In the right-angled trapezoid $AB CD$, $AB \parallel CD$, $m( \angle A) = 90°$, $AD = DC = a$ and $AB =2a$. On the perpendiculars raised in $C$ and $D$ on the plane containing the trapezoid one considers points $E$ and $F$ (on the same side of the plane) such that $CE = 2a$ and $DF = a$. Find the distance from the point $B$ to the plane $(AEF)$ and the measure of the angle between the lines $AF$ and $BE$.

For all positive integers $m\geq 1$, denote by $\mathcal{G}_m$ the set of simple graphs with exactly $m$ edges. Find the number of pairs of integers $(m,n)$ with $1<2n\leq m\leq 100$ such that there exists a simple graph $G\in\mathcal{G}_m$ satisfying the following property: it is possible to label the edges of $G$ with labels $E_1$, $E_2$, $\ldots$, $E_m$ such that for all $i\neq j$, edges $E_i$ and $E_j$ are incident if and only if either $|i-j|\leq n$ or $|i-j|\geq m-n$. $\textit{Note: }$ A graph is said to be $\textit{simple}$ if it has no self-loops or multiple edges. In other words, no edge connects a vertex to itself, and the number of edges connecting two distinct vertices is either $0$ or $1$.

In a triangle $ABC$, $\angle BAC =90^{\circ}$. Point $D$ lies on the side $BC$ and satisfies $\angle BDA=2\angle BAD$. Prove that \[\frac{2}{AD}=\frac{1}{BD}+\frac{1}{CD} \]

Let $ABC$ be a triangle. Show that there exists a lin $l$ in the plane of $ABC$ such that the overlapping area of $ABC$ and $A^{'}B^{'}C^{'}$, the symmetric of $ABC$ with respect to $l$, is greater than $\frac{2}{3}$ of area of $ABC$.

Twelve people go to a party. First, everybody with no friends at the party leave. Then, at the $i$-th hour, everybody with exactly $i$ friends left at the party leave. After the eleventh hour, what is the maximum number of people left? Note that friendship is mutual. [i]2017 CCA Math Bonanza Team Round #5[/i]

In an acute triangle $ABC$, the altitude from $C$ intersects $AB$ at $E$ and the altitude from $B$ intersects $AC$ at $D$. $CE$ and $BD$ intersect at a point $H$. A circle with diameter $DE$ intersects $AB$ and $AC$ at points $F,G$ respectively. $FG$ and $AH$ intersect at $K$. If $\overline{BC} = 25$, $\overline{BD} = 20$, and $\overline{BE} = 7$, the length of $AK$ is of the form $\frac{p}{q}$ , where $p, q$ are relatively prime positive integers. Find $p + q$.

Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.

Let $n\ge 4$ be an even integer. A regular $n$-gon and a regular $(n-1)$-gon are inscribed into the unit circle. For each vertex of the $n$-gon consider the distance from this vertex to the nearest vertex of the $(n-1)$-gon, measured along the circumference. Let $S$ be the sum of these $n$ distances. Prove that $S$ depends only on $n$, and not on the relative position of the two polygons.

Several triangles are [b]intersecting[/b] if any two of them have non-empty intersections. Show that for any two finite collections of intersecting triangles, there exists a line that intersects all the triangles. [i] Proposed by usjl[/i]

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $(f(x + y))^2 = f(x^2) + f(y^2)$ for all $x, y \in \mathbb{Z}$.

Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that \[f(x)+f\left(1-\frac1x\right)=\arctan x\] for all real $x\ne 0.$ (As usual, $y=\arctan x$ means $-\pi/2<y<\pi/2$ and $\tan y=x.$) Find \[\int_0^1f(x)\,dx.\]

Prove that for any positive integer $n$ the following identity holds $\frac{(2n)!}{n!}= 2^n \cdot (2n - 1)!!$

Suppose that $A$ be the set of all positive integer that can write in form $a^2+2b^2$ (where $a,b\in\mathbb {Z}$ and $b$ is not equal to $0$). Show that if $p$ be a prime number and $p^2\in A$ then $p\in A$. [i]Marcel Tena[/i]

We have $ n $ kinds of puddings. There are $ a_{i} $ puddings which are $ i $-th type and those $ S = a_{1}+\cdots+a_{n} $ puddings are distinct. Now, for a given arrangement of puddings: $ p_{1}, \dots, p_{n} $. Define $ c_{i} $ to be $$ \#\{1 \le j \le S-1 \ \mid \ p_{j}, p_{j+1} \text{ are the same type.}\} $$ Show that if $ S $ is composite, then the sum of $ \prod_{i=1}^{n}{c_{i}} $ over all possible arrangements is a multiple of $ S $.

Let $ c\in\mathbb C$ and $ A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}$. a) Prove that for each $ c\in C$, $ A_c$ is infinite. b) Prove that if $ p\in A_1$, and $ p(z_0) \equal{} 0$, then $ |z_0| < 1.7$. c) Prove that each element of $ A_c$ is odd or even. Let $ f_c \equal{} z^2 \plus{} c\in \mathbb C[z]$. We see easily that $ B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c \equal{} B_c$. d) $ |c| > 2$. e) $ c\in \mathbb Q\backslash\mathbb Z$. f) $ c$ is a non-algebraic number g) $ c$ is a real number and $ c\not\in [ \minus{} 2,\frac14]$.

Let $a$ be [i]good[/i] if the number of prime divisors of $a$ is equal to $2$. Do there exist $18$ consecutive good natural numbers?

Construct a triangle given the radii of the excircles.

Consider three parallelograms $P_1,~P_2,~ P_3$. Parallelogram $P_3$ is inside parallelogram $P_2$, and the vertices of $P_3$ are on the edges of $P_2$. Parallelogram $P_2$ is inside parallelogram $P_1$, and the vertices of $P_2$ are on the edges of $P_1$. The sides of $P_3$ are parallel to the sides of $P_1$. Prove that one side of $P_3$ has length at least half the length of the parallel side of $P_1$.

What is the least positive integer $x$ for which the expression $x^2 + 3x + 9$ has $3$ distinct prime divisors?

In the plane, six circles are given so that none of the circles contain one the center of the other. Show that there is no point that lies in all the circles.

Let $\triangle ABC$ be a triangle and $H$ its orthocenter. We draw the circumference $\mathcal{C}_1$ that passes through $H$ and its tangent to $BC$ at $B$ and the circumference $\mathcal{C}_2$ that passes through $H$ and its tangent to $BC$ at $C$. If $\mathcal{C}_1$ cuts line $AB$ again at $X$ and $\mathcal{C}_2$ cuts line $AC$ again at $Y$. Prove that $X,Y$ and $H$ are collinear.

Define the sequence of real numbers $\{x_n\}_{n \geq 1}$, where $x_1$ is any real number and \[x_n = 1 - x_1x_2\ldots x_{n-1} \text{ for all } n > 1.\] Show that $x_{2011} > \frac{2011}{2012}$.

A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? $\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$

If the six-digit number $\underline{2}\, \underline{0}\, \underline{2} \, \underline{1} \, \underline{a} \, \underline{b}$ is divisible by $9$, what is the greatest possible value of $a \cdot b$? $\textbf{(A) }18\qquad\textbf{(B) }20\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad\textbf{(E) }42$

Find all positive integers $x, y, z$ such that $45^x-6^y=2019^z$ [i]Proposed by Dorlir Ahmeti, Albania[/i]