$4^4 \cdot 9^4 \cdot 4^9 \cdot 9^9=$ $ \textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\textbf{(D)}\ 36^{36} \qquad\textbf{(E)}\ 1296^{26} $

Determine all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the set \[\left \{ \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}\] is finite, and for all $x \in \mathbb{R}$ \[f(x-1-f(x)) = f(x) - x - 1\]

A spinning turntable is rotating in a vertical plane with period $ 500 \, \text{ms} $. It has diameter 2 feet carries a ping-pong ball at the edge of its circumference. The ball is bolted on to the turntable but is released from its clutch at a moment in time when the ball makes a subtended angle of $\theta>0$ with the respect to the horizontal axis that crosses the center. This is illustrated in the figure. The ball flies up in the air, making a parabola and, when it comes back down, it does not hit the turntable. This can happen only if $\theta>\theta_m$. Find $\theta_m$, rounded to the nearest integer degree? [asy] filldraw(circle((0,0),1),gray(0.7)); draw((0,0)--(0.81915, 0.57358)); dot((0.81915, 0.57358)); draw((0.81915, 0.57358)--(0.475006, 1.06507)); arrow((0.417649,1.14698), dir(305), 12); draw((0,0)--(1,0),dashed); label("$\theta$", (0.2, 0.2/3), fontsize(8)); label("$r$", (0.409575,0.28679), NW, fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

Adam has a box with $15$ pool balls in it, numbered from $1$ to $15$, and picks out $5$ of them. He then sorts them in increasing order, takes the four differences between each pair of adjacent balls, and finds exactly two of these differences are equal to $1$. How many selections of $5$ balls could he have drawn from the box? [i]Proposed by Adam Bertelli[/i]

Let $ABCD$ be a square face of a cube with edge length $2$. A plane $P$ that contains $A$ and the midpoint of $\overline{BC}$ splits the cube into two pieces of the same volume. What is the square of the area of the intersection of $P$ and the cube?

What is the number of ways you can choose two distinct integers $a$ and $b$ (unordered i.e. choosing $(a, b)$ is same as choosing $(b, a)$ from the set $\{1, 2, \cdots , 100\}$ such that difference between them is atmost 10, i.e. $|a-b|\leq 10$ [list=1] [*] ${{100}\choose{2}} -{{90}\choose{2}}$ [*] ${{100}\choose{2}} -90$ [*] ${{100}\choose{2}} -{{90}\choose{2}}-100$ [*] None of the above [/list]

For $n \geq 1$, let $a_n$ be the number beginning with $n$ $9$'s followed by $744$; eg., $a_4=9999744$. Define $$f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \}$$, for $n\geq 1$. Find $f(1)+f(2)+f(3)+ \cdots + f(10)$.

Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.

In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?

Let $n$ be a natural number divisible by $3$. We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ($m > 1$), the number of black squares is not more than white squares. Find the maximum number of black squares.

Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions: (i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and (ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ? [i]Proposed by Igor I. Voronovich, Belarus[/i]

Let $E$ be the midpoint of side $[AB]$ of square $ABCD$. Let the circle through $B$ with center $A$ and segment $[EC]$ meet at $F$. What is $|EF|/|FC|$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ \dfrac{3}{2} \qquad\textbf{(C)}\ \sqrt{5}-1 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \sqrt{3} $

Let $ABC$ be an acute triangle with $AB < AC$. The points $A_1$ and $A_2$ are located on the line $BC$ so that $AA_1$ and $AA_2$ are the inner and outer angle bisectors at $A$ for the triangle $ABC$. Let $A_3$ be the mirror image $A_2$ with respect to $C$, and let $Q$ be a point on $AA_1$ such that $\angle A_1QA_3 = 90^o$. Show that $QC // AB$.

Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$. How many times does the digit $1$ occur in its decimal representation?

A PUMaC grader is grading the submissions of forty students $s_1, s_2, ..., s_{40}$ for the individual finals round, which has three problems. After grading a problem of student $s_i$, the grader either: $\bullet$ grades another problem of the same student, or $\bullet$ grades the same problem of the student $s_{i-1}$ or $s_{i+1}$ (if $i > 1$ and $i < 40$, respectively). He grades each problem exactly once, starting with the first problem of $s_1$ and ending with the third problem of $s_{40}$. Let $N$ be the number of different orders the grader may grade the students’ problems in this way. Find the remainder when $N$ is divided by $100$.

Let the incircle of $\triangle ABC$ meet $BC,CA,AB$ at $J,K,L$. Let $D(\ne B, J),E(\ne C,K), F(\ne A,L)$ be points on $BJ,CK,AL$. If the incenter of $\triangle ABC$ is the circumcenter of $\triangle DEF$ and $\angle BAC = \angle DEF$, prove that $\triangle ABC$ and $\triangle DEF$ are isosceles triangles.