Given triangle $ ABC$ with medians $ AE$, $ BF$, $ CD$; $ FH$ parallel and equal to $ AE$; $ BH$ and $ HE$ are drawn; $ FE$ extended meets $ BH$ in $ G$. Which one of the following statements is not necessarily correct? $ \textbf{(A)}\ AEHF \text{ is a parallelogram} \qquad \textbf{(B)}\ HE\equal{}HG \\ \textbf{(C)}\ BH\equal{}DC \qquad \textbf{(D)}\ FG\equal{}\frac{3}{4}AB \qquad \textbf{(E)}\ FG\text{ is a median of triangle }BFH$

Determine all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ f(2x+f(y))=x+y +f(f(x)) , \ \ \ \forall x,y \in \mathbb{R}^+.\]

How many ways are there to place 8 1s and 8 0s in a $4\times 4$ array such that the sum in every row and column is 2? \begin{tabular}{|c|c|c|c|} \hline 1 & 0 & 0 & 1 \\ \hline 0 & 1 & 1 & 0 \\ \hline 0 & 1 & 1 & 0 \\ \hline 1 & 0 & 0 & 1 \\ \hline \end{tabular} [i]2016 CCA Math Bonanza Individual #11[/i]

Let $ABCDEF $ be a regular hexagon. On the line $AF $ mark the point $X$so that $ \angle DCX = 45^o$ . Find the value of the angle $FXE$. (Vyacheslav Yasinsky)

For each positive integer $m$, be $P(m)$ the product of all the digits of $m$. It defines the succession $a_1,a_2, a_3\cdots, $ as follows: . $a_1$ is a positive integer less than 2018 . $a_{n+1}=a_n+P(a_n)$ for each integer $n\ge 1$ Prove that for every integer $n \ge1$ it is true that $a_n \le 2^{2018}.$

On an infinite triangular lattice, there is a single atom at a lattice point. We allow for four operations as illustrated in Figure 1. In words, one could take an existing atom, split it into three atoms, and place them at adjacent lattice points in one of the two displayed fashions (a “split”). One could also reverse the process, i.e. taking three existing atoms in the displayed configurations, and merge them into a single atom at the center (a “merge”). [center][img]https://cdn.artofproblemsolving.com/attachments/2/5/41abc4dc8fb8235e5eb0c98638f9e4a0896c05.png[/img][/center] Figure 1: The four possible operations on an atom. Assume that, after finitely many operations, there is again only a single atom remaining on the lattice. Show that this is possible if and only if the final atom is contained in the sublattice implied by Figure 2. [center][img]https://cdn.artofproblemsolving.com/attachments/b/4/7a7bd10a1862947c250fa07571c061367a5a71.png[/img][/center] Figure 2: The possible positions for the final atom is the green sublattice. The position of the original atom is marked in purple.

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$. [i]Proposed by Anton Trygub[/i]

A circle with circumference 6n units is given and 3n points divide the circumference in n intervals of 1 unit, n intervals of 2 units, and n intervals of 3 units. Prove that there is at least one pair of points that are diametrically opposite to each other.

$a$ and $b$ are real numbers that satisfy \[a^4+a^2b^2+b^4=900,\] \[a^2+ab+b^2=45.\] Find the value of $2ab.$ [i]Author: Ray Li[/i]

Given a $ 1 \times 25$ rectangle divided into $ 25$ "boxes" ($ 1 \times 1$), is it possible to write integers $ 1$ to $ 25$ so that the sum of any two adjacent "boxes" is a perfect square?

Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$

$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$. If $f(6) = 6$, then $f(2012) = ?$ $ \textbf{(A)}\ -2010 \qquad \textbf{(B)}\ -2000 \qquad \textbf{(C)}\ 2000 \qquad \textbf{(D)}\ 2010 \qquad \textbf{(E)}\ 2012$

Let $ N \equal{} \{1,2 \ldots, n\}, n \geq 2.$ A collection $ F \equal{} \{A_1, \ldots, A_t\}$ of subsets $ A_i \subseteq N,$ $ i \equal{} 1, \ldots, t,$ is said to be separating, if for every pair $ \{x,y\} \subseteq N,$ there is a set $ A_i \in F$ so that $ A_i \cap \{x,y\}$ contains just one element. $ F$ is said to be covering, if every element of $ N$ is contained in at least one set $ A_i \in F.$ What is the smallest value $ f(n)$ of $ t,$ so there is a set $ F \equal{} \{A_1, \ldots, A_t\}$ which is simultaneously separating and covering?

Isabel wants to partition the set $\mathbb{N}$ of the positive integers into $n$ disjoint sets $A_{1}, A_{2}, \ldots, A_{n}$. Suppose that for each $i$ with $1\leq i\leq n$, given any positive integers $r, s\in A_{i}$ with $r\neq s$, we have $r+s\in A_{i}$. If $|A_{j}|=1$ for some $j$, find the greatest positive integer that may belong to $A_{j}$.

Consider a $9 \times 9$ array of white squares. Find the largest $n \in\mathbb N$ with the property: No matter how one chooses $n$ out of 81 white squares and color in black, there always remains a $1 \times 4$ array of white squares (either vertical or horizontal).

Three fixed circles pass through the points $A$ and $B$. Let $X$ be a variable point on the first circle different from $A$ and $B$. The line $AX$ intersects the other two circles at $Y$ and $Z$ (with $Y$ between $X$ and $Z$). Show that the ratio $\frac{XY}{YZ}$ is independent of the position of $X$.

Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}$ . Find the least positive integer $k$ making the number $k!\cdot S_{2016}$ an integer.

Let be a convex polygon with $n >5$ vertices and area $1$. Prove that there exists a convex hexagon inside the given polygon with area at least $\dfrac{3}{4}$

Each of the cells of a $7 \times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the expected number of hidden edges is less than $3$.

Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$

There is a row of $100N$ sandwiches with ham. A boy and his cat play a game. In one action the boy eats the first sandwich from any end of the row. In one action the cat either eats the ham from one sandwich or does nothing. The boy performs 100 actions in each of his turns, and the cat makes only 1 action each turn; the boy starts first. The boy wins if the last sandwich he eats contains ham. Is it true that he can win for any positive integer $N{}$ no matter how the cat plays? [i]Ivan Mitrofanov[/i]

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right? [asy] size(340); int i, j; for(i = 0; i<10; i = i+1) { for(j = 0; j<5; j = j+1) { if(10*j + i == 11 || 10*j + i == 12 || 10*j + i == 14 || 10*j + i == 15 || 10*j + i == 18 || 10*j + i == 32 || 10*j + i == 35 || 10*j + i == 38 ) { } else{ label("$*$", (i,j));} }} label("$\leftarrow$"+"Dec. 31", (10.3,0)); label("Jan. 1"+"$\rightarrow$", (-1.3,4));[/asy]

Let $X,\ Y$ and $Z$ be the midpoints of sides $BC,\ CA$, and $AB$ of the triangle $ABC$, respectively. Let $P$ be a point inside the triangle. Prove that the quadrilaterals $AZPY,\ BXPZ$, and $CYPX$ have equal areas if, and only if, $P$ is the centroid of $ABC$.

Denote $\overline{w_a}, \overline{w_b}, \overline{w_c}$ the external angle-bisectors in triangle $ABC$, prove that $$\sum \limits_{cyc} \frac{1}{w_a}\leq \sqrt{\frac{(s^2 - r^2 - 4Rr)(8R^2 - s^2 - r^2 - 2Rr)}{8s^2R^2r}}$$