In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.

For any three real numbers $ a$, $ b$, and $ c$, with $ b\neq c$, the operation $ \otimes$ is defined by: \[ \otimes(a,b,c) \equal{} \frac {a}{b \minus{} c} \]What is $ \otimes(\otimes(1,2,3), \otimes(2,3,1),\otimes(3,1,2))$? $ \textbf{(A)}\ \minus{}\!\frac {1}{2}\qquad \textbf{(B)}\ \minus{}\!\frac {1}{4}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ \frac {1}{4}\qquad \textbf{(E)}\ \frac {1}{2}$

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

Let $ ABC$ be a triangle and let $ I$ be the incenter. $ Ia$ $ Ib$ and $ Ic$ are the excenters opposite to points $ A$ $ B$ and $ C$ respectively. Let $ La$ be the line joining the orthocenters of triangles $ IBC$ and $ IaBC$. Define $ Lb$ and $ Lc$ in the same way. Prove that $ La$ $ Lb$ and $ Lc$ are concurrent. Daniel

Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.

Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is [i]not[/i] a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.

Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$, $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers.

Let $f$ be a real function defined in the interval $[0, +\infty )$ and suppose that there exist two functions $f', f''$ in the interval $[0, +\infty )$ such that \[f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0.\] Let $g$ be a function for which \[g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}.\] Prove that $g$ is bounded.

Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle. [i]M. Kungojin[/i]

Find all positive integers $n$ such that $27^n- 2^n$ is a perfect square.

Find all triples of natural $ (x,y,n)$ satisfying the condition: \[ \frac {x! \plus{} y!}{n!} \equal{} 3^n \] Define $ 0! \equal{} 1$

$A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are two squares with their vertices arranged clockwise.The perpendicular bisector of segment $A_1B_1,A_2B_2,A_3B_3,A_4B_4$ and the perpendicular bisector of segment $A_2B_2,A_3B_3,A_4B_4,A_1B_1$ intersect at point $P,Q,R,S$ respectively.Show that:$PR\perp QS$.

In a certain country there are more than $101$ towns. The capital of this country is connected by direct air routes with $100$ towns, and each town, except for the capital, is connected by direct air routes with $10$ towns (if $A$ is connected with $B, B$ is connected with $A$). It is known that from any town it is possible to travel by air to any other town (possibly via other towns). Prove that it is possible to close down half of the air routes connected with the capital, and preserve the capability of travelling from any town to any other town within the country. (IS Rubanov)

Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. [b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.

Let $S$ be a set of $n>0$ elements, and let $A_1 , A_2 , \ldots A_k$ be a family of distinct subsets such that any two have a non-empty intersection. Assume that no other subset of $S$ intersects all of the $A_i.$ Prove that $ k=2^{n-1}.$

Prove that if $$ (a + b + c)^2 = 3 (ab + bc + ac - x^2 - y^2 - z^2),$$ where $ a $, $ b $, $ c $, $ x $, $ y $, $ z $ denote real numbers, then $ a = b = c $ and $ x = y = z = 0 $.

Let $a+\frac{1}{b}=8$ and $b+\frac{1}{a}=3$. Given that there are two possible real values for $a$, find their sum. $\text{(A) }\frac{3}{8}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }3\qquad\text{(D) }5\qquad\text{(E) }8$

Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.

Let $n\in\mathbb{N}\setminus\left\{0\right\}$ be a positive integer. Find all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying that : $$f(x+y^{2n})=f(f(x))+y^{2n-1}f(y),(\forall)x,y\in\mathbb{R},$$ and $f(x)=0$ has an unique solution.

$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?

Let $ABC$ be a triangle. Its incircle meets the sides $BC, CA$ and $AB$ in the points $D, E$ and $F$, respectively. Let $P$ denote the intersection point of $ED$ and the line perpendicular to $EF$ and passing through $F$, and similarly let $Q$ denote the intersection point of $EF$ and the line perpendicular to $ED$ and passing through $D$. Prove that $B$ is the mid-point of the segment $PQ$. Proposed by Karl Czakler

In a triangle two altitudes are not smaller than the sides on to which they are dropped. Find the angles of the triangle.

Determine the largest prime which divides both $2^{24}-1$ and $2^{16}-1$. [i]2017 CCA Math Bonanza Individual Round #6[/i]