Consider $n \geq 2$ distinct points in the plane $A_1,A_2,...,A_n$ . Color the midpoints of the segments determined by each pair of points in red. What is the minimum number of distinct red points?

Prove that if $a,b,c$ are positive numbers with $abc=1$, then \[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]

Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio \[\frac{AB \cdot CD}{AC \cdot BD}, \] and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)

The English alphabet, which has $26$ letters, is randomly permuted. Let $p_1$ be the probability that $\text{AB},$ $\text{CD},$ and $\text{EF}$ all appear as contiguous substrings. Let $p_2$ be the probability that $\text{ABC}$ and $\text{DEF}$ both appear as contiguous substrings. Compute $\tfrac{p_1}{p_2}.$

Consider a scalene triangle $ ABC $ with $ AB <AC <BC. $ The $ AB $ side mediator cuts the $ B $ side at the $ K $ point and the $ AC $ prolongation at the $ U. $ point. $ AC $ side cuts $ BC $ side at $ O $ point and $ AB $ side extension at $ G$ point. Prove that the $ GOKU $ quad is cyclic, meaning its four vertices are at same circumference

Given a $8$x$8$ square board a) Prove that: for any ways to color the board, we are always be able to find a rectangle consists of $8$ squares such that these squares are not colored. b) Prove that: we can color $7$ squares on the board such that for any rectangles formed by $\geq 9$ squares, there are at least $1$ colored square.

Let $ABC$ be an acute triangle. Let$H$ and $D$ be points on $[AC]$ and $[BC]$, respectively, such that $BH \perp AC$ and $HD \perp BC$. Let $O_1$ be the circumcenter of $\triangle ABH$, and $O_2$ be the circumcenter of $\triangle BHD$, and $O_3$ be the circumcenter of $\triangle HDC$. Find the ratio of area of $\triangle O_1O_2O_3$ and $\triangle ABH$.

Let $n$ be a positive integer which is not prime. Prove that there exist $k, a_{1},a_{2},...a_{k}>1$ positive integers such that $a_{1}+a_{2}+\cdots+a_{k}=n(\frac1{a_{1}}+\frac1{a_{2}}+\cdots+\frac1{a_{k}})$ Edit: the $a_{i}'s$ have to be grater than 1. Sorry, my mistake :blush:

There are several bears living on the $2025$ islands of the Arctic Ocean. Every bear sometimes swims from one island to another. It turned out that every bear made at least one swim in a year, but no two bears made equal swams. At the same time, exactly one swim was made between each two islands $A$ and $B$: either from $A$ to $B$ or from $B$ to $A$. Prove that there were no bears on some island at the beginning and at the end of the year. [i]A. Kuznetsov[/i]

Given $x_0\in\mathbb R$, $f,g:\mathbb R\to\mathbb R$, we define the $\emph{non-redundant binary tree}$ $T(x_0,f,g)$ in the following way: [list=1] [*]The tree $T$ initially consists of just $x_0$ at height $0$. [*]Let $v_0,\dots,v_k$ be the vertices at height $h$. Then the vertices of height $h+1$ are added to $T$ by: for $i=0,1,\dots,k$, $f(v_i)$ is added as a child of $v_i$ if $f(v_i)\not\in T$, and $g(v_i)$ is added as a child of $v_i$ if $g(v_i)\not\in T$. [/list] For example, if $f(x)=x+1$ and $g(x)=x-1$, then the first three layers of $T(0,f,g)$ look like: [asy] size(100); draw((-0.1,-0.2)--(-0.4,-0.8),EndArrow(size=3)); draw((0.1,-0.2)--(0.4,-0.8),EndArrow(size=3)); draw((-0.6,-1.2)--(-0.9,-1.8),EndArrow(size=3)); draw((0.6,-1.2)--(0.9,-1.8),EndArrow(size=3)); label("$0$",(0,0)); label("$1$",(-.5,-1)); label("$-1$",(.5,-1)); label("$2$",(-1,-2)); label("$-2$",(1,-2));[/asy] If $f(x)=1024x-2047\lfloor x/2\rfloor$ and $g(x)=2x-3\lfloor x/2\rfloor+2\lfloor x/4\rfloor$, then how many vertices are in $T(2016,f,g)$?

What is the least integer $n>2003$ such that $5^n + n^5$ is a multiple of $11$? $ \textbf{(A)}\ 2010 \qquad\textbf{(B)}\ 2011 \qquad\textbf{(C)}\ 2012 \qquad\textbf{(D)}\ 2014 \qquad\textbf{(E)}\ \text{None of the preceding} $

In the collection of all right circular cylinders of fixed volume $c$, what is the ratio $\frac hr$ of the cylinder which has the least total surface area?

Let $ABC$ be a triangle and $M$ be the midpoint of $AB$. If it is known that $\angle CAM + \angle MCB = 90^o$, show that triangle $ABC$ is isosceles or right.

A [i]derangement [/i] of the letters $ABCDEF$ is a permutation of these letters so that no letter ends up in the position it began such as $BDECFA$. An [i]inversion [/i] in a permutation is a pair of letters $xy$ where $x$ appears before $y$ in the original order of the letters, but $y$ appears before $x$ in the permutation. For example, the derangement $BDECFA$ has seven inversions: $AB, AC, AD, AE, AF, CD$, and $CE$. Find the total number of inversions that appear in all the derangements of $ABCDEF$.

Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$? [i]2018 CCA Math Bonanza Tiebreaker Round #3[/i]

Four lines are given in a plane so that any three of them determine a triangle. One of these lines is parallel to a median in the triangle determined by the other three lines. Prove that each of the other three lines also has this property.

The sequence $(z_n)$ of complex numbers satisfies the following properties: [list] [*]$z_1$ and $z_2$ are not real. [*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$. [*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$. [*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list] Find the product of all possible values of $z_1$.

An ellipse with focus $F$ is given. Two perpendicular lines passing through $F$ meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Prove that this quadrilateral is inscribed into a conic with focus $F$

Prove that there exist infinitely many pairs of real numbers $(x,y)$ such that $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$

Find all positive reals $a,b,c$ which fulfill the following relation \[ 4(ab+bc+ca)-1 \geq a^2+b^2+c^2 \geq 3(a^3+b^3+c^3) . \] created by Panaitopol Laurentiu.

Rectangle $R$ with area $20$ and diagonal of length $7$ is translated $2$ units in some direction to form a new rectangle $R'.$ The vertices of $R$ and $R'$ that are not contained in the other rectangle form a convex hexagon. Compute the maximum possible area of this hexagon.

Let $ \{a_n\}_{n=1}^\infty $ be an arithmetic progression with $ a_1 > 0 $ and $ 5\cdot a_{13} = 6\cdot a_{19} $ . What is the smallest integer $ n$ such that $ a_n<0 $?

Let $x,y,z,t$ be real numbers with $x,y,z$ not all equal such that \[x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.\] Find all possible values of $ t$ such that $xyz+t=0$.

For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$

A natural number $A$ is given. One may add to it one of its divisors $d$ ($1 < d < A$). One may then repeat this operation with the new number $A + d$ and so on. Prove that starting from $A = 4$ one can get any composite number by these operations. (M Vyalyi)