The length of the shortest trip from $A$ to $B$ along the edges of a tetrahedron shown is the length of 2 edges. How many different 2-edge trips are there from $A$ to $B$?

[asy]

size(4cm,4cm);

pair a1, b1, c1, d1;

a1=(1,1);
b1=(0,1);
c1=(0.5,1.6);
d1=(1,0);

draw(a1--b1--c1--a1);
draw(a1--d1--b1);
draw(d1--c1);

label("$A$",d1,SW);
label("$B$",c1,NE);

[/asy]


The numbers \(x\) and \(y\) are inversely proportional. Given that the sum \(x + y = 60\) and \(x\) is three times \(y\), find the value of \(y\) when \(x = 12\).


A flood in Melbourne, Australia, resulted in $45 million in damage, estimated in Australian dollars. At that time, 1.2 Australian dollars were equivalent to 1 American dollar. Calculate the damage in American dollars.


A polynomial of degree $17$ is divided by $d(x)$ to give a quotient of degree $10$ and a remainder of $5x^4 - 3x^3 + 2x^2 - x + 15$. What is $\deg d$?


Cookie Monster stumbles upon another mysterious cookie with a boundary defined by the equation $x^2 + y^2 + 17 = 6x + 10y$. He needs to determine if this is a jumbo-sized cookie or just a regular cookie. Calculate the radius of this cookie.


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


Given that the point $(9,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $3y=\frac{f(3x)}{3} + 3$. What is the sum of coordinates of that point?


Amanda is distributing pencils at a school event. She finds that when she distributes them in groups of 5, she has 4 left over. When she distributes them in groups of 6, she has 3 left over. When she distributes them in groups of 8, she has 5 left over. What is the least number of pencils that Amanda could have?


Solution: \boxed{-7 \cdot 2^{2005} + 3^{2007}}


In a math club, members are practicing problem-solving using a workbook with $1200$ pages. Two members, David and Emily, decide to split the workbook so that both spend the same amount of time practicing. David takes $60$ seconds to solve a page, while Emily takes $40$ seconds. They plan a similar approach where David starts from page $1$ and goes up to a certain page, and Emily starts from the next page and finishes the book. Determine the last page David should solve to ensure both spend the same amount of time on the workbook.


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

Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$  Find $m \times t.$


What is the greatest common divisor of \(12345\) and \(6789\)?


Convert \( \sqrt{2} e^{13\pi i/4} \) to rectangular form.

Solution: \boxed{-1 - i}


Solution: \boxed{1 - \frac{1}{\log_3 101}}


Below is the graph of $y = a \tan bx$ for some positive constants $a$ and $b$. The graph has roots at $\pm \frac{\pi}{4}$ and it passes through the point $\left(\frac{\pi}{8}, 1\right)$. Determine $a \cdot b$.


The equation $x^2 + ax = -12$ has only integer solutions for $x$. If $a$ is a positive integer, what is the greatest possible value of $a$?


In parallelogram $EFGH$, the measure of angle $EFG$ is 4 times the measure of angle $FGH$. Calculate the degree measure of angle $EHG$.


The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 34, b_{12}=150$, and for all $n \geq 3$, each $b_n$ is the arithmetic mean of the first $n-1$ terms. Find $b_2$.


The solution of the equation $9^{x+9} = 12^x$ can be expressed in the form $x = \log_c 9^9$. What is $c$?


The third and fifth terms of a geometric sequence are 8 and $\frac{27}{4}$, respectively. Determine the first term of the sequence.


Find the area of the triangle with vertices \(A = (5, -3)\), \(B = (0, 2)\), and \(C = (4, -4)\).

Solution: \boxed{\frac{10}{2} = 5}


Find the number of integers \( n \) that satisfy
\[ 15 < n^2 < 120. \]


Let $P$ be the parabola with equation $y=x^2 + 4$ and let $Q = (10, 5)$.
There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s.$ What is $r + s$?


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


The measures of the angles of a hexagon are in the ratio 2:2:2:3:3:4. What is the number of degrees in the measure of the largest angle?


What is the remainder when 13754 is divided by 11?


An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. What is the smallest abundant number that is not a multiple of 5?


A circle $S$ has a circumference of $16\pi$ inches, and segment $AB$ is a diameter. If the measure of angle $SAC$ is $45^{\circ}$, what is the length, in inches, of segment $AC$?


Suppose there exists a scalar $m$ such that for any vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ such that $\mathbf{a} + 2\mathbf{b} + \mathbf{c} = \mathbf{0},$ the equation
\[m (\mathbf{a} \times \mathbf{c}) + 2(\mathbf{b} \times \mathbf{c}) + \mathbf{c} \times \mathbf{a} = \mathbf{0}\]holds. Find $m.$


It is desired to construct a right triangle in the coordinate plane such that its legs are parallel to the $x$ and $y$ axes, and the medians to the midpoints of the legs lie on the lines $y = 5x + 1$ and $y = mx + 2$. Determine the number of different constants $m$ for which such a triangle exists.


Solution: \boxed{-5 \log_3 2}


What is the greatest common divisor of $987654$ and $876543$?


The polynomial \( Q(x) = 3x^3 + dx^2 + ex + f \) has the property that the product of its zeros, the sum of its zeros, and the sum of the coefficients are all equal. The \( y \)-intercept of the graph of \( y = Q(x) \) is 18. Determine the value of \( e \).


What is the smallest integer \( n \), greater than 1, such that \( n^{-1} \pmod{2310} \) is defined?


Two lines with slopes of $2$ and $3$ intersect at the point $P(2,8)$. Calculate the area of triangle $PQR$ formed where $Q$ and $R$ are points where these lines intersect the x-axis.


Let $x$ be a real number. Consider the following five statements:

1. $0 < x^3 < 1$
2. $x^3 > 1$
3. $-1 < x < 0$
4. $0 < x < 1$
5. $0 < x^2 - x^3 < 1$

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


How many positive integers smaller than $500,000$ are powers of $3$, but are not powers of $27$? Note that $3^6 = 729$.


Find the sum of $10_{10} + 23_{10}$, first by converting each number to base 3, performing the addition in base 3, and then converting the result back to base 10.


What is the value of $y$ in the equation $\frac{3}{y} + \frac{4}{y} \div \frac{6}{y} = 1.5$?


A regular hexagon has an area of $150\sqrt{3}$ cm². If each side of the hexagon is decreased by 3 cm, by how many square centimeters is the area decreased?


In right triangle $GHI$, we have $\angle G = 40^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find $GH$ to the nearest tenth. You may use a calculator for this problem.


The number $1001$ can be written as $17q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q-r$?


Find the sum of all positive divisors of $72$ that are also divisors of $24$.


A parallelogram has vertices at $(2,3)$, $(5,8)$, $(11,8)$, and $(8,3)$. Calculate the perimeter and the area of the parallelogram and find the sum of both values.

Solution: \boxed{2\sqrt{34} + 42}


Find the complex number \( z \) such that \( 3z + 4i\bar{z} = -6 - 8i \).


In a standard deck of 52 cards, there are four suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), each containing one Jack, Queen, and King among other ranks. Assuming that the deck is shuffled thoroughly, what is the probability that the top card is either a Jack, Queen, or King of $\heartsuit$?


What is the only integer whose cube is less than its triple?


A line with slope equal to -1 and a line with slope equal to -2 intersect at the point \( P(2,5) \). Determine the area of \( \triangle PQR \) where \( Q \) and \( R \) are the x-intercepts of the respective lines.


Four dice with faces numbered 1 through 6 are stacked as shown. Ten of the twenty-four faces are visible, leaving fourteen faces hidden (back, bottom, between). What is the total number of dots NOT visible in this view? Assume the visible numbers are 1, 1, 2, 2, 3, 3, 4, 5, 5, 6.


Jar A initially contains six red buttons and ten blue buttons. Carla removes the same number of red buttons as blue buttons from Jar A and places them in an empty Jar B such that Jar A retains $\frac{3}{4}$ of its original number of buttons. If Carla then randomly selects a button from Jar A and a button from Jar B, what is the probability that both selected buttons are red? Express your answer as a common fraction.


Calculate $\lfloor |{-5.7}| \rfloor^2 + |\lfloor -5.7 \rfloor| - \frac{1}{2}$.


What is the least positive integer that is divisible by the three smallest primes each greater than 7?


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


Express the number 2345 as a sum of distinct powers of 2. What is the least possible sum of the exponents of these powers?


For a positive integer $n > 3^3$, what is the greatest common divisor of $n^3 + 3^3$ and $n + 3$?

Solution: \boxed{n + 3}


In the diagram, three identical circles touch each other, and each circle has a circumference of 24. Calculate the perimeter of the shaded region within the triangle formed by the centers of the circles.


Given the function $f(x) = x^k$, where $k < 0$, what is the range of $f(x)$ on the interval $[2, \infty)$?

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


Given a convex quadrilateral \( ABCD \) with an interior point \( P \) such that \( P \) divides \( ABCD \) into four triangles \( ABP, BCP, CDP, \) and \( DAP \). Let \( G_1, G_2, G_3, \) and \( G_4 \) denote the centroids of these triangles, respectively. Determine the ratio of the area of quadrilateral \( G_1G_2G_3G_4 \) to the area of \( ABCD \).


Calculate $180 \div (8 + 9 \cdot 3 - 4)$.


Let $x$ and $y$ be positive real numbers such that $x^3 + y^3 = x + y.$ Simplify
\[\frac{x}{y} + \frac{y}{x} - \frac{2}{xy}.\]


Let $\theta$ be an angle such that $\sin 2\theta = \frac{1}{5}$. Compute $\sin^6 \theta + \cos^6 \theta$.


Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{v},$ $\mathbf{Q} \mathbf{v}$ is the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{9}{14} & -\frac{3}{14} & \frac{6}{14} \\ -\frac{3}{14} & \frac{1}{14} & -\frac{2}{14} \\ \frac{6}{14} & -\frac{2}{14} & \frac{4}{14} \end{pmatrix}}


The graph of the parabola $x = 3y^2 - 9y + 5$ has an $x$-intercept $(a,0)$ and two $y$-intercepts $(0,b)$ and $(0,c)$. Find $a + b + c$.


What is the units digit of the sum of the ten terms of the sequence $1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 9! + 9, \, 10! + 10$?


There exist constants $b_1, b_2, b_3, b_4, b_5, b_6$ such that
\[\cos^6 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta\]
for all angles $\theta.$ Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2.$


Two concentric spheres have radii of 5 units and 10 units, respectively. Calculate the volume, in cubic units, of the space within the larger sphere but outside the smaller sphere. Express your answer in terms of $\pi$.

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


How many two-digit numbers are there in which the tens digit is exactly two times the ones digit?


Imagine you have 8 shirts, 5 pairs of pants, and 2 different styles of jackets. For forming an outfit, one needs to choose a shirt and a pair of pants, and has the option to wear a jacket in one of the styles available or no jacket at all. How many potential outfits can you assemble?


In the diagram, the grid is made up of squares. A larger right triangle (points A at bottom left, B at bottom right, C at top right of the grid) is cut out. Within the larger triangle, a smaller triangle (points D at bottom left, E 12 units to the right along the base from D, F 3 units up from E) is shaded. Calculate the area of the shaded region. The base of the larger triangle is the width of the grid and its height is one unit less than the height of the grid. The dimensions of the entire grid are 15 units by 5 units.


If the roots of the quadratic equation $2x^2 + 14x + c = 0$ are $x = \frac{-14 \pm \sqrt{10}}{4}$, then what is the value of $c$?


What is the volume, in cubic inches, of a rectangular box, whose faces have areas of $36$ square inches, $18$ square inches, and $8$ square inches?


Determine how many perfect squares less than 20,000 can be expressed as the difference of squares of two consecutive even numbers.


What is the constant term of the expansion of $\left(9x + \frac{2}{3x}\right)^8$?


A rectangular prism measures 15-inches by 25-inches by 12-inches. Find the length of the diagonal connecting point A and point B in inches. Express your answer in simplest radical form.


Solve for $y$ in the equation $\frac{1}{6} + \frac{6}{y} = \frac{14}{y} + \frac{1}{14}$.


When the base-12 integer $1543_{12}$ is divided by $9$, what is the remainder?


Find the largest prime divisor of $2102012_7$.


I have 16 distinguishable socks in my drawer: 4 white, 4 brown, 4 blue, and 4 red. In how many ways can I choose a pair of socks, provided that one of the socks chosen must be red and the other sock must be of a different color?


What is two-thirds times three-fifths times four-sevenths times five-eighths?


How many pairs of positive integers $(x,y)$ satisfy $x^2 - y^2 = 40$?


Solution: \boxed{16 - 30i}


Determine how many values of \( c \) in the interval [0, 2000] make the equation \[8 \lfloor x \rfloor + 3 \lceil x \rceil = c\] have a solution for \( x \).


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


Solve for $x$ in the following equation: $5-3ix = -2 + 6ix$.


Suppose $532_6 = cde_{10}$, where $c$, $d$, and $e$ are base-10 digits. Determine $\frac{c \cdot e}{10}$.


Using the digits 1, 2, 3, 4, 5, how many three-digit numbers greater than 300 and divisible by 5 can be formed if each digit can be used more than once?


Let $x^2+px+q = 0$ be a quadratic whose roots are each three more than the roots of $2x^2-4x-5$. Determine the value of $q$.


Let $k,$ $p,$ and $q$ be real numbers, and let $A,$ $B,$ $C$ be points such that the midpoint of $\overline{BC}$ is $(k,0),$ the midpoint of $\overline{AC}$ is $(0,p),$ and the midpoint of $\overline{AB}$ is $(0,0).$ Find
\[\frac{AB^2 + AC^2 + BC^2}{k^2 + p^2}.\]


The quadratic $3x^2 + 6x - 9$ has two real roots. What is the sum of the squares of these roots? Express your answer as a common fraction in lowest terms.


Thirteen girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Bella, starts with the ball, skips the next four girls, and throws to the sixth girl, who then skips the next four girls and throws the ball to the eleventh girl. If this throwing pattern continues, including Bella's initial throw, how many total throws are necessary for the ball to return to Bella?


Suppose that $A$ and $B$ are digits in base $d > 7$ such that $\overline{AB}_d + \overline{AA}_d = 172_d$. Find $A_d - B_d$ in base $d$.


Below is the graph of an ellipse. (Assume that tick marks are placed every $1$ unit along the axes.)
[asy]
size(8cm);

int x, y;
for (y = -5; y <= 3; ++y) {
  draw((-1,y)--(7,y),gray(0.7));
}
for (x = -1; x <= 7; ++x) {
  draw((x,-5)--(x,3),gray(0.7));
}

draw(shift((3,-2))*xscale(1.5)*shift((-3,2))*Circle((3,-2),2.5));
draw((-1,0)--(7,0),EndArrow);
draw((0,-5)--(0,3),EndArrow);
[/asy]
Compute the coordinates of the focus of the ellipse with the greater \(x\)-coordinate given that the endpoints of the major axis of the ellipse are \((0,-2)\) and \((6,-2)\), and the endpoints of the minor axis are \((3,1)\) and \((3,-5)\).

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


Let $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $60<f(7)<70$, $80<f(8)<90$, $2000k < f(50)<2000(k+1)$ for some integer $k$. Determine the value of $k$.


Given that 12 is the arithmetic mean of the set $\{8, 15, 22, 5, x\}$, what is the value of $x$?


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


Find the smallest sum of six consecutive prime numbers that is divisible by 5.


For a given positive integer $n > 3^3$, what is the greatest common divisor (GCD) of $n^3 + 27$ and $n + 3$?

Solution: \boxed{n + 3}


For all values of $x$ for which it is defined, $g(x) = \cot \frac{x}{6} - \cot x$ can be written as
\[g(x) = \frac{\sin kx}{\sin \frac{x}{6} \sin x}.\]Find the value of $k.$


Given pentagon $ABCDE,$ side $\overline{AB}$ is extended past $B$ to $A'$ such that $A'B = AB$. Points $B'$, $C'$, $D'$, and $E'$ are similarly constructed for the remaining sides of the pentagon.

After this construction, the original points $A$, $B$, $C$, $D$, and $E$ are erased. You know only the locations of points $A'$, $B'$, $C'$, $D'$, and $E'$ and need to reconstruct pentagon $ABCDE$.

There exist real numbers $p, q, r, s, t$ such that
\[\overrightarrow{A} = p \overrightarrow{A'} + q \overrightarrow{B'} + r \overrightarrow{C'} + s \overrightarrow{D'} + t \overrightarrow{E'}.\]
Find the ordered quintuple $(p, q, r, s, t)$.

Solution: \boxed{\left(\frac{1}{31}, \frac{2}{31}, \frac{4}{31}, \frac{8}{31}, \frac{16}{31}\right)}


The surface area of a sphere is $256\pi\text{ cm}^2$. Calculate the volume of the sphere in cubic centimeters and leave your answer in terms of $\pi$.

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


In a right-angled triangle, the sum of the squares of the three side lengths is 2500. What is the length of the hypotenuse of this triangle?


A square and four circles, each with a radius of 7 inches, are arranged as shown. What is the area, in square inches, of the square?
[asy]
unitsize(1mm);
defaultpen(linewidth(0.7pt));

draw((0,0)--(28,0)--(28,28)--(0,28)--cycle);
draw(Circle((7,7),7));
draw(Circle((21,7),7));
draw(Circle((7,21),7));
draw(Circle((21,21),7));
[/asy]

Solution: \boxed{784\text{ in}^2}


What is the value of $$\frac{1}{3}\times9\times\frac{1}{27}\times81\times\frac{1}{243}\times729\times\frac{1}{2187}\times6561\times\frac{1}{19683}\times59049?$$


The points \(A\), \(B\), and \(C\) are positioned on the surface of a sphere with center \(O\) and radius \(30\). The side lengths of the triangular configuration defined by these points are \(AB = 30\), \(BC = 40\), and \(CA = 50\). It is also given that the distance from \(O\) to triangle \(ABC\) is expressible in the form \(\frac{p\sqrt{q}}{r}\), where \(p\), \(q\), and \(r\) are positive integers, \(p\) and \(r\) are relatively prime, and \(q\) is not divisible by the square of any prime. Determine \(p+q+r\).


What is the area enclosed by the region defined by the equation \(x^2 + y^2 + 6x + 16y + 18 = 0\)?


Let $f(x) = \frac{2x + 6}{x}.$ The sequence $(f_n)$ of functions is defined by $f_1 = f$ and
\[ f_n = f \circ f_{n - 1} \]
for all $n \ge 2$. For example,
\[ f_2(x) = f(f(x)) = \frac{\frac{2x + 6}{x} + 6}{\frac{2x + 6}{x}} = \frac{8x + 6}{2x + 6},\]
\[ f_3(x) = f(f_2(x)) = \frac{\frac{8x + 6}{2x + 6} + 6}{\frac{8x + 6}{2x + 6}} = \frac{14x + 42}{8x + 6}.\]
Let $S$ be the set of all real numbers $x$ such that
\[ f_n(x) = x \]
for some positive integer $n$. Find the number of elements in $S$.


Let \(a, b, c, d\) be nonzero integers such that
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2 = \begin{pmatrix} 12 & 0 \\ 0 & 12 \end{pmatrix}.
\]
Find the smallest possible value of \(|a| + |b| + |c| + |d|\).


Consider a fair icosahedral die with numbers $1, 2, 3, ..., 20$ on its faces. What is the expected number of digits in the number obtained when the die is rolled? Express your answer as a decimal.


In a drawer, there are 5 white socks, 3 brown socks, and 4 blue socks. How many ways can you choose a pair of socks such that the socks are of different colors and at least one of them is blue?

Solution: \boxed{20 + 12 = 32}


John's quiz scores so far are: 85, 88, 90, 92, and 83. What score does he need to get on his sixth quiz to make the arithmetic mean of the six scores equal to 90?


The equation $x^2 + 10x = 45$ has two solutions. The positive solution has the form $\sqrt{a} - b$ for positive natural numbers $a$ and $b$. What is $a + b$?


What fraction of $\frac{3}{4}$ is $\frac{1}{6}$?


A custom deck of cards has 60 cards divided into 4 suits, two of which are red and the other two are black. Each suit now contains 15 cards. After shuffling the deck, in how many ways can we pick two different cards such that the second card picked is of a different color than the first?


Find the measure of the largest angle in a triangle where the side lengths are $\sqrt{8u-4}$, $\sqrt{8u+4}$, and $4\sqrt{u}$. Is this triangle a right triangle?


Find the value of $b$ so that the lines with the given equations are perpendicular:
\[
y = -3x + 4
\]
\[
2y + bx = 10
\]


Consider the polynomials \( f(x) = 2 - 10x + 4x^2 - 5x^3 + 7x^4 \) and \( g(x) = 5 - 3x - 8x^3 + 11x^4 \). Find \( c \) such that the polynomial \( f(x) + c g(x) \) has degree 3.


What is the sum of all integer solutions to \( |n| < |n+4| < 15 \)?


Compute $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}^{2019}$.

Solution: \boxed{\begin{pmatrix} 1 & 0 \\ 4038 & 1 \end{pmatrix}}


If \( p \), \( q \), and \( r \) are positive integers such that \( \gcd(p, q) = 300 \) and \( \gcd(p, r) = 450 \), then what is the smallest possible value of \( \gcd(q, r) \)?


Calculate the sum of the coefficients in the polynomial $2(4x^{8} - 5x^5 + 9x^3 - 6) + 8(x^6 - 4x^3 + 6)$ when it is fully simplified.


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


What are the rightmost three digits of $7^{1997}$?


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


The club has grown to 30 members. Now, 11 of them are left-handed, and 20 of them like jazz music. 4 of the club members are right-handed and dislike jazz music. How many club members are left-handed and like jazz music, assuming people are either left-handed or right-handed?


What is the remainder when $3^{17}$ is divided by $7$?


Circle \( D \) has radius 8 cm. What is the area of the largest possible inscribed triangle that uses one diameter of circle \( D \) as one side and is a right triangle?

Solution: \boxed{64 \text{ square centimeters}}


At a conference with $30$ businessmen, fifteen businessmen drank coffee, twelve businessmen drank tea, and eight businessmen drank juice. Six businessmen drank both coffee and tea, four drank both coffee and juice, and two drank both tea and juice. One businessman drank all three beverages. How many businessmen drank neither coffee, tea, nor juice?


In a 16-inch by 20-inch rectangle, there are two circles, each with a diameter of 8 inches. The circles are placed such that they do not extend beyond the rectangular region. What is the greatest possible distance (in inches) between the centers of these two circles?

Solution: \boxed{4\sqrt{13}\text{ inches}}


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


A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles needed to completely cover a rectangular region that is 3 feet by 6 feet, including a square area of 1 foot by 1 foot in one corner that has already been covered?


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

Solution: \boxed{\cos^2 t}


Moving along a specific line in the Cartesian plane, when the $x$-value increases by 2 units, the $y$-value increases by 5 units. If the $x$-value instead increases by 8 units, how many units will the $y$-value increase?


Determine \( g(x) \), where the terms should be arranged in order of decreasing degree, if
\[ 2x^5 + 4x^3 - 5x + 3 + g(x) = 3x^4 - 7x^3 + x^2 - 2. \]

Solution: \boxed{-2x^5 + 3x^4 - 11x^3 + x^2 + 5x - 5}


On this 7 by 7 grid of points, what fraction of the larger square's area is inside a shaded rhombus centered within the grid, where each side of the rhombus aligns with two points diagonally spaced apart (like the previous problem, but bigger grid)?

[asy]
size(180);
fill((3,3)--(4,4)--(3,5)--(2,4)--cycle,gray(0.7));
for(int i=0; i<=6; ++i){
for(int j=0; j<=6; ++j){
dot((i,j));
}
}
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((3,3)--(4,4)--(3,5)--(2,4)--cycle);
[/asy]


In 1970, there were 300,000 cases of a certain disease reported in a country. By 2000, the cases had reduced to 600. How many cases of this disease would have been reported in 1990 if the number of cases reported from 1970 to 2000 decreased linearly?


Jenna collects stamps. She puts the same number of stamps on each page and then inserts each page into one of her three stamp books. One of her stamp books has a total of 1200 stamps, another has 1800 stamps, and the third has 2400 stamps. What is the largest number of stamps that Jenna could be putting on each page?


The image of the point with coordinates $(2,3)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(10,7)$. Find $m+b$.


Emily surveyed a group of people about their knowledge of bats. To the nearest tenth of a percent, she found that $74.5\%$ of the people surveyed thought bats carried diseases. Of the people who thought bats carried diseases, $52.3\%$ said that bats frequently carried Lyme disease. Since bats do not frequently carry Lyme disease, these 34 people were mistaken. How many total people did Emily survey?


The curve $y = \sin x$ intersects the horizontal line $y = \sin 50^\circ$. If these intersections occur at successive $x$ values alternating in a specific pattern, what are the ratios of the lengths of these segments? Assume the ratio alternates as $r:s:r:s\dots$ with $r < s$. Compute the ordered pair of relatively prime positive integers $(r, s)$.

Solution: \boxed{(2, 7)}


Solution: \boxed{5 - 2i}


A school needs to transport 520 students for a field trip. If each of five available buses can hold a maximum of 45 students per trip, how many minimum trips are required to transport all the students?


A box contains 5 white balls and 5 black balls. If I draw them out of the box one at a time, what is the probability that all my draws alternate colors?


My friend Carla only likes numbers that are divisible by 8, such as 16, or 8,024. How many different ones digits are possible in numbers that Carla likes?


The algae in a lake doubles in coverage every day. By the end of 24 days, the lake is fully covered in algae. Determine on which day of the month was only 12.5% of the lake covered with algae.


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


A square carpet of side length 12 feet is designed with one large shaded square and twelve smaller, congruent shaded squares, as shown. If the ratios \(12:S\) and \(S:T\) are both equal to 4, and \(S\) and \(T\) are the side lengths of the shaded squares, what is the total shaded area?

Solution: \boxed{15.75 \text{ square feet}}


In 1980, there were 600,000 cases of a certain disease reported in Country X. By 2000, the number of reported cases decreased to 1,000. Calculate the number of cases that would have been reported in 1990 if the number of cases decreased linearly over this period.


How many integers between $700$ and $1000$ contain both the digits $5$ and $6$?


The graph of the rational function $\frac{1}{q(x)}$ is shown with vertical asymptotes at \(x = -2\), \(x = -1\), and \(x = 1\). If \(q(x)\) is a cubic polynomial and \(q(3) = 8\), find \(q(x)\).

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


The angles of a hexagon are in the ratio of 3:3:3:4:5:6. What is the number of degrees in the measure of the largest angle?


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


Find all solutions to the inequality \[\frac{(2x-5)(x-3)}{x} \ge 0.\] (Give your answer in interval notation.)

Solution: \boxed{(0, \frac{5}{2}] \cup [3, \infty)}


A square is evenly divided into four congruent rectangles, as shown in a similar way. If the perimeter of each of these four rectangles is 32 inches, what is the perimeter of the square, in inches?

[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((0,1)--(4,1));
draw((0,2)--(4,2));
draw((0,3)--(4,3));
[/asy]

Solution: \boxed{51.2 \text{ inches}}


The numbers $p,$ $q,$ $r,$ $s$ are equal to 1, 3, 5, 7, in some order. Find the largest possible value of
\[pq + qr + rs + sp.\]


On a checkerboard with 100 squares, each square $k$ is filled with $3^k$ grains of rice. Calculate how many more grains of rice are placed on the $15^{th}$ square than on the first $10$ squares combined.


Determine how many four-digit perfect squares are palindromes.


How many perfect squares less than 5000 have a ones digit of 1, 2 or 4?


Determine the domain of the expression $\frac{\sqrt{2x-4}}{\sqrt{9-3x} + \sqrt{x-1}}$.

Solution: \boxed{[2, 3]}


A regular octahedron is formed by joining the midpoints of the edges of a regular tetrahedron. Calculate the ratio of the volume of this octahedron to the volume of the original tetrahedron.


For a certain value of $k$, the system
\[
\begin{align*}
x + ky + 4z &= 0, \\
4x + ky + z &= 0, \\
3x + 5y - 2z &= 0
\end{align*}
\]
has a solution where $x,$ $y,$ and $z$ are all nonzero. Find $\frac{xz}{y^2}$.


Three cubes are rolled. Each cube has faces numbered 1 to 6. What is the probability that the sum of the numbers on the top faces of the three cubes is even? Express your answer as a common fraction.


If John flips 9 coins, what is the probability that he gets more heads than tails?


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


Let $\triangle PQR$ be a right triangle with $Q$ as a right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, find the length of $RS$.


The bagel shop has expanded and now has enough benches to sit $321_7$ people. If $3$ people are supposed to sit on one bench, how many benches does the shop have?


If the ratio of $3x + 5$ to $4x + 4$ is equal to the ratio of $5x + 4$ to $10x + 5$, find the product of all real solutions for $x$.


A right triangle with angles of 45°, 45°, and 90° has an inscribed circle with a radius of 8 cm. What is the length of the hypotenuse of the triangle? Express your answer in simplest radical form.

Solution: \boxed{16(\sqrt{2} + 1)}


The point $(2,3,4)$ is rotated $90^\circ$ about the $z$-axis, then reflected through the $xy$-plane, rotated $90^\circ$ about the $x$-axis, and finally reflected through the $yz$-plane. Find the coordinates of the point now.


Determine all values of $x$ that satisfy the following inequalities:
$$\frac{x}{2} \leq 3 + x < -3(1 + x).$$
Express the solution in interval notation.

Solution: \boxed{[-6, -\frac{3}{2})}


The Euler family has four girls aged 12, twin boys each aged 9, and two girls aged 15 and 17. What is the mean (average) of the ages of the children?


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


In the diagram, the smaller circles touch the larger circle and each other at the center of the larger circle. The radius of the larger circle is $8.$ What is the area of the shaded region?

[asy]
size(100);
import graph;
filldraw(Circle((0,0),2.5),mediumgray);
filldraw(Circle((-1.25,0),1.25),white);
filldraw(Circle((1.25,0),1.25),white);
[/asy]


Three of the four vertices of a rectangle are $(5, 11)$, $(16, 11)$, and $(16, -2)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 5)^2 + (y + 2)^2 = 16$? Express your answer in terms of $\pi$.


The area of the base of a hemisphere is $144\pi$. Calculate both the total surface area and the volume of the hemisphere. Express your answers in terms of $\pi$.


Compute
\[
\left( 1 - \frac{1}{\sin 30^\circ} \right) \left( 1 + \frac{1}{\cos 60^\circ} \right) \left( 1 - \frac{1}{\cos 30^\circ} \right) \left( 1 + \frac{1}{\sin 60^\circ} \right).
\]


In a new diagram, the grid is made up of squares with different dimensions. What is the area of the shaded region? The grid has the following structure:
- The bottom rectangle is 3 units high and 5 units wide.
- The middle rectangle is 4 units high and 7 units wide.
- The top rectangle is 5 units high and 12 units wide.
The shaded region is the entire grid minus a right-angled triangle with a base of 12 units and a height of 5 units.


Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2?


Find the sum of all integral values of $c$ with $c \leq 40$ for which the equation $y = x^2 - 9x - c^2$ has two rational roots.


Find the quotient of the division $(4z^5 - 3z^4 + 2z^3 - 5z^2 + 9z - 4)/(z + 3)$.

Solution: \boxed{4z^4 - 15z^3 + 47z^2 - 146z + 447}


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


Let $N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Here a minimally intersecting triple $(A, B, C)$ is defined by the condition that $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. Find the remainder when $N$ is divided by $1000$.


Let \[f(x) = \left\{
\begin{array}{cl}
2x + 4 & \text{if } x < -3, \\
-x^2 + x + 3 & \text{if } x \ge -3.
\end{array}
\right.\]Find the sum of all values of $x$ such that $f(x) = 0.$


Jonah’s five cousins are visiting and there are four identical rooms for them to stay in. If any number of cousins can occupy any room, how many different ways can the cousins be arranged among the rooms?


What is the units digit of the sum of the twelve terms of the sequence \( 1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 11! + 11, \, 12! + 12 \)?


A library has eight identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least one book is in the library and at least one is checked out?


A box contains eight cards. Four of the cards are blue on both sides, two cards are blue on one side and yellow on the other, and two of the cards are yellow on both sides. You pick a card randomly from the box and look at a random side. Given that the side you see is yellow, what is the probability that the other side is also yellow?


Suppose the point $(2,4)$ is on the graph of $y = 3f(x)$ where $f(x)$ is a function. Find the sum of the coordinates of one point that must lie on the graph of $y = \frac{1}{3}f^{-1}(x)$.


Margo walks to her friend's house in 15 minutes. After a 5-minute stop, it takes Margo 25 minutes to return home using the same route. If her average walking rate for the entire trip, excluding the stop, is 3 miles per hour, how many total miles did she walk?

Solution: \boxed{2 \text{ miles}}


Let $p, q, r$, and $s$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}p^2+q^2&=&r^2+s^2&=&3024,\\ pr&=&qs&=&1500.\end{array}\]
Compute the value of $\lfloor T \rfloor$, where $T=p+q+r+s$.


Given that $-2 - 5\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx + 49 = 0\] and that $a$ and $b$ are rational numbers, compute $a.$


Consider a rectangle where line segments are drawn through it. Initially, no line segment is present. For 5 line segments (including a fundamental line $PQ$), what is the largest number of sections into which the rectangle can be divided?


Let $Q$ be the plane passing through the origin with normal vector $\begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$. Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{v}$, $\mathbf{Q} \mathbf{v}$ is the projection of $\mathbf{v}$ onto plane $Q$.

Solution: \boxed{\begin{pmatrix} \frac{5}{9} & \frac{2}{9} & -\frac{4}{9} \\ \frac{2}{9} & \frac{8}{9} & \frac{2}{9} \\ -\frac{4}{9} & \frac{2}{9} & \frac{5}{9} \end{pmatrix}}


Determine how many solutions the following equation has:
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-2^2)(x-4^2)(x-6^2)\dotsm(x-24^2)} = 0
\]


What is the sum of the digits of the base-2 expression for $435_{10}$?


Simplify $20 \cdot \frac{14}{15} \cdot \frac{2}{18} \cdot \frac{5}{4}$.


In a different scenario, Sandra calculates sales data and uses the equation: \[ fp - w = 15000 \]
Given that \( f = 10 \) and \( w = 10 + 250i \), what is the value of \( p \)?

Solution: \boxed{1501 + 25i}


Let $\zeta$ be a nonreal root of $z^4 = 1$. Find the number of ordered pairs $(a,b)$ of integers such that $|a\zeta + b| = 1$.


In triangle $ABC$, $\angle C=90^\circ$, $AC=9$ and $BC=12$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED=90^\circ$. If $DE=6$, then what is the length of $BE$?


For how many different negative values of $x$ is $\sqrt{x + 200}$ a positive integer?


Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 4$ and $\frac{\cos x}{\cos y} = \frac{1}{3}$. Find the value of
\[\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}.\]


Find the sum of all angles \( x \) in the range \([0^\circ, 360^\circ]\) that satisfy:
\[
\sin^3 x + \cos^3 x = \frac{1}{\cos x} + \frac{1}{\sin x}
\]


Suppose a parabola has vertex $\left(\frac{3}{2}, -\frac{1}{4}\right)$ and the equation $y = ax^2 + bx + c$ where $a > 0$ and $2a + b + 3c$ is an integer. Find the smallest possible value of $a$.


Determine the smallest positive integer $n$ such that $6^n \equiv n^5 \pmod{4}$.


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 1:3$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is 25, determine the area of $\triangle ABC$.


Tamika selects two different numbers at random from the set $\{7, 11, 14\}$ and multiplies them. Carlos takes two different numbers at random from the set $\{2, 4, 7\}$ and adds them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.


The image of the point with coordinates $(2, -4)$ under the reflection across the line $y = mx + b$ is the point with coordinates $(-4, 4)$. Find $m+b$.


What is the base-10 integer 729 when expressed in base 7?


The value of $y$ varies inversely as $x^2$ and when $x=4$, $y=2$. What is the value of $x$ when $y=8$?


A triangular region is bounded by the two coordinate axes and the line given by the equation $3x + y = 9$. What is the area of this region, in square units?


How many different four-letter arrangements can be formed using the seven letters $A, B, C, D, E, F$, and $G$, if the first letter must be $D$, one of the other letters must be $E$, and the last letter must be $A$, given that no letter can be used more than once in the arrangement?


One of the roots of $z^2 = -75 + 65i$ is $4 + 9i.$ What is the other root?

Solution: \boxed{-4 - 9i}


Point \((x,y)\) is randomly picked from the rectangular region with vertices at \( (0,0), (2016,0), (2016,2017), \) and \( (0,2017) \). What is the probability that \( x > 3y \)? Express your answer as a common fraction.


What is the base \(2\) representation of \(125_{10}\)?


In her previous five attempts, Joy reached times of 82, 93, 99, 85, and 95 seconds in a 100-meter sprint. After her sixth attempt, she managed to lower her median time to 90 seconds. What was her time, in seconds, for her sixth attempt?

Solution: \boxed{87 \text{ seconds}}


Let $\star (x)$ be the sum of the digits of a positive integer $x$. $\mathcal{S}$ is the set of positive integers such that for all elements $n$ in $\mathcal{S}$, we have that $\star (n)=15$ and $0\le n< 10^{8}$. If $m$ is the number of elements in $\mathcal{S}$, compute $\star(m)$.


A conference hall is setting up seating for a lecture. Each complete row can contain $15$ chairs, and currently, there are $225$ chairs in total. If $140$ participants are expected to attend, how many chairs should be removed to have full rows with the least number of empty seats?


How many times does the digit 8 appear in the list of all integers from 1 to 700?


If \(x + \frac{1}{x} = 8\), find the value of \(x^2 + \frac{1}{x^2}\) and subsequently \(x^3 + \frac{1}{x^3}\).

Solution: \boxed{x^3 + \frac{1}{x^3} = 488}


It is currently 3:00:00 PM, as shown on a 12-hour digital clock. In 300 hours, 55 minutes, and 30 seconds, what will the time be and what is the sum of the hours, minutes, and seconds?


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


Simplify the following expression in $x$ and $y$:
\[
3x + 5x^2 - 4y - (6 - 3x - 5x^2 + 2y) - (4y^2 - 8 + 2x^2 - y).
\]
Express your answer in the form $ax^2 + by^2 + cx + dy + e$, where $a$, $b$, $c$, $d$, and $e$ are numbers.

Solution: \boxed{8x^2 - 4y^2 + 6x - 5y + 2}


Let $a \bowtie b = a + \sqrt{b + \sqrt{b + \sqrt{b + ...}}}$ with the process repeated $n$ times. If $5 \bowtie h = 10$ and the process is repeated 4 times, find the value of $h$.


Let $n$ be a positive integer and $a$, $b$ be integers such that each is its own inverse modulo $n$. What is the remainder when the product $a \cdot b$ is divided by $n$?


**The equation $x^2 + 16x = 100$ has two solutions. Find the positive solution expressed in the form $\sqrt{a} - b$. What is $a + b$?**


How many integers $n$ satisfy the condition $200 < n < 300$ and the condition $n$ has the same remainder when it is divided by $7$ or by $9$?


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


Three of the roots of the equation \[px^4 + qx^3 + rx^2 + sx + t = 0\] are $4$, $-3$, and $0$. Given that $p \neq 0$, compute $\frac{q+r}{p}$.


A $200\times 420\times 480$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes?


Solution: \boxed{-\frac{6}{17} - \frac{10}{17}i}


Five packages are delivered to five houses, one to each house. What is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction.


Catherine 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 1500th 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 1498th, 1499th, and 1500th digits, in that order).


Suppose that $A$, $B$, and $C$ are non-zero distinct digits less than $7$, and suppose we have ${AB_7}+{C_7}={C}0_7$ and ${AB_7}+{BA_7}={CC_7}$. Find the three-digit number ${ABC}$. (Numbers are expressed in base-7, interpret similarly to the initial problem description).


Determine the number of possible prime values of $p$ such that $2017_p + 504_p + 217_p + 203_p + 14_p = 245_p + 415_p + 531_p$.


How many perfect squares less than 500 have a ones digit of 2, 3, or 4?


A gasoline tank is $\frac{3}{4}$ full. After $15$ gallons are used, it is $\frac{1}{3}$ full. How many more gallons must be used to empty the tank?


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


If $m$ and $n$ are positive integers such that $\gcd(m,n) = 18$, what is the smallest possible value of $\gcd(20m, 25n)$?


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


Examining the properties of the roots of the function $x^2 - tx + q$, where the roots $r_1$ and $r_2$ satisfy the condition that $r_1 + r_2 = r_1^2 + r_2^2 = \dots = r_1^{2010} + r_2^{2010}$. Calculate the maximum possible value of $\dfrac1{r_1^{2010}}+\dfrac1{r_2^{2010}}$.


If $a$, $b$, $c$, and $d$ are real numbers satisfying:
\begin{align*}
a+b+c &= 3, \\
a+b+d &= -2, \\
a+c+d &= 8, \text{ and} \\
b+c+d &= -1,
\end{align*}
what is $ab + cd$?


Find the common ratio of the infinite geometric series: $$\frac{7}{8} - \frac{14}{27} + \frac{56}{243} - \dots$$


In a triangle, two of the side lengths are 9 and 10, and the angle between them is $150^\circ$. Find the length of the third side.

Solution: \boxed{\sqrt{181 + 90\sqrt{3}}}


In pentagon $ABCDE$, angles $A$ and $B$ are $75^\circ$ and $95^\circ$, respectively. The two identical angles $C$ and $D$ are equal, and the fifth angle $E$ is triple the measure of angle $C$ plus $20^\circ$. Find the measure of the largest angle.


Suppose $56 \cdot 89 \cdot 94 \equiv m \pmod{25}$, where $0 \le m < 25$.


Suppose \(a\), \(b\), and \(c\) are three positive numbers that satisfy \(abc = 1\), \(a + \frac{1}{c} = 7\), and \(b + \frac{1}{a} = 34\). Find \(c + \frac{1}{b}\).


Points $P$ and $Q$ are selected on the graph of $y = -\frac{1}{2}x^2$ so that triangle $POQ$ is equilateral. Find the length of one side of triangle $POQ$, where $O$ is the origin $(0,0)$.


Find the value of $b$ such that the lines given by the equations below are perpendicular:
\begin{align*}
y &= 3x - 7 \\
4y + bx = 12.
\end{align*}


Triangle $ABC$ is a right triangle where $AB = AC$. Points $D$, $E$, and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. Point $G$ is the midpoint of segment $\overline{DF}$, and point $H$ is the midpoint of segment $\overline{DE}$. Determine the ratio of the area inside triangle $DEF$ to the area outside triangle $DEF$ but inside triangle $ABC$. Express your answer as a common fraction.


What is the remainder when $97^{45}$ is divided by 100?


A four-digit integer has digits that sum up to 16. The sum of the two middle digits is ten, and the thousands digit minus the units digit is two. If the integer is divisible by 11, find the integer.


A certain type of grass spreads across an empty lot at an exponential rate, doubling its covered area every day. On day $24$, the lot is completely covered by the grass. Determine on which day only $1/8$ of the lot was covered by the grass.


Solution: \boxed{7x^4 - 7x^3 - 42x^2 + 82x + 110}


What is the positive difference between the median and the mode of the data given in the stem and leaf plot below? In this plot, $7|3$ represents $73.$

\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
4 & $7 \hspace{2mm} 8 \hspace{2mm} 8 \hspace{2mm} 9 \hspace{2mm} 9$ \\ \hline
5 & $0 \hspace{2mm} 1 \hspace{2mm} 1 \hspace{2mm} 1 \hspace{2mm} 2$ \\ \hline
6 & $5 \hspace{2mm} 6 \hspace{2mm} 7 \hspace{2mm} 8 \hspace{2mm} 9$ \\ \hline
7 & $1 \hspace{2mm} 2 \hspace{2mm} 3 \hspace{2mm} 3 \hspace{2mm} 4$ \\ \hline
8 & $0 \hspace{2mm} 2 \hspace{2mm} 5 \hspace{2mm} 7 \hspace{2mm} 9$ \\\hline
\end{tabular}


What is the sum of the tens digit and the ones digit of the integer form of $(2+4)^{15}$?


How many integers from 1 through 9999, inclusive, do not contain any of the digits 1, 2, 3, or 4?


For some positive integer $j$, when 75 is divided by $j^2$, the remainder is 3. What is the remainder when 130 is divided by $j$?


In Ms. Green's history class, there are 16 boys and 20 students overall brought their notebooks to class. If 11 of the students who brought notebooks are girls, how many boys did not bring their notebooks to class?


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

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

Solution: \boxed{(2, 4)}


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

Solution: \boxed{9\sqrt{13} + 81}


A wooden bridge is supported by two identical metal supports shaped like triangular wedges as shown in the diagram:
[asy]
import olympiad;
import math;

// Draw triangles
pair A = (0, 1);
pair B = (-cos(1.3962), 0);
pair C = (cos(1.3962), 0);
pair D = (2, 1);
pair E = (2-cos(1.3089), 0);
pair F = (2+cos(1.3089), 0);
draw(A--B--C--cycle);
draw(D--E--F--cycle);
draw(A--D);
label('$A$',A,N);
label('$B$',B,S);
label('$C$',C,S);
label('$D$',D,N);
label('$E$',E,S);
label('$F$',F,S);
[/asy]
If $\triangle ABC$ and $\triangle DEF$ are isosceles with $AB = AC$ and $DE = DF,$ with angles $\angle BAC = 25^\circ$ and $\angle EDF = 40^\circ,$ what is $\angle DAC + \angle ADE$?


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.


Determine how many integer values of \( b \) are possible if there are exactly three integers \( x \) satisfying the inequality
\[ x^2 + bx - 2 \leq 0. \]


A truncated cone has horizontal bases with radii 12 and 3. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?


Let $
f(n) =
\begin{cases}
n^2 - n + 1 & \text{if }n\text{ is odd} \\
\dfrac{n}{2} & \text{if }n\text{ is even}
\end{cases}.
$

For how many integers $n$ from 1 to 128, inclusive, does $f ( f (\dotsb f (n) \dotsb )) = 1$ for some number of applications of $f$?


Let $f(z) = (2 + i)z^3 + (4 + i)z^2 + \alpha z + \gamma$ be defined for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers, and $i^2 = -1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $|\alpha| + |\gamma|$?


Determine how many integers satisfy the following inequalities:
\[
\begin{array}{cccc}
(1) & -4x & \geq & x+9 \\
(2) & -3x & \leq & 15 \\
(3) & -5x & \geq & 3x+21
\end{array}
\]


Derek buys hamburgers that come in packages of 10, and he buys hamburger buns that come in packages of 15. What is the smallest number of hamburger packages he can buy in order to have an equal number of hamburgers and hamburger buns?


In a press conference before the World Series Game, nine All-Stars will be taking questions. Three are Cubs, two are Red Sox, two are Yankees, and two are Dodgers. If teammates insist on sitting together, how many ways can the nine All-Stars be seated in a row?


Let \(a\) and \(b\) be positive real numbers such that \(a + 3b = 1.\) Find the minimum value of
\[
\frac{1}{a} + \frac{1}{b}.
\]

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


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


Expand the following product and simplify: $\frac{3}{4}\left(\frac{8}{y} - 6y^2 + 3y\right)$.

Solution: \boxed{\frac{6}{y} - \frac{9y^2}{2} + \frac{9y}{4}}


A certain integer has $3$ digits when written in base $6$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$?


On the Cartesian plane, a dog is tied to a post at the point $(5,1)$ using a 7-foot rope. An impenetrable obstacle lies exactly at the point $(6,3)$. Ignoring the influence of the obstacle, what would be the greatest distance the dog can be from the origin?

Solution: \boxed{\sqrt{26} + 7}


Find the largest interval including the point $x=2$ on which the function $g(x) = 3x^2 - 9x - 4$ is invertible.

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


A parallelogram is generated by the vectors $\begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 2 \\ 2 \end{pmatrix}.$ Calculate the cosine of the angle $\theta$ between the diagonals of the parallelogram.


Let \(a\) and \(b\) be positive real numbers such that \(a + b = 3.\) Find the set of all possible values of \(\frac{1}{a} + \frac{1}{b}.\)

Solution: \boxed{\left[\frac{4}{3}, \infty\right)}


What is the value of $x$ for which $\frac{1}{\text{log}_3x} + \frac{1}{\text{log}_4x} + \frac{1}{\text{log}_5x} = 2$?


A postal service in a fictional country charges an extra $0.15 in postage if the ratio of the length of an envelope, in centimeters, to its height, in centimeters, is either less than $1.5$ or greater than $2.8$. Determine for how many of these envelopes the extra $0.15 in postage must be paid:
\[
\begin{tabular}[t]{ccc}
Envelope & Length in cm & Height in cm\\\hline
X &7 &5\\
Y &10 &4\\
Z &8 &5\\
W &14 &5
\end{tabular}
\]


Given that \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are unit vectors in three-dimensional space, find the maximum value of
\[
\|\mathbf{u} - \mathbf{v}\|^2 + \|\mathbf{u} - \mathbf{w}\|^2 + \|\mathbf{v} - \mathbf{w}\|^2 + \|\mathbf{u} + \mathbf{v} + \mathbf{w}\|^2.
\]


A traffic light has the following cycle: green for 50 seconds, yellow for 5 seconds, and then red for 40 seconds. Leah picks a random five-second time interval to watch the light. What is the probability that the color changes while she is watching?


What is the greatest common divisor of 128, 144, and 360?


The line $y = \frac{2}{3} x + 5$ is parameterized by the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -3 \\ s \end{pmatrix} + t \begin{pmatrix} l \\ -6 \end{pmatrix}.\]
Enter the ordered pair $(s,l).$

Solution: \boxed{(3, -9)}


A rectangular prism has 6 faces (2 rectangular and 4 square faces), 12 edges, and 8 vertices. Choose one of its square faces as the base of a new pyramid. The numbers of external faces, vertices, and edges of the resulting shape (the combination of the prism and pyramid) are added. What is the maximum value of this sum?


Consider a sequence $x_1,$ $x_2,$ $x_3,$ $\dots$ defined by
\begin{align*}
x_1 &= \sqrt[4]{4}, \\
x_2 &= (\sqrt[4]{4})^{\sqrt[4]{4}},
\end{align*}and in general,
\[x_n = (x_{n - 1})^{\sqrt[4]{4}}\]for $n > 1.$   What is the smallest value of $n$ for which $x_n$ is an integer?


Evaluate $\frac{8}{29} - \frac{5}{87}$ and reduce the result to its simplest form.


How many distinct arrangements of the letters in the word "basics" are there, specifically those beginning with a vowel?


Given the functions $p(x) = |x| - 3$ and $q(x) = -|x|$, evaluate $q(p(x))$ at $x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5$ and find the sum of these eleven values.


A box contains ten cards. Four of the cards are black on both sides, three cards are black on one side and red on the other, and three of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other side is also red?


Triangle $ABC$ has vertices at $A(0, 8)$, $B(2, 0)$, and $C(10, 0)$. A horizontal line with equation $y = t$ intersects line segment $\overline{AB}$ at $T$ and line segment $\overline{AC}$ at $U$, forming triangle $\triangle ATU$ with area 20. Compute $t$.

Solution: \boxed{8 - 2\sqrt{10}}


Let $P$ be a point outside of circle $O.$ A segment is drawn from $P$ such that it is tangent to circle $O$ at point $T.$ Meanwhile, a secant from $P$ intersects $O$ at $A$ and $B,$ such that $PA < PB.$ If $PA = 4$ and $PT = 2(AB - PA),$ then what is $PB$?

Solution: \boxed{\frac{17 + \sqrt{33}}{2}}


Suppose the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(3,3)$, $(6,9)$, and $(9,18)$. Find one point where the graphs of $y=h(3x)$ and $y=3j(x)$ must intersect. What is the sum of the coordinates of that point?


Let $a\star b = \dfrac{\sqrt{a+b}}{\sqrt{a-b}}$. If $x \star 36 = 9$, find $x$.


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


Find the value of $\sec 150^\circ + \tan 225^\circ$.

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


Consider the quadratic function $f(x) = 3x^2 - 6x - 9$. Determine the largest interval containing the point $x=2$ where $f(x)$ is invertible.

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


A rectangle has a length of 16 and a width of 8. Given that a piece of string snugly fits around the perimeter of this rectangle, rounded to the nearest whole number, what is the area of the largest circle that can be formed from this piece of string?


If $\overline{AD} \| \overline{FG}$, how many degrees are in angle $EFG$?

[asy]
import olympiad;

pair A = (-15,20);
pair B = (-12,35);
pair C = (35,50);
pair D = (35,20);
pair E = (14,20);
pair F = (0,0);
pair G = (40,0);

draw(F--G);
draw(F--C);
draw(A--D);
draw(B--E);

label("F", F, W);
label("G", G, ENE);
label("C", C, N);
label("A", A, W);
label("D", D, ENE);
label("E", E, SE);
label("B", B, NW);

draw(scale(20)*anglemark(G, F, C));
draw(shift(E)*scale(35)*shift(-E)*anglemark(B, E, A));
draw(shift(E)*scale(20)*shift(-E)*anglemark(C, E, B));

label("$x$", (6,20), NW);
label("$3x$", (13,25), N);
label("$2x$", (5,0), NE);
[/asy]


Expand the expression $(2x + 3y + 4)(5x + 6y + 7)$ and find the sum of the coefficients of the terms which contain a nonzero power of "y".


The measure of angle $ACB$ is 50 degrees. If ray $CA$ is rotated 540 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?

Solution: \boxed{50 \text{ degrees}}


In a circle with center $O$, point $O$ is the center of the circle circumscribed about $\triangle ABC$. Given $\angle BOC = 150^\circ$ and $\angle AOB = 130^\circ$, find the degree measure of $\angle ABC$.


Maria has 12 different kinds of lunch meat and 11 different kinds of cheese. If she wants to make a sandwich with two kinds of meat, two kinds of cheese, and two kinds of toppings from 8 options, how many different sandwiches could she make?


Let $p, q, r, s, t, u, v, w$ be distinct elements in the set $\{-8, -6, -4, -1, 1, 3, 5, 14\}$. What is the minimum possible value of $(p+q+r+s)^2 + (t+u+v+w)^2$?


Regular square $ABCDE$ and regular octagon $AEFGHIJK$ are drawn on opposite sides of line segment $AE$ such that they are coplanar. What is the degree measure of exterior angle $DEF$?


The volume of a certain rectangular solid is $512\text{ cm}^3$, its total surface area is $384\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.


What is the smallest whole number $b$ such that 90 can be expressed in base $b$ using only three digits?


The mean of one set of three numbers is 15, and the mean of a separate set of five numbers is 20. Additionally, it is known that the largest number in the first set is equal to the smallest number in the second set, which is 12. Find the mean of the set of all eight numbers.


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


Let $a$ and $b$ be angles such that:
\[\sin (a + b) = \sin a + \sin b.\]
Find the maximum value of $\sin a.$


Simplify the product \[\frac{6}{3} \cdot \frac{9}{6} \cdot \frac{12}{9} \dotsm \frac{3n+3}{3n} \dotsm \frac{2982}{2979}.\]


Evaluate the munificence of the monic quadratic polynomial $p(x) = x^2 - 2x - 1$ over the interval $-1 \le x \le 1$.


What is the value of $2^4 \times 3^2 \times 5^2 \times 11$?


What is the least positive multiple of 7 that is not a lucky integer?


Emily paid for a $\$2$ sandwich using 50 coins consisting of pennies, nickels, and dimes, and received no change. How many dimes did Emily use?


If $\frac{72}{2^4 \cdot 5^7}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?


Let $x$ and $y$ be two positive real numbers such that $x + y = 27.$  Find the ordered pair $(x,y)$ for which $x^3 y^4$ is maximized.

Solution: \boxed{\left(\frac{81}{7}, \frac{108}{7}\right)}


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?


Consider the function \( g(x) = x^3 + 3x^{1/2} \). Evaluate \( 3g(3) - 2g(9) \).

Solution: \boxed{-1395 + 9\sqrt{3}}


Find $x^2 + y^2$ if $x$ and $y$ are positive integers such that:
\[
xy + x + y = 17 \quad \text{and} \quad x^2y + xy^2 = 72.
\]


The sum of the positive divisors of a positive integer of the form $2^i5^j$ is equal to $930$. What is $i + j$?


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 $g(13)=13$, $g(31)=31$ and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Find the unique number that is not in the range of $g$.


The sides of a triangle have lengths of $13$, $14$, and $15$. Find the length of the shortest altitude.


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


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


Calculate the sum of \( \left(\frac{1}{4}\right) + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \left(\frac{1}{4}\right)^4 + \left(\frac{1}{4}\right)^5 \).


Consider an ellipse defined by $\frac{x^2}{25} + \frac{y^2}{d^2} = 1$ and a hyperbola defined by \[\frac{x^2}{169} - \frac{y^2}{64} = \frac{1}{36}\]. If the foci of both shapes coincide, find $d^2$.


Compute $3\tan 30^\circ + 6\sin 30^\circ.$

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


Two different natural numbers are selected from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction.


The Grunters play the Screamers 5 times. The Grunters are $60\%$ likely to win any given game. What is the probability that the Grunters will win at least 4 out of the 5 games? Express your answer as a common fraction.


A square pyramid has a base edge of 24 inches and an altitude of 18 inches. A smaller square pyramid, whose altitude is one-third of the original pyramid's altitude, is cut away from the apex. What fractional part of the original pyramid's volume does the remaining frustum represent?


The sum of all the positive factors of integer $x$ is 18. If one of the factors is 2, what is the value of $x$?


If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$, what is the angle between the vectors $2\mathbf{a}$ and $-\mathbf{b}$?


The gravitational force that a planet exerts on an object is inversely proportional to the square of the distance between the center of the planet and the object. If Anna stands 5,000 miles from the center of this planet and experiences a gravitational force of 500 Newtons, what would be the gravitational force exerted on her when she stands 200,000 miles from the center of the same planet? Express your answer as a fraction.


How many positive divisors does 8! have?


John decides to draw a regular six-pointed star (hexagram) in the sand, and at each of the 6 outward-pointing and 6 inward-pointing points, he places one of twelve different sea shells. How many ways can he place the shells if reflections and rotations of an arrangement are considered equivalent?


A square carpet of side length 12 feet contains one large shaded circle and eight smaller, congruent shaded squares, as illustrated. The ratios $12 : \text{R}$ and $\text{R} : \text{S}$ are both equal to 4, where $\text{R}$ is the diameter of the shaded circle and $\text{S}$ is the side length of each shaded square. Determine the total shaded area.

Solution: \boxed{\frac{9\pi}{4} + \frac{9}{2}}


Find the largest possible value of $x$ in the simplified form $x=\frac{a+b\sqrt{c}}{d}$ if $\frac{7x}{5} + 3 = \frac{4}{x}$, where $a, b, c,$ and $d$ are integers. What is $\frac{acd}{b}$?


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?


A license plate in a certain state consists of 3 digits and 3 letters. The letters are not necessarily distinct and must appear next to each other. These six characters may appear in any order. How many distinct license plates are possible?


Tim is organizing a week-long series of pranks. On Monday, he gets his friend Joe to help. On Tuesday, he can choose between two friends, either Ambie or John. For Wednesday, there are four new people willing to help. However, on Thursday, none of these previous individuals can participate, but Tim has convinced five different friends to help. On Friday, Tim decides to go solo again. Additionally, this time, whoever helps on Wednesday cannot help on Thursday. How many different combinations of people could Tim involve in his pranks during the week?


In a parallelogram, the lengths of the sides are $12$, $5y-3$, $3x+2$, and $9$, consecutively. Find the value of $x+y$.


Determine the area enclosed by $|4x| + |3y| = 12$.


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


Two rays with a common endpoint $O$ form a $45^\circ$ angle. Point $A$ lies on one ray, and point $B$ lies on the other ray with $AB=1$. What is the maximum possible length of $\overline{OB}$?


The point $(1,1,1)$ is first reflected through the $yz$-plane, then rotated $90^\circ$ about the $y$-axis, followed by a reflection through the $xz$-plane, another rotation of $90^\circ$ about the $y$-axis, and finally reflected through the $xy$-plane. Find the coordinates of the new position of the point.

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


In a rectangular coordinate system, what is the distance from the origin to the point $(20, -21)$?


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


Frank has an unfair eight-sided die. The probabilities are such that rolling an 8 has a probability of $\frac{3}{8}$, and each of the numbers from 1 to 7 has the same probability of rolling. What is the expected value of the number shown when this die is rolled?


Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$?


Humanity finds 12 habitable planets, of which 6 are Earth-like and 6 are Mars-like. Earth-like planets require 3 units of colonization resources, while Mars-like need 1 unit. If 18 units of colonization resources are available, how many different combinations of planets can be colonized, assuming each planet is unique?


Let $M$ be the midpoint of the segment joining vertices $A$ and $B$ of triangle $ABC$, and let $Q$ be an arbitrary point. Then there exists a constant $k$ so that
\[ QA^2 + QB^2 + QC^2 = k \cdot QM^2 + MA^2 + MB^2 + MC^2. \]
Find $k$.


Seven distinct integers are picked at random from the set {1,2,3,...,12}. What is the probability that among those selected, the second smallest number is 4?


Find the $2 \times 2$ matrix $\mathbf{M}$ such that $\mathbf{M} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 7 \\ 5 \end{pmatrix}$ and $\mathbf{M} \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 18 \\ -1 \end{pmatrix}.$

Solution: \boxed{\mathbf{M} = \begin{pmatrix} \frac{61}{14} & -\frac{4}{7} \\ \frac{1}{7} & \frac{11}{7} \end{pmatrix}}


Define the operation \(\oplus\) by
\[a \oplus b = \frac{a + b}{1 + ab}.\]
Compute
\[10 \oplus (9 \oplus (8 \oplus (\ldots \oplus (2 \oplus 1) \ldots))).\]


What is the least integer whose square is 80 more than its triple?


When rolling a fair 8-sided die, what is the probability of rolling a number that is either a prime or a multiple of 3?


Ben twice chooses a random integer between 1 and 60, inclusive (and he may choose the same integer both times). What is the probability that at least one of the numbers Ben chooses is a multiple of 5, and the sum of the numbers chosen is even?


Given \(x + \frac{1}{x} = 7\), calculate the value of \(x^6 + \frac{1}{x^6}\).


How many even perfect square factors does $3^6 \cdot 5^8$ have?


Find a monic cubic polynomial \(Q(x)\) with integer coefficients such that
\[Q(\sqrt[3]{5} + 2) = 0.\]

Solution: \boxed{x^3 - 6x^2 + 12x - 13}


Given that the point \((3, 10)\) is on the graph of \( y = g(x) \), determine a point that must lie on the graph of \( 3y = 4g(3x) + 6 \). What is the sum of the coordinates of that point?


What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $C(3, 0)$ and $D(4, 5)$?


A right circular cylinder with its height equal to its diameter is inscribed in a right circular cone. The cone has a diameter of 16 units and an altitude of 20 units. The axes of the cylinder and the cone coincide. Calculate the radius of the cylinder and express your answer in a simplified fraction form.


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


Find the integer $n$, $10 \le n \le 20$, such that \[n \equiv 7882 \pmod{7}.\]


Let \( y \) be a real number. Consider the following five statements:

1. \( 0 < y^3 < 1 \)
2. \( y^3 > 1 \)
3. \( -1 < y < 0 \)
4. \( 0 < y < 1 \)
5. \( 0 < y^2 - y^3 < 1 \)

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


Suppose $a$ and $b$ are nonzero real numbers, and the equation $x^2 + 2ax + b = 0$ has solutions $a$ and $b$. Find the ordered pair $(a, b)$.

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


Alice tells Bob, "My number is 36." Bob responds, "Every prime factor of your number is a prime factor of my number, plus I have one additional distinct prime factor."


Let $M$ be the product of all odd primes less than $2^5$. What remainder does $M$ leave when divided by $2^5$?


Let \(b_n = 7^n + 9^n\). Determine the remainder when \(b_{86}\) is divided by \(50\).


Three dice with faces numbered from 1 to 6 are rolled. What is the probability that the sum of the numbers on the top faces is a multiple of 3? Express your answer as a common fraction.


A prism has 18 edges. How many faces does the prism have?


In right triangle $ABC$, we have $\angle A = 30^\circ$, $\angle B = 90^\circ$, and $BC = 12$. Find $AB$ to the nearest tenth.


A number in the set $\{40, 41, 42, 43, ... , 999\}$ is randomly selected. What is the probability that it is either a number less than 60 or a multiple of 10? Express your answer as a common fraction.


Let $\mathbf{a}$ and $\mathbf{b}$ be nonzero vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1$ and $\|\mathbf{a} + 2 \mathbf{b}\| = 1$. Find the cosine of the angle between $\mathbf{a}$ and $\mathbf{b}$.


Let set $\mathcal{A}$ be a 100-element subset of $\{1,2,3,\ldots,120\}$, and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$.


If the consecutive integers from $100$ to $1$ were written as $$100999897...,$$ what would be the $190^{\text{th}}$ digit to be written?


Compute the product:
\[\left( 1 + \cos \frac{\pi}{12} \right) \left( 1 + \cos \frac{5\pi}{12} \right) \left( 1 + \cos \frac{7\pi}{12} \right) \left( 1 + \cos \frac{11\pi}{12} \right).\]


Let $x = (2 + \sqrt{2})^{10}$, let $n = \lfloor x \rfloor$, and let $f = x - n$. Find
\[x(1 - f).\]


Given the number
\[e^{11\pi i/60} + e^{23\pi i/60} + e^{35 \pi i/60} + e^{47\pi i /60} + e^{59 \pi i /60},\]
express it in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. Find $\theta$.


Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = SQ = 8$ and $SR = 7$, then what is the volume of pyramid $SPQR$?


What is the largest possible median for the six-number set $\{x, 2x, 3, 2, 5, 7\}$ if $x$ can be any integer?


Consider a hyperbola where the center is at $(3, -1)$, one focus is at $(3 + \sqrt{53}, -1)$, and one vertex is at $(7, -1)$. The equation of this hyperbola can be written as:
\[
\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1.
\]
Find $h + k + a + b$.

Solution: \boxed{6 + \sqrt{37}}


Consider a constant $k$ and a point $C = (0, c)$ such that for all chords $\overline{AB}$ of the parabola $y = x^2 + k$ passing through $C$, the value
\[ t = \frac{1}{AC^2} + \frac{1}{BC^2} \]
is a fixed constant. Find the constant $t$.


What is the sum of all the solutions of $x = \left|3x-|80-3x|\right|$?


The line $y = \frac{3}{4} x + 5$ is parameterized by the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -3 \\ s \end{pmatrix} + t \begin{pmatrix} m \\ -7 \end{pmatrix}.\]
Find the ordered pair $(s,m).$

Solution: \boxed{\left(\frac{11}{4}, -\frac{28}{3}\right)}


I want to choose a license plate which is 4 characters long, where the first character is a letter, the last character is a digit, and the two middle characters are either both letters or both digits. Additionally, the two middle characters must be the same, and the first character must be different from the last. How many ways are there for me to choose such a license plate?


It can be shown that for any positive integer $n,$
\[
\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{pmatrix},
\]
where $F_n$ denotes the $n$th Fibonacci number.

Compute $F_{1000} F_{1002} - F_{1001}^2.$


Calculate the largest integer \( m \) for which \( \binom{10}{4} + \binom{10}{5} = \binom{11}{m} \).


Determine the sum of all integers from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ that are primitive roots modulo 11.


The image of the point with coordinates $(2, -2)$ under the reflection across the line $y = mx + b$ is the point with coordinates $(8, 4)$.  Find $m+b$.


If \( m = 2^{12} \cdot 3^{10} \cdot 5^9 \cdot 7^6 \), how many of the natural-number factors of \( m \) are multiples of 630?


A 10-foot by 12-foot floor is tiled with square tiles of size 2 feet by 2 feet. Each tile has a pattern consisting of four white quarter circles of radius 1 foot centered at each corner of the tile, and a white square of side 1 foot in the center. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

Solution: \boxed{90 - 30\pi \text{ square feet}}


6 15-sided dice are rolled. What is the probability that three of the dice show a number from 1 to 9 and the other three show a number from 10 to 15? Express your answer as a common fraction.


What is the area enclosed by the graph of $|2x| + |5y| = 10$?


Solution: \boxed{4(t^2 - 34)}


Solve for $x$ in the equation
\[ 4^{(\sqrt{16^x})} = 16^{(\sqrt{2^x})}. \]


Let $b_n = 7^n + 9^n$. Determine the remainder upon dividing $b_{95}$ by $64$.


Evaluate: $6 - 8\left(9 - 4^2\right) \div 2 - 3.$


Find the value of $k$ such that the lines $y=6x+4$, $y=-3x-30$, and $y=4x+k$ intersect at the same point.


A cube has side length $8$ cm. Its vertices are alternately colored black and purple. What is the volume of the tetrahedron whose corners are the purple vertices of the cube?

Solution: \boxed{\frac{512}{3} \text{ cm}^3}


Determine all values of $m$ so that the domain of the function
\[ c(x) = \frac{3x + m}{3x^2 + mx + 7} \]
is the set of all real numbers.

Solution: \boxed{\left(-2\sqrt{21}, 2\sqrt{21}\right)}


A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. Find the solution to the equation $504 \,\diamondsuit\, (12\,\diamondsuit\, x)=50.$


Express the quotient $2314_5 \div 21_5$ in base $5$.


Consider the hyperbolas \[\frac{x^2}{16} - \frac{y^2}{25} = 1\] and \[\frac{y^2}{100} - \frac{x^2}{M} = 1.\] They have the same asymptotes. Find \(M\).


The function \( g(x) \) satisfies
\[ g(xy) = xg(y) + 1 \] for all real numbers \( x \) and \( y \). If \( g(1) = 24 \), find \( g(80) \).


Given that $-2 - 5\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx + 45 = 0\] and that $a$ and $b$ are rational numbers, compute $a.$


In a diagram with two concentric circles, a chord AB of length 100 units is tangent to the smaller circle. If the radius of the smaller circle is 40 units, what is the radius of the larger circle? Additionally, calculate the area of the shaded region between the two circles. Express your answer in terms of $\pi$.


Solution: \boxed{328 - 52\sqrt{34}}


The ratio of the areas of two squares is $\frac{300}{147}$. Find the simplified form of the ratio of their side lengths, expressed as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Additionally, if the perimeter of the larger square is 60 units, determine the side length of the smaller square.


In triangle $PQR,$ points $M$ and $N$ are on $\overline{PQ}$ and $\overline{PR},$ respectively, such that $PM = 2$ and $MQ = 6$, while $PN = 3$ and $NR = 9$. The angle bisector $\overline{PS}$ of $\angle P$ intersects $\overline{MN}$ at point $L$. Compute $\frac{PL}{PS}.$

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


An 8-sided die has numbers from 1 to 8 on its faces (each number appears on one face). How many ways can I paint two faces of the die red, such that the numbers on these two faces do not add up to a sum greater than 7?


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


Calculate the product $\left(\frac{5}{10}\right)\left(\frac{10}{15}\right)\left(\frac{15}{20}\right)\cdots\left(\frac{5005}{5010}\right)$. Express your answer as a common fraction.


Let $g(x)$ be a function defined by \[g(x) = \left\{
\begin{array}{cl}
-x & x\le 0, \\
2x-43 & x>0.
\end{array}
\right.\] If $a$ is negative, find $a$ such that $g(g(g(13)))=g(g(g(a)))$.

Solution: \boxed{a = -30}


Let $p$ and $q$ be the roots of the equation $6x^2 - 7x - 20 = 0$. Calculate $(p-q)^2$.


In triangle $XYZ$, the measure of $\angle X$ is $72^\circ$. The measure of $\angle Y$ is $10$ degrees more than four times the measure of $\angle Z$. What is the measure, in degrees, of $\angle Z$?


**Begin with the integer 1,000,000,000. The sequence is formed by repeatedly dividing the number by 3. What is the last integer in this sequence?**


In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 12. What is the greatest possible perimeter of the triangle?


Given two triangles $\triangle XYZ$ and $\triangle PQR$, where side $XY = 9$ cm, side $YZ = 21$ cm, and side $XZ = 15$ cm. Triangle $\triangle PQR$ has side $PQ = 3$ cm, side $QR = 7$ cm, and the angle $\angle PQR$ is the same as $\angle XYZ$. What is the length of segment $PR$?

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


A function \(f(x)\) is defined for all real numbers \(x\). For all non-zero values \(x\), we have
\[3f\left(x\right) + f\left(\frac{1}{x}\right) = 15x + 8.\]
Let \(S\) denote the sum of all of the values of \(x\) for which \(f(x) = 2004\). Compute the integer nearest to \(S\).


Find the product of the greatest common divisor and the least common multiple of \(75\) and \(125.\)


In polar coordinates, the point $\left(-3, \frac{\pi}{8}\right)$ is equivalent to what other point, in the standard polar coordinate representation? Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi$.

Solution: \boxed{(3, \frac{9\pi}{8})}


What is the smallest positive value of $x$ such that $x + 7890$ results in a palindrome?


Let $a$, $b$, $c$, and $d$ be positive real numbers. What is the smallest possible value of $(a+b+c+d)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+d}+\frac{1}{c+d}\right)$?


If $2x + 3y = 9$ and $xy = -12$, what is the value of $4x^2 + 9y^2$?


What is the smallest positive integer $n$ such that $8820$ is a factor of $n!$?


Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u}$, $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto the vector $\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} \frac{1}{9} & \frac{2}{9} & \frac{2}{9} \\ \frac{2}{9} & \frac{4}{9} & \frac{4}{9} \\ \frac{2}{9} & \frac{4}{9} & \frac{4}{9} \end{pmatrix}}


How many positive integers less than 1,000 have at most two different digits?


If you multiply the base-10 numbers 123 and 78, subtract 156, and then write this result in base-6, what is the units digit of this base-6 representation?


Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$, where $1 \le n \le 50$, is $f(n)$ prime and further, the number $n$ can be expressed as the sum of two squares?


Starting with the prime number 3, add subsequent prime numbers in order: $3$, $3 + 5$, $3 + 5 + 7$, etc. How many of the first 15 such sums are also prime?


A deck of 60 cards, divided into 5 suits of 12 cards each, is shuffled. In how many ways can we pick three different cards in sequence? (Order matters, so picking card A, then card B, then card C is different from picking card B, then card A, then card C.)


The product of two positive integers plus their sum is 135. The integers are relatively prime, and each is less than 30. What is the sum of the two integers?


In rectangle $ABCD$, $AB = 8$ and $BC = 4$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF = 2$ and $GC = 3$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.


In a diagram, a smaller circle is inscribed within a larger circle. The gray area exposed around the smaller circle within the larger circle is equal to five times the area of the white circular region of the smaller circle. Determine the ratio of the radius of the small circle to the radius of the large circle.


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


Below is the graph of $y = a \sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ The graph completes exactly 5 full oscillations as $x$ varies from $0$ to $2\pi$. Find the value of $b$.

[asy]
import TrigMacros;

size(400);

real f(real x)
{
	return 2*sin(5*x + pi) + 1;
}

draw(graph(f,0,2*pi,n=700,join=operator ..),red);
trig_axes(0,2*pi,-3,4,pi/5,1);
layer();
rm_trig_labels(-4,4, 2);

label("$1$", (0,1), E);
label("$2$", (0,2), E);
label("$3$", (0,3), E);
label("$-1$", (0,-1), E);
label("$-2$", (0,-2), E);
label("$-3$", (0,-3), E);
[/asy]


In how many ways can the digits of $40,\!752$ be arranged to form a 5-digit number, where numbers cannot begin with 0?


Determine the area enclosed by the region defined by the equation \(x^2 + y^2 - 10x + 24y = -144\).


Determine the values of $x$ for which $x^2 - 34x + 225 \leq 9$.

Solution: \boxed{[17-\sqrt{73}, 17+\sqrt{73}]}


Let $\mathbf{a} = \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}.$ Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = 2(\mathbf{a} \times \mathbf{b}).$

Solution: \boxed{\begin{pmatrix} 7 \\ 2 \\ 2 \end{pmatrix}}


The first term of a sequence is $2157$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the ${2157}^{\text{th}}$ term of the sequence?


Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 160^\circ & -\sin 160^\circ \\ \sin 160^\circ & \cos 160^\circ \end{pmatrix}^n = \mathbf{I}.\]


Solution: \boxed{1 + i}


A circle is tangent to the lines $5x - 2y = 40$ and $5x - 2y = 10.$ The center of the circle lies on the line $3x - 4y = 0.$ Find the center of the circle.

Solution: \boxed{\left(\frac{50}{7}, \frac{75}{14}\right)}


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


When $q(x) = Dx^6 + Ex^4 + Fx^2 + 5$ is divided by $x - 2,$ the remainder is 17. Find the remainder when $q(x)$ is divided by $x + 2.$


Find the cross product of vectors $\begin{pmatrix} 3 \\ a \\ 4 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 2 \\ a+1 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} a^2 + a - 8 \\ -3a - 7 \\ 6 + a \end{pmatrix}}


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


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


The function $f$ maps nonnegative integers to real numbers, given that $f(1) = 4,$ and for all nonnegative integers $m \geq n,$
\[ f(m + n) + f(m - n) = \frac{f(2m) + f(2n)}{2}. \]
Find the sum of all possible values of $f(10)$.


Given triangle $ABC$ with an incircle centered at $D$. If $\angle BAC = 54^\circ$ and $\angle ABC = 72^\circ$, find the measure of $\angle BDC$.


The expression $x^2 + 14x + 45$ can be written as $(x + a)(x + b),$ and the expression $x^2 - 19x + 90$ can be written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?


What is the distance between the points $(3,7)$ and $(-5,3)$? Also, determine if the line segment between these points crosses the x-axis at $x = -1$.


Determine the remainder of the binary number $101101011010_2$ when divided by 8, and provide your answer in base 10.


Let $x_1, x_2, x_3$ be positive real numbers such that $x_1 + 3x_2 + 5x_3 = 100.$ Find the smallest possible value of $x_1^2 + x_2^2 + x_3^2.$


My friend Alex has a fascination with numbers that are divisible by 8. How many different last digits are possible in numbers that Alex likes?


Find the greatest common divisor of $8!$ and $(6!)^2$.


A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside this cylinder?


Determine the three real numbers $x$ that are excluded from the domain of the function given by $$g(x) = \frac{1}{2+\frac{1}{2+\frac{1}{x}}}.$$ What is the sum of those three numbers?


Two right triangles share a side as follows: Triangle $ABF$ and triangle $ABG$ are right triangles with $AB = BF = 8$ units and $AB = BG = 12$ units respectively. Point $H$ is the midpoint of $BF$. Calculate the area of $\triangle ABH$.


A polynomial of degree $17$ is divided by a polynomial \(d(x)\) to give a quotient of degree $10$ and a remainder of $5x^4+3x^3-2x+8$. What is \(\deg(d)\)?


What is the product of all real numbers that are tripled when added to their reciprocals?


The Greater Eighteen Hockey League has three divisions, with six teams in each division. Each team plays each of the other teams in its own division three times and every team in the other divisions twice. How many league games are scheduled?


If $5x \equiv 15 \pmod{25}$ and $2x \equiv 10 \pmod{25}$, what is the remainder when $x^2$ is divided by $25$?


A box contains 7 white balls and 8 black balls. Five balls are drawn out of the box at random. What is the probability that four of them are white?


An ellipse is tangent to the $x$-axis at $(1, 0)$ and tangent to the line $y = -4$, with its foci located at $(1, \sqrt{3} - 2)$ and $(1, -\sqrt{3}-2)$. Find the length of the major axis.


Suppose that $b$ is a positive integer greater than or equal to $2.$ When $256$ is converted to base $b$, the resulting representation should have $4$ digits. What is the number of possible values for $b$?


Regular octagon $ABCDE$ and regular decagon $AEFGHIJKLM$ are drawn on opposite sides of line segment $AE$ such that they are coplanar. What is the degree measure of exterior angle $DEF$?


Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 25 + |x_1 + x_2 + \dots + x_n|.\] What is the smallest possible value of $n$?


Our football team has increased to 12 members, of which 4 are strong enough to play offensive lineman. Additionally, only 2 are trained to play as kickers, while all other positions can be played by anyone. In how many ways can we choose a starting lineup consisting of a quarterback, a running back, an offensive lineman, a kicker, and a wide receiver?


In a diagram, there exists a larger circle and a smaller circle centered within it. The gray area visible between the circles is equal to twice the area of the smaller circle. Determine the ratio of the radius of the smaller circle to the radius of the larger circle and the fraction of the total area of the larger circle that is occupied by the smaller circle. Express your answers as common fractions.


Calculate $8! - 7 \cdot 7! + 2 \cdot 6!$.


What is the diameter of the circle inscribed in triangle $DEF$ if $DE = 10$, $DF = 5$, and $EF = 9$?


Find the value of the expression \[(2^{1500} + 5^{1501})^2 - (2^{1500} - 5^{1501})^2\] and express it as $k \cdot 10^{1500}$, where $k$ is a positive integer.


Find the sum of the squares of the solutions to the equation
\[
\left| x^2 - x + \frac{1}{2024} \right| = \frac{1}{2024}.
\]


A $120\times 150\times 375$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes, but 10% of these cubes are missing at random. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes that are not missing?


The equation of a parabola is $x^2 - 4x + 3y + 8 = 0.$ Find the vertex of the parabola.

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


Suppose $a$ and $b$ are nonzero real numbers, and the equation $x^2 + ax + b = 0$ has solutions $-2a$ and $b$. Find the ordered pair $(a, b)$ if it is known that one root is twice the negative of the other.

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


Suppose in a right triangle $LMN$ with a right angle at $M$, $\sin N = \frac{3}{5}$. If $LM = 18$, determine the length of $LN$.


The local basketball club saw a 15% increase in participation this year compared to last year. The number of male members increased by 10% and the number of female members increased by 25%. Last year, there were 30 male members. What fraction of the club's members are female this year? Express your answer as a common fraction.


When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 75. If one integer is 60, what is the smallest possible value of the other integer?


I had $\$45$ in allowance money and spent it as indicated in the pie graph shown. How many dollars did I spend on books?

[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(.707,.707);
B=(-.966,.259);
C=(-.707,-.707);
D=(.342,-.940);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(-.1,.5);
label("Games", W, N);
label("$\frac{2}{9}$", W, S);
X=(-.55, 0);
label("Books", X, S);
Y=(-.17,-.7);
label("Snacks", Y, N);
label("$\frac{1}{3}$", Y, S);
Z=(.5, -.15);
label("Toys", Z, N);
label("$\frac{1}{5}$", Z, S);
[/asy]

Solution: \boxed{11 \text{ dollars.}}


The greatest common divisor of 35 and some number between 90 and 100 is 7. What is the number?


A stock investment went up by 40% in 2006. Starting at this increased value, what percent would it have to go down in 2007 to return to its original price at the beginning of 2006?


Solution: \boxed{27^{x+1} = 27 \cdot 17^{3/2}}


What is the base-10 integer 1023 when expressed in base 7?


Lucy has $45$ dollars and $75$ cents. She wants to buy as many popsicles as she can with her money. Each popsicle normally costs $2$ dollars and $25$ cents each, but there is a discount of $5$ cents per popsicle if she buys more than 10 popsicles. How many popsicles can she buy maximum?


Consider the system of equations
\[
4x - 3y = a,
\]
\[
6y - 8x = b.
\]
If this system has a solution, find $\frac{a}{b}$, assuming $b \neq 0$.

Solution: \boxed{\frac{a}{b} = -\frac{1}{2}}


In an expanded professional bowling tournament, the top 6 bowlers have a playoff. First, #6 bowls #5. The loser receives the $6^\text{th}$ prize and the winner bowls #4. The loser of this game receives the $5^\text{th}$ prize, and the winner faces #3. The loser of this subsequent game gets the $4^\text{th}$ prize, and the winner bowls #2. The loser of this game gets the $3^\text{rd}$ prize, leaving the winner to bowl against #1. Finally, the winner of this last game gets the $1^\text{st}$ prize and the loser takes the $2^\text{nd}$ prize. In how many orders can bowlers #1 through #6 receive the prizes?


Let $x$ be a real number. Consider the following five statements:

1. $0 < x^3 < 1$
2. $x^3 > 1$
3. $-1 < x < 0$
4. $1 < x < 2$
5. $0 < 3x - x^3 < 2$

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


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


In an isosceles triangle, the vertex angle measures 100 degrees. How many obtuse interior angles does this triangle contain?


Let the first term of a geometric sequence be $\frac{1}{2}$, and let the second term be $30$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by one million?


In right triangle $DEF$, $DE=15$, $DF=9$, and $EF=12$ units. What is the length of the altitude from $F$ to $DE$?


Each letter of the alphabet is given a random integer value, where $Z = 15$. If $QUIZ$ scores 60 points, $QUIET$ scores 75 points, and $QUIT$ scores 50 points, what is the value of $E$?


Let $a$ and $b$ be real numbers such that $a^2 + b^2 \geq 1$. Find the maximum value of $a \sin \theta + b \cos \theta$, for $\theta \in [0, 2\pi]$.

Solution: \boxed{\sqrt{a^2 + b^2}}


Ms. Hamilton's eighth-grade class has expanded its annual basketball tournament format. Initially, there are twenty-four teams entered into a preliminary single-game knockout round where only 50% of the teams proceed to the main tournament. From then on, the tournament is a standard single-elimination format. How many total games will be played to determine the winner?


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


Harry and Sandy now stand at coordinates $(10, -5)$ and $(-4, 12)$ respectively. They wish to meet exactly halfway between them. At what coordinates will they meet?

Solution: \boxed{(3, 3.5)}


Let $0 \le a \le 1,$ $0 \le b \le 2,$ $0 \le c \le 1,$ and $0 \le d \le 1.$ Find the maximum value of
\[
\sqrt[4]{abcd} + \sqrt{(1 - a)(1 - b)(1 - c)(1 - d)}.
\]


How many three-digit whole numbers have no 5's and no 8's as digits?


The quadratic equation $ax^2 + 30x + c = 0$ has exactly one solution. If $a+c = 35$, and $a < c$, find the ordered pair $(a, c)$.

Solution: \boxed{\left(\frac{35 - 5\sqrt{13}}{2}, \frac{35 + 5\sqrt{13}}{2}\right)}


Determine the area of the triangle with vertices at \(A = (1, 2)\), \(B = (-1, -1)\), and \(C = (4, -3)\). Additionally, verify if this triangle is a right triangle.


A zoo has five pairs of different animals - each pair consisting of one male and one female. If the zookeeper decides to start feeding the male lion, and must finish feeding with the female bear, how many ways can he feed all animals ensuring that he alternates between genders each time?


The average of the numbers $1, 2, 3,\dots, 50,$ and $x$ is $51x$. What is $x$?


Suppose $a, b, c$ are integers taken from the set of positive integers less than $7$ such that:
\begin{align*}
abc &\equiv 2 \pmod{7}, \\
3c &\equiv 1 \pmod{7}, \\
4b &\equiv 2 + b \pmod{7}.
\end{align*}
What is the remainder when $a+b+c$ is divided by $7$?


There are \(15n\) members in the Saxbridge parade troupe, and when they line up in rows of 24, there are 3 members left over. If \(n\) is an integer and there are fewer than 1200 members in the troupe, what is the maximum number of people that could be in the Saxbridge parade troupe?


How many odd integers between 2000 and 3000 have three different digits, where the units digit must be odd?


Solution: \boxed{-1 + \frac{1}{2^{2252}}}


Calculate \(7 \cdot 6! + 6 \cdot 5! + 2 \cdot 5!\).


A street has parallel curbs 60 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 25 feet and each stripe is 70 feet long. Find the distance, in feet, between the stripes.

Solution: \boxed{\frac{150}{7} \text{ feet}}


Point $C(0,p)$ lies on the $y$-axis. Point $D(10, 10)$ and $O(0,0)$ are also given in the coordinate plane. Determine the expression for the area of $\triangle COD$ in terms of $p$. Simplify your answer as much as possible.


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


If I roll 6 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number?


Consider a circle whose equation in a shifted coordinate system where origin is at $(1,-2)$ is given by $(x-1)^2 + (y+2)^2 = 4x + 12y + 6$. Determine the value of $h+k$ where $(h,k)$ is the center of the circle in the standard coordinate system.

Modified


Calculate the value of the expression $[2a-(b-2c)] - [(2a-b) - 2c] + 3(a - c)$ when $a = 8$, $b = 10$, and $c = 3$.


Brand X soda now advertises, "We offer 25% more soda than Brand Y and, if you purchase three cans, the total price is 15% less than three cans of Brand Y!" What is the ratio of the unit price of Brand X soda to the unit price of Brand Y soda when purchasing three cans of each? Express your answer as a common fraction.


A circle with center $C$ is tangent to the positive $x$-axis and positive $y$-axis and is externally tangent to the circle centered at $(5,0)$ with a radius of 3. Find the sum of all possible radii of the circle with center $C$.


A sector is removed from a circular piece of paper with a radius of 6 inches, resulting in a $240^\circ$ sector. This sector is then transformed into a right circular cone by bringing the cut edges together. Determine the circumference of the base of the cone. Provide your answer in terms of $\pi$.


Determine the number of functions \( f : \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \),
\[ f(x + f(y)) = x + y + k \]
where \( k \) is a real constant.


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


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

Solution: \boxed{x^2 + 1}


Let $Q$ be a point on line segment $\overline{CD}$ such that $CQ:QD = 4:1$ and the magnitude of $\overrightarrow{D}$ is twice the magnitude of $\overrightarrow{C}$. Then express $\overrightarrow{Q}$ as a linear combination of $\overrightarrow{C}$ and $\overrightarrow{D}$ using constants $m$ and $n$. Enter the ordered pair $(m,n)$.

Solution: \boxed{\left(\frac{1}{5}, \frac{4}{5}\right)}


Let $\triangle ABC$ be an isosceles triangle such that $BC = 40$ and $AB = AC$. The incenter $I$ of $\triangle ABC$ satisfies $IC = 24$. Determine the length of the inradius of $\triangle ABC$.


When four positive integers are divided by $53$, the remainders are $31$, $45$, $17$, and $6$, respectively.

When the sum of the four integers is divided by $53$, what is the remainder?


In right triangle $XYZ$ where $\angle X = 90^\circ$, the lengths of sides $XY = 8$ and $YZ = 17$. Find $\cos Z$.


Find the least odd prime factor of $2047^4 + 1$.