What is the least possible value ofwhere  is a real number?

The numbers $1$, $2$, $\cdots$, $9$ are randomly placed into the $9$ squares of a $3\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?

A sphere with center $O$ has radius $6$. A triangle with sides of length $15$, $15$, and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?

Real numbers between $0$ and $1$, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is $0$ if the second flip is heads and $1$ if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,\ 1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $\mid x-y\mid > \frac{1}{2}$?

Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number , then Todd must say the next two numbers ( and ), then Tucker must say the next three numbers (, , ), then Tadd must say the next four numbers (, , , ), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number  is reached. What is the th number said by Tadd?

Let , , and  be the distinct roots of the polynomial . It is given that there exist real numbers , , and  such thatfor all . What is ?

For how many integers $n$ between $1$ and $50$, inclusive, is $$\frac{(n^2-1)!}{(n!)^n}$$

The area of a pizza with radius $4$ is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$?

Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$?

Positive real numbers $x\ne 1$ and $y\ne 1$ satisfy $\log_2x=\log_y16$ and $xy=64$. What is $\left(\log_2\frac{x}{y}\right)^2$?

How many ways are there to paint each of the integers $2$, $3$, $\cdots$, $9$ either red, green, or blue so that each number has a different color from each of its proper divisors?

For a certain complex number $c$, the polynomial $$P(x)=(x^2-2x+2)(x^2-cx+4)(x^2-4x+8)$$

has exactly $4$ distinct roots. What is $\mid c\mid$?

Positive real numbers $a$ and $b$ have the property that $$\sqrt{\log a}+\sqrt{\log b} +\log\sqrt{a} + \log\sqrt{b}=100$$

and all four terms on the left are positive integers, where $\log$ denotes the base-$10$ logarithm. What is $ab$?

Let $s_k$ denote the sum of the $k^{th}$ powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1}=as_k+bs_{k-1}+ck_{k-2}$ for $k=2$, $3$, $\cdots$. What is $a+b+c$?

In $\triangle{ABC}$ with integer side lengths, $$\cos{A}=\frac{11}{16},\qquad\cos{B}=\frac{7}{8},\qquad\text{and}\qquad\cos{C}=-\frac{1}{4}$$

What is the least possible perimeter for $\triangle{ABC}$?

Let $$z=\frac{1+i}{\sqrt{2}}$$

What is $$\left(z^{1^2}+z^{2^2}+z^{3^2}+\cdots+z^{12^2}\right)\left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\cdots+\frac{1}{z^{12^2}}\right)$$

Circles $\omega$ and $\gamma$, both centered at $O$, have radii $20$ and $17$, respectively. Equilateral triangle $ABC$, whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$, has vertex $A$ on $\omega$, and the line containing side $\overline{BC}$ is tangent to $\gamma$, Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$, and $\frac{BP}{CP}=3$. Find the length of  $AB$.

Define binary operations $\diamondsuit$ and $\heartsuit$ by $$a\diamondsuit b=a^{\log_7(b)}\qquad\text{and}\qquad a\heartsuit b=a^{\frac{1}{\log_7(b)}}$$

for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3=3\heartsuit 2$ and $$a_n=(n\heartsuit (n-1))\diamondsuit a_{n-1}$$

for all integers $n\ge 4$. To the nearest integer, what is $\log_7(a_{2019})$?

Let $\triangle{A_0B_0C_0}$ be a triangle whose angle measures are exactly $59.999^{\circ}$, $60^{\circ}$, and $60.001^{\circ}$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$& to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle{A_nB_nC_n}$ is obtuse?

Alicia had two containers. The first was $\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement.

$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

In a high school wit $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?

All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?

$\textbf{(A) } (-1,2) \qquad\textbf{(B) } (0,1) \qquad\textbf{(C) } (1,-2) \qquad\textbf{(D) } (1,0) \qquad\textbf{(E) } (1,2)$

Triangle $ABC$ lies in the first quadrant. Points& $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is not always true?

$\textbf{(A) }$ Triangle $A'B'C'$" lies in the first quadrant.

$\textbf{(B) }$ Triangles $ABC$ and $A'B'C'$ have the same area.

$\textbf{(C) }$ The slope of line $AA'$ is $-1$.

$\textbf{(D) }$ The slopes of lines $AA'$ and $CC'$ are the same.

$\textbf{(E) }$ Lines $AB$ and $A'B'$ are perpendicular to each other.

There is a real $n$ such that $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$?