Consider $n\geq 3$ points in the plane, no three of which are collinear. For every convex polygon with vertices among the $n$ points, place $k\cdot 2^k$ coins in every one of its vertices, where $k$ is the number of points strictly in the interior of the polygon. Show that in total, no matter the configuration of the $n$ points, there are at most $n(n+1)\cdot 2^{n-3}$ placed coins. [i]Cristi Săvescu[/i]

For any $ n\ge 2 $ natural, show that the following inequality holds: $$ \sum_{i=2}^n\frac{1}{\sqrt[i]{(2i)!}}\ge\frac{n-1}{2n+2} . $$

On the sides $AB, AC ,BC$ of the triangle $ABC$, the points $M, N, K$ are selected, respectively, such that $AM = AN$ and $BM = BK$. The circle circumscribed around the triangle $MNK$ intersects the segments $AB$ and $BC$ for the second time at points $P$ and $Q$, respectively. Lines $MN$ and $PQ$ intersect at point $T$. Prove that the line $CT$ bisects the segment $MP$.

The $m \times n$ chessboard is colored by black and white. In one step, two neighbouring squares are selected (squares with a common side) and their color changes according to the follwing way: - white becomes black, - black become red, - Red becomes white. For which $m$ and $n$, these steps can change the colors of all the initial squares from white to black and from black to white?

Let $S$ be the set of all n-tuples $(X_1,...,X_n)$ of subsets of the set $\{1,2,..,1000\}$, not necessarily different and not necessarily nonempty. For $a = (X_1,...,X_n)$ denote by $E(a)$ the number of elements of $X_1\cup ... \cup X_n$. Find an explicit formula for the sum $\sum_{a\in S} E(a)$

Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that \[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\] where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$

Given four spheres with their radii equal to $2,2,3,3$ respectively, each sphere externally touches the other spheres. Suppose that there is another sphere that is externally tangent to all those four spheres, determine the radius of this sphere.

Gary plays the following game with a fair $n$-sided die whose faces are labeled with the positive integers between $1$ and $n$, inclusive: if $n = 1$, he stops; otherwise he rolls the die, and starts over with a $k$-sided die, where $k$ is the number his $n$-sided die lands on. (In particular, if he gets $k = 1$, he will stop rolling the die.) If he starts out with a $6$-sided die, what is the expected number of rolls he makes?

Square $ABCD$ has side length $1$; circle $\Gamma$ is centered at $A$ with radius $1$. Let $M$ be the midpoint of $BC$, and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\Gamma$. Compute $MN$. [i]2018 CCA Math Bonanza Individual Round #11[/i]

Let $\triangle{ABC}$ be an equilateral triangle. Point $D$ is on segment $\overline{BC}$ such that $BD=1$ and $DC=4.$ Points $E$ and $F$ lie on rays $\overrightarrow{AC}$ and $\overrightarrow{AB},$ respectively, such that $D$ is the midpoint of $\overline{EF}.$ Compute $EF.$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(xy) = yf(x) + x + f(f(y) - f(x)) \] for all $x,y \in \mathbb{R}$.

Let $\omega=\cos\frac{\pi}{5}+\text{i}\sin\frac{\pi}{5}$, which equation has roots $\omega,\omega^3,\omega^7,\omega^9$? $\text{(A)}x^4+x^3+x^2+x+1=0\qquad\text{(B)}x^4-x^3+x^2-x+1=0$ $\text{(C)}x^4-x^3-x^2+x+1=0\qquad\text{(D)}x^4+x^3+x^2-x+1=0$

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\tfrac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of his points against the other nine of the ten). What was the total number of players in the tournament?

What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?

A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$. Find $D$. [i]Proposed by Levi Iszler[/i]

For $a,b,c$ positive reals find the minimum value of \[ \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}. \]

In concurrent computing, two processes may have their steps interwoven in an unknown order, as long as the steps of each process occur in order. Consider the following two processes: \begin{tabular}{c|cc} Process & $A$ & $B$\\ \hline Step 1 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 2 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$\\ Step 3 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 4 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$ \end{tabular} One such interweaving is $A1$, $B1$, $A2$, $B2$, $A3$, $B3$, $B4$, $A4$, but $A1$, $A3$, $A2$, $A4$, $B1$, $B2$, $B3$, $B4$ is not since the steps of $A$ do not occur in order. We run $A$ and $B$ concurrently with $x$ initially valued at $6$. Find the minimal possible value of $x$ among all interweavings.

Let $n>1$ be a positive integer. Find the number of binary strings $(a_1, a_2, \ldots, a_n)$, such that the number of indices $1\leq i \leq n-1$ such that $a_i=a_{i+1}=0$ is equal to the number of indices $1 \leq i \leq n-1$, such that $a_i=a_{i+1}=1$.

Let $x$ and $y$ be integers from $-10$ to $10$, inclusive, with $xy \ne1$. Compute the number of ordered pairs $(x, y) $ such that $$\left| \frac{x + y}{1 - xy} \right|\le 1.$$

Let $ \triangle ABC $ be an acute triangle with circumcenter $ O $ and centroid $ G .$ Let $ D $ be the midpoint of $ BC $ and $ E\in \odot (BC) $ be a point inside $ \triangle ABC $ such that $ AE \perp BC . $ Let $ F=EG \cap OD $ and $ K, L $ be the point lie on $ BC $ such that $ FK \parallel OB, FL \parallel OC . $ Let $ M \in AB $ be a point such that $ MK \perp BC $ and $ N \in AC $ be a point such that $ NL \perp BC . $ Let $ \omega $ be a circle tangent to $ OB, OC $ at $ B, C, $ respectively $ . $ Prove that $ \odot (AMN) $ is tangent to $ \omega $

Show that there exist no integers $a$ and $b$ such that $a^3+a^2b+ab^2+b^3=2001$.

In a village with a population of $1000$, two hundred people have been infected by a disease. A diagnostic test can be done to check whether a person is infected, but the result could be erroneous. That is, there is a $5\%$ probability that the test result of an infected person shows that they are not infected and a $5\%$ probability that the test result of a healthy person shows that they are infected. We randomly choose someone from the population of this village and take the diagnostic test from him. What is the probability that the test result declares that person is infected?

Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is: $\textbf{(A)}\ \dfrac{16}{9} \qquad \textbf{(B)}\ \dfrac{16}{3}\qquad \textbf{(C)}\ \dfrac{4}{9} \qquad \textbf{(D)}\ \dfrac{4}{3} \qquad \textbf{(E)}\ -\dfrac{4}{3}$

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{n+1}=a_{n}+\lfloor \sqrt{a_{n}}\rfloor.\] Show that $a_{n}$ is a square if and only if $n=2^{k}+k-2$ for some $k \in \mathbb{N}$.