$N$ oligarchs built a country with $N$ cities with each one of them owning one city. In addition, each oligarch built some roads such that the maximal amount of roads an oligarch can build between two cities is $1$ (note that there can be more than $1$ road going through two cities, but they would belong to different oligarchs). A total of $d$ roads were built. Some oligarchs wanted to create a corporation by combining their cities and roads so that from any city of the corporation you can go to any city of the corporation using only corporation roads (roads can go to other cities outside corporation) but it turned out that no group of less than $N$ oligarchs can create a corporation. What is the maximal amount that $d$ can have?

In $\Delta ABC$, $AD \perp BC$ at $D$. $E,F$ lie on line $AB$, such that $BD=BE=BF$. Let $I,J$ be the incenter and $A$-excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$, and $\Delta PEI \sim \Delta QFJ$ .

Find the number of ordered quadruplets $(a_1,a_2,a_3,a_4)$ of integers, for which $a_1\geq 1$, $a_2\geq 2$, $a_3\geq 3$, and $-10\leq a_4\leq 10$ and $a_1+a_2+a_3+a_4=2011$ .

Find all real solutions to the following system of equations. Carefully justify your answer. \[ \left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y \\ \\ \displaystyle\frac{4y^2}{1+4y^2} = z \\ \\ \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right. \]

A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and \[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}. \] Determine $ S_{1024}.$

The sets of rational numbers $A = \{a_1, \dots, a_5\}$ and $B = \{b_1, \dots, b_5\}$ both contain $0$ and satisfy the condition that $$ \{a_i + b_j\}_{i,j} = \{0, 1, 2, \dots, 23, 24\}. $$ Determine these sets. (The set $\{a_i + b_j\}_{i,j}$ consists of all possible sums between an element of $A$ and an element of $B$)

Percy buys $3$ apples for $6$ dollars, $4$ pears for $16$ dollars, and $1$ watermelon for $5$ dollars. Assuming the rates stay the same, how much would it cost to buy $10$ apples, $3$ pears, and $2$ watermelons? $\textbf{(A) } 38\qquad\textbf{(B) } 39\qquad\textbf{(C) } 40\qquad\textbf{(D) } 41\qquad\textbf{(E) } 42$

Prove that the equation $x^{19} + x^{17} = x^{16 }+ x^7 + a$ for any $a \in R$ has at least two imaginary roots

Consider a regular $24$-gon $\mathcal{P}.$ A quadrilateral is said to be inscribed in $\mathcal{P}$ if its vertices are among those of $\mathcal{P}.$ We consider two inscribed quadrilaterals equivalent if one can be obtained from the other via a rotation about the center of $\mathcal{P}.$ How many distinct (i.e. not equivalent) quadrilaterals can be inscribed in $\mathcal{P}$?

Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$. Prove that: there is exactly one point $T$ inside $\triangle ABC$ such that $\angle BTC=\angle CTA=\angle ATB=120^{\circ}$. ($T$ is called [i]Fermat-Torricelli[/i] point of $\triangle ABC$)

There are $n$ sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions? [i](Proposed by Bogdan Rublov)[/i]

Points $A,B,C,D,E,F$ is on the circle $O.$ A line $\ell$ is tangent to $O$ at $E$ is parallel to $AC$ and $DE>EF.$ Let $P,Q$ be the intersection of $\ell$ and $BC,CD$ ,respectively and let $R,S$ be the intersection of $\ell$ and $CF,DF$ ,respectively. Show that $PQ=RS$ if and only if $QE=ER.$

Monty wants to play a game with you. He shows you five boxes, one of which contains a prize and four of which contain nothing. He allows you to choose one box but not to open it. He then opens one of the other four boxes that he knows to contain nothing. Then, he makes you switch and choose a different, unopened box. However, Monty sketchily reveals the contents of another (empty) box, selected uniformly at random from the two or three closed boxes (that you do not currently have chosen) that he knows to contain no prize. He then offers you the chance to switch again. Assuming you seek to maximize your return, determine the probability you get a prize.

Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet? $ \textbf{(A)}\ 0.10 \qquad \textbf{(B)}\ 0.15 \qquad \textbf{(C)}\ 0.20 \qquad \textbf{(D)}\ 0.25 \qquad \textbf{(E)}\ 0.30$

Suppose that $\xi \ne 1$ is a root of the polynomial $f(x) = x^{167} -1$. Compute $$\left|\sum_{0<a<b<167} \xi^{a^2+b^2} \right|.$$ In the above summation $a,b$ are integers

$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that \[ \sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2 \]

Evaluate $\sum_{n=1}^\infty \dfrac{n^5}{n!}.$

$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$. (1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for any line on the plane, there are two rational points on it, then it passes through all the $ 100$ sets. (2) Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.

Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$ $ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $

A rectangle is divided into $2\times 1$ and $1\times 2$ dominoes. In each domino, a diagonal is drawn, and no two diagonals have common endpoints. Prove that exactly two corners of the rectangle are endpoints of these diagonals.

Let $P$, $A$, $B$, $C$, $D$ be points on a plane such that $PA = 9$, $PB = 19$, $PC = 9$, $PD = 5$, $\angle APB = 120^\circ$, $\angle BPC = 45^\circ$, $\angle CPD = 60^\circ$, and $\angle DPA = 135^\circ$. Let $G_1$, $G_2$, $G_3$, and $G_4$ be the centroids of triangles $PAB$, $PBC$, $PCD$, $PDA$. $[G_1G_2G_3G_4]$ can be expressed as $a\sqrt{b} + c\sqrt{d}$. Find $a+b+c+d$. [i]2022 CCA Math Bonanza Individual Round #15[/i]

Ruby has a non-negative integer $n$. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or $0$. (e.g. $86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6$) [i]Proposed by Wong Jer Ren[/i]

Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that \[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]

Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular. (Vyacheslav Yasinsky)

Let $ABCDEF$ be a regular hexagon with side length $ 1$. Now, construct square $AGDQ$. What is the area of the region inside the hexagon and not the square?