Let $ABC$ be a triangle with $BC=a$ $AC=b$ $AB=c$. For each line $\Delta$ we denote $d_{A}, d_{B}, d_{C}$ the distances from $A,B, C$ to $\Delta$ and we consider the expresion $E(\Delta)=ad_{A}^{2}+bd_{B}^{2}+cd_{C}^{2}$. Prove that if $E(\Delta)$ is minimum, then $\Delta$ passes through the incenter of $\Delta ABC$.

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and \[a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}\] for each $n \in \mathbb{N}$.

Let $K_1, K_2, K_3, K_4, K_5$ be 5 distinguishable keys, and let $D_1, D_2, D_3, D_4, D_5$ be $5$ distinguishable doors. For $1 \leq i \leq 5$, key $K_i$ opens doors $D_{i}$ and $D_{i+1}$ (where $D_6 = D_1$) and can only be used once. The keys and doors are placed in some order along a hallway. Key\$ha walks into the hallway, picks a key and opens a door with it, such that she never obtains a key before all the doors in front of it are unlocked. In how many orders can the keys and doors be placed such that Key\$ha can open all of the doors? [i]Author: Mitchell Lee[/i] [hide="Clarifications"] [list=1][*]The doors and keys are in series. In other words, the doors aren't lined up along the side of the hallway. They are blocking Key\$ha's path to the end, and the only way she can get past them is by getting the appropriate keys along the hallway. [*]The doors and keys appear consecutively along the hallway. For example, she might find $K_1 D_1 K_2 D_2 K_3 D_3 K_4 D_4 K_5 D_5$ down the hallway in that order. Also, by "she never obtains a key before all the doors in front of it are unlocked," we mean that she cannot obtain a key before all the doors appearing before the key are unlocked. In essence, it merely states that locked doors cannot be passed. [*]The doors and keys do not need to alternate down the hallway.[/list][/hide]

Let $ABC$ be an acute-angled triangle. Suppose that the altitude of $\vartriangle ABC$ at $B$ intersects the circle with diameter $AC$ at $P$ and $Q$, and the altitude at $C$ intersects the circle with diameter $AB$ at $M$ and $N$. Prove that $P, Q, M$ and $N$ lie on a circle.

Let $n$ be a positive integer and let $p, q>n$ be odd primes. Prove that the positive integers $1, 2, \ldots, n$ can be colored in $2$ colors, such that for any $x \neq y$ of the same color, $xy-1$ is not divisible by $p$ and $q$.

Given an ordered list of $3N$ real numbers, we can trim it to form a list of $N$ numbers as follows: We divide the list into $N$ groups of $3$ consecutive numbers, and within each group, discard the highest and lowest numbers, keeping only the median. \\ Consider generating a random number $X$ by the following procedure: Start with a list of $3^{2021}$ numbers, drawn independently and unfiformly at random between $0$ and $1$. Then trim this list as defined above, leaving a list of $3^{2020}$ numbers. Then trim again repeatedly until just one number remains; let $X$ be this number. Let $\mu$ be the expected value of $\left|X-\frac{1}{2} \right|$. Show that \[ \mu \ge \frac{1}{4}\left(\frac{2}{3} \right)^{2021}. \]

Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$, the decimal representation of the number $c^n+2014$ has digits all less than $5$. [i]Proposed by Evan Chen[/i]

Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.

Compute \[\sum_{i=0}^{\frac{q-1}{2}}\left\lfloor\frac{ip}{q}\right\rfloor+\sum_{j=0}^{\frac{p-1}{2}}\left\lfloor\frac{jq}{p}\right\rfloor\] if $p=51$ and $q=81$. [i]2018 CCA Math Bonanza Team Round #7[/i]

Let $f(n)=\dfrac{x_1+x_2+...+x_n}{n}$, where $n$ is a positive integer. If $x_k=(-1)^k,k=1,2,...,n$, the set of possible values of $f(n)$ is: $\textbf{(A)}\ \{0\} \qquad \textbf{(B)}\ \{\dfrac{1}{n}\} \qquad \textbf{(C)}\ \{0,-\dfrac{1}{n}\} \qquad \textbf{(D)}\ \{0,\dfrac{1}{n}\} \qquad \textbf{(E)}\ \{1,\dfrac{1}{n}\}$

Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle ABC$? [asy] for(int i = 0; i < 6; ++i){ for(int j = 0; j < 6; ++j){ draw(sqrt(3)*dir(60*i+30)+dir(60*j)--sqrt(3)*dir(60*i+30)+dir(60*j+60)); } } draw(2*dir(60)--2*dir(180)--2*dir(300)--cycle); label("A",2*dir(180),dir(180)); label("B",2*dir(60),dir(60)); label("C",2*dir(300),dir(300)); [/asy] $ \textbf {(A) } 2\sqrt{3} \qquad \textbf {(B) } 3\sqrt{3} \qquad \textbf {(C) } 1+3\sqrt{2} \qquad \textbf {(D) } 2+2\sqrt{3} \qquad \textbf {(E) } 3+2\sqrt{3} $

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

Find the largest real number $ C_1 $ and the smallest real number $ C_2 $, such that, for all reals $ a,b,c,d,e $, we have \[ C_1 < \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a} < C_2 \]

Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$. Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle. [color=blue]Should the first sentence read: Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ [u]with respect to[/u] $M$. ? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.[/color]

Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

Investigate the behaviour as $t\to+\infty$ of solutions of the systems \[\begin{cases} \dot{x}=y\\ \dot{y}=2\sin y-y-x\end{cases}\] \[\begin{cases} \dot{x}=y\\ \dot{y}=2x-x^{3}-x^{2}-\epsilon y\end{cases}\] where $\epsilon\ll 1$.

David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? $\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$

Find the real numbers $x$ and $y$ such that $$(x^2 -x +1)(3y^2-2y + 3) -2=0.$$

Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex? $ \textbf{(A)}\ \frac {5}{256} \qquad \textbf{(B)}\ \frac {21}{1024} \qquad \textbf{(C)}\ \frac {11}{512} \qquad \textbf{(D)}\ \frac {23}{1024} \qquad \textbf{(E)}\ \frac {3}{128}$

Let $a$ and $b$ be two positive integers and $A$ be a subset of $\{1, 2,…, a + b \}$ that has more than $ \frac {a + b} {2}$ elements. Show that there are two numbers in $A$ whose difference is $a$ or $b$.

In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$AA,\, AB ,\, AC, \, BB , \, BC ,\, AD , \, CC, \, BD, \, CD, \, DD$$ in which $AA$ means two seniors, $AB$ means a senior and a junior, etc. Determine numerical values to assign to $A,B,C$ and $D$ so that the set of numbers $A+A, A+B, A+C, B+B, \ldots $ corresponding to the order above will be in descending order. Find the general solution and the solution in least positive integers.

Let $ a$, $ b$, $ c$ are sides of a triangle. Find the least possible value $ k$ such that the following inequality always holds: $ \left|\frac{a\minus{}b}{a\plus{}b}\plus{}\frac{b\minus{}c}{b\plus{}c}\plus{}\frac{c\minus{}a}{c\plus{}a}\right|<k$ [i](Vitaly Lishunov)[/i]

Logan is constructing a scaled model of his town. The city's water tower stands $ 40$ meters high, and the top portion is a sphere that holds $ 100,000$ liters of water. Logan's miniature water tower holds $ 0.1$ liters. How tall, in meters, should Logan make his tower? $ \textbf{(A)}\ 0.04\qquad \textbf{(B)}\ \frac{0.4}{\pi}\qquad \textbf{(C)}\ 0.4\qquad \textbf{(D)}\ \frac{4}{\pi}\qquad \textbf{(E)}\ 4$

$8712$ is an integral multiple of its reversal, $2178$, as $8712=4 * 2178$. Find another $4$-digit number which is a non-trivial integral multiple of its reversal.