What is the area, in square units, of triangle $ABC$ if $A(-5,4)$, $B(1,7)$, and $C(4,-3)$?


Find the largest positive integer $n$ such that
\[\sin^n x + \cos^n x \ge \frac{2}{n}\]for all real numbers $x.$


Let \(\mathbf{u}, \mathbf{v}, \mathbf{w},\) and \(\mathbf{z}\) be vectors such that \(\|\mathbf{u}\| = 2,\) \(\|\mathbf{v}\| = 6,\) \(\|\mathbf{w}\| = 3,\) and \(\|\mathbf{z}\| = 1,\) and
\[\mathbf{u} + \mathbf{v} + \mathbf{w} + \mathbf{z} = \mathbf{0}.\] Compute \(\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} + \mathbf{u} \cdot \mathbf{z} + \mathbf{v} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{z} + \mathbf{w} \cdot \mathbf{z}.\)


The planning committee at school has 12 members. Exactly 5 of these members are teachers. A four-person subcommittee with at least two teachers must be formed from the members of the planning committee. How many distinct subcommittees are possible?


What perfect-square integer is closest to 350 that is also a multiple of 4?


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


The function $f(x)$ satisfies
\[f(x + y) = f(x) + f(y)\]for all real numbers $x$ and $y,$ and $f(6) = 3.$  Find $f(7).$


Below is a portion of the graph of a function, $y=v(x)$:
[asy]
import graph; size(5.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=3.25,ymin=-3.25,ymax=3.25;

pen cqcqcq=rgb(0.75,0.75,0.75);

/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);

Label laxis; laxis.p=fontsize(10);

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f2(real x){return -x^2+3*sin(x*pi/3);}
draw(graph(f2,-3.25,3.25),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]

What is the exact value of $v(-2.5) + v(-1) + v(1) + v(2.5)$?


What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (8, 6) and the midpoint of the segment with endpoints at (5, 0) and (5, 9)?


Calculate the value of $(8^8 \div 8^5) \cdot 2^{10} \cdot 2^3$.


If a deck of 60 cards, consisting of 52 playing cards and 8 instruction cards, is distributed as evenly as possible among 9 people, how many people will end up with exactly 6 playing cards?


Let $\mathbf{p}$ and $\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $45^\circ.$ Find the area of the parallelogram whose diagonals correspond to $\mathbf{p} + 3\mathbf{q}$ and $3\mathbf{p} + \mathbf{q}.$


Sixty ping-pong-balls are numbered 1, 2, 3, ..., 59, 60. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of 6 or 8 or both, and is also an odd number? Express your answer as a common fraction.

Solution: \boxed{\frac{0}{60} = 0}


A $140 \times 320 \times 360$ rectangular solid is constructed from $1 \times 1 \times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of these unit cubes?


Find the coefficient of $x$ when $5(x - 6) + 7(8 - 3x^2 + 6x) - 9(3x - 2)$ is simplified.


Find the largest value of $n$ such that $5x^2 + nx + 110$ can be factored as the product of two linear factors with integer coefficients.


Alexio has 120 cards numbered from 1 to 120, inclusive, and places them in a box. He then randomly picks a card. What is the probability that the number on the card is a multiple of 2, 4, or 6? Express your answer as a common fraction.


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0 < y \le 10$, then what is the largest possible value of $a+b+c$?


What is the greatest integer less than 200 for which the greatest common factor of that integer and 24 is 2?


Calculate $\theta_1 + \theta_2 + \theta_3 + \theta_4$ for the complex equation $z^4 = 16i$, where each solution can be expressed as:
\[
z_k = r_k (\cos \theta_k + i \sin \theta_k),
\]
and $r_k > 0$ and $0^\circ \leq \theta_k < 360^\circ$ for $k=1, 2, 3, 4$.


The fifth and seventeenth terms of an arithmetic sequence are $\frac{3}{8}$ and $\frac{7}{12}$, respectively. What is the eleventh term?


Suppose the numbers \[2 - \sqrt{3},\; -2 - \sqrt{3},\; 3 + 2\sqrt{5},\; 3 - 2\sqrt{5}\] are roots of a nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?


In a class of 150 students, all taking the same test, what is the maximum number of students who can receive a score above the class average?


A rectangular prism has dimensions 3 units, 4 units, and 5 units along its length, width, and height, respectively. Each rectangular face has two diagonals. How many total diagonals does this rectangular prism have, including the longest diagonal of the prism itself?


Given that $q(x) + (2x^6 + 4x^4 + 5x^2 + 2) = (x^6 + 12x^5 + 13x^4 + 14x^3 + 17x + 3)$, express $q(x)$ as a polynomial with the degrees of the terms in decreasing order.

Solution: \boxed{-x^6 + 12x^5 + 9x^4 + 14x^3 - 5x^2 + 17x + 1}


Let $n = 2^{12} \cdot 3^{15} \cdot 5^9 \cdot 7^3$. How many of the natural-number factors of $n$ are multiples of $210$?


In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Charlie, and Dave, refuse to sit in four consecutive seats?


If $(x+3)(x-4) = 20$, find the sum of the possible values of $x$.


Three dice, each with faces numbered from 1 to 6, are rolled. What is the probability that the sum of the numbers on the top faces is exactly 12? Express your answer as a common fraction.


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


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


A circle centered at $O$ is circumscribed around $\triangle ABC$. The central angles $\angle AOC$ and $\angle BOC$ measure $130^\circ$ and $120^\circ$ respectively. Determine the measure of $\angle BAC$.


Let \( b > 0 \), and let \( Q(x) \) be a polynomial with integer coefficients such that
\[ Q(2) = Q(4) = Q(6) = Q(8) = b \]
and
\[ Q(1) = Q(3) = Q(5) = Q(7) = -b. \]
What is the smallest possible value of \( b \)?


Let the line $q$ be the perpendicular bisector of points $P = (12, 9)$ and $Q = (4, 6)$. Given that $PQ$ meets $q$ at point $R = (x, y)$, what is $3x + 2y$?


Simplify $\sqrt[3]{48a^5b^8c^{14}}$ and determine the sum of the exponents of the variables that are outside the radical.


Suppose I have 7 shirts, 5 pairs of pants, and 4 ties. If an outfit requires a shirt and pants, and can either have a tie or not have a tie, but no two items (shirt, pants, tie) can be of the same color, how many outfits can I make?


Find the domain of the rational function $f(x) = \frac{x^3 - 3x^2 + 6x - 8}{x^2 - 5x + 6}$.

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


Find the number of integers $n$ that satisfy
\[20 < n^2 < 150.\]


It is currently $8\!:\!00\!:\!00 \text{ a.m.}$. What time will it be in $102930$ seconds?  (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)

Solution: \boxed{12\!:\!35\!:\!30 \text{ p.m.}}


Convert the hexadecimal number $ABABA_{16}$ to binary and determine how many base-2 digits (bits) it contains.


The Gnollish language now consists of 4 words: "splargh," "glumph," "amr," and "krack." In a sentence, neither "splargh" can come directly before "glumph," nor can "amr" come directly before "krack." How many valid three-word sentences are there in this expanded Gnollish language?


In right $\Delta DEF$, $\angle DFE$ is a right angle. Point $N$ is the midpoint of $\overline{EF}$. What is the number of centimeters in the length of median $\overline{DN}$? Additionally, calculate the length of the altitude from point $D$ to $\overline{EF}$. The lengths of sides $\overline{DE}$ and $\overline{DF}$ are 6 cm and 8 cm respectively. [asy] pair D,E,F,N;
D = (0,0); E = (8,0); F = (0,6); N = (E+F)/2;
draw(N--D--E--F--D);
label("$D$",D,W); label("$E$",E,E);
label("$F$",F,W); label("$N$",N,NE);
label("6 cm",D--F,W); label("8 cm",D--E,S);
[/asy]

Solution: \boxed{DN = 5 \text{ cm}, \text{ Altitude } = 4.8 \text{ cm}}


A plane intersects a right circular cylinder of radius $2$ forming an ellipse. If the major axis of the ellipse is $40\%$ longer than the minor axis, what is the length of the major axis?


The matrix for reflecting over a specific line $\ell,$ which passes through the origin, is given by
\[
\begin{pmatrix} \frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & -\frac{3}{5} \end{pmatrix}.
\]
Determine the direction vector for line $\ell$. Enter your answer in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a, b$ are integers, $a > 0$, and $\gcd(|a|,|b|) = 1$.

Solution: \boxed{\begin{pmatrix} 2 \\ 1 \end{pmatrix}}


What is the arithmetic mean of all positive three-digit multiples of 6?


The fencing around three congruent equilateral triangular gardens is used to create one larger equilateral triangular garden. Determine the ratio of the total area of the three small gardens to the area of the new larger garden.


In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 15$, $\tan D = 2$, and $\tan B = 3$, then what is $BC$?


The graph of the function $g(x)$ is shown below. How many values of $x$ satisfy $g(g(x)) = 5$? Assume g(x) behaves similarly but with adapted values as shown below. [asy]
import graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;

for(int i = -4; i <= 5; ++i) {
	draw((i,-1)--(i,6), dashed+mediumgrey);
}

for(int i = 1; i <= 6; ++i) {
	draw((-4,i)--(5,i), dashed+mediumgrey);
}

Label laxis; laxis.p=fontsize(10);

xaxis("$x$",-4.36,5.56,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis("$y$",-0.92,6.12,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw((xmin,(-(0)-(-2)*xmin)/-2)--(-1,(-(0)-(-2)*-1)/-2),linewidth(1.2)); draw((-1,3)--(3,5),linewidth(1.2)); draw((3,(-(-16)-(2)*3)/2)--(xmax,(-(-16)-(2)*xmax)/2),linewidth(1.2));

label("$g(x)$",(-3.52,4.6),SE*lsf);

dot((-4.32,4.32),ds); dot((5.56,2.44),ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);

[/asy]


What integer $n$ satisfies $0 \le n < 31$ and $$-250\equiv n\pmod{31}~?$$


Three of the four vertices of a rectangle are \((1, 7)\), \((14, 7)\), and \((14, -4)\). What is the area of the intersection of this rectangular region and the region inside the graph of the equation \((x - 1)^2 + (y + 4)^2 = 16\)?


Consider the functions:
$$
\begin{array}{ccc}
f(x) & = & 7x^3 - \frac{1}{x} + 5\\
g(x) & = & x^3 - k
\end{array}
$$
Find the value of $k$ if $f(3) - g(3) = 5$.

Solution: \boxed{k = -\frac{485}{3}}


Suppose that $b$ is a positive integer greater than or equal to $2$. When $256$ is converted to base $b$, the resulting representation has $3$ digits. What is the number of possible values for $b$?


What is the perimeter, in cm, of quadrilateral \( EFGH \) if \( \overline{EF} \perp \overline{FG} \), \( \overline{HG} \perp \overline{FG} \), \( EF = 15 \) cm, \( HG = 6 \) cm, and \( FG = 20 \) cm?

Solution: \boxed{41 + \sqrt{481} \text{ cm}}


Let \( r_1, r_2, r_3 \) be the roots of the polynomial
\[ x^3 - 3x^2 + 8 = 0. \]
Find the monic polynomial, in \( x \), whose roots are \( 3r_1, 3r_2, \) and \( 3r_3 \).

Solution: \boxed{x^3 - 9x^2 + 216}


The integers $r$ and $k$ are randomly selected, where $-5 < r < 7$ and $2 < k < 10$. What is the probability that the division $r \div k$ results in an integer value? Express your answer as a common fraction.


Consider two shapes made up of unit squares. The first shape is a rectangle 4 units long and 3 units wide. Inside this rectangle, an additional vertical line divides one of its halves into two rectangles each 2 units long. The second shape is a simple 6x1 rectangle. What is the positive difference in their perimeters?


If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are unit vectors such that $\mathbf{a} \cdot \mathbf{b} = 0$ and $\mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c} = \frac{1}{2}$, find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2.
\]


What is the smallest abundant number that is not a multiple of 4?


Let $p$ be the smallest positive, three-digit integer congruent to 7 (mod 13). Let $q$ be the smallest positive, four-digit integer congruent to 7 (mod 13). What is $q-p$?


If $3x + 4$ is a factor of $9x^3 + kx^2 + 16x + 64$, find $k$.


If \(\sum_{n = 0}^{\infty} \cos^{2n} \theta = \frac{9}{2}\), what is the value of \(\cos{4\theta}\)?


Find the coordinates of the center of the hyperbola \[\frac{(4y+8)^2}{7^2} - \frac{(5x-5)^2}{3^2} = 1.\]

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


Let \[q(x,y) =
\begin{cases} 2x + 3y &\quad \text{if } x \ge 1 \text{ and } y \ge 1, \\
x + y^2 &\quad \text{if } x < 0 \text{ and } y < 0, \\
4x - 2y &\quad \text{otherwise}.
\end{cases}
\]
What is $q(q(2,-2),q(0,0))$?


Find all $x$ which satisfy $$\frac{x}{4} \leq 3 + 2x < -3(1 + x).$$ Express your answer in interval notation, simplifying any fractions which occur in your answer.

Solution: \boxed{\left[-\frac{12}{7}, -\frac{6}{5}\right)}


Calculate $\sqrt{\sqrt[4]{0.008}}$. Express your answer as a decimal to the nearest hundredth.


Assume that $h$ and $k$ are functions for which $h^{-1}(k(x)) = 3x-1$. Find $k^{-1}(h(-10))$.


Consider two right triangles ABC and ABD, where D is moved to make the length $AD$ longer, such that $AD = 30$ units. All other measurements remain the same as before except $BD$ which needs recalculations as below. Calculate the new length of segment $BC$ in units.
```asy
size(150);
pair A, B, C, D, X;
A = (0, 0);
B = (0, 12);
C = (-16, 0);
D = (-35, 5);
draw(A--B--D--A);
draw(B--C);
draw((0, 1.5)--(-1.5, 1.5)--(-1.5, 0));
label("$16$", (A + C)/2, S);
label("A", A, SE);
label("B", B, NE);
label("C", C, S);
label("D", D, SW);
```


Tony makes $60\%$ of his initial 30 basketball shots. He then takes an additional 10 shots and increases his shooting percentage to $62\%$. How many of the last 10 shots did he make?


Let $\{b_n\}$ be an arithmetic sequence such that the sum of the first 50 terms is 150, i.e.,
$$
b_1 + b_2 + \cdots + b_{50} = 150,
$$
and the sum of the terms from the 51st to the 100th term is 300, i.e.,
$$
b_{51} + b_{52} + \cdots + b_{100} = 300.
$$
Determine the value of $b_{10} - b_1$.


Define a function $g(n)$ as follows:
\[ g(n) =
\begin{cases}
n^2 - 2n + 2 & \text{if } n \text{ is odd} \\
2n & \text{if } n \text{ is even}
\end{cases}.
\]
For how many integers $n$ from 1 to 100, inclusive, does $g( g (\dotsb g (n) \dotsb )) = 2$ for some number of applications of $g$?


Jessica's grandmother's age is fifteen times Jessica's age. If Jessica's grandmother was 60 years old when Jessica was born, how many years old is Jessica now?


According to the given data, what was the average daily high temperature in Brookdale from October 1st, 2023 through October 7th, 2023, inclusive? Daily high temperatures in degrees Fahrenheit during that period were as follows: 51, 67, 64, 61, 50, 65, 47.


Consider a convex pentagon $ABCDE$. Let $P_A, P_B, P_C, P_D,$ and $P_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCD,$ and $ABCE$, respectively. Compute $\frac{[P_A P_B P_C P_D P_E]}{[ABCDE]}$.


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

Solution: \boxed{y = 0 \text{ or } y = \pi}


Find the largest common value less than 1000 between the arithmetic progressions $\{4, 9, 14, 19, \ldots\}$ and $\{5, 14, 23, 32, \ldots \}$.


A circle having center $(0,k)$, with $k>8$, is tangent to the lines $y=x$, $y=-x$ and $y=8$. What is the radius of this circle?

Solution: \boxed{8\sqrt{2} + 8}


Find $5273_{8} - 3614_{8}$. Express your answer in base $8$.


Below is a portion of the graph of a function, \( y = F(x) \):

[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.5,xmax=4.5,ymin=-0.99,ymax=8.5;

pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);

Label laxis; laxis.p=fontsize(10);

xaxis("",xmin,xmax, Ticks(laxis,Step=1.0,Size=2, NoZero), Arrows(6), above=true); yaxis("",ymin,ymax, Ticks(laxis,Step=1.0,Size=2,NoZero), Arrows(6), above=true);
real f2(real x){return sqrt(abs(x+2))+(10/pi)*atan(sqrt(abs(x-1)))+1;}
draw(graph(f2,xmin,xmax),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=F(x)$",(xmax+0.25,f2(xmax)),E);
[/asy]

Given \( y = F(x) = \sqrt{|x + 2|} + \frac{10}{\pi} \arctan(\sqrt{|x - 1|}) + 1 \), what is \( F(2) \)?


In a shooting match, six clay targets are arranged in two hanging columns of two targets each and one column of two taller targets. A marksman is to break all the targets according to the following rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the lowest remaining target in the chosen column.
If the rules are followed, in how many different orders can the six targets be broken?


Find \(a+b+c\) if the graph of the equation \(y=ax^2+bx+c\) is a parabola with a vertex at (4, 4), vertical axis of symmetry, and contains the point (3, 0).


Simplify the expression $3x + 2(2 - x) - 5(\cos(\pi/6) - 3x) + x(2 + 5x)$.

Solution: \boxed{5x^2 + 18x + 4 - \frac{5\sqrt{3}}{2}}


Determine the number of solutions to
\[3\cos^3 x - 7\cos^2 x + 3\cos x = 0\]
in the range \(0 \leq x \leq 2\pi\).


A square $ABCD$ has a side length of 36 inches. An equilateral triangle $BFC$ with the same side length is cut from the square. This triangle is then rotated 180 degrees around point B and attached to create a new shaded region $ABFCDE$. Calculate the perimeter of this shaded region.

Solution: \boxed{216 \text{ inches}}


Find the number of triples $(x, y, z)$ of real numbers that satisfy:
\begin{align*}
x &= 2023 - 2024 \operatorname{sign}(y + z), \\
y &= 2023 - 2024 \operatorname{sign}(x + z), \\
z &= 2023 - 2024 \operatorname{sign}(x + y).
\end{align*}


The graphs of $y=|x|$ and $y=-x^2-4x-3$ are drawn. For every $x$, a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments.


Fully factor the following expression: $6x^3 - 54x$

Solution: \boxed{6x(x + 3)(x - 3)}


How many times does the digit 8 appear in the list of all integers from 1 to 800?


Find the sum of $642_8$ and $157_8$ in base $8$.


In the diagram, $\triangle ABC$ is right-angled at $A$, with $AB = 12$ and $AC = 16$. The point $D$ is on $BC$ such that $AD$ is perpendicular to $BC$. Determine the lengths of $DB$ and $DC$. [asy]
pair A, B, C, D;
B=(0,0);
A=(0,12);
C=(20,0);
D=(0,0);
draw(A--B--C--A--D);
label("B", B, SW);
label("A", A, N);
label("D", D, S);
label("C", C, SE);
label("12", (A+B)/2, NW);
label("16", (A+C)/2, NE);
[/asy]


Two circles, one with a radius of 7 inches and the other with a radius of 3 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one along the larger circle at $4\pi$ inches per minute, the other along the smaller circle at $3\pi$ inches per minute. How many minutes will it take before their next meeting at point P, and how far will each bug have traveled by then?


The function $g$ defined by $g(x) = \frac{px+q}{rx+s}$, where $p$, $q$, $r$, and $s$ are nonzero real numbers, has the properties $g(5) = 5$, $g(13) = 13$, and $g(g(x)) = x$ for all values except $\frac{-s}{r}$. Find the unique number that is not in the range of $g$.


Find the minimum value of the expression
\[(15 - x)(8 - x)(15 + x)(8 + x).\]


Suppose that all four of the numbers \[3 - \sqrt{8}, \;5+\sqrt{11}, \;16 - 2\sqrt{10}, \;-\sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^3+ax^2+bx+c$ and $x^3+bx^2+cx+a$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^3+ax^2+bx+c$ and $x^3+bx^2+cx+a$ is $x^4-3x^3-2x^2+6x+9$.  Find $a+b+c$.


Alice rolls a fair eight-sided die every morning. If Alice rolls a perfect square number, she drinks coffee. If she rolls a prime number, she drinks tea. If she rolls a number that is neither, she drinks water. In a non-leap year, what is the expected value of the difference between the number of days Alice drinks coffee and the number of days she drinks tea?


Determine the solution to $15y + 4 \equiv 7 \pmod{18}$ where $y \equiv b \pmod{n}$ for some positive integers $n \geq 2$ and $b < n$. Find $b+n$.


If \( a \), \( b \), and \( c \) are digits and \( 0.abc \) can be expressed as \( \frac{1}{y} \) where \( y \) is an integer such that \( 0 < y \le 15 \), then what is the largest possible value of \( a+b+c \)?


Solve the inequality
\[\frac{x^2 - 16}{x^2 + 10x + 25} < 0.\]

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


In how many ways can you arrange the digits of 1250 to get a four-digit multiple of 5?


What are the last two digits in the sum of factorials of the first 15 positive integers?


Alice chooses randomly two integers between 1 and 100, inclusive. What is the probability that at least one of the integers Alice chooses is a multiple of 4?


For how many positive integers $n \geq 2$ is $1111_n$ a prime number?


Calculate $3421_6 + 4525_6$. Express your answer in base $6$.


Let $n$ be a positive integer and $a, b, c$ be integers such that $a \equiv b^{-1} \pmod{n}$ and $c \equiv b \pmod{n}$. What is the remainder when $abc$ is divided by $n$?


What is the smallest prime number whose digits sum to $23$?


Determine the area, in square units, of the square with vertices at \(P (1, 1)\), \(Q (-4, 2)\), \(R (-3, 7)\), and \(S (2, 6)\).

Solution: \boxed{26 \text{ square units}}


The operation $\star$ is defined as $\frac{m}{n}\star\frac{p}{q} = (m)(p)\left(\frac{n}{q}\right)$. What is the simplified value of $\frac{5}{9}\star\frac{6}{4}$?


Consider the lines given by the equations:
\[
a: y = 4x + 3, \quad b: 2y = 3x + 4, \quad c: y = 8x - 1, \quad d: 2y = x + 6, \quad e: 4y = 2x - 8.
\]
Determine how many pairs of these lines are either parallel or perpendicular to each other.


Given the system of equations
\[
4x - 2y = a,
\]
\[
6y - 12x = b,
\]
find $\frac{a}{b},$ assuming $b \neq 0.$


Determine the range of the function $p(x) = x^4 - 4x^2 + 4$ over the domain $[0, \infty)$.

Solution: \boxed{[0, \infty)}


Find the remainder when \(x^4 + 1\) is divided by \(x^2 - 4x + 6.\)

Solution: \boxed{16x - 59}


Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{w},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{v}.$  If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{4},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{v}\|}.$


In a library, there are $1452$ books stacked into piles, each containing $42$ books.

A librarian is instructed to redistribute these books such that each new pile contains $43$ books. After redistributing as many full piles as possible, how many books remain?


Determine how many unique products can be calculated by multiplying two or more distinct members of the set $\{2, 3, 5, 11, 13, 17\}$.


Let $x$, $y$, and $z$ be real numbers such that
\[\cos (x+\pi/4) + \cos (y+\pi/4) + \cos (z+\pi/4) = \sin (x+\pi/4) + \sin (y+\pi/4) + \sin (z+\pi/4) = 0.\]
Find the sum of all possible values of $\cos (2x) + \cos (2y) + \cos (2z).$


Kevin Kangaroo begins his journey on a number line at 0 with a goal of reaching a rock located at 1. However, he hops only $\frac{1}{4}$ of the distance toward the rock with each leap. After each leap, he tires, reducing his subsequent hop to $\frac{1}{4}$ of the remaining distance to the rock. Calculate the total distance Kevin hops after seven leaps. Express your answer as a common fraction.


Let $g(x) = x^2 - x + 2007$. What is the greatest common divisor of $g(102)$ and $g(103)$?


The line segment connecting points $(1,3,6)$ and $(6,-2,12)$ in 3D space can be parameterized by:
\begin{align*}
x &= at + b, \\
y &= ct + d, \\
z &= et + f,
\end{align*}
where $0 \le t \le 1,$ and $t = 0$ corresponds to the point $(1,3,6)$. Find the sum $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$.


In a right triangle $RPT$, where $\angle RTP = 90^\circ$, suppose $\sin \angle PRQ = \frac{3}{5}$. If $Q$ lies on line $PT$ extended such that $\angle PRQ = 180^\circ - \angle RPT$, what is $\cos \angle RPQ$?


Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 7\}$. What is the probability that the greatest common divisor of these two numbers is one? Express your answer as a common fraction.


Let \( p, q, r, s, t, u \) be positive real numbers such that \( p + q + r + s + t + u = 10 \). Find the minimum value of
\[ \frac{1}{p} + \frac{9}{q} + \frac{25}{r} + \frac{49}{s} + \frac{81}{t} + \frac{121}{u}. \]


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


In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $j+k$?


A store sells a special edition toy and discovers that if the toy's price is \( p \) dollars (where \( p \leq 30 \)), then the number of toys sold per month is \( 150 - 4p \). What price should the store charge to maximize its revenue?


For a number $3\,452\,X\,Y2$ to be divisible by $9$, where $X$ and $Y$ each represent a single digit, find the sum of all possible values of $X+Y$.


A circle is circumscribed about an equilateral triangle with side lengths of $12$ units each. In the same plane, a square has been drawn with its side length equal to the height of this equilateral triangle. Calculate the area of the circle and the square, expressing the area of the circle in terms of $\pi$.


Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ 2 \end{pmatrix}.$

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


An equilateral triangle has an area of $81\sqrt{3}$ $\text{cm}^2$. If each side of the triangle is decreased by 3 cm, by how many square centimeters is the area decreased?

Solution: \boxed{24.75\sqrt{3} \text{ cm}^2}


Determine the standard spherical coordinates of the point $\left(5, \frac{15 \pi}{4}, \frac{3 \pi}{4}\right)$. Enter your answer in the form $(\rho, \theta, \phi)$, where $\rho > 0$, $0 \le \theta < 2 \pi$, and $0 \le \phi \le \pi$.

Solution: \boxed{\left( 5, \frac{7 \pi}{4}, \frac{3 \pi}{4} \right)}


Let $(v_n)$ be a sequence of real numbers defined by the recurrence relation
\[v_{n+2} = 3v_{n+1} - v_n\]
with initial conditions $v_3 = 16$ and $v_6 = 211$. Determine $v_5$.


Consider the graph of \( y = g(x) \), with \( 1 \) unit between grid lines, where \( g(x) = \frac{(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{720} - 2.5 \), defined only on the shown domain.

Determine the sum of all integers \( c \) for which the equation \( g(x) = c \) has exactly \( 4 \) solutions.

[asy]
size(150);
real f(real x) {return (x-4)*(x-2)*x*(x+2)*(x+4)*(x+6)/720-2.5;}
draw(graph(f,-6.5,6.5), red+linewidth(1));
xaxis("$x$",EndArrow);
yaxis("$y$",EndArrow);
real[] ys = {-3, -2.5, -2};
for(real y : ys) draw((-6.5,y)--(6.5,y), dashed+green);
[/asy]


How many integers $-12 \leq n \leq 12$ satisfy $(n-3)(n+5)(n+9)<0$?


What is the sum of all integer values $n$ such that $\binom{30}{15} + \binom{30}{n} = \binom{31}{16}$?


How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?


The least common multiple of $x$, $12$ and $18$ is $108$. What is the greatest possible value of $x$?


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


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


Find the number of positive integers $n \leq 500$ such that $21n$ is a perfect square.


Compute $\arctan ( \tan 75^\circ - 2 \tan 30^\circ )$. (Express your answer in degrees as an angle between $0^\circ$ and $180^\circ$.)


Find the matrix $\mathbf{N}$ such that
\[\mathbf{N} \mathbf{u} = \left( \begin{pmatrix} -3 \\ 6 \\ 1 \end{pmatrix} + \begin{pmatrix} 4 \\ -2 \\ 5 \end{pmatrix} \right) \times \mathbf{u}\]
for all vectors $\mathbf{u}.$

Solution: \boxed{\begin{pmatrix} 0 & -6 & 4 \\ 6 & 0 & -1 \\ -4 & 1 & 0 \end{pmatrix}}


Find the smallest positive integer that is both an integer power of 7 and is not a palindrome.


What is the probability that the square root of a randomly selected two-digit whole number is less than 7? Express your answer as a common fraction.


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.

Solution: \boxed{k = 6}


The diameter, in inches, of a sphere with three times the volume of a sphere of radius 6 inches can be expressed in the form $a\sqrt[3]{b}$ where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$.


Find the number of even digits in the base-7 representation of $528_{10}$.


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


A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction.

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


How many prime numbers have squares between 2500 and 4900?


A number is chosen at random from the set of consecutive natural numbers $\{1, 2, 3, \ldots, 30\}$. What is the probability that the number chosen is a factor of $5!$? Express your answer as a common fraction.


We are allowed to remove exactly one integer from the list $$-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,$$ 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?


If 25,197,624 hot dogs are packaged in sets of 6, how many will be left over?


Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\heartsuit$ and the second card is a king?


Suppose that the angles of triangle $PQR$ satisfy
\[\cos 2P + \cos 2Q + \cos 2R = 1.\]
Two sides of the triangle have lengths 7 and 24. Find the maximum length of the third side.


If $x$ is a real number and $\lceil x \rceil = 12,$ how many possible values are there for $\lceil x^2 \rceil$?


Suppose $a$ and $b$ are positive integers where $a$ has $4$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, what is the least possible value of $b$?


What is the base $2$ representation of $123_{10}$?


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


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


Simplify the following expression into a polynomial:
$$(3x^2 - 2x + 5)(x - 2) - (x - 2)(2x^2 + 5x - 8) + (2x - 3)(x - 2)(x + 4).$$

Solution: \boxed{3x^3 - 8x^2 + 5x - 2}


Let $\mathbf{Q}$ be the matrix for reflecting over the vector $\begin{pmatrix} 4 \\ -2 \end{pmatrix}.$ Find $\mathbf{Q}^2.$

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


How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?


What is the smallest prime whose digits sum to $23$?


Calculate the sum of the first 5 terms of the geometric series: $\left(\dfrac{1}{4}\right) + \left(\dfrac{1}{4}\right)^2 + \left(\dfrac{1}{4}\right)^3 + \left(\dfrac{1}{4}\right)^4 + \left(\dfrac{1}{4}\right)^5$.


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


Consider the polynomial equation
\[z^{12} - z^9 + z^6 - z^3 + 1 = 0.\]
Among its roots, determine the maximum value of the imaginary part, expressed as \(\sin \theta\) where \(-90^\circ \leq \theta \leq 90^\circ\). Find \(\theta\).


Find the sum of all $x$ that satisfy the equation $\frac{-15x}{x^2-4} = \frac{3x}{x+2} - \frac{9}{x-2}.$


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


In right triangle $ABC$, we have $\sin A = \frac{3}{5}$ and $\sin B = 1$. Find $\sin C$.


Stuart has drawn a pair of concentric circles. He draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the small one. If $m\angle ABC=60^\circ$, then how many segments will he draw before returning to his starting point at $A$?


A student, Alex, is required to do a specified number of homework assignments to earn homework points using a different system: for the first four points, each point requires one homework assignment; for the next four points (points 5-8), each requires two homework assignments; then, for points 9-12, each requires three assignments, etc. Calculate the smallest number of homework assignments necessary for Alex to earn a total of 20 homework points.


How many three-digit numbers are multiples of neither 5 nor 9?


Suppose a customized deck of 60 cards has 15 ranks (Ace, 2 to 10, Jack, Queen, King, Ace+1, Ace+2) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), each containing one card of each rank. There are two black suits ($\spadesuit$, $\clubsuit$) and two red suits ($\heartsuit$, $\diamondsuit$). The deck is shuffled randomly. Determine the probability that the top card is a red suit and the second card is a black suit.


Express as a common fraction: $(0.\overline{08})(0.\overline{3})$.


For a nonnegative integer $n$, let $r_7(3n)$ represent the remainder when $3n$ is divided by $7$. Determine the $22^{\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 4~.$$


In Mr. Jenkins' 30-student mathematics class, no two students have the same initials, and each student's first name and last name begin with the same letter. Considering the letter ``Y'' as a consonant in this scenario, what is the probability of randomly picking a student whose initials are vowels? Each vowel (A, E, I, O, U) has two students representing it, while every other consonant only has one student. Express your answer as a common fraction.


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


Express $\sqrt{x} \div \sqrt{y}$ as a common fraction, given:

$$ \frac{ \left( \frac{1}{3} \right)^2 + \left( \frac{1}{4} \right)^2 }{ \left( \frac{1}{5} \right)^2 + \left( \frac{1}{6} \right)^2 } = \frac{25x}{61y} $$


Let $g(x) = \left\lceil\dfrac{1}{x+3}\right\rceil$ for $x > -3$, and $g(x) = \left\lfloor\dfrac{1}{x+3}\right\rfloor$ for $x < -3$. ($g(x)$ is not defined at $x = -3$.) Which integer is not in the range of $g(x)$?


How many non-similar triangles have angles whose degree measures are distinct positive integers in a geometric progression?


Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than or equal to 10?


Evaluate the expression
\[e^{11\pi i/80} + e^{31\pi i/80} + e^{51 \pi i/80} + e^{71\pi i /80} + e^{91 \pi i /80}\]
and express it in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. Find $\theta$.

Solution: \boxed{\theta = \frac{51\pi}{80}}


A bag contains 6 red marbles, 7 white marbles, 8 blue marbles, and 4 green marbles. Three marbles are drawn from the bag (without replacement). What is the probability that at least two of them are of the same color?


How many positive $3$-digit numbers are multiples of $30$, but not of $75$?


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


The equation $x^2 + ax = -30$ has only integer solutions for $x$. If $a$ is a positive integer, determine the greatest possible value of $a$.


Add $5_6 + 23_6.$ Express your answer in base $6.$


Write the following expression as a polynomial: $$(3x^2+2x-5)(x-2)-(x-2)(x^2-5x+28)+(4x-7)(x-2)(x+4).$$


A box of 150 personalized pencils costs $\$40$. If someone wants to buy 3000 pencils, and the price per pencil decreases by $5\%$ for orders over 1000 pencils, how much would it cost?


Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 3/5 chance that she will toss it to Bob and a 2/5 chance that she will keep the ball. If Bob has the ball, there is a 1/3 chance that he will toss it to Alice, and a 2/3 chance that he will keep it. Alice starts with the ball. What is the probability that Alice has the ball again after two turns?


Let $\alpha$ be a real number such that $\alpha \neq 0$ and $\beta$ be a real number where $|\beta| = 1$. Find the minimum value of
\[
\left| \frac{\beta + \alpha}{1 + \alpha \beta} \right|.
\]


What is the sum of the tens digit and the units digit in the decimal representation of $11^{2010}$?


Consider the vectors $\mathbf{a} = \begin{pmatrix} 4 \\ 1 \\ -3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}$. There exist scalars $s$, $t$, and $u$ such that
\[
\begin{pmatrix} 5 \\ 2 \\ -7 \end{pmatrix} = s \mathbf{a} + t \mathbf{b} + u (\mathbf{a} \times \mathbf{b}).
\]
Find $u$.


An infinite geometric series has a first term of $512$ and a sum of $3072$. What is its common ratio?


Compute $\binom{7}{3}$ and then compute $3!$.


Find all solutions to
\[x^2 + 6x + 6x \sqrt{x + 4} = 24.\]


Let \(x,\) \(y,\) and \(z\) be nonzero complex numbers such that \(x + y + z = 15\) and
\[
(x - y)^2 + (x - z)^2 + (y - z)^2 = 2xyz.
\]
Find \( \frac{x^3 + y^3 + z^3}{xyz}. \)


Find the sum of the first eight terms in the geometric sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$.


For how many integer values of $n$ between 1 and 200 inclusive does the decimal representation of $\frac{n}{180}$ terminate?


How many four-digit positive integers have at least one digit that is a 3 or a 4?


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


How many positive multiples of $6$ that are less than $800$ end with the digit $4$?


The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ where each term is the sum of the two preceding ones. What is the remainder when the $150^{\mathrm{th}}$ term of the sequence is divided by 9?


How many positive divisors of $180$ are neither divisible by $3$ nor by $5$?


Consider the three-digit number 864 where altering the middle digit can potentially affect divisibility rules. If you change the middle digit \(D\) (in place of 6) so that the new number \(8D4\) is divisible by 4, how many possibilities are there for \(D\)?


Larry now only likes numbers that end with two digits making the whole number divisible by 4. He has some favorite numbers like 120, 1156, and 504. How many different pairs of the last two digits are possible in numbers that Larry likes?


At the HappyTail Training Center, dogs can learn to perform three tricks: jump, fetch, and bark. Of the dogs at the center:
\begin{tabular}{l@{\qquad}l}
65 dogs can jump & 25 dogs can jump and fetch \\
40 dogs can fetch & 20 dogs can fetch and bark \\
45 dogs can bark & 23 dogs can jump and bark \\
15 dogs can do all three & 12 dogs can do none
\end{tabular}
How many dogs are at the center?


A storage container in the shape of a cylinder is 5 inches in diameter and 8 inches high and is priced at $\$$1.50. If another storage container with a diameter of 10 inches and a height of 10 inches is purchased, how much would it cost if buying larger containers offers a 10% discount on the rate per volume unit compared to smaller containers?


Sara rolls six fair 10-sided dice, each numbered from 1 to 10. What is the probability that exactly three of the dice show a prime number?


Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 20.\]
Find the largest possible value of $d.$

Solution: \boxed{\frac{5 + \sqrt{105}}{2}}


How many prime numbers between 50 and 100 have a prime remainder when divided by 7?


Consider the graph of the equation \[\sqrt{x^2+y^2} + |y-2| = 4.\] Determine the distance between the vertices of the parabolas formed by this equation.


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


Let \( a, b, \) and \( c \) be the roots of the polynomial
\[ x^3 - 3x^2 + 4x - 1 = 0. \]
Find the monic polynomial in \( x \) whose roots are \( a + 3, b + 3, \) and \( c + 3 \).

Solution: \boxed{x^3 - 12x^2 + 49x - 67}


Given that $b$ is a multiple of $1632$, find the greatest common divisor of $b^2 + 11b + 30$ and $b + 6$.


Determine the number of integer solutions $(x, y)$ to the equation $2^{2x} - 3^{2y} = 79$.


What is the domain of the function $f(x) = \log_5(\log_2(\log_3(\log_4(\log_6 x))))$?

Solution: \boxed{(6^{64}, \infty)}


Consider four function graphs, labelled (2) through (5). The domain of function (3) is now $$\{-5, -4, -3, -2, -1, 0, 1, 2, 3\}.$$ Determine the product of the labels of the functions which are invertible. The function expressions and domains for the other functions are:
- Function (2): $f(x) = x^2 - 4x + 3$ over the domain $[-1, 4]$
- Function (4): $g(x) = -\tan(x)$ over the domain $(-\frac{\pi}{2}, \frac{\pi}{2})$
- Function (5): $h(x) = 5/x$ over the domain $(-\infty, -0.2) \cup (0.2, \infty)$


Express the quotient $3213_4 \div 13_4$ in base 4.


In rectangle $ABCD$, $AB = 9$ cm, and $BC = 12$ cm. It's given that $DE = DF$. The area of triangle $DEF$ is one-third the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.

Solution: \boxed{12 \text{ cm}}


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


There are 3 math teams in the region, with 6, 9, and 10 students respectively. Each team now has three co-captains. If I randomly select a team, and then randomly select three members of that team to give a copy of $\emph{Advanced Algebra}$, what is the probability that all of the people who receive books are co-captains?


In the diagram, $G$ and $H$ are the midpoints of $\overline{PQ}$ and $\overline{QR}$ respectively, where $P(0,7)$, $Q(0,0)$, and $R(10,0)$. Determine the area of $\triangle GHR$.


Given that $\sec x + \tan x = \frac{5}{2},$ find all possible values of $\sin x.$


Find the integer $n$, $0 \le n \le 10$, such that \[n \equiv 123456 \pmod{11}.\]


Let $\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ meets side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, then what is $RS$?


Express the quotient $1302_4 \div 12_4$ in base 4.


For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n^2 x)$ on the interval $[0, 2\pi]$. What is $\sum_{n=2}^{100} G(n)$?


If two distinct numbers are selected at random from the first ten prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction.


What is the nonnegative difference between the roots of the equation $x^2 + 40x + 300 = -100$? Additionally, calculate the product of the roots.


In a book, the pages are numbered from 1 through $n$. When summing the page numbers, one page number was mistakenly added three times instead of once, resulting in an incorrect total sum of $2046$. Identify the page number that was added three times.


Segment $s_1$ has endpoints at $(3, -2)$ and $(9, 6)$. Segment $s_2$ is obtained by translating $s_1$ by $4$ units to the left and $2$ units down, then rotating $90^\circ$ counterclockwise about its midpoint. Find the coordinates of the new midpoint of segment $s_2$.

Solution: \boxed{(2, 0)}


A region $R$ in the complex plane is defined by
\[
R = \{x + iy: -2 \leq x \leq 2, -2 \leq y \leq 2\}.
\]
A complex number $z = x + iy$ is chosen uniformly at random from $R$. What is the probability that $\left(\frac{1}{2} + \frac{1}{2}i\right)z$ is also in $R$?


Given an ellipse and a hyperbola,\[\frac{x^2}{16} - \frac{y^2}{25} = 1\]and \[\frac{x^2}{K} + \frac{y^2}{25} = 1\], have the same asymptotes. Find the value of $K$.


Let $a_1, a_2, \dots$ be a sequence of positive real numbers such that
\[ a_n = 7a_{n-1} - 2n \]
for all $n > 1.$ Find the smallest possible value of $a_1.$


A circle with center $D$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(5,0)$ with radius $1$. What is the sum of all possible radii of the circle with center $D$?


How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one?


In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1500$ and $AD = 3000$. Let $\angle A = 30^\circ$, $\angle D = 60^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Determine the length $PQ$.


If \(x\) is real, compute the maximum integer value of
\[
\frac{4x^2 + 8x + 19}{4x^2 + 8x + 9}.
\]


Let
\[\mathbf{B} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}.\]
Compute $\mathbf{B}^{48}.$

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


If 60 cards are dealt to 9 people as evenly as possible, how many people will end up with fewer than 7 cards?


Find $q(x)$ if the graph of $\frac{x^4 - x^3 - 6x^2 + x + 6}{q(x)}$ has vertical asymptotes at $1$, $-1$, and $-3$, and $q(2) = 24$.

Solution: \boxed{\frac{8}{5}(x^3 + 3x^2 - x - 3)}


Consider the function $y = v(x)$ defined by:
$$ v(x) = -x + \frac{3}{2}\sin\left(\frac{x \pi}{2}\right). $$
Find the exact value of $v(-3.14) + v(-1) + v(1) + v(3.14)$.


A repair crew needs to reach the center $C$ of a circular fountain. They place a 20-foot plank from $A$ to $B$ and then a 12-foot plank from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$. Calculate the area of the circular base of the fountain. Express your answer in terms of $\pi$.

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


Define the operation $a \nabla b = 3 + b^a$. What is the value of $(2 \nabla 3) \nabla 4$?


Given that $ABC$ is a triangle with $A=(0,0)$ and $B=(48,18)$, and both the coordinates of $C$ are integers, find the minimum area of $\triangle ABC$.


The quadratic equation $ax^2 + 36x + c = 0$ has exactly one solution. If $a + c = 37$, and $a < c$, find the ordered pair $(a, c)$.

Solution: \boxed{\left(\frac{37 - \sqrt{73}}{2}, \frac{37 + \sqrt{73}}{2}\right)}


Determine the number of $x$-intercepts for the graph of the parabola $x = -3y^2 + 2y + 3$.


An ellipse has major and minor axes of lengths 12 and 6 respectively. A circle is drawn with one focus as a center, and it is just tangent to the ellipse inside it. Find the radius of this circle.

Solution: \boxed{6 - 3\sqrt{3}}


The five-fold sum of a positive number and its square is 780. What is the number?


An octahedron consists of two square-based pyramids glued together along their square bases, forming a polyhedron with eight faces. Suppose an ant starts walking from the bottom vertex and randomly picks one of the four adjacent vertices (middle ring) and calls it vertex A. From vertex A, the ant then randomly selects one of its four adjacent vertices and calls it vertex B. What is the probability that vertex B is the top vertex of the octahedron? Express your answer as a common fraction.


For how many integer values of \( k \) between 10 and 230 inclusive does the decimal representation of \( \frac{k}{330} \) terminate?


Let \( T \) be a positive integer whose only digits are 0s and 1s. If \( X = T \div 15 \) and \( X \) is an integer, what is the smallest possible value of \( X \)?


In the diagram, three identical circles each touch the other two. The circumference of each circle is 48. Calculate both the perimeter of the shaded region and the area of the triangle formed by joining the centers of the three circles. [asy]
defaultpen(1);
path p = (1, 0){down}..{-dir(30)}dir(-60){dir(30)}..{dir(-30)}((2, 0) + dir(-120)){-dir(-30)}..{up}(1, 0)--cycle;
fill(p, gray(0.75));
draw(unitcircle);
draw(shift(2 * dir(-60)) * unitcircle);
draw(shift(2) * unitcircle);
[/asy]

Solution: \boxed{\frac{576 \sqrt{3}}{\pi^2}}


In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 15$, $\tan D = 2$, and $\tan B = 3$, then what is $BC$?


If $\frac{7}{36}$ is expressed in decimal form, what digit is in the 101st place to the right of the decimal point?


In hexagon $ANGLES$, $\angle A \cong \angle G \cong \angle E$, $\angle N$ is supplementary to $\angle S$, and $\angle L$ is right angle. Find the measure of $\angle E$.


A line with slope of $-3$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(10,0)$ and the $y$-axis at $D$. The lines intersect at $E(5,5)$. What is the area of the shaded quadrilateral $OBEC$?


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 $50$, and $BC=8$. Find the volume of the tetrahedron.


Let $g(x) = x^2 - 3x + 2023$. What is the greatest common divisor of $g(50)$ and $g(52)$?


If \(k\) and \(\ell\) are positive 4-digit integers such that \(\gcd(k, \ell) = 5\), what is the smallest possible value for \(\mathop{\text{lcm}}[k, \ell]\)?


**Find the remainder when \(x^4 + 2\) is divided by \(x^2 + 7x + 3.\)**

Solution: \boxed{-301x - 136}


For what values of \(z\) is \(z^2 - 40z + 350 \leq 6\)? Express your answer in interval notation.

Solution: \boxed{[20 - 2\sqrt{14}, 20 + 2\sqrt{14}]}


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


What is the remainder when 111,222,333 is divided by 37?


The volume of a given sphere is $72\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\pi$.

Solution: \boxed{36\pi \sqrt[3]{4}}


Consider a rectangle where points $A$, $B$, $C$, and $D$ are the midpoints of its sides. $AB$ is twice as long as $BC$. If the area of the rectangle is 72, what is the area of the rhombus formed by points $A$, $B$, $C$, and $D$?


Determine the smallest abundant number that is not a multiple of 5.


Piravena must make a trip from \(A\) to \(B\), then from \(B\) to \(C\), then from \(C\) to \(A\). Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with \(C\) a distance of 3500 km from \(A\) and with \(B\) a distance of 3700 km from \(A\). To take a bus, it costs Piravena \$0.20 per kilometer. To take an airplane, it costs her a \$120 booking fee, plus \$0.12 per kilometer. Determine the distance she travels for her complete trip, assuming she uses only the airplane for the longest leg of the trip.


What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\pmod{72}$ and $a^{-1}$ is also undefined $\pmod{45}$?


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 $t_{50}$ is divided by $7$?


Find the largest positive integer $n$ such that
\[\sin^n x + \cos^n x \ge \frac{2}{n}\]
for all real numbers $x$.


Let \(a\) and \(b\) be positive real numbers such that \(5a + 3b < 90.\) Find the maximum value of
\[
ab (90 - 5a - 3b).
\]


A basketball team has 12 players. I need to choose a starting lineup consisting of 2 forwards, 2 guards, and 1 center. How many different starting lineups can I form if I can assign any player to any position?


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


If the three solutions of $x^3 = -27$ are written in the form $a+bi$, where $a$ and $b$ are real numbers, find the product of those solutions with $a > 0.$


Let $\mathbf{a}$ and $\mathbf{b}$ be nonzero vectors such that
\[\|\mathbf{a}\| = 2, \|\mathbf{b}\| = 3, \|\mathbf{a} + \mathbf{b}\| = \sqrt{13}.\]
Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.


In a 45-45-90 triangle, the hypotenuse measures 16 inches. A median is drawn from the right-angle vertex to the midpoint of the hypotenuse. Calculate the area of one of the two triangular sections formed by this median.

Solution: \boxed{32 \text{ square inches}}


The sum of two positive integers \(a\) and \(b\) is 1023. What is the largest possible value of \(\gcd(a, b)\)?


For all positive integers $n$, let $g(n)=\log_{3003} n^3$. Compute $g(7) + g(11) + g(13)$.


Find the number of functions \(g : \mathbb{R} \to \mathbb{R}\) such that
\[g(x + y) g(x - y) = (g(x) - g(y))^2 - 6x^2 g(y)\] for all real numbers \(x\) and \(y.\)


On a Cartesian plane where each unit is one meter, a cat is tied to a pole located at $(5,1)$ with a 15-meter leash. Determine the farthest distance the cat can go from the origin $(0,0)$.

Solution: \boxed{\sqrt{26} + 15}


In triangle $DEF$, angle $D$ is thrice as large as angle $E$. If angle $E$ measures $18^\circ$, what is the measure of angle $F$?


Alice starts to make a list, in increasing order, of the positive integers that have a first digit of 2. She writes $2, 20, 21, 22, \ldots$ but by the 1000th digit she (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits she wrote (the 998th, 999th, and 1000th digits, in that order).


How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 10\}$ satisfy the following two conditions?
1. No two consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$.


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


For \(-1 < r < 1\), let \( S(r) \) denote the sum of the geometric series
\[ 15 + 15r + 15r^2 + 15r^3 + \cdots. \]
Let \( a \) between \(-1\) and \( 1\) satisfy \( S(a)S(-a) = 2025 \). Find \( S(a) + S(-a) \).


Let $r$, $s$, and $t$ be solutions of the equation $x^3 - 7x^2 + 11x = 13$.

Compute $\frac{r+s}{t} + \frac{s+t}{r} + \frac{t+r}{s}$.


Find the product of all constants $t$ such that the quadratic $x^2 + tx - 12$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers.


Let $x,$ $y,$ and $z$ be nonzero complex numbers such that $x + y + z = 10$ and
\[
2((x - y)^2 + (x - z)^2 + (y - z)^2) = xyz.
\]
Find $\frac{x^3 + y^3 + z^3}{xyz}$.


What is the smallest positive value of $m$ so that the equation $15x^2 - mx + 630 = 0$ has integral solutions?


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


Simplify $\frac{4^5 + 4^3}{4^4 - 4^2 - 4}$ and express your answer as a common fraction.


Find the area of the triangle with vertices $(2, -3),$ $(1, 4),$ and $(-3, -2).$


Determine the transformation matrix $\mathbf{N}$ that takes square $WXYZ$ and sends it to square $W'X'Y'Z'$, such that $WXYZ$ is rotated $90^{\circ}$ counter-clockwise around $W$, then translated 2 units right and 3 units up.

Solution: \boxed{\begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & 3 \\ 0 & 0 & 1 \end{pmatrix}}


At a school cafeteria, Jenny wants to buy a meal consisting of one main dish, one drink, one dessert, and one side dish. The list below contains Jenny's preferred choices available:

\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Main Dishes} & \textbf{Drinks} & \textbf{Desserts} & \textbf{Side Dishes} \\
\hline
Spaghetti & Water & Cookie & Salad \\
\hline
Turkey Sandwich & Juice & Cake & Fruit Cup \\
\hline
Veggie Burger &  &  & Chips \\
\hline
Mac and Cheese &  &  &  \\
\hline
\end{tabular}

How many distinct possible meals can Jenny arrange from these options?


An ellipse has foci at $(15, 30)$ and $(15, 90)$ in the $xy$-plane and is tangent to the y-axis. What is the length of its major axis?


Jackie and Phil each have a fair coin and a biased coin. The biased coin comes up heads with probability $\frac{3}{5}$. Each flips their set of coins once. Calculate the probability that Jackie gets the same number of heads as Phil, and express this probability in simplest form as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution: \boxed{19 + 50 = 69}


Let \(a, b, c\) be real numbers such that \(a + b + c = 3.\) Find the set of all possible values of \(ab + ac + bc.\)

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


In the diagram, the grid is composed of rectangular areas totaling 60 squares. The entire grid has a width of 15 units and a height of 5 units. There is a shaded region that covers part of the grid, except for two right-angled triangles. The larger triangle has a base along the bottom grid line spanning the entire width of the grid and a height of 5 units. A smaller triangle is located in the top left corner with a base of 3 units and a height of 2 units. Calculate the area of the shaded region.


Find the product of all positive integral values of $n$ such that $n^2 - 33n + 272 = p$ for some prime number $p$. There is at least one such $n$.


Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $324\sqrt{2}$ square feet and $648\sqrt{2}$ square feet. The two planes are $12$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?

Solution: \boxed{24 + 12\sqrt{2}}


Let \[f(x) = \left\{
\begin{array}{cl}
x^2 - 1 & \text{ if } x < 10, \\
x - 20 & \text{ if } x \geq 10.
\end{array}
\right.\]
What is $f(f(f(25)))$?


Starting with the number 2, the first prime, alternately add and subtract subsequent prime numbers: $2$, $2 + 3$, $2 + 3 - 5$, $2 + 3 - 5 + 7$, and so on. Determine how many of the first 15 such sums are prime.


Compute the smallest base-10 positive integer greater than 10 that is a palindrome when written in both base 2 and base 4.


Find the coefficient of \(x^{60}\) in the expansion of
\[(x - 1)(x^2 - 2)(x^3 - 3) \dots (x^{10} - 10).\]


Let $g(n)$ be the sum of the positive integer divisors of $n$. How many values of $n$, where $1 \le n \le 30$, is $g(n)$ prime?


You are given a sequence of $37$ terms; each term has the form $Q+n$ where $Q$ is the product $2 \times 3 \times 5 \times\ldots \times 53$ of all prime numbers less than or equal to $53$, and $n$ takes successively the values $4, 5, 6,\ldots, 40$. Let $M$ be the number of primes appearing in this sequence. Determine $M$.


If $y < 0$, find the range of all possible values of $y$ such that $\lceil y \rceil \cdot \lfloor y \rfloor = 72$. Express your answer using interval notation.

Solution: \boxed{(-9, -8)}


Let $a,b,c$ be distinct real numbers such that
\[\frac{a}{b-c} + \frac{b}{c-a} + \frac{c}{a-b} = -1.\]
Find all possible values of
\[\frac{a^3}{(b-c)^2} + \frac{b^3}{(c-a)^2} + \frac{c^3}{(a-b)^2}.\]


In the diagram, each circle is divided into two equal halves, and $O$ is the center of the larger circle. The area of the larger circle is $100\pi.$ What is the total area of the shaded regions when the smaller circle's diameter equals half the radius of the larger circle?
[asy]
size(100);
import graph;
fill(Arc((0,0),3,180,360)--cycle,mediumgray);fill(Arc((0,1.5),1.5,0,180)--cycle,mediumgray);
draw(Circle((0,0),3));
draw(Circle((0,1.5),1.5));

dot((0,0)); label("$O$",(0,0),N);
draw((-3,0)--(3,0)); draw((-1.5,1.5)--(1.5,1.5));
[/asy]


The function $g(x)$ satisfies
\[g(x + y) = g(x) + g(y)\] for all real numbers $x$ and $y,$ and $g(3) = 4.$ Find $g(6)$ and $g(7).$


For real numbers $s,$ the intersection points of the lines $2x - 3y = 4s + 6$ and $2x + y = 3s + 1$ are plotted. All these points lie on a particular line. Determine the slope of this line.


Consider $n$ as a positive integer. Determine the maximum possible value for the greatest common divisor of $12n+5$ and $7n+3$.


Fully simplify the following expression: $[(3+4+5+6)\div3] + [(3\cdot6+9)\div4]$.


If
\[
\frac{\cos^6 \alpha}{\cos^3 \beta} + \frac{\sin^6 \alpha}{\sin^3 \beta} = 1,
\]
then find the sum of all possible values of
\[
\frac{\sin^6 \beta}{\sin^3 \alpha} + \frac{\cos^6 \beta}{\cos^3 \alpha}.
\]


For all even integers $n$, determine the largest integer that always divides the difference between $n$ and the cube of $n$.


Simplify the product \[\frac{6}{2}\cdot\frac{10}{6}\cdot\frac{14}{10} \dotsm \frac{4n+2}{4n} \dotsm \frac{1004}{1000}.\]


Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 17 + |x_1 + x_2 + \dots + x_n|.\]What is the smallest possible value of $n$?


Determine how many integer values of $b$ there are such that the inequality
\[ x^2 + bx + 3 \leq 0 \]
has exactly three integer solutions $x$.


Fifteen girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Ami, starts with the ball, skips the next five girls and throws to the seventh girl, who then skips the next five girls and throws the ball to the thirteenth girl. If the throwing pattern continues, including Ami's initial throw, how many total throws are necessary for the ball to return to Ami?


What is the sum of the proper divisors of 343?


What three-digit integer is equal to the sum of the factorials of its digits, where one of the digits is `3`, contributing `3! = 6` to the sum?


In triangle $ABC$, $AB$ is congruent to $AC$, and the measure of angle $ABC$ is $60^{\circ}$. Segment $BD$ bisects angle $ABC$ with point $D$ on side $AC$. Point $E$ is on side $BC$ such that segment $DE$ is parallel to side $AB$, and point $F$ is on side $AC$ such that segment $EF$ is parallel to segment $BD$ and $EF$ bisects $\angle DEC$. How many isosceles triangles are present?


The line joining $(1,3)$ and $(4,1)$ divides the square shown into two parts. What fraction of the area of the square is above this line? The square has vertices at $(1,1)$, $(1,4)$, $(4,4)$, and $(4,1)$.


If Lily rolls three fair six-sided dice, what is the probability that she rolls more 2's than 5's?


The function \( f(x) \) satisfies
\[ f(x) + 3f(1 - x) = 4x^3 \]
for all real numbers \( x \). Find \( f(4) \).


Eugene, Claire, and Liam are planning a cycling trip. Eugene cycles at a rate of 6 miles per hour. If Claire cycles $\frac{3}{4}$ as fast as Eugene, and Liam cycles $\frac{4}{3}$ as fast as Claire, how fast does Liam cycle?


Betty is hiking in a national park. She walks northward five miles, then turns $30$ degrees eastward and walks eight miles. How far is she from her starting point? Express your answer in simplest radical form.

Solution: \boxed{DF (Distace from starting point) = \sqrt{89 + 40\sqrt{3}}}


Determine the smallest positive integer $n$ such that all the roots of the equation $z^6 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity.


Dr. Geo has a four-digit house number $WXYZ$ formed by combining two distinct two-digit primes, `$WX$` and `$YZ$`. Each prime number must be less than 60. How many such house numbers are possible, given that no digits in $WXYZ$ are zeroes?


If $y > 0$, find the range of all possible values of $y$ such that $\lceil{y}\rceil\cdot\lfloor{y}\rfloor = 72$. Express your answer using interval notation.

Solution: \boxed{(8, 9)}


A botanical garden map shows that one inch represents 1200 feet on the ground. A line segment on the map is 6.5 inches long. An error in the map indicates a part of the path that doesn’t exist, which is shown as a 0.75-inch line segment. How many feet on the ground does the correct path cover?

Solution: \boxed{6900 \text{ feet}}


A right cylinder with a base radius of 4 units is inscribed in a sphere of radius 6 units. Calculate the total volume, in cubic units, of the space inside the sphere and outside the cylinder, denoted as $X\pi$. Find $X$ as a common fraction.

Solution: \boxed{X = 288 - 64\sqrt{5}}


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


Suppose we have an equilateral triangle with vertices at \( (0, 0) \), \( (6, 0) \), and \( (3, 3\sqrt{3}) \), and we place 36 points equally spaced along the perimeter (including the vertices and dividing each side into 12 congruent sections). If \( P \), \( Q \), and \( R \) are any three non-collinear points among these, how many different possible positions are there for the centroid of \( \triangle PQR \)?


Find the last two digits of the sum $$6! + 1 + 12! + 1 + 18! + 1 + \cdots + 96! + 1.$$


Suppose that the roots of $x^3 + 4x^2 + 5x - 13 = 0$ are $p$, $q$, and $r$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $p+q$, $q+r$, and $r+p$. Find $w$.


If \(x\) is real, compute the maximum integer value of
\[
\frac{4x^2 + 12x + 19}{4x^2 + 12x + 9}.
\]


A club consists of six executives and a number of regular members. Each year, the executives are replaced, and every regular member invites one new person to join as a regular member. There are initially eighteen people in the club. Calculate the total number of people in the club after five years.


Let $\mathbf{m},$ $\mathbf{n},$ and $\mathbf{p}$ be unit vectors such that the angle between $\mathbf{m}$ and $\mathbf{n}$ is $\alpha,$ and the angle between $\mathbf{p}$ and $\mathbf{m} \times \mathbf{n}$ is also $\alpha.$ If $\mathbf{n} \cdot (\mathbf{p} \times \mathbf{m}) = \frac{1}{2\sqrt{2}},$ find the smallest possible value of $\alpha,$ in degrees.


Two chords, $XY$ and $WZ$, intersect inside a circle at point $O$. Given that $XO = 5$ and $WO = 7$, determine the ratio $\frac{ZO}{YO}$.


It takes 18 men working steadily 3 days to build a wall. If 3 of the men work twice as efficiently as the others, how many days would it take 30 men to build the same wall, assuming 5 of them are twice as efficient as the others?


Let \(a,\) \(b,\) and \(c\) be real numbers such that
\[a^3 + b^3 + c^3 - 3abc = 8.\]Find the minimum value of \(a^2 + b^2 + c^2.\)


What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{127}$?


The real number $y$ satisfies $y^2 - 7y + 12 < 0.$  Find all possible values of $y^2 + 7y + 14.$

Solution: \boxed{(44, 58)}


In a similar game setup, there are 30 boxes, each containing one of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$1,000\\\hline\$1&\$5,000\\\hline\$5&\$10,000\\\hline\$10&\$25,000\\\hline\$25&\$50,000\\\hline\$50&\$75,000\\\hline\$75&\$100,000\\\hline\$100&\$200,000\\\hline\$200&\$300,000\\\hline\$300&\$400,000\\\hline\$400&\$500,000\\\hline\$500&\$750,000\\\hline\$750&\$1,000,000\\\hline\end{tabular} What is the minimum number of boxes a participant needs to eliminate to have at least a 50% chance of holding a box containing no less than $200,000?


What is the greatest three-digit multiple of 27?


Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers where the common difference is positive. If $b_5b_6 = 21$, what is $b_4b_7$?


Find the value of $b$ such that the following equation in base $b$ is true:
$$\begin{array}{c@{}c@{}c@{}c@{}c@{}c@{}c}
&&8&7&3&6&4_b\\
&+&9&2&4&1&7_b\\
\cline{2-7}
&1&8&5&8&7&1_b.
\end{array}$$


Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15} \dotsm \frac{3n+6}{3n} \dotsm \frac{3003}{2997}.\]


Below is the graph of $y = a \sin (bx + c) + d$ for some positive constants $a, b, c,$ and $d$. The graph oscillates between 5 and -3. Find $d$.


How many four-digit numbers satisfy the property that the second digit is the average of the first and third digits, and the last digit is twice the first digit?


Determine \(\sqrt[5]{161051}\) without a calculator.


Let $A,$ $R,$ $M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_{10} (AL) + \log_{10} (AM) &= 3, \\
\log_{10} (ML) + \log_{10} (MR) &= 4, \\
\log_{10} (RA) + \log_{10} (RL) &= 5.
\end{align*}
Compute the value of the product $ARML.$


For real numbers $s,$ the point of intersection of the lines $3x + y = 5s + 6$ and $2x - 3y = 3s - 5$ is plotted. All the plotted points lie on a line. Find the slope of this line.


Find values of \(A\) and \(B\) such that the polynomial \(x^{103} + Ax^2 + B\) is divisible by \(x^2 + x + 1\), and determine \(A + B\).


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.


What is the smallest positive integer $n$ for which $n^2$ is divisible by 24 and $n^3$ is divisible by 540?


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


A square and a regular hexagon are coplanar and share a common side $\overline{AD}$. Determine the degree measure of exterior angle $BAC$. Use a diagram for reference if needed.


Eight 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal?


What are the largest and smallest possible medians for the five number set $\{x, x+1, 4, 3, 6\}$ if $x$ can be any real number?


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


Quadrilateral $ABCD$ is a square, and segment $AE$ is perpendicular to segment $ED$. If both $AE$ and $DE$ equal $10$ units, what is the area of pentagon $AEDCB$, in square units?
[asy]
size(150);
pair A, B, C, D, E;
A=(0,12);
B=(0,0);
C=(12,0);
D=(12,12);
E=(6,6);
draw(A--B--C--D--A);
draw(A--E--D);
label("A", A, NW);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("E", E, S);
[/asy]


In writing the integers from 100 through 199 inclusive, how many times is the digit 1 written?


A region is bounded by four quarter-circle arcs constructed at each corner of a square whose sides measure $4/\pi$. Calculate the perimeter of this region.


Solution: \boxed{3 - 9x}


A pentagon is obtained by joining, in order, the points \((0,0)\), \((1,2)\), \((3,3)\), \((4,1)\), \((2,0)\), and back to \((0,0)\). The perimeter of the pentagon can be written in the form \(a + b\sqrt{c} + d\sqrt{e}\), where \(a\), \(b\), \(c\), \(d\), and \(e\) are whole numbers. Find \(a+b+c+d+e\).


A frustum of a right circular cone was generated by slicing a smaller cone from the top of a larger cone. The larger cone originally had a height of 30 cm before the top was cut off to create the frustum. Now, the frustum that remains has a height of 18 cm and the radius of its upper base is 6 cm, while the radius of its lower base is 10 cm. Determine the altitude of the smaller cone that was cut off.

Solution: \boxed{12 \text{ cm}}


Let $a,$ $b,$ and $c$ be constants, and suppose that the inequality \[\frac{(x-a)(x-b)}{x-c} \le 0\] is true if and only if either $x < -6$ or $|x-31| \le 1.$ Given that $a < b,$ find the value of $a + 2b + 3c.$


My friend Julia likes numbers that are divisible by 4. How many different last two digits are possible in numbers that Julia likes?


A solid rectangular block is composed of $N$ congruent 1-cm cubes glued together face to face. If this block is viewed so that three of its faces are visible, exactly $143$ of the 1-cm cubes are not visible. Determine the smallest possible value of $N$.


A polynomial $p(x)$ of degree $17$ is divided by a polynomial $g(x)$, resulting in a quotient polynomial of degree $9$ and a remainder of $5x^5 + 2x^3 - 3x + 7$. What is $\deg g$?


In a circle with radius 5, diameters $AB$ and $CD$ intersect at right angles at the center of the circle, denoted as $O$. Calculate the area of the shaded region formed by two right triangles and two sectors, where triangles $AOB$ and $COD$ are shaded, as well as sectors $AOC$ and $DOB$.

Solution: \boxed{25 + \frac{25\pi}{2}}


A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard?


Let $\mathbf{u}_0$ be a vector. The vector $\mathbf{u}_0$ is projected onto $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$, resulting in the vector $\mathbf{u}_1$. The vector $\mathbf{u}_1$ is then projected onto $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$, resulting in the vector $\mathbf{u}_2$. Find the matrix that takes $\mathbf{u}_0$ to $\mathbf{u}_2$.

Solution: \boxed{\begin{pmatrix} \frac{28}{65} & \frac{14}{65} \\ \frac{42}{65} & \frac{21}{65} \end{pmatrix}}


If
\[
(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 89^\circ) = 2^m,
\]
then find $m.$


The measure of each exterior angle of a regular polygon is $24$ degrees. What is the number of sides and the sum of the measures of the interior angles of this polygon?


A list of five positive integers has all of the following properties:
- The only integer in the list that occurs more than once is $9$,
- Its median is $10$, and
- Its average (mean) is $11$.

What is the largest possible integer that could appear in the list?


What is the largest integer $n$ for which $\binom{10}{3} + \binom{10}{4} = \binom{11}{n}$?


The set of all solutions of the system
$$
\begin{cases}
x+y\leq 4 \\
2x+y\geq 1 \\
x\geq 0 \\
y\geq 0
\end{cases}
$$
is a quadrilateral region. Find the number of units in the length of the longest side. Express your answer in simplest radical form.


Six equally sized rectangles are arranged to form a larger rectangle $PQRS$. If the length of each small rectangle is one-and-a-half times its breadth and the total area of $PQRS$ is 5400, determine the length, $x$, of each smaller rectangle, rounded off to the nearest integer.


How many triangles with positive area can be formed where each vertex is at point $(i,j)$ in the coordinate grid, with integers $i$ and $j$ ranging from $1$ to $4$ inclusive?


Define $X\star Y$ as $X\star Y = \frac{(X+Y)}{4}$. What is the value of $(3\star 11) \star 6$?


Solution: \boxed{\sqrt{18 - 24\sqrt{2}}}


Fully simplify the following expression: $[(1+2+3+6)\div3] + [(3 \cdot 6 + 9)\div4]$.


A circular pie with a diameter of 20 cm is cut into four equal-sized sector-shaped pieces. Let $l$ be the length in centimeters of the longest line segment that can be drawn within one of these pieces. Calculate $l^2$.


The square root of \( 3x \) is greater than 4 and less than 6. How many integer values of \( x \) satisfy this condition?


Samantha has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 7 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal.


In triangle $XYZ$, where $\angle X = 90^\circ$, suppose that $\sin Y = \frac{3}{5}$. Determine $\cos Z$.


Solution: \boxed{108 \cdot 2^{\frac{1}{3}}}


Solution: \boxed{117 + 44i}


A circle rests in the interior of the parabola with equation \( y = x^2 + 1 \), so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?


A 60-foot by 20-foot rectangular garden is enclosed by a fence. To make the garden larger, while using the same amount of fencing, its shape is changed to a circle. How many square feet larger than the old garden is the new garden?


Calculate the arithmetic mean of the reciprocals of the first five prime numbers.


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


In triangle \(DEF\), \(DE = 24\) and \(EF = 18\). Find the largest possible value of \(\tan D\).


Let $p$ and $q$ be the solutions of the equation $3x^2 + 9x - 21 = 0$. What is the value of $(3p-4)(6q-8)$?


A portion of the graph of $y = H(x)$ is shown in blue. The distance between grid lines is $1$ unit.

Compute $H(H(H(H(H(2)))))$.

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}
};
rr_cartesian_axes(-5,7,-4,10);
real h(real x) {return ((x-2)*(x-2)/2 + 2);}
draw(graph(h,2-sqrt(2*13),2+sqrt(2*13),operator ..), blue);
[/asy]


What is the least integer value of $x$ such that $\lvert3x + 4\rvert \le 25$?


Let \( \theta \) be an acute angle, and let
\[
\cos \frac{\theta}{2} = \sqrt{\frac{x}{2x + 1}}.
\]
Express \( \sin \theta \) in terms of \( x \).

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


Let \( a, b, c \) be positive real numbers. Determine the set of all possible values of
\[ g(a, b, c) = \frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{c + a}. \]

Solution: \boxed{(1, 2)}


A $44$-gon $Q_1$ is constructed in the Cartesian plane, and the sum of the squares of the $x$-coordinates of the vertices equals $176$. The midpoints of the sides of $Q_1$ form another $44$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $44$-gon, $Q_3$. Find the sum of the squares of the $x$-coordinates of the vertices of $Q_3$.


Find the smallest composite number that has no prime factors less than 15.


If $x^{3y} = 8$ and $x = 2$, what is the value of $y$? Express your answer as a common fraction.


Calculate $\tan \left( \frac{15\pi}{4} \right)$.


Let $a,$ $b,$ $c,$ $d$ be real numbers, none of which are equal to $-2,$ and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$ If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{\omega},
\]
then find
\[
\frac{1}{a + 2} + \frac{1}{b + 2} + \frac{1}{c +2} + \frac{1}{d + 2}.
\]


Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$


Consider a quadratic equation of the form \( x^2 + cx + \frac{1}{4} \), where \( c \) is a specific negative number. By completing the square, the equation is rewritten as \( (x+n)^2 + \frac{1}{16} \). Determine the value of \( c \).


Find the minimum value of the maximum of \( |x^2 - 2xy| \) over \( 0 \leq x \leq 1 \) for \( y \) in \( \mathbb{R} \).

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


Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\log_2(T)$?


After John takes 30 shots, he has made $60\%$ of his shots. After he takes 10 more shots, he raises his percentage to $62\%$. How many of the last 10 shots did he make?


Find the number of positive solutions to
\[
\cos (\arcsin( \cot (\arccos \sqrt{x}) ) ) = x.
\]


The areas of three squares are 25, 64, and 121. What is the average (mean) of their side lengths and the average of their perimeters?


Let $b_1, b_2, \dots$ be a sequence defined by $b_1 = b_2 = 1$ and $b_{n+2}=b_{n+1}+b_n$ for $n \geq 1$. Find \[
\sum_{n=1}^\infty \frac{b_n}{3^{n+1}}.
\]


Bob has saved up $7777_8$ dollars for a trip to Japan. A round-trip airline ticket costs $1500_{10}$ dollars, and he plans to spend another $250_{10}$ dollars on meals. In base ten, how many dollars will he have left for lodging?


A point $(x,y)$ is a distance of 8 units from the $x$-axis. It is a distance of 7 units from the point $(2,5)$. It is a distance $m$ from the origin. Given that $x>2$, what is $m$?

Solution: \boxed{\sqrt{108 + 8\sqrt{10}}}


Given matrix \[\mathbf{B} = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix},\] compute \(\mathbf{B}^{15}\).

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


The quadratic \(x^2 + 2600x + 2600\) can be written in the form \((x+d)^2 + e\), where \(d\) and \(e\) are constants. What is \(\frac{e}{d}\)?


Fully factor the following expression: $3x^2-75$

Solution: \boxed{3(x + 5)(x - 5)}


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


Find the nonconstant polynomial \( Q(x) \) such that
\[
Q(Q(x)) = (x^2 + 2x + 2) Q(x).
\]

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


Find the integers $m$ and $n$, where $0 \le m \le 17$ and $0 \le n \le 13$, such that:
\[ m \equiv 98765 \pmod{18} \]
\[ n \equiv 98765 \pmod{14} \]


When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 44. If one integer is 60, what is the smallest possible value of the other integer?


Determine the number of ways to arrange the letters of the word MOREMOM.


Susan has exactly one of each of the first 22 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. Based on a historical graph, 7 states joined the union during the decade from 1800 through 1809. What fraction of Susan's 22 quarters represents states that joined the union in this decade? Express your answer as a common fraction.


A point is selected at random from the portion of the number line from 0 to 7. What is the probability that the point is closer to 6 than to 0? Express your answer as a decimal to the nearest tenth.


Seventy cards, numbered 1-70, are placed in a box. One card is randomly selected. What is the probability that the number on the card is prime and is a multiple of 11? Express your answer as a common fraction.


Below is the graph of $y = A \sin (Bx + C) + D$ for some positive constants $A,$ $B,$ $C,$ and $D.$ The graph oscillates between 5 and 1. Find $A.$


Find the maximum value of
\[
\cos(\theta_1 + \alpha) \sin (\theta_2 + \alpha) + \cos(\theta_2 + \alpha) \sin (\theta_3 + \alpha) + \cos(\theta_3 + \alpha) \sin (\theta_4 + \alpha) + \cos(\theta_4 + \alpha) \sin (\theta_5 + \alpha) + \cos(\theta_5 + \alpha) \sin (\theta_1 + \alpha),
\]
over all real numbers $\theta_1,$ $\theta_2,$ $\theta_3,$ $\theta_4,$ $\theta_5,$ and for a fixed real number $\alpha.$


A club has $4$ members from each of $4$ different schools, totaling $16$ members. Determine the number of ways to arrange a presidency meeting under the following conditions:

i. The club must choose one of the $4$ schools at which to host the meeting.

ii. The host school sends $3$ representatives to the meeting, and each of the other three schools sends $1$ representative.


The number of games won by five volleyball teams are represented in a chart, but the names are not shown. Use the clues below to identify how many games the Falcons won:

1. The Falcons won fewer games than the Comets but more than the Astros.
2. The Raptors won more games than the Falcons and less than the Javelins.
3. The Comets won at least 30 games.

The win chart shows the following numbers: 26, 32, 36, 40, 44. Determine how many games the Falcons won.


A half-marathon is 13 miles and 192.5 yards. One mile equals 1760 yards.
Maria has completed six half-marathons. If the total distance Maria covered in these half-marathons is $m$ miles and $y$ yards, where $0 \leq y < 1760$, what is the value of $y$?


Let $a,$ $b,$ and $c$ be the roots of the polynomial
\[ x^3 - 4x^2 + 6x - 8 = 0. \]
Find the monic polynomial, in $x,$ whose roots are $a + 3,$ $b + 3,$ and $c + 3.$

Solution: \boxed{x^3 - 13x^2 + 57x - 89}


Consider a modified sequence where each term is of the form $Q + m$, where $Q$ is the product of all prime numbers less than or equal to $53$ (i.e., $Q = 2 \times 3 \times 5 \times \ldots \times 53$), and $m$ takes successive values $2, 3, 4, \ldots, 55$. Determine the number of prime numbers, $M$, that appear in this sequence.


Both roots of the quadratic equation \( x^2 - 107x + k = 0 \) are odd prime numbers. Find the number of possible values of \( k \).


Let $\triangle ABC$ be an isosceles triangle such that $BC = 40$ and $AB = AC.$ The incenter of $\triangle ABC$ is denoted by $I$, and $IC = 26.$ What is the length of the inradius of the triangle?


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, 6);
B = (2, -1);
L = extension(A, B, (0,8), (1,8));
R = extension(A, B, (0,-8), (1,-8));

draw(L--R, red);

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


Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac{1}{z} = 4,$ and $y + \frac{1}{x} = 20.$ Find $z + \frac{1}{y}.$


What is the remainder when 1,493,829 is divided by 7?


Determine the range of the function
\[ g(x) = \frac{3(x + 5)(x - 4)}{x + 5}. \]

Solution: \boxed{(-\infty, -27) \cup (-27, \infty)}


For how many integer values of \( n \) between 1 and 160 inclusive does the decimal representation of \( \frac{n}{160} \) terminate?


How many positive perfect square integers are factors of the product $\left(2^{14}\right)\left(3^{18}\right)\left(7^{21}\right)$?


How many positive factors of 1800 are both perfect squares and perfect cubes?


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 12 and an altitude of 15. The axes of the cylinder and the cone coincide. Find the radius of the cylinder.


A box of 150 personalized pencils costs $\$45$. How many dollars does it cost to buy 3200 pencils?


What is the minimum value of $3x^2 - 18x + 2023$?


Suppose we have 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, then what is $FQ$?


A $\textit{composite number}$ is a number that has two or more prime factors. The number 105 can be expressed as the sum of two composite numbers. What is the minimum positive difference between two such numbers?


Express $\sin 3x + \sin 7x$ as a product of trigonometric functions.

Solution: \boxed{2 \sin 5x \cos 2x}


Determine the number of ways to arrange the letters of the word EXCEED.


A square park contains a circular pond at its center, surrounded by a ring-shaped picnic area and then a cycling track around it. The cycling track is 4 feet wide everywhere, the picnic area is 10 feet wide everywhere, and the pond has a diameter of 16 feet. What is the diameter, in feet, of the circle that forms the outer boundary of the cycling track?

Solution: \boxed{44 \text{ feet}}


Calculate the median of the set \( \{m, m + 7, m + 8, m + 12, m + 20\} \). If the median is 15, what is the mean of the set?


What power of 7 does the expression $\dfrac{\sqrt[4]{7}}{\sqrt[7]{7}}$ equal?


How many different triangles can be formed with a perimeter of 8 units if each side must have an integral length and at least one side must be 4 units or less?


In $\triangle ABC$, we have $\angle A = 90^\circ$ and $\tan C = 3$. If $BC = 150$, determine the length of segment $AB$.


Consider a cubic polynomial \( x^3 - sx^2 + px - q \) whose roots \( r_1, r_2, \) and \( r_3 \) satisfy \( r_1 + r_2 + r_3 = r_1^2 + r_2^2 + r_3^2 \). Calculate the maximum possible value of \( \frac{1}{r_1^3} + \frac{1}{r_2^3} + \frac{1}{r_3^3} \).


Let $Q =  2\left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{4} \right) \left( 1 - \frac{1}{5} \right) \dotsm \left( 1 - \frac{1}{n} \right)$, with an added factor of 2 at the start of the product. What is the value of $Q$ if $n=2007$? Express your answer as a common fraction.


Compute $\cos \left( \arctan \frac{3}{4} \right)$ and $\tan \left( \arctan \frac{3}{4} \right)$.


Fully factor the following expression: $3x^2 + 3x - 18$.

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


Given six positive integers from a list of ten positive integers as $3, 5, 1, 7, 9, 6$, what is the largest possible value of the median of this list of ten positive integers?


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


Solution: \boxed{86x^4 - 8x^5}


By partial fractions, decompose the rational function
\[
\frac{1}{x(x+1)(x+2)(x+3)(x+4)(x+5)}
\]
as the sum of simpler fractions:
\[
\frac{A}{x} + \frac{B}{x+1} + \frac{C}{x+2} + \frac{D}{x+3} + \frac{E}{x+4} + \frac{F}{x+5}
\]
and find \( A + B + C + D + E + F \).


Find five consecutive integers greater than 25 and less than 50, each of which is not a prime number. Additionally, the sum of these five integers must be a multiple of 10. What is the largest of these five integers?


The third and fifteenth terms of an arithmetic sequence are $\frac{3}{8}$ and $\frac{7}{9}$, respectively. What is the ninth term?


Suppose we have 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, then what is the length of $FQ$?


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


If $18 = x^4 + \frac{1}{x^4}$, then what is the value of $x^2 + \frac{1}{x^2}$?

Solution: \boxed{x^2 + \frac{1}{x^2} = \sqrt{20}}


Suppose that $g(x)$ is a function such that
\[ g(xy) + 2x = xg(y) + g(x) \]for all real numbers $x$ and $y.$ If $g(-1) = -3$, then compute $g(-1001)$.


What is the sum of the squares of the solutions of the quadratic equation $-42 = -2x^2 - 4x?$


The mean of one set containing seven numbers is 15, and the mean of another set containing eight numbers is 21. If an additional number is included in the first set, making the mean of the augmented set 16, what is the mean of all the numbers combined in both sets?


Solve for $x$: $\frac{x+7}{x-4} = \frac{x-1}{x+6}$, and verify if $x \neq -6$ or $x \neq 4$.


On a grid with dots spaced one unit apart, both horizontally and vertically, a polygon has been traced with the following vertices in sequence: (0,0), (5,0), (5,5), (0,5), (5,10), (0,10), (0,0). Calculate the number of square units enclosed by this polygon.


In right triangle $ABC$ with $\angle C = 90^\circ$, we have $AB = 15$ and $AC = 5$.  Find $\sin A$ and $\cos A$.


The graphs of five functions are labelled from **(1) through (5)**. Provided below are descriptions of three:
   1. The domain of function (2) is from $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ It is graphed as a set of discrete points.
   2. Function (4) is defined by the equation $$y = x^3$$ and is graphed from $$x = -3$$ to $$x = 3$$.
   3. Function (5) is a rational function defined by $$y = \frac{5}{x}$$, excluding the origin from its domain.

Determine the product of the labels of the functions that are invertible.


There are 30 members in the book club. 12 of them are left-handed. 18 of them like sci-fi books. 4 of them are right-handed and do not like sci-fi books. How many club members are left-handed and like sci-fi books?


Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(1) = -1$ and
\[g(x^2 - y^2) = (x - y) (g(x) - g(y))\] for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(3),$ and let $t$ be the sum of all possible values of $g(3).$ Find $m \times t.$


Solution: \boxed{\log_{16} 6 = \frac{\log_4 6}{2}}


What is the least common multiple of the numbers 1320 and 924?


At the Clevercat Training Center, cats can learn to do three tricks: jump, fetch, and spin. Of the cats at the center:
- 60 cats can jump
- 35 cats can fetch
- 40 cats can spin
- 20 cats can jump and fetch
- 15 cats can fetch and spin
- 25 cats can jump and spin
- 10 cats can do all three
- 8 cats can do none

How many cats are in the center?


The function \( g(x) \), defined for \( 0 \le x \le 1 \), has the following properties:
(i) \( g(0) = 0 \).
(ii) If \( 0 \le x < y \le 1 \), then \( g(x) \le g(y) \).
(iii) \( g(1 - x) = 1 - g(x) \) for all \( 0 \le x \le 1 \).
(iv) \( g \left( \frac{x}{4} \right) = \frac{g(x)}{2} \) for \( 0 \le x \le 1 \).

Find \( g \left( \frac{3}{16} \right) \).


What is the sum of all the positive two-digit integers divisible by both the sum and product of their digits, and by the square of the difference of their digits?


Let $a$ and $b$ be real numbers. One of the roots of $x^3 + ax + b = 0$ is $2 + i \sqrt{2}.$ Find $a + b.$


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


A biased die with numbers 1 to 6 gives a number 1 with a probability of $\frac{1}{4}$, and any other number with equal probability. If rolling a 1 wins you \$8, while rolling any other number costs you \$3, compute the expected value, in dollars, of your winnings after one roll.


Let \(x\), \(y\), and \(z\) be positive real numbers such that \(x + y + z = 2\) and \(x^2 + y^2 + z^2 = xz + yz + xy\). Find the maximum possible value of \(xyz\).


What is the total number of digits used when the first 2500 positive even integers are written?


Find the focus of the parabola \( y = 2(x-1)^2 \).

Solution: \boxed{(1, \frac{1}{8})}


Let $x_1,$ $x_2,$ $x_3$ be positive real numbers such that $2x_1 + 3x_2 + 4x_3 = 120.$ Find the smallest possible value of
\[x_1^2 + x_2^2 + x_3^2.\]


What is the sum of all integer solutions to $4 < (x-3)^2 < 36$?


A piece of cheese is located at $(15, 12)$ in a coordinate plane. A mouse is at $(3, -3)$ and is running up the line $y = -4x + 9$. At which point $(a, b)$ does the mouse start getting farther from the cheese rather than closer to it? What is $a + b$?


During a year when Valentine's Day, February 14, falls on a Tuesday, what day of the week is Cinco de Mayo (May 5) and how many days are between February 14 and May 5 inclusively?


In convex quadrilateral \(EFGH\), \(\angle E \cong \angle G\), \(EF = GH = 200\), and \(EH \neq FG\). The perimeter of \(EFGH\) is \(720\). Find \(\lfloor 1000 \cos E \rfloor\).


For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. For example, $6! = 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1 = 720$. What is the greatest common divisor of $8!$ and $10!$?


Given that $a$ is an odd multiple of $17$, find the greatest common divisor of $2a^2 + 33a + 85$ and $a + 17$.


The sum of the first 3000 terms of a geometric sequence is 500. The sum of the first 6000 terms is 950. Find the sum of the first 9000 terms.


Let \[p(x,y) =
\begin{cases}
2x + 3y &\quad \text{if } x \ge 0 \text{ and } y \ge 0, \\
x^2 - y &\quad \text{if } x < 0 \text{ and } y < 0, \\
4x + 2y &\quad \text{otherwise}.
\end{cases}
\]
What is $p(p(2,-3),p(-4,-3))$?


Consider the quadratic function \( g(x) = 3x^2 - 9x + 4 \). Determine the largest interval that includes the point \( x = 2 \) to make \( g(x) \) invertible.

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


The Johnson family has 5 sons and 4 daughters. How many ways can they be seated in a row of 9 chairs such that at least 2 sons sit next to each other?


The values of $x$ and $y$ are always positive, and $3x^2$ and $y$ vary inversely. If $y$ is 15 when $x$ is 3, find $x$ when $y$ is 6750.


How many degrees are in the measure of the smaller angle formed by the hour and minute hands of a clock when the time is 8:30 p.m.?


Determine the value of $a$ such that the quadratic expression $4x^2 + 18x + a$ is the square of a binomial.


Find the minimum value of the function
\[ h(x) = x^2 + \frac{1}{x^2} + \frac{1}{x^2 + \frac{1}{x^2}} \]
for \( x > 0 \).


Consider the expression $x^2 + 13x + 40$ which can be factored as $(x + a)(x + b)$, and the expression $x^2 - 19x + 88$ which can be factored as $(x - c)(x - d)$, where $a$, $b$, $c$, and $d$ are integers. Calculate $a + b + c + d$.