

Solution: \boxed{\frac{3500}{3} \pi}


A square \(PQRS\) has a segment \(PT\) that is perpendicular to segment \(TR\), forming a right triangle within the square. If \(PT = 12\) units and \(TR = 9\) units, find the area of pentagon \(PTRSQ\) in square units.

Solution: \boxed{58.5 \text{ square units}}


A parabola has its vertex at $(1, -3)$ and passes through the point $(3, 5)$. Determine the difference between the zeros of the quadratic equation of this parabola.


A fenced, rectangular field measures 40 meters by 60 meters. An agricultural researcher has 2400 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using some of the 2400 meters of fence?


Express as a common fraction: $$\cfrac{ \frac{3}{7}+\frac{5}{8} }{ \frac{5}{12}+\frac{2}{9}}.$$


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


If \( b \) and \( n \) are positive integers with \( b, n \leq 18 \), what is the greatest number of positive factors \( b^n \) can have?


Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number, with the tens digit being 2.


Let $x$ and $y$ be real numbers such that $3(x^2 + y^2) = x - y.$ Find the maximum value of $x + y.$


How many positive four-digit integers are divisible by $9$?


A rectangle with integer dimensions has an area that is numerically five times the number of units in its perimeter. What is the number of units in the perimeter of this rectangle if one of its sides is greater than 10 units?


In a set of seven consecutive integers, the largest integer is less than three times the average of these integers. What is the smallest integer that could be in the set?


Find the maximum value of $x$ such that $-2x^2 + 9x - 7 \leq 0$.


What is the value of the following expression: $3 - 8 + 13 - 18 + 23 - \cdots - 98 + 103 - 108 + 113$ ?


If $n = 2^{12} \cdot 3^{15} \cdot 5^{7}$, how many of the natural-number factors of $n$ are multiples of 180?


The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $15$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.


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


Points \( P \) and \( Q \) are selected on the graph of \( y = -x^2 \) such that triangle \( PQO \) is equilateral, where \( O \) is the origin. Find the length of one side of triangle \( PQO \).


A triangle with sides in the ratio 1:1:√2 is inscribed in a circle of radius 3. What is the area of this triangle?


In a classroom, there are 36 books. Each book is either hardcover or paperback. If the number of paperback books exceeds the number of hardcover books by a composite number of books, what is the greatest possible number of hardcover books?


I have 7 marbles numbered 1 through 7 in a bag. Suppose I take out two different marbles at random. What is the expected value of the sum of the numbers on the marbles?


Let $F_1 = (1,0)$ and $F_2 = (5,0)$. The set of points $P$ such that $PF_1 + PF_2 = 10$ form an ellipse. The equation of this ellipse can be written as
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.\]
Find $h + k + a + b.$

Solution: \boxed{8 + \sqrt{21}}


For a class of 20 students, the frequency histogram of days missed is as shown below. Determine how many days is the mean number of days missed per student greater than the median number of days missed per student. Express your answer as a common fraction.

[asy]
draw((0,0)--(0,7),linewidth(2));
draw((0,0)--(7,0),linewidth(2));

draw((0,0)--(1,0)--(1,2)--(0,2)--(0,0)--cycle,linewidth(2)); // 2 students missed 0 days
draw((1,0)--(2,0)--(2,3)--(1,3)--(1,0)--cycle,linewidth(2)); // 3 students missed 1 day
draw((2,0)--(3,0)--(3,6)--(2,6)--(2,0)--cycle,linewidth(2)); // 6 students missed 2 days
draw((3,0)--(4,0)--(4,5)--(3,5)--(3,0)--cycle,linewidth(2)); // 5 students missed 3 days
draw((4,0)--(5,0)--(5,2)--(4,2)--(4,0)--cycle,linewidth(2)); // 2 students missed 4 days
draw((5,0)--(6,0)--(6,2)--(5,2)--(5,0)--cycle,linewidth(2)); // 2 students missed 5 days

label("0",(0.5,0),S);
label("1",(1.5,0),S);
label("2",(2.5,0),S);
label("3",(3.5,0),S);
label("4",(4.5,0),S);
label("5",(5.5,0),S);

label("1",(0,1),W);
label("2",(0,2),W);
label("3",(0,3),W);
label("4",(0,4),W);
label("5",(0,5),W);
label("6",(0,6),W);

label("Number of School Days Missed",(3.5,8));
label("by Ms. Jane's Students",(3.5,7));

label("$\#$ of Days of School Missed",(3,-2));
label(rotate(90)*"$\#$ of Students",(-2,2));
[/asy]

Solution: \boxed{\frac{2}{5}\text{ days}}


The volume of a cylinder is $54\pi$ cm³, and it has a height three times its radius. Calculate the total volume of a cone and a sphere both having the same radius as the cylinder. Express your answer in terms of $\pi$.


How many positive integers less than 300 are divisible by 4, 3, and 5?


A rectangular base pyramid $PABCD$ has dimensions where $AB = 10$ and $BC = 6$. $\overline{PA}$ is perpendicular to $\overline{AD}$ and $\overline{AB}$, and the length of the line segment $PB$ equals 20 units. Determine the volume of the pyramid $PABCD$.


Find $\left \lceil \frac{15}{7} \cdot \frac{-27}{3}\right\rceil - \left\lfloor \frac{15}{7} \cdot \left \lceil \frac{-27}{3}\right \rceil \right \rfloor$.


Simplify $\sqrt{12} \times \sqrt{27} - 3$.


A cylindrical container of honey with a diameter of 4 inches and a height of 5 inches costs $\$0.80$. At the same rate, what would be the price for a container that is 8 inches in diameter and 10 inches high?


Dr. Wise has a four-digit house number $ABCD$ containing no zeroes. This number is represented by two distinct two-digit prime numbers, denoted $AB$ and $CD$. Each prime number is less than 50. How many unique house numbers can Dr. Wise potentially have?


Suppose two distinct integers are chosen from between 4 and 20, inclusive. What is the probability that their product is even and neither of the integers is 13?


A point $P$ is chosen at random from the interior of a square $ABCD$. What is the probability that the triangle $ABP$ has a greater area than each of the triangles $BCP$, $CDP$, and $DAP$?


Increasing the radius of a cylinder by $5$ units increased its volume by $x$ cubic units. Increasing the height of the cylinder by $4$ units, keeping the radius the same, also increases the volume by $x$ cubic units. If the original height is $2$, find the original radius.

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


Find the area in the plane contained by the graph of
\[|x + y| + |x - y| \le 8.\]


Maria wants to save money for a down payment on a house and decides to invest in a savings account that compounds semiannually with an annual interest rate of \(8\%\). She needs a total of $\$100,\!000$ at the end of \(10\) years. To the nearest dollar, how much should she initially invest?


A scalene triangle has side lengths which are prime numbers. Its perimeter is also required to be a prime number. Determine the smallest possible perimeter of such a triangle if no side length is less than 11.


How many integers between 500 and 800 have both 12 and 7 as factors?


What is the distance, in units, between the points $(1, -3, 4)$ and $(6, 2, -3)$?


What is the sum of all two-digit primes greater than 20 but less than 80 that remain prime when their digits are reversed?


A rectangle and an isosceles triangle share a common side, which lies along the $x$-axis, with the right vertex of the rectangle and the left vertex of the triangle located at $(15, 0)$. The dimensions of the rectangle are $15$ units wide (along the $x$-axis) and $10$ units high, while the base and height of the triangle are both $10$ units. A segment is drawn from the top left vertex of the rectangle to the top vertex of the triangle. Calculate the area of the triangle formed by this segment and part of the $x$-axis.

Solution: \boxed{75 \text{ sq. units}}


Calculate the modulo residue of $250\cdot 15 - 337\cdot 5 + 22 \pmod{13}$.


What is the area, in square units, of a triangle that has sides of $7, 4,$ and $4$ units? Express your answer in simplest radical form.


What is the smallest whole number $b$ such that 78 can be expressed in base $b$ using only three digits?


Suppose the graph of a function, $y = f(x)$, has the property that if it is shifted $30$ units to the right, the resulting graph is identical to the original graph of $y = f(x)$.

What is the smallest positive $b$ such that if the graph of $y = f\left(\frac{x}{3}\right)$ is shifted $b$ units to the right, then the resulting graph is identical to the original graph of $y = f\left(\frac{x}{3}\right)$?


A spherical buoy was frozen within an ice sheet leaving a visible circular hole $30$ cm across at the top and now $12$ cm deep. After the buoy is removed leaving a hole, what is the radius of the buoy (in centimeters)?


There are 6 blue chips, 5 red chips, and 4 yellow chips in a bag. One chip is drawn from the bag. That chip is then placed back into the bag, and a second chip is drawn. Determine the probability that the two selected chips are of different colors. Express your answer as a common fraction.


A square pyramid has a base edge of 40 inches and an altitude of 20 inches. A smaller square pyramid, whose altitude is one-third of the original altitude, is cut away at the apex of the original pyramid. What fractional part of the volume of the original pyramid is the volume of the remaining frustum?


Determine the number of digits in the value of $2^{15} \times 5^{10} \times 3$.


A band has 90 members who will all be marching during halftime. They need to march in rows with the same number of students per row. If there must be between 4 and 15 students per row, in how many possible row-lengths can the band be arranged?


Square $ABCD$ has sides of length 4. Set $T$ is the set of all line segments that have length 4 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $T$ enclose a region whose area to the nearest hundredth is $m$. Find $100m$.


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


Natasha has more than $\$10$ but less than $\$20$ worth of dimes. When she puts her dimes in stacks of 6, she has 2 left over. When she puts them in stacks of 7, she has 2 left over. When she puts them in stacks of 8, she also has 2 left over. How many dimes does Natasha have?


A penny, fifty-cent coin, dime, and quarter are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.


Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$'', where the digits $A$, $B$, $C$, and $D$ are not necessarily distinct. If each of the two-digit primes is between 40 and 60, how many such house numbers are possible?


What is the greatest three-digit number that is one less than a multiple of 11 and four more than a multiple of 7?


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

a) When divided by 4, the remainder is 3.

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

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

d) When divided by 9, the remainder is 8.


An author writes a book containing 60,000 words in 100 hours. For the first 20% of the time, the author writes with 50% increased productivity. How many words per hour did the author write on average?


If $a$ and $b$ are odd integers, how many terms in the expansion of $(a+b)^8$ are odd?


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


Faces \(ABC\) and \(BCD\) of tetrahedron \(ABCD\) meet at an angle of \(45^\circ\). The area of face \(ABC\) is \(150\), the area of face \(BCD\) is \(90\), and \(BC=12\). Find the volume of the tetrahedron.


In $\triangle ABC$, we have $AC = BC = 10$ and $AB = 4$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 11$. What is $BD$?


The asymptotes of a hyperbola are $y = x + 2$ and $y = -x + 4$. Furthermore, the hyperbola passes through the point $(4, 2)$. Find the distance between the foci of this hyperbola.


Given a point with rectangular coordinates $(-3, -4, 2)$ and corresponding spherical coordinates $(\rho, \theta, \phi)$, find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta + \pi, \phi)$.

Solution: \boxed{(3, 4, 2)}


Suppose I now have 8 shirts, 5 pairs of pants, 4 ties, and 2 belts. An outfit consists of one shirt, one pair of pants, and optionally one tie and/or one belt. How many different outfits can I create considering all these options?


We are allowed to remove exactly one integer from the list $$-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,$$ and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12?


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


Let \(x\) and \(y\) be positive real numbers such that
\[\frac{1}{x + 1} + \frac{1}{y + 1} = \frac{1}{2}.\]
Find the minimum value of \(x + y\).


Compute $\arctan \frac{3}{7} + \arctan \frac{7}{3}.$  Express your answer in radians.


**A rectangular prism measures 12-inches by 15-inches by 8-inches. Calculate the length, in inches, of the diagonal connecting vertices A and B, and additionally, find the total surface area of the prism.**
[asy]
unitsize(0.75cm);
defaultpen(linewidth(0.7pt)+fontsize(10pt));
dotfactor=4;

draw((0,1)--(1,1)--(1,0)--(0,0)--(0,1)--(1,2)--(2,2)--(1,1));
draw((1,0)--(2,1)--(2,2));
dot((0,1));
label("$A$",(0,1),W);
dot((2,1));
label("$B$",(2,1),E);
[/asy]

Solution: \boxed{AB = \sqrt{433} \text{ inches, Total Surface Area} = 792 \text{ square inches}}


What three-digit number with units digit 5 and hundreds digit 6 is divisible by 11?


Simplify the expression $\dfrac{36}{19}\cdot\dfrac{57}{40}\cdot\dfrac{95}{171}$.


Compute the sum:
\[
\sin^2 10^\circ + \sin^2 20^\circ + \sin^2 30^\circ + \dots + \sin^2 90^\circ.
\]


Mr. Thompson recorded the following information from a science test taken by $100$ students in his classes:
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&4\\\hline
90&10\\\hline
80&30\\\hline
70&19\\\hline
60&20\\\hline
50&10\\\hline
40&4\\\hline
30&2\\\hline
20&1\\\hline
\end{tabular}

Using this data, what was the average percent score for these $100$ students?


Bob has a seven-digit phone number and a five-digit postal code. The sum of the digits in his phone number and the sum of the digits in his postal code are the same. Bob's phone number is 346-2789. What is the largest possible value for Bob's postal code, given that no two digits in the postal code are the same?


Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, determine the combined area of the football-shaped regions II and III. Express your answer as a decimal to the nearest tenth.


The median of a set of consecutive even integers is 154. If the greatest integer in the set is 168, what is the smallest integer in the set?


In convex quadrilateral $ABCD$, $\angle A = \angle C$, $AB=CD=150$, and $AD \ne BC$. The perimeter of $ABCD$ is 560. Find $\cos A$.


Express as a common fraction: $(0.\overline{15})(0.\overline{35})$.


How many degrees are in the measure of the smaller angle that is formed by the hour-hand and minute-hand of a clock when it is 2:30?


What is the domain of the function $$f(x) = \frac{1}{(x-3)^2 + (x-6)}~?$$ Express your answer in interval notation.

Solution: \boxed{(-\infty, \frac{5 - \sqrt{13}}{2}) \cup (\frac{5 - \sqrt{13}}{2}, \frac{5 + \sqrt{13}}{2}) \cup (\frac{5 + \sqrt{13}}{2}, \infty)}


If \( A \diamond B \) is defined as \( A \diamond B = 4A + 3B \times 2 \), what is the value of \( A \) for which \( A \diamond 5 = 64 \)?


What is the greatest common divisor of $2024$ and $1728$?


Let $b_n = \frac{5^n - 1}{4}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. What is the maximum possible value that $e_n$ can take on?


Find the number of integers \( n \) that satisfy
\[ 50 < n^2 < 200. \]


Find the reflection of \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) over the vector \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}.\)

Solution: \boxed{\begin{pmatrix} \frac{11}{5} \\ -\frac{2}{5} \end{pmatrix}}


In triangle $ABC$, $\angle C = 90^\circ$, $AB = 13$, and $BC = 5$. What is $\tan A$?


A square EFGH is centered at (4, -6) and has an area of 16 square units. The top side of the square is horizontal. The square is dilated from a new center at (2, -2) with a scale factor of 3. Determine the coordinates of the vertex of the image of square EFGH that is farthest from the center of dilation.

Solution: \boxed{(14, -20)}


In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form.


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

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


Two 24-sided dice have the following configurations: 5 purple sides, 8 blue sides, 10 red sides, and 1 gold side. What is the probability that when both dice are rolled, they will show the same color?


Now, consider that each of the 3 math clubs in the school district receives one new member, bringing their membership to 6, 8, and 9, respectively. With the same configuration of two co-presidents per club, if I randomly select a club, and then randomly select four members of that club to give a copy of $\emph{Introduction to} \allowbreak\ \emph{Counting and} \allowbreak\ \emph{Probability}$, what is the probability that two of the people who receive books are co-presidents?


A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. Find the solution to the equation $2024 \,\diamondsuit\, (8\,\diamondsuit\, y)=200.$


Solution: \boxed{-1 + \frac{1}{3^{16}}}


How many integers $n$ satisfy the inequality $-5\pi \leq n \leq 12\pi$?


A child has a collection of 120 distinct blocks. Each block is one of 3 materials (plastic, wood, metal), 2 sizes (small, large), 5 colors (blue, green, red, yellow, orange), and 4 shapes (circle, square, triangle, hexagon). How many blocks in the set differ from the 'metal large green hexagon' in exactly 2 ways?


A green point and a purple point are chosen at random on the number line between 0 and 1. What is the probability that the purple point is greater than the green point but less than three times the green point?


A bin has 10 black balls and 5 white balls. Five balls are drawn at random. What is the probability of drawing 3 black balls and 2 white balls?


Consider the equation $x^2 + 16x = 100$. The positive solution can also be written in the form $\sqrt{a} - b$ for positive natural numbers $a$ and $b$. Find $a + b$.


If \(G\,243\,F16\) is divisible by \(9\), where \(G\) and \(F\) each represent a single digit, what is the sum of all possible values of the sum \(G+F\)?


What is the value of $\sqrt{5} \times 5^{\frac{1}{3}} + 15 \div 5 \times 3 - 9^{\frac{5}{2}}$?

Solution: \boxed{5^{\frac{5}{6}} - 234}


In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length 10, sides $\overline{CD}$ and $\overline{DA}$ both have length 17, and the measure of angle $ADC$ is $120^\circ$. What is the length of diagonal $\overline{AC}$?

Solution: \boxed{AC = \sqrt{867}}


If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn?


Find the polynomial $p(x)$, with real coefficients, such that $p(3) = 10$ and
\[p(x) p(y) = p(x) + p(y) + p(xy) - 2\] for all real numbers $x$ and $y.$

Solution: \boxed{x^2 + 1}


The polynomial $3x^3 + dx + 9$ has a factor of the form $x^2 + qx + 1.$ Find $d.$


Two fair 6-sided dice are rolled. What is the probability that at least one die shows a 3 on the first die or a 4 on the second die?


In $\triangle XYZ$, medians $\overline{XU}$ and $\overline{YV}$ intersect at right angles at centroid $K$. Given that $XU = 12$ and $YV = 18$, determine the area of $\triangle XYZ$.

Solution: \boxed{144 \text{ square units}}


Determine all possible values of $a,$ given that there exist integers $a,$ $b,$ and $c$ such that
\[(x - a)(x - 5) + 3 = (x + b)(x + c).\]

Solution: \boxed{1, 9}


Kevin Kangaroo starts his journey on a number line at position 0. He wants to reach the position 1, but he has a unique way of hopping: each time, he hops exactly $\frac{1}{4}$ of the remaining distance towards his goal. Each subsequent hop covers only $\frac{1}{4}$ of the remaining distance from his current position. How far has he hopped after six hops? Express your answer as a common fraction.


Calculate $\left|\left(1+2i + 3 - \sqrt{3}i\right)^3\right|$.

Solution: \boxed{\left(23 - 4\sqrt{3}\right)^{3/2}}


Find the area of the triangle bounded by the $y$-axis and the lines $y - 4x = 3$ and $2y + x = 15$.


There are $18$ different complex numbers $z$ such that $z^{18} = 1$. For how many of these is $z^9$ a real number?


A triangle is constructed using sticks of lengths 12, 20, and 24 inches joined end-to-end. Determine the smallest integral length of a piece that can be cut from each of the sticks so that the remaining pieces do not form a triangle.


How many positive integers less than 300 are divisible by 4, 6, and 10?


Find all rational roots of the equation \[3x^5 - 2x^4 + 5x^3 - x^2 - 7x + 2 = 0.\]


A pyramid has its base as an $8\times 10$ rectangle. The edges connecting the apex to each corner of this rectangle are all $15$ units long. Compute the volume of this pyramid.


Ryosuke drives from his home to the airport to pick up a friend. The odometer reads 85,412 miles when he leaves his house, and it reads 85,443 miles when he arrives at the airport. Ryosuke's car gets 25 miles per gallon, and the price of one gallon of gas is $\$3.95$. What is the cost of gas that Ryosuke used to drive to the airport? (Express your answer in dollars and round to the nearest cent.)


Let $m$ be a positive integer such that in base 8, $m$ is expressed as $\overline{XYZ}_8$, and in base 12, $m$ is expressed as $\overline{ZYX}_{12}$. Find the largest possible value of $m$ in base 10.


A circular park has its diameter increased by a factor of 3. Simultaneously, the circumference is noted to be thrice its original length as well. What is the ratio of the original area to the enlarged area, and by what factor does the circumference increase?


Sue owns 15 pairs of shoes divided into four colors: seven identical black pairs, four identical brown pairs, two identical gray pairs, and two identical red pairs. If Sue picks two shoes at random, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? Express your answer as a common fraction.


Let \( p \) and \( q \) be real numbers. One of the roots of the polynomial \( x^3 + px + q = 0 \) is \( 2 + i \sqrt{2} \). Find \( p + q \).


Fifteen points are marked on a circle. How many distinct convex polygons of four or more sides can be drawn using some (or all) of these fifteen points as vertices?


A line through the points $(3, -7)$ and $(k, 20)$ is parallel to the line $5x + 4y = 28$. This line also intersects the line $x - 3y = 5$ at a point. What is the value of $k$?


Express the decimal $0.7\overline{56}$ as a common fraction.


Find the greatest common divisor of 18,222, 24,546, and 66,364.


In a similar race setup, all runners start at point $A$, must touch any part of a 1500-meter wall, and stop at point $B$. Given that the distances from $A$ to the closest point on the wall is 400 meters and from $B$ to the closest point on the wall is 600 meters, what is the minimum distance a participant must run? Round your answer to the nearest meter.


Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 500, what is the sum of $a$ and $b$?


A geometric sequence of positive integers is formed where the first term is 6 and the fourth term is 768. What is the second term of this sequence?


Four vertices of a rectangle include the points $(2, 3)$, $(2, 15)$, and $(13, 3)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 13)^2 + (y - 3)^2 = 16$?


You receive a fortune cookie containing four two-digit positive integers as your lucky numbers. The first three numbers are 28, 46, and 59, but unfortunately, the last number is partially obscured. If the sum of the digits of all four numbers equals $\frac{1}{4}$ of the total of the four numbers, what is the smallest possible value for the fourth number?


What is the period of \(y = \cos\left(\frac{x}{4} + \frac{\pi}{4}\right) + 2\)?


The graphs of \(y = 4 - 2x + x^2\) and \(y = 2 + 2x + x^2\) intersect at multiple points. Find the maximum difference between the \(y\)-coordinates of these intersection points.


Let $w, x, y, z, t$ be nonnegative numbers such that $w + x + y + z + t = 120$. Find the maximum possible value of the expression
\[ wx + xy + yz + zt. \]


Let \[f(x) =
\begin{cases}
2x^3 + 4&\text{if } x\le 2, \\
bx + 5 &\text{if } x>2.
\end{cases}
\]Find \( b \) if the graph of \( y=f(x) \) is continuous.


The polynomial $x^{103} + Cx + D$ is divisible by $x^2 - x + 1$ for some real numbers $C$ and $D.$ Find $C + D.$


Maria sells four enchiladas and five tacos for \$3.30 and she sells five enchiladas and three tacos for \$3.60. Assuming a fixed price per item, what is the cost, in dollars, of six enchiladas and six tacos? Express your answer as a decimal to the nearest hundredth.


How many positive multiples of 7 that are less than 150 have a units digit of 3?


The quadratic \( x^2 + 1600x + 1607 \) can be written in the form \((x+b)^2 + c\), where \( b \) and \( c \) are constants. What is \( \frac{c}{b} \)?


A modified deck of 54 cards includes 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) in each of the four suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), plus two jokers, both considered as black cards. This deck is randomly arranged. What is the probability that the top card is red and the second card is black?


In how many ways can I arrange 5 different math books and 4 different history books on my bookshelf, if I require there to be a history book on both ends?


The sequence $b_1,$ $b_2,$ $b_3,$ $\dots$ satisfies $b_1 = 15,$ $b_{10} = 105,$ and for all $n \ge 3,$ $b_n$ is the arithmetic mean of the first $n - 1$ terms.  Find $b_2.$


Consider the geometric sequence starting with the first term as $4$ and the common ratio of $5/3$. Find the tenth term of this sequence. Express your answer as a common fraction.


What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\pmod{63}$ and $a^{-1}$ is also undefined $\pmod{66}$?


Simplify $2\sqrt{50x^3} \cdot \sqrt{45x^5} \cdot \sqrt{98x^7}$. Express your answer in simplest radical form in terms of $x$.


Let $Q$ be a point outside of circle $C.$ A segment is drawn from $Q$ such that it is tangent to circle $C$ at point $U.$ Meanwhile, a secant from $Q$ intersects $C$ at points $M$ and $N,$ such that $QM < QN.$ Given that $QM = 4$ and $QU = MN - QM,$ find the length of $QN.$


Three identical cylindrical pipes, each with a diameter of 12 cm, are packed in a triangular arrangement inside a crate. The base of the crate is visible, showing two pipes touching each other and a third pipe resting on top of the first two, forming a triangle as shown below:

[asy]
draw(circle((10,10),12),black+linewidth(1));
draw(circle((34,10),12),black+linewidth(1));
draw(circle((22,30),12),black+linewidth(1));
draw((50,0)--(50,42),black+linewidth(1));
draw((49,0)--(51,0),black+linewidth(1));
draw((49,42)--(51,42),black+linewidth(1));
label("$h$",(50,0)--(50,42),E);
[/asy]

Determine the height, $h,$ of this arrangement of pipes.

Solution: \boxed{12 + 6\sqrt{3} \text{ cm}}


In the expanded Gnollish language, there are 4 words: "splargh," "glumph," "amr," and "blarg." A sentence in this language cannot have "splargh" directly before "glumph" and cannot have "amr" directly before either "glumph" or "blarg." How many valid 3-word sentences can be formed under these rules?


Yann and Camille go to a restaurant that has 12 items on the menu, and each of them orders one main dish. Additionally, they choose one appetizer from a separate list of 5 appetizers to share. How many different combinations of meals can Yann and Camille order, considering that it matters who orders which main dish?


$ABCDEFGH$ is a cube with volume 8. Find the volume of pyramid $ACDH$.


Consider the function $y = f(x)$ defined as follows:
\[ f(x) = \begin{cases}
x + 3 & \text{if } x \leq 0 \\
-x + 3 & \text{if } 0 < x \leq 3 \\
0 & \text{if } x > 3
\end{cases} \]

For certain constants $a$, $b$, and $d$,
\[ h(x) = af(bx) + d. \]
The graph of $y = h(x)$ results from stretching $f(x)$ vertically by a factor of 2, horizontally by a factor of 3, and shifting it up by 5 units. Find the ordered triple $(a, b, d)$.

Solution: \boxed{(2, \frac{1}{3}, 5)}


In the diagram, \( O \) is the center of a circle with radii \( OP=OQ=7 \). Determine the perimeter of the shaded region if \( \widehat{PQ} \) subtends an angle of \( 270^\circ \).

Solution: \boxed{14 + \frac{21\pi}{2}}


Consider the digits $1, 2, 3, 4, 5$, and $6$. How many $6$-digit positive integers with distinct digits can be formed where the digit $1$ is to the left of the digit $2$ and to the right of the digit $3$?


Suppose \( a \equiv 23 \pmod{37} \) and \( b \equiv 58 \pmod{37} \). Determine the integer \( n \) in the set \{150, 151, 152, ..., 190, 191\} such that
\[ a - b \equiv n \pmod{37} \].


Monica tosses a fair 8-sided die. If the roll is a prime number, then she wins that amount in dollars. If the roll is a perfect square, she wins double that amount in dollars. Otherwise, she wins nothing. What is the expected value of her winnings on one die toss? Express your answer as a dollar value to the nearest cent.


On the 6 by 6 square grid below, each dot is 1 cm from its nearest horizontal and vertical neighbors. Find the product of the area of square $PQRS$ (in cm$^2$) and the perimeter of square $PQRS$ (in cm) where the vertices $PQRS$ are positioned as follows on the grid: $P$ at (1,5), $Q$ at (5,6), $R$ at (6,2), and $S$ at (2,1). Express your answer in simplest radical form.


The pages of a book are numbered from 1 to $n$. If one of the page numbers is added twice by mistake, the resulting sum of all the numbered pages is found to be $3050$. Determine the number of the page that was mistakenly added twice.


Solution: \boxed{\frac{3C(B - 1)}{16C^2 - 1}}


Find the quotient when $x^5 - 16x^3 + 11x^2 - 9x + 10$ is divided by $x-3$.

Solution: \boxed{x^4 + 3x^3 - 7x^2 - 10x - 39}


A triangle on a Cartesian coordinate plane has vertices at (3, 1), (3, 6), and (8, 6). Calculate the area of this triangle and express the answer in square units.


If two numbers are randomly chosen without replacement from the set $\{5, 6, 9, 10\}$, what is the probability that their product will be a multiple of 15?


What is the largest integer less than $\log_3 \frac{3}{2} + \log_3 \frac{4}{3} + \cdots + \log_3 \frac{3010}{3009} + \log_3 \frac{3011}{3010}$?


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

Solution: \boxed{(4, 5]}


In right triangle $ABC$, angles $A$ and $B$ are congruent while the hypotenuse $AB = 10$. Calculate the length of $AC$ and determine the circumradius of $\triangle ABC$.


I am playing a similar walking game. On move 1, I do nothing, but on move $n$ where $2 \le n \le 30$, I take one step forward if $n$ is prime and three steps backwards if the number is composite. After all 30 moves, I stop and walk back to my original starting point. How many steps long is my walk back?


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


Margo walks to her friend's house in 15 minutes and takes 30 minutes using the same, but slippery, route to get back home. Her average walking rate for the outbound trip is 5 miles per hour and 3 miles per hour on the return trip. What are the total miles Margo walked, if her average speed for the entire trip was 3.6 miles per hour?

Solution: \boxed{2.75 \text{ miles}}


How many three-digit numbers remain if we exclude all three-digit numbers in which there are exactly two identical digits that are not adjacent and also all numbers with three distinct digits?


The expression $x^2 + 17x + 70$ can be written as $(x + a)(x + b)$, and the expression $x^2 - 19x + 84$ can be written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?


What is the units digit of the product of all the odd positive integers between 20 and 130?


Olympus Corporation has released a new version of its popular Vowel Soup. In this version, each vowel (A, E, I, O, U) appears 7 times in each bowl. How many six-letter "words" can be formed from this Olympus Vowel Soup? Note: Words do not need to be actual words in the English language.


The Gnollish language has recently expanded and now includes 4 words: "splargh," "glumph," "amr," and "florp." In constructing a sentence, "florp" can never immediately follow "glumph." How many valid 3-word sentences can be formed under this new rule?


Two triangles $\triangle FGH \sim \triangle IJK$, where $GH = 30\text{ cm}$. If the length of $IJ$ is $18\text{ cm}$ and side $JK$ is $15\text{ cm}$, find the length of $FG$. Assume $J$ lies between $I$ and $K$.

Solution: \boxed{36\text{ cm}}


Let $T$ be a positive integer consisting solely of 0s and 1s. If $X = T \div 60$ and $X$ is an integer, what is the smallest possible value of $X$?


Cookie Monster finds another cookie described by the equation $x^2 + y^2 + 16 = 6x + 14y$. He wonders if this cookie is a perfect bite-sized circle. Determine the radius of this cookie.


Determine the total number of pieces required to create a ten-row triangle using unit rods and connectors. Each n-row triangle uses rods in the pattern of the arithmetic sequence $3n$ and connectors forming an $(n+1)$-row triangle.


When the base-14 integer $1725_{14}$ is divided by $9$, what is the remainder?


Express \( 4.\overline{8} \) as a common fraction.


Chewbacca has 25 pieces of cherry gum and 35 pieces of grape gum. Some of the pieces are in complete packs, while others are loose. Each complete pack contains exactly $x$ pieces of gum. If Chewbacca loses one pack of cherry gum, then the ratio of the number of pieces of cherry gum he has to the number of pieces of grape gum will be the same as if he instead finds 6 packs of grape gum. Find $x$.


Let $a$, $b$, $c$, $u$, $v$, and $w$ be real numbers satisfying the three equations:
\begin{align*}
  17u +  bv +  cw &= 0 \\
   au + 29v +  cw &= 0 \\
   au +  bv + 56w &= 0.
\end{align*}
Suppose that $ a \ne 17$ and $ u \ne 0$. What is the value of
\[ \frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 56} \, ?\]


Allison, Brian, and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 4. The faces on Brian's cube are numbered 1, 2, 3, 4, 5, and 6. Three of the faces on Noah's cube have a 1 and three of the faces have a 5. All three cubes are rolled. What is the probability that Allison's roll is greater than each of Brian's and Noah's? Express your answer as a common fraction.


Let $D$ be the circle with the equation $x^2 + 16y + 89 = -y^2 - 12x$. Find the center $(p, q)$ and radius $s$ of $D$ and calculate $p+q+s$.

Solution: \boxed{-14 + \sqrt{11}}


For what values of $x$ is $x^2 - 50x + 625 \leq 25$?

Solution: \boxed{[20, 30]}


Find the remainder when $5^{5^{5^5}}$ is divided by 500.


Calculate \(\left(\frac{4}{5}\right)^{10} \cdot \left(\frac{2}{3}\right)^{-4}\).


John guesses randomly on a 10-question multiple choice test in which each problem has two choices, a correct one and an incorrect one. What is the probability that he gets at least 60% of the questions correct? Express your answer as a common fraction.


The positive integer divisors of 245, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. Calculate the sum of the two integers adjacent to 7.


Consider a right triangle $DEF$ with the right angle at $E$ such that $DF = \sqrt{85}$ and $DE = 7.$ A circle is drawn with its center on $DE$ such that the circle is tangent to $DF$ and $EF.$ If $Q$ is the point where the circle and side $DF$ meet, find the length of $FQ$.


Without using a calculator, determine the largest prime factor of $18^4 + 2 \times 18^2 + 1 - 17^4$.


Calculate the sum of the 1000 fractions of the form $\frac{2}{n(n+2)}$ where $n$ ranges from 1 to 1000. Provide your answer as a decimal rounded to the nearest thousandth.


Find the product of the solutions of the equation: $|x| = 3(|x|-1)$.


Find the distance between the points \((4, -3)\) and \((-6, 5)\), and determine the coordinates of their midpoint.

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


Consider a wooden cube, whose edges are two centimeters long, resting on a horizontal surface. A point source of light is positioned `x` centimeters directly above an upper vertex of the cube. This set-up results in the cube casting a shadow on the horizontal surface, the area of which, excluding the area beneath the cube, is 200 square centimeters. Determine the greatest integer that does not exceed `100x`.


Each row of a seating arrangement can seat either 6, 8, or 9 people. Fifty-eight people are to be seated. How many rows seat exactly 9 people if every seat is occupied and there are no more than 7 rows?


A basketball team consists of 12 players, including two pairs of twins, Alex and Brian, and Chloe and Diana. In how many ways can we choose a team of 5 players if no pair of twins can both be in the team lineup simultaneously?


There are $30n$ members in the Marlington orchestra, and when they line up in rows of 31, there are 5 members left over. If $n$ is an integer and there are fewer than 1500 orchestra members, what is the maximum number of people that could be in the Marlington orchestra?


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


Calculate $(2 \cos 30^\circ + 2i \sin 30^\circ)^8.$ Provide your answer in rectangular form.

Solution: \boxed{-128 - 128i\sqrt{3}}


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


If the three points $(2,a,b), (a,3,b), (a,b,4)$ are collinear, what is the value of $a + b$?


Determine the inverse of the function $h(x) = 3 - 7x$.

Solution: \boxed{\frac{3 - x}{7}}


After Sally takes 30 shots, she has made $60\%$ of her shots. After she takes 10 more shots, she raises her overall shooting percentage to $62\%$. How many of the last 10 shots did she make?


Calculate the number of odd digits in the base-4 representation of $157_{10}$.


The median of a set of consecutive odd integers is 107. If the greatest integer in the set is 121, what is the smallest integer in the set?


Determine the integers $p$ and $q$ such that $\log_{10} 158489$ lies between them. Find $p+q$.


Ten distinct integers are picked at random from $\{1,2,3,\ldots,15\}$. What is the probability that, among those selected, the third smallest is $5$?


Find the sum of the coefficients in the polynomial \(5(2x^8 - 3x^5 + 9x^3 - 4) + 3(4x^6 - 6x^4 + 7)\) when it is fully simplified.


If ten apples cost the same as five bananas, and five bananas cost the same as seven cucumbers, how many cucumbers can Joyce buy for the price of 20 apples?


In how many ways can 8 people be seated in a row of chairs if two of the people, Alice and Bob, refuse to sit next to each other, and another pair, Cindy and Dave, also refuse to sit next to each other?


A student must choose a program of five courses from an expanded list of courses that includes English, Algebra, Geometry, Calculus, Biology, History, Art, and Latin. This program must include both English and at least two mathematics courses. In how many ways can this program be chosen?


What is the smallest positive integer $n$ for which $n^2$ is divisible by 24 and $n^3$ is divisible by 864?


A quadratic equation has the form $x^2 + bx + \frac{1}{4}$, where $b$ is a specific negative number. Using the method of completing the square, the quadratic is rewritten as $(x+n)^2 + \frac{1}{16}$. What is the value of $b$?


Compute $\arccos (\cos 8).$ All functions are in radians.

Solution: \boxed{8 - 2\pi}


Compute $4 \begin{pmatrix} 3 \\ -9 \end{pmatrix} + 3 \begin{pmatrix} -1 \\ 6 \end{pmatrix} - 5 \begin{pmatrix} 0 \\ -2 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 9 \\ -8 \end{pmatrix}}


On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{3}$, meaning if the length of the object was originally $x$, then it is now $x + \frac{1}{3} x$. On the second day, he increased the object's longer length by $\frac{1}{4}$; on the third day by $\frac{1}{5}$, and so on. On the $n^{\text{th}}$ day, Barry will make the object's length exactly 150 times its original length. What is the value of $n$?


A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 4 meters from its base. When Leah walks 3 meters from the base of the pole towards the point where the cable is attached to the ground, her head just touches the cable. Leah is 1.7 meters tall. How tall is the pole?


The slant height of a cone is 15 cm, and the height from the vertex to the center of the base is 13 cm. What is the number of cubic centimeters in the volume of the cone? Express your answer in terms of $\pi$.

Solution: \boxed{\frac{728}{3} \pi}


If $0 \le p \le 1$ and $0 \le q \le 1$, define $F(p, q)$ by
\[
F(p, q) = -3pq + 4p(1-q) + 2(1-p)q - 5(1-p)(1-q).
\]
Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $G(p)$?


Compute without using a calculator: $9! - 8! - 7! + 6!$


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

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


Determine the smallest value of the function $\lvert x + 3\rvert + \lvert x + 6\rvert + \lvert x + 8\rvert + \lvert x + 10\rvert$.


How many two-digit numbers have digits whose sum is a triangular number?


In quadrilateral $ABCD$, $AB = BC = 15$, $CD = DA = 30$, and $\angle BDC = 60^\circ$. Points $M$ and $N$ are the midpoints of $\overline{BC}$ and $\overline{DA}$ respectively. Compute $MN$.


Determine the point of intersection of the asymptotes of the graph of
\[y = \frac{x^2 - 4x + 3}{x^3 - 4x^2 + 4x}.\]


If the quadratic $x^2 + 6nx + 2n$ has exactly one real root, find the positive value of $n$.


Determine the first term in the geometric sequence, where the fifth term is 72 and the eighth term is 576.


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

Solution: \boxed{x = \sqrt{5\sqrt{5} + 1}}


Find the radius of the circle with equation $x^2 - 10x + y^2 - 8y + 29 = 0$.


A point in space $(x, y, z)$ is randomly selected such that $x^2 + y^2 \le 4$ and $-2 \le z \le 2$. What is the probability that $x^2 + y^2 + z^2 \le 4$?


What is the probability that when two 8-sided dice are rolled, the sum is a perfect square or a perfect cube?


Calculate the value of $\dfrac{11!}{7!4!} \times 2$ without using a calculator.


Given the function \( f(x) = \frac{x^2 + 3x + 1}{x^2 - 5x + 6} \), determine the values of \( x \) where there are vertical asymptotes.

Solution: \boxed{x = 2 \text{ and } x = 3}


How many ways are there to put 5 distinguishable balls into 2 indistinguishable boxes such that neither box is empty?


A solid box measures 18 cm by 12 cm by 10 cm. A new solid is formed by removing a cube of side 4 cm from each corner of this box. What percent of the original volume is removed?


Determine the value of $2^{-1} + 4^{-1} \pmod{17}$ and express your answer as an integer from $0$ to $16$.


An equilateral triangle has an area of $100\sqrt{3}$ $\text{cm}^2$. If each side of the triangle is decreased by 3 cm, by how many square centimeters is the area decreased?


Luke wants to fence a rectangular piece of land with an area of at least 450 square feet. The length of the land is 1.5 times the width. What should the width of the rectangle be if he wants to use the least amount of fencing?


Compute the unique positive integer \( n \) such that
\[
3 \cdot 2^3 + 4 \cdot 2^4 + 5 \cdot 2^5 + \dots + n \cdot 2^n = 2^{n + 11}.
\]


Convert $157_{10}$ to base 13. Represent $10$ as $A$, $11$ as $B$, and $12$ as $C$, if necessary.


Pyramid $PABCD$ has a rectangular base $ABCD$. If $AB = 10$, $AD = 6$, $PA$ is perpendicular to plane $ABCD$, and $PC = 25$, determine the volume of pyramid $PABCD$.


John now has 15 marbles, including two red, two green, and two blue marbles. In how many ways can he choose 5 marbles, if at least two of the chosen marbles should be from the red, green, or blue group?


Find the smallest positive integer $b$ for which $x^2 + bx + 1764$ factors into a product of two polynomials, each having integer coefficients.


Find the area of the triangle bounded by the $y$-axis and the lines $y-2x=-1$ and $4y+x=16$.


Let $a,$ $b,$ $c$ be the roots of $x^3 - 3px + p = 0.$ Compute the determinant
\[\begin{vmatrix}
   a & b & c \\
   c & a & b \\
   b & c & a
\end{vmatrix}.\]


How many ways are there to arrange the letters of the word $\text{C}_1\text{O}_1\text{M}_1\text{M}_2\text{U}_1\text{N}_1\text{I}_1\text{T}_1$, in which the two M's are considered different?


Tom's graduating class has 360 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 12 rows and at least 18 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$?


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 317^\circ.$


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


Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 2.$  Find the minimum value of
\[\frac{x + y}{xyz}.\]


Lucas has 1500 dollars to invest. He invests part of the money at Alpha Bank, which compounds annually at 4 percent, and the rest at Beta Bank, which compounds annually at 6 percent. After one year, Lucas has a total of $1584.50. How much did Lucas originally invest at Alpha Bank, in dollars?


The base 7 number $45x2_7$ is divisible by 9. What is the digit $x$?

Solution: \boxed{x = 4}


From the associated diagram, find the positive difference in the $x$-coordinates when lines $l$ and $m$ reach $y = 20.$ Assume line $l$ passes through $(0,5)$ and $(3,0),$ and line $m$ passes through $(0,2)$ and $(7,0).$


The ratio of \(3x + 5\) to \(4x + 6\) is equal to the ratio of \(6x + 4\) to \(10x + 8\). Find the sum of all real \(x\) which satisfy this equation.


What is the sum of all the solutions of $x = \left|2x - |72 - 2x|\right|$?

Solution: \boxed{24 + 14.4 + 72 = 110.4}


Suppose 5 different integers are randomly chosen from between 10 and 59, inclusive.  What is the probability that they each have a different tens digit?


Suppose that $x$ is inversely proportional to $y$. If $x = 5$ when $y = 10$, find $x$ when $y = 40$.


The difference between two perfect squares is 143. What is the smallest possible sum of the two perfect squares?


Determine how many perfect squares less than 12,100 are the difference between the squares of two consecutive integers.


Let \(x\), \(y\), and \(z\) be complex numbers such that:
\[
xy + 3y = -9, \\
yz + 3z = -9, \\
zx + 3x = -9.
\]

Find all possible values of \(xyz\).


A spiral staircase makes a $225^\circ$ turn as it rises 12 feet. The radius of the staircase is 4 feet. Calculate the length of the handrail, rounding your answer to the nearest tenth of a foot.


Two circles are externally tangent at point T. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common external tangents with points $A$ and $A'$ on the smaller circle, and $B$ and $B'$ on the larger circle. If $PA = AB = 5$, calculate the area of the smaller circle.


How many rows of Pascal's Triangle contain the number $53$?


Determine the value of $n$ such that \(10^n = 10^{-7} \times \sqrt{\frac{10^{85}}{0.0001}}\).


How many of the 729 smallest positive integers written in base 9 use 7 or 8 (or both) as a digit?


In right triangle $XYZ$ with $\angle X = 90^\circ$, let $XY = 8$ and $YZ = 17$. Find $\cos Z$ and $\sin X$.

Solution: \boxed{\cos Z = \frac{15}{17} \text{ and } \sin X = 1}


If \(0 \le p \le 1\) and \(0 \le q \le 1\), define \(F(p, q)\) by
\[
F(p, q) = -4pq + 5p(1-q) + 2(1-p)q - 3(1-p)(1-q).
\]
Define \(G(p)\) to be the maximum of \(F(p, q)\) over all \(q\) (in the interval \(0 \le q \le 1\)). What is the value of \(p\) (in the interval \(0 \le p \le 1\)) that minimizes \(G(p)\)?


If $a$, $b$, $c$, and $d$ are real numbers satisfying:
\begin{align*}
a+b+c &= 3, \\
a+b+d &= -1, \\
a+c+d &= 8, \text{ and} \\
b+c+d &= 0,
\end{align*}
what is $ab + cd$?


Quadrilateral $EFGH$ is a parallelogram. $\angle FGH$ is given as $75^\circ$. Additionally, $\angle EFG$ is given as $105^\circ$. What is the degree measure of $\angle E$?


Consider the expressions:
\[ A = 1 \times 2 + 3 \times 4 + \cdots + 49 \times 50 + 51 \]
\[ B = 1 + 2 \times 3 + 4 \times 5 + \cdots + 48 \times 49 + 50 \times 51 \]
Calculate the positive difference between $A$ and $B$.


Find the distance between the foci of the ellipse
\[\frac{x^2}{45} + \frac{y^2}{5} = 9.\]


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 $T(b)T(-b) = 2250$. Find $T(b)+T(-b)$.


In triangle $PQR$, $PQ = 20$, $PR = 21$, and $QR = 19$. Points $F$ and $G$ are on $PQ$ and $PR$ respectively, such that line $FG$ is parallel to $QR$ and passes through the centroid of triangle $PQR$. Find the length of $FG$ in the simplest form $a/b$, where $a$ and $b$ are co-prime integers. Find $a+b$.


What is the minimum value of \(x^2 + y^2 -8x + 6y + xy + 20\) for real \(x\) and \(y\)?


Emily's quiz scores so far are: 85, 92, 88, 90, and 93. What score does she need to get on the sixth quiz to make the arithmetic mean of the six scores equal to 91?


The sum of the first $25$ positive even integers is equivalent to the sum of four consecutive odd integers. What is the largest of these four integers?


Cara is sitting with her seven friends in a line at a table. How many different combinations of friends can sit immediately next to her on the right?


The arithmetic progressions $\{2, 5, 8, 11, \ldots\}$ and $\{4, 12, 20, 28, \ldots \}$ have some common values. What is the largest value less than 1000 that they have in common?


Let \(a,\) \(b,\) and \(c\) be nonnegative real numbers such that \(a^2 + b^2 + c^2 = 1.\) Find the maximum value of:
\[2ac \sqrt{2} + 2ab.\]


Sides $\overline{AB}$ and $\overline{GH}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$?


How many non-similar triangles have angles whose degree measures are distinct positive integers in an arithmetic progression, such that the middle angle is a multiple of 20?


How many positive integers, not exceeding 200, are multiples of 5 or 7 but not 10?


Find $P(0)$ for the unique polynomial $P(x)$ of degree $4$ with rational coefficients and leading coefficient $1$, which has $\sqrt{3}+\sqrt{7}$ as a root.


In a specific hyperbola, the center is located at $(1+i,0)$, one focus is at $(1+i, 8i),$ and one vertex is at $(1+i, -4i)$. The equation of this hyperbola can be written as
\[
\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1.
\]
Find the sum of the real parts of $h, k, a,$ and $b$.

Solution: \boxed{5 + 4 \sqrt{3}}


Determine the coefficients $(p, q)$ if the polynomial $px^4 + qx^3 + 20x^2 - 10x + 15$ has a factor of $5x^2 - 3x + 3.$

Solution: \boxed{(0, \frac{25}{3})}


Find an ordered pair $(x, y)$ that solves the system: \begin{align*} 7x &= 10 - 3y,\\ 4x &= 5y - 14. \end{align*}

Solution: \boxed{\left(\frac{8}{47}, \frac{138}{47}\right)}


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 = (10, 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("$10$", 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]


Let $f(x) = x^2 - 9x + 20$ and let $g(f(x)) = 3x + 1$. What is the sum of all possible values of $g(10)$?


Compute the sum
\[
\sin^2 0 + \sin^2\left(\frac{\pi}{180}\right) + \sin^2\left(\frac{2\pi}{180}\right) + \dots + \sin^2\left(\frac{90\pi}{180}\right).
\]


How many odd positive multiples of 7 that are less than 200 have a units digit of 1?


The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:3:5$ which is expressed as \[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]. Determine the sum of all possible values of $n+k$.


In the function machine shown, the input is 15. What is the output?

[asy]
size(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;
real height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;
real widthC = 20,heightC = 6;
real widthE = 10, lengthE = 4.5,angleE = 60;
draw(a,(0,0)--(width1,0)--(width1,height)--(0,height)--cycle); label(a,"$\mbox{In}\mbox{put}$ = 15",(width1/2,height/2));
draw(b,(0,0)--(width2,0)--(width2,height)--(0,height)--cycle); label(b,"Multiply by 3",(width2/2,height/2));
draw(c, (widthC/2,0)--(0,heightC/2)--(-widthC/2,0)--(0,-heightC/2)--cycle);
label(c,"Compare with 30",(0,0));
draw(d,(0,0)--(width3,0)--(width3,height)--(0,height)--cycle); label(d,"Add 10",(width1/2,height/2));
draw(e,(0,0)--(width4,0)--(width4,height)--(0,height)--cycle); label(e,"Divide by 2",(width1/2,height/2));
draw(f,(0,0)--(widthE,0)--(widthE,0)+lengthE*dir(angleE)--lengthE*dir(angleE)--cycle);
label(f,"$\mbox{Out}\mbox{put}$ = ?",lengthE/2*dir(angleE) + (widthE/2,0));
add(shift(width1/2*left)*a); draw((0,0)--(0,-2),EndArrow(4));
add(shift(5*down + width2/2*left)*b);
add(shift((7+heightC/2)*down)*c); draw((0,-5)--(0,-7),EndArrow(4));
pair leftpt = (-widthC/2,-7-heightC/2), rightpt = (widthC/2,-7-heightC/2);
draw("$\le 30$?",leftpt--(leftpt + 2.5W)); draw((leftpt + 2.5W)--(leftpt + 2.5W+2S),EndArrow(4));
draw("$> 30?$",rightpt--(rightpt + 2.5E),N); draw((rightpt + 2.5E)--(rightpt + 2.5E+2S),EndArrow(4));
rightpt = rightpt + 2.5E+2S;
leftpt = leftpt + 2.5W+2S;
add(shift(leftpt+height*down+.3*width3*left)*d);
add(shift(rightpt+height*down+.7*width4*left)*e);
rightpt = rightpt+.75height*down+.7*width4*left;
leftpt = leftpt+.75height*down+.7*width3*right;
draw(leftpt--rightpt);
pair midpt = (leftpt+rightpt)/2;
draw(midpt--(midpt+2down),EndArrow(4));
add(shift(midpt+.65widthE*left+(2+lengthE*Sin(angleE))*down)*f);[/asy]


$\triangle XYZ$ is similar to $\triangle MNP$. The side $\overline{YZ}$ measures 10 cm, and the side $\overline{XZ}$ measures 7 cm. In $\triangle MNP$, the side $\overline{MN}$ measures 4.2 cm. Find the length of $\overline{NP}$, rounding your answer to the nearest tenth of a cm.

Solution: \boxed{6.0 \text{ cm}}


In triangle $ABC$, $AB = 6$, $BC = 8$, $AC = 10$, and $CD$ is the angle bisector of angle $ACB$. Find the length of $CD$.

Solution: \boxed{\frac{8 \sqrt{10}}{3}}


A bridge is constructed by placing a wooden plank between two triangular wedges each with equal heights as depicted in the following diagram:
[asy]
import olympiad;
import math;

// Draw triangles
pair A = (0, 1);
pair B = (-cos(1.3962), 0);
pair C = (cos(1.3962), 0);
pair D = (2, 1);
pair E = (2-cos(1.3089), 0);
pair F = (2+cos(1.3089), 0);
draw(A--B--C--cycle);
draw(D--E--F--cycle);
draw(A--D);
label('$A$',A,N);
label('$B$',B,S);
label('$C$',C,S);
label('$D$',D,N);
label('$E$',E,S);
label('$F$',F,S);
[/asy]
If $AB = AC$ and $DE = DF$, $\angle BAC = 25^\circ$ and $\angle EDF = 35^\circ$, find $\angle DAC + \angle ADE$.


Tamika selects two different numbers at random from the set $\{7, 8, 9, 10\}$ and multiplies them. Carlos takes three different numbers at random from the set $\{2, 3, 5, 6\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result?  Express your answer as a common fraction.


In triangle $ABC$, $BC = 20$ units and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ at $D$ (midpoint) and $AC$ at $E$. Find the length of $DE$.


An eight-sided die, numbered from 1 to 8, is rolled. If $Q$ represents the product of the numbers on the seven sides that are visible, what is the largest number that is certain to divide $Q$?


Rodney is trying to guess a secret number based on these clues:
1. It is a two-digit integer.
2. The tens digit is even.
3. The units digit is odd.
4. The number is greater than 75.

Calculate the probability that Rodney guesses the correct number, expressed as a common fraction.


Calculate how many numbers are in the sequence $$ 2, 5, 8, \ldots, 2011, 2014. $$


At the CleverCat Academy, cats can learn to do three tricks: jump, play dead, and fetch. Of the cats at the academy:
- 60 cats can jump
- 35 cats can play dead
- 40 cats can fetch
- 20 cats can jump and play dead
- 15 cats can play dead and fetch
- 22 cats can jump and fetch
- 10 cats can do all three tricks
- 12 cats can do none of the tricks

How many cats are in the academy?


A solid box has dimensions 18 cm by 12 cm by 10 cm. From each corner of this box, a cube measuring 4 cm on each side is removed. What percentage of the original volume is removed?


A cube with an edge length of 5 units has the same volume as a square-based pyramid with base edge lengths of 10 units. What is the height \( h \) of the pyramid?


What percent of the square $EFGH$ is shaded? All angles in the diagram are right angles, and the side length of the square is 8 units. In this square:
- A smaller square in one corner measuring 2 units per side is shaded.
- A larger square region, excluding a central square of side 3 units, occupying from corners (2,2) to (6,6) is shaded.
- The remaining regions are not shaded.


Find the smallest value of $x$ that satisfies the equation $x|x| = 3x + 2$.


Find the value of $b$ so that the lines with the given equations are perpendicular:
\begin{align*}
y &= 3x + 5 \\
4y + bx = 8.
\end{align*}


Let $\alpha$, $\beta$, $\gamma$ be the roots of the cubic polynomial $x^3 - 3x - 2 = 0.$  Find
\[
\alpha(\beta - \gamma)^2 + \beta(\gamma - \alpha)^2 + \gamma(\alpha - \beta)^2.
\]


Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \ne 1$. Compute
\[\omega^{12} + \omega^{15} + \omega^{18} + \omega^{21} + \omega^{24}.\]


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


If $m = 2^{12} \cdot 3^{15} \cdot 5^{9}$, how many of the natural-number factors of $m$ are multiples of 180?


Let $s$ be the positive real solution to $x^3 - \frac{3}{7}x - 1 = 0$. Find the exact numerical value of
\[
S = s^3 + 2s^6 + 3s^9 + 4s^{12} + \dotsb.
\]


Five concentric circles are drawn with radii of 2, 4, 6, 8, and 10. The innermost circle is painted white, the next ring is black, the following ring is white, the next is black, and the outermost ring is white. What is the ratio of the black area to the white area? Express your answer as a common fraction.


In how many distinct ways can I arrange my six keys on a keychain, if I want to put my house key next to my car key, and assume that the keychain treats all arrangements as equivalent under rotation and reflection? Additionally, I want my office key and bike key to stay next to each other as well.


Find the inverse of the matrix
\[\begin{pmatrix} 5 & -3 \\ -2 & 1 \end{pmatrix}.\]
If the inverse does not exist, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} -1 & -3 \\ -2 & -5 \end{pmatrix}}


The sequence of numbers is arranged in rows such that each row $n$ contains $2n$ copies of the number $2n$. Given this arrangement, find the value of the $60^{\mathrm{th}}$ number if the sequence were continued accordingly.


A laptop's original price was first discounted by 20%, then it was further reduced by 30% during a seasonal sale, and finally, during a clearance event, it was discounted by another 10%. What is the total percent reduction from the original price?


A right cone has a base with a circumference of $20\pi$ inches and a height of 40 inches. The height of the cone is reduced while the circumference stays the same. After reduction, the volume of the cone is $400\pi$ cubic inches. What is the ratio of the new height to the original height, and what is the new volume?


Solution: \boxed{59049 \cdot \sqrt[3]{9}}


Consider Abby the alpaca is tethered to the corner of a $4\text{ m}$ by $6\text{ m}$ barn on a $5\text{ m}$ leash. How much area (in square meters) can Abby roam if she can only walk around the barn?

Solution: \boxed{19\pi\text{ m}^2}


What is the value of $\frac{15!}{6!9!}$?


Mr. Huang has 12 grandchildren. Assuming that each grandchild is equally likely to be a boy or a girl independently of the others, what is the probability that Mr. Huang has neither an equal number of grandsons and granddaughters nor all grandchildren being of the same gender?


What is the measure, in units, of the hypotenuse of a right triangle with leg lengths of 140 and 210 units?

Solution: \boxed{70\sqrt{13} \text{ units}}


Solution: \boxed{1 - \frac{1}{100\sqrt{2}}}


In Mr. Petrov's 30-student history class, each student’s initials are uniquely represented by the same letter for both the first and last name. Besides the typical vowels A, E, I, O, U, the letter "X" is also considered a vowel in this peculiar setting. What is the probability of randomly selecting a student whose initials are vowels, given this modification?


A circle is tangent to the lines \(5x - 4y = 40\) and \(5x - 4y = -20.\) The center of the circle lies on the line \(3x - y = 0.\) Find the center of the circle.

Solution: \boxed{\left(-\frac{10}{7}, -\frac{30}{7}\right)}


Let $AB$ be a diameter of a circle centered at $O$. Let $F$ be a point on the circle such that the tangent at $B$ intersects the tangent at $F$ and $AF$ at points $C$ and $D$, respectively. If $\angle BAF = 30^\circ$, find $\angle CFD$, in degrees.


Given nine distinguishable rings, let $m$ be the number of possible five-ring arrangements on the five fingers of one hand (including the thumb). The order of rings on each finger is significant, and it is not required that each finger have a ring. Find the leftmost three nonzero digits of $m$.


Let $a,$ $b,$ $c,$ $d$ be the roots of the equation $3x^4 - 8x^3 + x^2 + 4x - 10 = 0.$ Find $abcd.$


Given three-dimensional vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$, the volume of the parallelepiped they determine is 4. Find the volume of the parallelepiped determined by the vectors $\mathbf{a} + \mathbf{b}$, $\mathbf{b} + 2 \mathbf{c}$, and $\mathbf{c} - 5 \mathbf{a}$.


Eight students participate in a fruit eating contest, where they choose between apples and bananas. The graph shows the number of apples eaten by four students (Sam, Zoe, Mark, Anne) and the number of bananas eaten by another four students (Lily, John, Beth, Chris). In the apple contest, Sam ate the most apples and Mark ate the fewest. In the banana contest, Lily ate the most and Beth ate the fewest. Determine how many more apples than Mark did Sam eat, and how many more bananas than Beth did Lily eat?

[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[8];
for( int i = 0 ; i < 8 ; ++i ){
    yaxis[i] = (0,i);
    draw(yaxis[i]--yaxis[i]+(17,0));
}
draw((0,0)--(0,7));
draw((17,7)--(17,0));

// Apples
fill((1,0)--(1,5)--(2,5)--(2,0)--cycle,yellow);
fill((3,0)--(3,2)--(4,2)--(4,0)--cycle,yellow);
fill((5,0)--(5,3)--(6,3)--(6,0)--cycle,yellow);
fill((7,0)--(7,1)--(8,1)--(8,0)--cycle,yellow);

// Bananas
fill((9,0)--(9,2)--(10,2)--(10,0)--cycle,orange);
fill((11,0)--(11,4)--(12,4)--(12,0)--cycle,orange);
fill((13,0)--(13,6)--(14,6)--(14,0)--cycle,orange);
fill((15,0)--(15,1)--(16,1)--(16,0)--cycle,orange);

label("0",yaxis[0],W);
label("1",yaxis[1],W);
label("2",yaxis[2],W);
label("3",yaxis[3],W);
label("4",yaxis[4],W);
label("5",yaxis[5],W);
label("6",yaxis[6],W);
label("7",yaxis[7],W);
label("Students (Apples/Bananas) ",(8.5,0),S);
label("Results of a Fruit Eating Contest",(8.5,7),N);
label(rotate(90)*"$\#$ of Fruits Eaten",(-1,3.5),W);
[/asy]


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 $200$ and $PR = 40$, find the length of $QS$.


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


Simplify $\left( \frac{3x}{2} \right)^{-2} - \left( \frac{x^3}{4} \right)(2x)$.

Solution: \boxed{\frac{8 - 9x^6}{18x^2}}


Find the remainder when \( 8x^5 - 10x^4 + 6x^3 - 2x^2 + 3x - 35 \) is divided by \( 2x - 8 \).


What is the sum of all the odd integers from 21 through 51, inclusive?


The graphs of a function $f(x) = 4x + b$ and its inverse function $f^{-1}(x)$ intersect at the point $(-4, a)$. Given that $b$ and $a$ are both integers, what is the value of $a$?


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


Solve for $z$ in the equation: $5 + 2iz = 4 - 6iz$.

Express your answer in standard form.


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

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


Triangle $PAB$ is formed by two secants meeting at $P$ outside circle $O$ such that segments $PA$ and $PB$ intersect the circle. Points $A$ and $B$ are on the circle and $\angle APB = 60^\circ$. Find $\angle AOB$.


A circular pie with diameter $18\text{ cm}$ is cut into four equal-sized sector-shaped pieces. Let $m$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $m^2$?

Solution: \boxed{m^2 = 162}


The image of the point with coordinates $(2,3)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(8,-1)$. Find $m+b$.


A square is initially divided into sixteen equal smaller squares. The center four squares are then each divided into sixteen smaller squares of equal area, and this pattern continues indefinitely. In each set of sixteen smaller squares, the four corner squares are shaded. What fractional part of the entire square is shaded?


What is the sum of the digits of the base-2 expression for $345_{10}$?


The function \( \lfloor x\rfloor \) is defined as the largest integer less than or equal to \( x \). Given this, calculate the range of the function
$$f(x) = \lfloor 2x\rfloor - 2x.$$
Express your range in interval notation.


Given $J_k = 10\ldots 0128$, where there are $k$ zeros between the $1$ and the $128$, determine the maximum value of the number of factors of $2$ in the prime factorization of $J_k$.


All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list?


For what values of \( x \) is \( x^2 - 40x + 400 \leq 10 \)?

Solution: \boxed{[20 - \sqrt{10}, 20 + \sqrt{10}]}


Let $p,$ $q,$ $r$ be the roots of the cubic polynomial $x^3 - 2x - 2 = 0.$ Find
\[p(q - r)^2 + q(r - p)^2 + r(p - q)^2.\]


Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,50\pi)$.


How many distinct diagonals of a convex nonagon (9-sided polygon) can be drawn such that none of the diagonals is parallel to any side of the polygon?


Consider a cube with vertices at $(0,0,0), (0,0,6), (0,6,0), (0,6,6), (6,0,0), (6,0,6), (6,6,0),$ and $(6,6,6)$. A plane cuts the edges of this cube at the points $A = (0,3,0), B = (2,0,0),$ and $C = (2,6,6),$ among others. Determine the distance between the intersection points on the edges $(0,0,6)$ to $(6,0,6)$ and $(0,6,0)$ to $(0,6,6)$.


Given that the point $(3, 9)$ is on the graph of $y=g(x)$, determine a point that must be on the graph of $3y=4g(3x) - 6$. What is the sum of the coordinates of that point?


In a triangle \( ABC \), where \( AB = 25 \) and \( BC = 20 \). Find the largest possible value of \( \tan A \).


In the arrangement of two concentric circles, the radius of the outer circle is three times the radius of the inner circle. Calculate the area of the space between the two circles (the gray region) if the width of this region is 4 feet. Express your answer in terms of $\pi$.


If $g(x) = g(3-x)$ for all $x$, then what line is necessarily an axis of symmetry of the graph of $y=g(x)$?

Solution: \boxed{x = 1.5}


Determine the greatest common divisor of all six-digit integers formed by repeating an even three-digit integer. For example, forming 484,484 from 484 or 252,252 from 252.


My friend Julia finds numbers fascinating only if they are divisible by 3. How many different last digits are possible in numbers that Julia likes?


For a French class, I need to master a list of 600 vocabulary words for an upcoming test. The score on the test is based on the percentage of words I recall correctly. In this class, I have noticed that even when guessing the words I haven't studied, I have about a 10% chance of getting them right due to my prior knowledge. What is the minimum number of words I need to learn in order to guarantee at least a 90% score on this test?


Let \( A = (2,0) \) and \( B = (8,6) \). Let \( P \) be a point on the parabola \( y^2 = 8x \). Find the smallest possible value of \( AP + BP \).


Sandy thinks of a number. She triples it, adds 20, squares the result, and the final outcome is 2500. What was her original number?


If $5x - 3y = 27$ and $3x + 5y = 1$, find the value of $x + y$.


We have a triangle $\triangle DEF$ and a point $L$ on $EF$ such that $DL$ is an altitude of $\triangle DEF$. If $DE = 12$, $EL = 9$, and $EF = 15$, then what is the area of $\triangle DEF$?


Triangle $XYZ$ is an isosceles triangle where $\angle X = 40^\circ$. Determine the measure of the largest interior angle of triangle $XYZ$.


A line is parameterized by a parameter $t$, such that the vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$, and the vector on the line at $t = 4$ is $\begin{pmatrix} 5 \\ -7 \end{pmatrix}$. Find the vector on the line at $t = -2$.

Solution: \boxed{\begin{pmatrix} -1 \\ 17 \end{pmatrix}}


Out of the set of numbers 1742, 1865, 1907, 2003, 2091, and 2157, six numbers are given, and five of these have a mean (average) of 1952. What is the mean (average) of the remaining number?


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 20 and altitude 24, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.


Five concentric circles are drawn with radii of 1, 3, 5, 7, and 9. The inner circle is painted black, the first ring around it is white, the second ring is black, the third ring is white, and the outermost ring is black. What is the ratio of the black area to the white area? Express your answer as a common fraction.


Given the line segment connecting points $(1,-3)$ and $(6,12)$ that can be parameterized by
\begin{align*}
x &= at + b, \\
y &= ct + d,
\end{align*}
where $0 \le t \le 1$ and $t = 0$ corresponds to $(1,-3)$, find the values $a+c^2+b^2+d^2.$


Let $A = (0, 5)$ and $B = (0, 15)$. Points $A'$ and $B'$ are on the line $y = x$, and lines $\overline{AA'}$ and $\overline{BB'}$ intersect at point $C = (3, 7)$. Find the length of $\overline{A'B'}$.


Factor $x^2 - 6x + 9 - 64x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form \((ax^2+bx+c)(dx^2+ex+f)\), with \(a<d\).

Solution: \boxed{(-8x^2 + x - 3)(8x^2 + x - 3)}


Solution: \boxed{16x^{18} - 32x^{11} + 40x^3 + 24}


Calculate the sum of the first 5 terms of the geometric series $1 + \left(\frac{1}{4}\right) + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \left(\frac{1}{4}\right)^4$. Express your answer as a common fraction.


What is the greatest common factor of 120, 180, and 300?


Given $g(x)$ is a function defined only for $-1 \le x \le 2$, and $g(x) = cx+d$ for constants $c$ and $d$ where $c > 0$, what is the range of $g$ over its domain? Express your answer in interval notation.

Solution: \boxed{[-c+d, 2c+d]}


There are $30n$ members in the Hamilton marching band, and when they line up in rows of 34, there are 2 band members left over. If $n$ is an integer and there are fewer than 1500 band members, what is the maximum number of people that could be in the Hamilton marching band?


Solve for $x$ in the following equation: $2 + ix = -2 - 2ix$ (where $i^2 = -1$).


For what value of $k$ does the equation $(x+5)(x+2) = k + 3x$ have exactly one real solution? Express your answer as a common fraction.


The operation $\star$ is defined as $a \star b = a^3 \div b$. For how many integer values of $x$ will the value of $27 \star x$ be a positive integer?


Solution: \boxed{2^{4/3} - 30}


What is the 7th term of an arithmetic sequence of 25 terms where the first term is 5 and the last term is 80?


Two numbers are independently selected from the set of positive integers less than or equal to 6. Exactly one of the numbers must be even. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction.


Mike climbs up a flight of stairs of $n$ steps in two ways. When he does it $6$ steps at a time, there are $5$ steps left at the top. When he does it $7$ steps at a time, there are $3$ steps left at the top. What is the smallest possible value of $n$ that is greater than $20$?


What is the positive difference between the two largest prime factors of $195195$?


When \(\frac{7}{5000}\) is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit?


If the system of equations
\[
4x - 2y = c,
\]
\[
6y - 12x = d.
\]
has a solution, find \( \frac{c}{d}, \) assuming \( d \neq 0. \)


Rationalize the denominator of the expression $\frac{3}{4\sqrt{6} + 5\sqrt{7}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms, and all radicals are in simplest radical form. What is $A+B+C+D+E$?


The polynomial
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
has coefficients that are all integers, with roots \( -3, 4, 8, \) and \( -1/4 \). If \( e \) is a positive integer, find its smallest possible value.


A $\textit{palindrome}$ is a number which reads the same forward and backward. For example, 121 and 2332 are palindromes. What is the least natural number that can be added to 52,679 to create a palindrome?


A newly formed society consists of 12 members and a similar hierarchical leadership structure as described in the original problem. The society seeks to appoint a chief, followed by two supporting chiefs (Supporting Chief A and Supporting Chief B). Supporting Chief A will oversee 3 inferior officers, while Supporting Chief B will oversee 2 inferior officers. Determine the number of different ways to establish this leadership configuration.


A point in space has rectangular coordinates $(3, -3, 2\sqrt{2})$. It is known that these are derived from its spherical coordinates $(\rho, \theta, \phi)$. Find the rectangular coordinates of the point whose spherical coordinates are $(\rho, \theta + \frac{\pi}{2}, -\phi)$.

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


The function $f(x)$ satisfies
\[ f(xy) = \frac{f(x)}{y} \]
for all positive real numbers $x$ and $y.$ If $f(45) = 15,$ find $f(60).$


"**For how many integer values of \(n\) between 1 and 510 inclusive does the decimal representation of \(\frac{n}{2125}\) terminate?**"


In a diagram, rectangle $ABCD$ is divided into six identical squares. If $ABCD$ has a perimeter of 160 cm, what is its area, in square centimeters?

Solution: \boxed{1536 \text{ square centimeters}}


In triangle $XYZ$, where $\angle X = 90^\circ$, $YZ = 25$, and $\tan Z = 7\cos Y$. What is $XY$?


Determine all \(y \in [0, \frac{\pi}{2}]\) for which \[\cos(x+y) \geq \cos x + \cos y - 1\] for every \(x\) between \(0\) and \(\frac{\pi}{2}\), inclusive.

Solution: \boxed{y = 0}


Suppose that $b$ is a positive integer greater than or equal to $2.$ Determine the number of possible values for $b$ if $256$ is converted to base $b$, and the resulting representation has $4$ digits.


Suppose I have 5 shirts, 3 ties, and 4 pairs of pants. If an outfit requires a shirt and pants, and can either have a tie or not have a tie, how many outfits can I create?


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x+y) + x f(y) - 2xy - x^2 + 2\] for all real numbers $x$ and $y.$

Let $n$ be the number of possible values of $f(1),$ and let $s$ be the sum of all possible values of $f(1).$ Find $n \times s.$


The function $g(x)$ satisfies the equation
\[xg(y) = 2yg(x)\] for all real numbers $x$ and $y$. If $g(10) = 30$, find $g(2)$.


Let $n$ represent the smallest integer that satisfies the following conditions:
$\frac{n}{2}$ is a perfect square.
$\frac{n}{3}$ is a perfect cube.
$\frac{n}{5}$ is a perfect fifth.
How many divisors does $n$ have that are not multiples of 6?


The line $y = 5x - 7$ is parameterized by the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} s \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ m \end{pmatrix}.\]
Enter the ordered pair $(s,m)$.


Yan is positioned between his home and a stadium. He has two choices to reach the stadium: either he can walk directly there, or he can walk back home and then ride his bicycle to the stadium. Riding his bicycle is 5 times faster than walking. Both options take the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?


A whole number is defined as ''7-shifty" if the remainder when the number is divided by 7 is greater than 2. What is the least three-digit 7-shifty whole number?


My cousin reads five times as fast as I do. If it takes me 3 hours to read a book, how many minutes will it take my cousin to read a version of the book that is 50% longer than the one I read?

Solution: \boxed{54 \text{ min}}


What is the remainder when $98^{50}$ is divided by 100?


How many positive integers less than $151$ are multiples of either $6$ or $8$, but not both at once?


Let the ordered triples $(a,b,c)$ of complex numbers that satisfy
\begin{align*}
a + bc &= 9, \\
b + ac &= 15, \\
c + ab &= 15.
\end{align*}
be $(a_1,b_1,c_1),$ $(a_2,b_2,c_2),$ $\dots,$ $(a_n,b_n,c_n).$  Find $a_1 + a_2 + \dots + a_n.$


There are two **positive** integers $d$ for which the equation $$8x^2 + 16x + d = 0$$ has rational solutions. What is the product of those two values of $d$?


All triangles have the same value, and all circles have the same value. Determine the sum of two circles and a triangle. The equations are as follows:
\begin{align*}
\Delta + 3 \bigcirc &= 18 \\
2 \Delta + \bigcirc &= 14 \\
\Delta + 2 \bigcirc &= \ ?
\end{align*}

Solution: \boxed{x + 2y = \frac{68}{5}}


Suppose we want to divide 12 dogs into three groups, one with 4 dogs, one with 5 dogs, and one with 3 dogs. How many ways can we form the groups such that Fluffy is in the 4-dog group and Nipper is in the 5-dog group?


Given the expression \( c \cdot a^b - d \), where each of \( a \), \( b \), \( c \), and \( d \) are distinct numbers from the set {0, 1, 2, 3} and \( d \neq 0 \), determine the maximum possible value of the expression.


An integer greater than 99 and less than 1000 is randomly chosen. What is the probability that its digits are all different?


What is the greatest integer less than or equal to \[\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}?\]


In triangle $XYZ,$ the midpoint of $\overline{YZ}$ is $(2,4,1),$ the midpoint of $\overline{XZ}$ is $(-1,3,-2),$ and the midpoint of $\overline{XY}$ is $(3,2,5).$ Find the coordinates of vertex $X.$

Solution: \boxed{(0, 1, 2)}


How many integers are there between $\sqrt{10}$ and $\sqrt{40}$?


What is the sum of the two solutions to the equation $-3x^2 - 27x + 54 = 0$?


Express the quotient $2034_5 \div 23_5$ in base $5$.


A deck of fifty cards consists of five $1$'s, five $2$'s,..., and five $10$'s. A matching triplet (three 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 also form a pair, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$


Triangle $DEF$ has a perimeter of 3010 units. The sides have lengths that are all integer values with $DE < EF \leq FD$. What is the smallest possible value of $EF - DE$?


Express $0.\overline{2} + 0.\overline{02} + 0.\overline{0002}$ as a common fraction.


Simplify $(8 \times 10^7) \div (4 \times 10^2)$.


Find the radius of the circle given by the equation $4x^2 - 8x + 4y^2 + 16y + 20 = 0.$


Find the largest three-digit integer that is divisible by each of its distinct, non-zero digits, where the hundreds digit is 8.


For what base is the representation of $329_{10}$ a four-digit number whose final digit is even?


I have 6 marbles numbered 1 through 6 in a bag. Suppose I take out two different marbles at random. What is the expected value of the sum of the numbers on the marbles?


The least common multiple of two numbers is 5040, and the greatest common divisor is 24. Given that one of the numbers is 240, what is the other number?


Solution: \boxed{-x^2 + 23x - 3}


There are $48$ different complex numbers \( z \) such that \( z^{48} = 1 \). For how many of these is \( z^{12} \) a real number?


Consider the line \( n \) defined by the equation \( y - 2x = 1 \) and the line \( p \) defined by the equation \( 3x + y = 6 \). Calculate the intersection point of these two lines. Additionally, determine the angle between these two lines.


How many different counting numbers will each leave a remainder of 6 when divided into 53?


Without using a calculator, compute $1005^2 - 995^2 - 1002^2 + 996^2$.


Compute
$$\sum_{k=1}^{500} k\left(\lceil \log_3{k}\rceil- \lfloor \log_3{k} \rfloor\right).$$


I had \$50 in allowance money and spent it as indicated in the pie graph shown. How many dollars did I spend on games?

[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(1,0);
B=(0,1);
C=(-1,0);
D=(0,-1);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(0.5,0.3);
label("Books", W, N);
label("$\frac{1}{4}$", W, S);
X=(-0.7, 0);
label("Games", X, S);
Y=(0,-0.6);
label("Snacks", Y, N);
label("$\frac{1}{5}$", Y, S);
Z=(-0.2, 0.7);
label("Apps", Z, N);
label("$\frac{3}{10}$", Z, S);
[/asy]


Determine the domain of the function $f(x) = \tan(\arcsin(x^2)).$

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


Solve for $x$:
$$\frac{x+4}{x-3} = \frac{x-2}{x+2}.$$
Express your answer as a fraction.


The infinite sequence \(T = \{t_1, t_2, t_3, \ldots\}\) is defined by \(t_1 = 3\) and \(t_n = 3^{t_{n-1}}\) for each integer \(n > 1\). What is the remainder when \(t_{100}\) is divided by 7?


In a shooting competition, nine clay targets are set up in three hanging columns. The first column contains four targets, the second column contains three targets, and the third column contains two targets. A sharpshooter must break all the targets under the following conditions:
1) The sharpshooter first selects any one of the columns from which a target is to be broken.
2) The sharpshooter must then break the lowest remaining target in the chosen column.
What are the different possible orders in which the sharpshooter can break all nine targets?


The faces of a decagonal die are labeled with digits $1$ through $10$. What is the probability, expressed as a common fraction, of rolling a sum of $18$ with a pair of such decagonal dice?


Find the modular inverse of $15$ modulo $16$.


There are six unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed to these six people, with each person getting one letter, what is the probability that none of the six people receives the correct letter?


If $a, b, c$ are non-negative integers less than $9$ such that
\[
a + 2b + 3c \equiv 1 \pmod{9},
\]
\[
2a + 3b + c \equiv 2 \pmod{9},
\]
\[
3a + b + 2c \equiv 3 \pmod{9},
\]
then determine the remainder when $abc$ is divided by $9$.