Let $A_1, A_2, A_3, \ldots , A_{12}$ be the vertices of a regular $12-$gon (dodecagon). Find the number of points in the plane that are equidistant to at least $3$ distinct vertices of this $12-$gon.

$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.

Let $f(x) = x^2 + ax + b$, where $a$ and $b$ are real numbers. If $f(f(1)) = f(f(2)) = 0$, then find $f(0)$.

The circles $\mathcal{K}_1,\mathcal{K}_2$ and $\mathcal{K}_3$ are pairwise externally tangent to each other; the point of tangency betwwen $\mathcal{K}_1$ and $\mathcal{K}_2$ is $T$. One of the external common tangents of $\mathcal{K}_1$ and $\mathcal{K}_2$ meets $\mathcal{K}_3$ at points $P$ and $Q$. Prove that the internal common tangent of $\mathcal{K}_1$ and $\mathcal{K}_2$ bisects the arc $PQ$ of $\mathcal{K}_3$ which is closer to $T$.

Define the sequnce ${(a_n)}_{n\ge1}$ by $a_1=1$ and $a_n=5a_{n-1}+3^{n-1}$ for $n\ge2$. Find the greatest power of $2$ that divides $a_{2^{2019}}$.

There are $10$ objects of total weight $20$, each of the weights being a positive integers. Given that none of the weights exceeds $10$ , prove that the ten objects can be divided into two groups that balance each other when placed on 2 pans of a balance.

If $a,b,c$ are the roots of $x^3+20x^2+1x+5$, compute $(a^2+1)(b^2+1)(c^2+1)$. [i]2015 CCA Math Bonanza Tiebreaker Round #2[/i]

Farmer James has three types of cows on his farm. A cow with zero legs is called a $\textit{ground beef}$, a cow with one leg is called a $\textit{steak}$, and a cow with two legs is called a $\textit{lean beef}$. Farmer James counts a total of $20$ cows and $18$ legs on his farm. How many more $\textit{ground beef}$s than $\textit{lean beef}$s does Farmer James have? [i]Proposed by James Lin[/i]

We divide $999\times 999$ square into the angles with $3$ cells. Prove, that number of ways is divided by $2^7$.( Angle is a figure, that we can get if we remove one cell from $2 \times 2$ square).

Let $\omega$ be a circle, and let $A$ and $B$ be two points in its interior. Prove that there exists a circle passing through $A$ and $B$ that is contained in the interior of $\omega$.

Celia chooses a number $n$ and writes the list of natural numbers from $1$ to $n$: $1, 2, 3, 4, ..., n-1, n.$ At each step, it changes the list: it copies the first number to the end and deletes the first two. After $n-1$ steps a single number will be written. For example, for $n=6$ the five steps are: $$ 1,2,3,4,5,6 \to 3,4,5,6,1 \to 5,6,1,3 \to 1,3,5 \to 5,1 \to 5$$ and the number $5$ is written. Celia chose a number $n$ between $1000$ and $3000$ and after $n-1$ steps the number $1$ was written. Determine all the values of $n$ that Celia could have chosen. Justify why those values work, and the others do not.

Find all functions $f$ , defined and having values in the set of integer numbers, for which the following conditions are satisfied: (a) $f(1) = 1$; (b) for every two whole (integer) numbers $m$ and $n$, the following equality is satisfied: $$f(m+n)·(f(m)-f(n)) = f(m-n)·(f(m)+ f(n))$$

Let $g:(0,\infty) \rightarrow (0,\infty)$ be a differentiable function whose derivative is continuous, and such that $g(g(x)) = x$ for all $x> 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.

A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$ $\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}$

Let $ABC$ be a triangle. Draw circles $\omega_A$, $\omega_B$, and $\omega_C$ such that $\omega_A$ is tangent to $AB$ and $AC$, and $\omega_B$ and $\omega_C$ are defined similarly. Let $P_A$ be the insimilicenter of $\omega_B$ and $\omega_C$. Define $P_B$ and $P_C$ similarly. Prove that $AP_A$, $BP_B$, and $CP_C$ are concurrent. [i]Tom Lu.[/i]

Let $\omega_1$ and $\omega_2$ be two circles that intersect at two points: $A$ and $B$. Let $C$ and $E$ be on $\omega_1$, and $D$ and $F$ be on $\omega_2$ such that $CD$ and $EF$ meet at $B$ and the three lines $CE$, $DF$, and $AB$ concur at a point $P$ that is closer to $B$ than $A$. Let $\Omega$ denote the circumcircle of $\triangle DEF$. Now, let the line through $A$ perpendicular to $AB$ hit $EB$ at $G$, $GD$ hit $\Omega$ at $J$, and $DA$ hit $\Omega$ again at $I$. A point $Q$ on $IE$ satisfies that $CQ=JQ$. If $QJ=36$, $EI=21$, and $CI=16$, then the radius of $\Omega$ can be written as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of a prime, and $\gcd(a, c) = 1$. Find $a+b+c$. [i]Proposed by Kevin Zhao[/i]

In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?

The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$ by Mahdi Etesami Fard

Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$, such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube.

Grogg’s favorite positive integer is $n\ge2$, and Grogg has a lucky coin that comes up heads with some fixed probability $p$, where $0<p<1$. Once each day, Grogg flips his coin, and if it comes up heads, he does two things: [list=1] [*] He eats a cookie. [/*] [*] He then flips the coin $n$ more times. If the result of these $n$ flips is $n-1$ heads and $1$ tail (in any order), he eats another cookie. [/*] [/list] He never eats a cookie except as a result of his coin flips. Find all possible values of $n$ and $p$ such that the expected value of the number of cookies that Grogg eats each day is exactly $1$.

Circle of radius $r$ is inscribed in a triangle. Tangent lines parallel to the sides of triangle cut three small triangles. Let $r_1,r_2,r_3$ be radii of circles inscribed in these triangles. Prove that \[ r_1 + r_2 + r_3 = r. \]

The vertices of a convex pentagon $ABCDE$ lie on a circle $\gamma_1$. The diagonals $AC , CE, EB, BD$, and $DA$ are tangents to another circle $\gamma_2$ with the same centre as $\gamma_1$. (a) Show that all angles of the pentagon $ABCDE$ have the same size and that all edges of the pentagon have the same length. (b) What is the ratio of the radii of the circles $\gamma_1$ and $\gamma_2$? (The answer should be given in terms of integers, the four basic arithmetic operations and extraction of roots only.)

Consider an octogon with equal angles and rational side lengths. Prove that it has a symmetry center.

Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$. Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.

Every cell of a $4\times 4$ square grid net, has $1\times 1$ size. Is it possible to represent this net as a union of the following sets: a) Eight broken lines of length five each? b) Five broken lines of length eight each?