Sophia writes an algorithm to solve the graph isomorphism problem. Given a graph $G=(V,E)$, her algorithm iterates through all permutations of the set $\{v_1, \dots, v_{|V|}\}$, each time examining all ordered pairs $(v_i,v_j)\in V\times V$ to see if an edge exists. When $|V|=8$, her algorithm makes $N$ such examinations. What is the largest power of two that divides $N$?

Find the maximum possible value of $a + b + c ,$ if $a,b,c$ are positive real numbers such that $a^2 + b^2 + c^2 = a^3 + b^3 + c^3 .$

We only know that the password of a safe consists of $7$ different digits. The safe will open if we enter $7$ different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than $7$ attempts? [i](5 points for Juniors and 4 points for Seniors)[/i]

Let $AB$ be the diameter of a circle $k$ and let $E$ be a point in the interior of $k$. The line $AE$ intersects $k$ a second time in $C \ne A$ and the line $BE$ intersects $k$ a second time in $D \ne B$. Show that the value of $AC \cdot AE+BD\cdot BE$ is independent of the choice of $E$.

Find $2 \times (2 + 3)$ $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }30$

The sequence $\{x_n\}$ is defined by $x_1=5$ and $x_{k+1}=x_k^2-3x_k+3$ for $k=1,2,3\cdots$. Prove that $x_k>3^{2^{k-1}}$ for any positive integer $k$.

The products of the opposite sidelengths of a cyclic quadrilateral $ABCD$ are equal. Let $B'$ be the reflection of $B$ about $AC$. Prove that the circle passing through $A,B', D$ touches $AC$

Prove that there is no function $f:\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x)^2 \geq f(x+y)(f(x)+y)$ for all $x,y \in \mathbb{R}^{+}$.

The number of solutions in positive integers of $2x+3y=763$ is: $\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 254\qquad \textbf{(C)}\ 128 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 0$

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

Positive numbers $ x_1, x_2, . . ., x_{2009}$ satisfy the equalities $$x^2_1 - x_1x_2 +x^2_2 =x^2_2 -x_2x_3+x^2_3=x^2_3 -x_3x_4+x^2_4= ...= x^2_{2008}- x_{2008}x_{2009}+x^2_{2009}= x^2_{2009}-x_{2009}x_1+x^2_1$$. Prove that the numbers $ x_1, x_2, . . ., x_{2009}$ are equal.

Consider a polygon with $m + n$ sides where $m, n$ are positive integers. Colour $m$ of its vertices red and the remaining $n$ vertices blue. A side is given the number $2$ if both its end vertices are red, the number $1/2.$ if both its end vertices are blue and the number $1$ otherwise. Let the product of these numbers be $P$. Find the largest possible value of $P$.

A circle centered at $ A$ with a radius of 1 and a circle centered at $ B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is [asy] size(220); real r1 = 1; real r2 = 3; real r = (r1*r2)/((sqrt(r1)+sqrt(r2))**2); pair A=(0,r1), B=(2*sqrt(r1*r2),r2); dot(A); dot(B); draw( circle(A,r1) ); draw( circle(B,r2) ); draw( (-1.5,0)--(7.5,0) ); draw( A -- (A+dir(210)*r1) ); label("$1$", A -- (A+dir(210)*r1), N ); draw( B -- (B+r2*dir(330)) ); label("$4$", B -- (B+r2*dir(330)), N ); label("$A$",A,dir(330)); label("$B$",B, dir(140)); draw( circle( (2*sqrt(r1*r),r), r )); [/asy] $ \displaystyle \textbf{(A)} \ \frac {1}{3} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {5}{12} \qquad \textbf{(D)} \ \frac {4}{9} \qquad \textbf{(E)} \ \frac {1}{2}$

Let $a \spadesuit b = \frac{a^2-b^2}{2b-2a}$ . Given that $3 \spadesuit x = -10$, compute $x$.

An acute angle between the diagonals of a cyclic quadrilateral is equal to $\phi$. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than $\phi$.

Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$. [i]Mihai Pitticari & Sorin Rǎdulescu[/i]

Let $\omega=e^{\frac{2\pi i}{2017}}$ and $\zeta = e^{\frac{2\pi i}{2019}}$. Let $S=\{(a,b)\in\mathbb{Z}\,|\,0\leq a \leq 2016, 0 \leq b \leq 2018, (a,b)\neq (0,0)\}$. Compute $$\prod_{(a,b)\in S}(\omega^a-\zeta^b).$$

A magical castle has $n$ identical rooms, each of which contains $k$ doors arranged in a line. In room $i, 1 \leq i \leq n - 1$ there is one door that will take you to room $i + 1$, and in room $n$ there is one door that takes you out of the castle. All other doors take you back to room $1$. When you go through a door and enter a room, you are unable to tell what room you are entering and you are unable to see which doors you have gone through before. You begin by standing in room $1$ and know the values of $n$ and $k$. Determine for which values of $n$ and $k$ there exists a strategy that is guaranteed to get you out of the castle and explain the strategy. For such values of $n$ and $k$, exhibit such a strategy and prove that it will work.

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

Prove that there is a set $ A $ consisting of $2002$ different natural numbers satisfying the condition: for each $ a \in A $, the product of all numbers from $ A $, except $ a $, when divided by $ a $ gives the remainder $1$.

A set $C$ of positive integers is called good if for every integer $k$ there exist distinct $a, b \in C$ such that the numbers $a+k$ and $b+k$ are not relatively prime. Prove that if the sum of the elements of a good set $C$ equals $2003$, then there exists $c \in C$ such that the set $C-\{c\}$ is good.

[b]8.[/b] Find all integers $a>1$ for which the least (integer) solution $n$ of the congruence $a^{n} \equiv 1 \pmod{p}$ differs from 6 (p is any prime number). [b](N. 9)[/b]

Let $ACD$ and $BCD$ be acute-angled triangles located in different planes. Let $G$ and $H$ be the centroid and the orthocenter respectively of the $BCD$ triangle; Similarly let $G'$ and $H'$ be the centroid and the orthocenter of the $ACD$ triangle. Knowing that $HH'$ is perpendicular to the plane $(ACD)$, show that $GG' $ is perpendicular to the plane $(BCD)$.

Compute the largest possible number of distinct real solutions for $x$ to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where $a$, $b$, and $c$ are real numbers. [i]Proposed by Tristan Shin

If $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $CDX$ is equilateral, compute $\angle{AXE}$ in degrees. [i]2020 CCA Math Bonanza Lightning Round #1.3[/i]