Your answer to this problem will be an integer between $0$ and $100$, inclusive. From all the teams who submitted an answer to this problem, let the average answer be $A$. Estimate the value of $\left\lfloor \frac23 A \right\rfloor$. If your estimate is $E$ and the answer is $A$, your score for this problem will be \[\max\left(0,\lfloor15-2\cdot\left|A-E\right|\right \rfloor).\] [i]Proposed by Andrew Zhao[/i]

Let $ f: Z \to Z$ be such that $ f(1) \equal{} 1, f(2) \equal{} 20, f(\minus{}4) \equal{} \minus{}4$ and $ f(x\plus{}y) \equal{} f(x) \plus{}f(y)\plus{}axy(x\plus{}y)\plus{}bxy\plus{}c(x\plus{}y)\plus{}4 \forall x,y \in Z$, where $ a,b,c$ are constants. (a) Find a formula for $ f(x)$, where $ x$ is any integer. (b) If $ f(x) \geq mx^2\plus{}(5m\plus{}1)x\plus{}4m$ for all non-negative integers $ x$, find the greatest possible value of $ m$.

There are $64$ towns in a country and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected or not. Our aim is to determine whether it is possible to travel from any town to any other by a sequence of roads. Prove that there is no algorithm which enables us to do so in less than $2016$ questions. (Proposed by Konstantin Knop)

In this problem, we consider integers consisting of $5$ digits, of which the rst and last one are nonzero. We say that such an integer is a palindromic product if it satises the following two conditions: - the integer is a palindrome, (i.e. it doesn't matter if you read it from left to right, or the other way around); - the integer is a product of two positive integers, of which the first, when read from left to right, is equal to the second, when read from right to left, like $4831$ and $1384$. For example, $20502$ is a palindromic product, since $102 \cdot 201 = 20502$, and $20502$ itself is a palindrome. Determine all palindromic products of $5$ digits.

$PQR$ is an acute scalene triangle. The altitude $PL$ and the bisector $RK$ of $\angle QRP$ meet at $H$ ($L$ on $QR$ and $K$ on $PQ$). $KM$ is the altitude of triangle $PKR$; it meets $PL$ at $N$. The circumcircle of $\triangle NKR$ meets $QR$ at $S$ other than $Q$. Prove that $SHK$ is an isosceles triangle.

Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence \[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \] forms an arithmetic progression. Prove that the terms of the sequence are equal.

For each $a$, $1<a<2$, the graphs of functions $y=1-|x-1|$ and $y=|2x-a|$ determine a figure. Prove that the area of this figure is less than $\frac13$.

A dart board looks like three concentric circles with radii of 4, 6, and 8. Three darts are thrown at the board so that they stick at three random locations on then board. The probability that one dart sticks in each of the three regions of the dart board is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.

Let $n\geq 2$ and $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $$\{\text{rank}(A^k)\mid k\geq 1\}=\{\text{rank}(B^k)\mid k\geq 1\}.$$ Prove that $\text{rank}(A^k)=\text{rank}(B^k)$ for all $k\geq 1$. [i]Cristi Săvescu[/i]

([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 $a_1, a_2, \dots , a_n$, \[ \sum_{i,j=1}^n \frac{a_ia_j}{i+j-1}\ge 0.\]