The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively. Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.

Let \(a_1, a_2, \ldots, a_n\) be integers such that \(a_1 > a_2 > \cdots > a_n > 1\). Let \(M = \operatorname{lcm} \left( a_1, a_2, \ldots, a_n \right)\). For any finite nonempty set $X$ of positive integers, define \[ f(X) = \min_{1 \leqslant i \leqslant n} \sum_{x \in X} \left\{ \frac{x}{a_i} \right\}. \] Such a set $X$ is called [i]minimal[/i] if for every proper subset $Y$ of it, $f(Y) < f(X)$ always holds. Suppose $X$ is minimal and $f(X) \geqslant \frac{2}{a_n}$. Prove that \[ |X| \leqslant f(X) \cdot M. \]

Positive real numbers $a$ and $b$ satisfy the following conditions: the function $f(x)=x^3+ax^2+2bx-1$ has three different real roots, while the function $g(x)=2x^2+2bx+a$ doesn't have real roots. Prove that $a-b>1$. [i](V. Karamzin)[/i]

Let $P$ be a finite set of primes, $A$ an infinite set of positive integers, where every element of $A$ has a prime factor not in $P$. Prove that there exist an infinite subset $B$ of $A$, such that the sum of elements in any finite subset of $B$ has a prime factor not in $P$.

Find the last eight digits of the binary development of $27^{1986}.$

Let trapezoid $ABCD$ inscribed in a circle $O$, $AB||CD$. Tangent at $D$ wrt $O$ intersects line $AC$ at $F$, $DF||BC$. If $CA=5, BC=4$, then find $AF$.

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?