([b]4[/b]) Let $ a$, $ b$ be constants such that $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$. Determine the pair $ (a,b)$.

[u]Super Mario 64![/u] Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with $4$ doors. This room is the first in a sequence of $2$ indistinugishable rooms. In each room, $1$ door leads to the next room in the sequence (or, for the second room, into Bowser's level), while the other $3$ doors lead to the first room. [b]p9.[/b] Suppose that in every room, Mario randomly picks a door to walk through. What is the expected number of doors (not including Mario's initial entrance to the first room) through which Mario will pass before he reaches Bowser's level? [b]p10.[/b] Suppose that instead there are $6$ rooms with $4$ doors. In each room, $1$ door leads to the next room in the sequence (or, for the last room, Bowser's level), while the other $3$ doors lead to the first room. Now what is the expected number of doors through which Mario will pass before he reaches Bowser's level? [b]p11.[/b] In general, if there are $d$ doors in every room (but still only $1$ correct door) and $r$ rooms, the last of which leads into Bowser's level, what is the expected number of doors through which Mario will pass before he reaches Bowser's level?

Let $ABC$ be a triangle with circumscribed and inscribed circles $(O)$ and $(I)$ respectively. $AA'$,$BB'$,$CC'$ are the bisectors of triangle $ABC$. Prove that $OI$ passes through the the isogonal conjugate of point $I$ with respect to triangle $A'B'C'$.

For each non-negative integer $n$, $F_n(x)$ is a polynomial in $x$ of degree $n$. Prove that if the identity \[F_n(2x)=\sum_{r=0}^{n} (-1)^{n-r} \binom nr 2^r F_r(x)\] holds for each n, then \[F_n(tx)=\sum_{r=0}^{n} \binom nr t^r (1-t)^{n-r} F_r(x)\]

In the rectangle $ABCD$, the ratio of the lengths of sides $BC$ and $AB$ is equal to $\sqrt{2}$. Point $X$ is marked inside this rectangle so that $AB=BX=XD$. Determine the measure of angle $BXD$.

$10$ rectangles have their vertices lie on a circle. The vertices divide the circle into $40$ equal arcs. Prove that two of the rectangles are congruent. [i]Proposed by Evan Chang (squareman), USA[/i]

Let $ABC$ be an arbitrary triangle. On the perpendicular bisector of $AB$, there is a point $P$ inside of triangle $ABC$. On the sides $BC$ and $CA$, triangles $BQC$ and $CRA$ are placed externally. These triangles satisfy $\vartriangle BPA \sim \vartriangle BQC \sim \vartriangle CRA$. (So $Q$ and $A$ lie on opposite sides of $BC$, and $R$ and $B$ lie on opposite sides of $AC$.) Show that the points $P, Q, C$ and $R$ form a parallelogram.

Find the sum of all two-digit prime numbers whose digits are also both prime numbers. [i]Proposed by Nathan Xiong[/i]

A segment $d$ is said to divide a segment $s$ if there is a natural number $n$ such that $s = nd = d+d+ ...+d$ ($n$ times). (a) Prove that if a segment $d$ divides segments $s$ and $s'$ with $s < s'$, then it also divides their difference $s'-s$. (b) Prove that no segment divides the side $s$ and the diagonal $s'$ of a regular pentagon (consider the pentagon formed by the diagonals of the given pentagon without explicitly computing the ratios).

Find the period of small oscillations of a water pogo, which is a stick of mass m in the shape of a box (a rectangular parallelepiped.) The stick has a length $L$, a width $w$ and a height $h$ and is bobbing up and down in water of density $\rho$ . Assume that the water pogo is oriented such that the length $L$ and width $w$ are horizontal at all times. Hint: The buoyant force on an object is given by $F_{buoy} = \rho Vg$, where $V$ is the volume of the medium displaced by the object and $\rho$ is the density of the medium. Assume that at equilibrium, the pogo is floating. $ \textbf{(A)}\ 2\pi \sqrt{\frac{L}{g}} $ $ \textbf{(B)}\ \pi \sqrt{\frac{\rho w^2L^2 g}{mh^2}} $ $ \textbf{(C)}\ 2\pi \sqrt{\frac{mh^2}{\rho L^2w^2 g}} $ $\textbf{(D)}\ 2\pi \sqrt{\frac{m}{\rho wLg}}$ $\textbf{(E)}\ \pi \sqrt{\frac{m}{\rho wLg}}$

Let $ABC$ be an acute triangle with $AB < AC$, let $\Gamma$ be its circumcircle and let $D$ be the foot of the altitude from $A$ to $BC$. Take a point $E$ on the segment $BC$ such that $CE=BD$. Let $P$ be the point on $\Gamma$ diametrically opposite to vertex $A$. Prove that $PE$ is perpendicular to $BC$.

Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$

A nondegenerate rhombus has side length $l$, and its area is twice that of its inscribed circle. Find the radius of the inscribed circle.

A cone with semivertical angle $30^{\circ}$ is half filled with water. What is the angle it must be tilted by so that water starts spilling?

Prove that, for any real numbers $a_1, a_2, \dots , a_n$, \[ \sum_{i,j=1}^n \frac{a_ia_j}{i+j-1}\ge 0.\]

Find all real functions $f$ defined on the real numbers except zero, satisfying $f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$

Walter rolls four standard six-sided dice and finds that the product of the numbers on the upper face is 144. Which of the following could NOT be on the sum of the upper four faces? $ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 18$

A [i]permutation[/i] of a finite set $S$ is a one-to-one function from $S$ to $S$. Given a permutation $f$ of the set $\{ 1, 2, \ldots, 100 \}$, define the [i]displacement[/i] of $f$ to be the sum \[ \sum_{i = 1}^{100} \left\lvert f(i) - i \right\rvert . \] How many permutations of $\{ 1, 2, \ldots, 100 \}$ have displacement 4?

A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.

Let $p$ be an odd prime. Determine the positive integers $x$ and $y$ with $x\leq y$ for which the number $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is non-negative and as small as possible.

Let $S$ be the set of fractions of the form $\frac{{\text {lcm}}(A, B)}{A+B}$, where $A$ and $B$ are positive integers and ${\text{lcm}}(A, B)$ is the least common multiple of $A$ and $B$. What is the smallest number exceeding 3 in $S$?

Let $a,b,c$ be three positive real numbers with $a+b+c=3$. Prove that \[ (3-2a)(3-2b)(3-2c) \leq a^2b^2c^2 . \] [i]Robert Szasz[/i]

What quadratic polynomial whose coefficient of $x^2$ is $1$ has roots which are the complex conjugates of the solutions of $x^2 -6x+ 11 = 2xi-10i$? (Note that the complex conjugate of $a+bi$ is $a-bi$, where a and b are real numbers.)

The cells of an $n\times n$ table ($n\ge 3$) are painted black and white in the chess-like manner. Per move one can choose any $2\times 2$ square and reverse the color of the cells inside it. Find all $n$ for which one can obtain a table with all cells of the same color after finitely many such moves.

Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles? [asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw(Circle((0,0),1)); draw(Circle((0,3),2)); draw(Circle((5,3),3)); label("A",(0.2,0),W); label("B",(0.2,2.8),NW); label("C",(4.8,2.8),NE); label("D",(5,0),SE); label("5",(2.5,0),N); label("3",(5,1.5),E); [/asy] $\textbf{(A) }3.5\qquad\textbf{(B) }4.0\qquad\textbf{(C) }4.5\qquad\textbf{(D) }5.0\qquad \textbf{(E) }5.5$