Let $\mathcal{R}$ denote the circular region in the $xy-$plane bounded by the circle $x^2+y^2=36$. The lines $x=4$ and $y=3$ divide $\mathcal{R}$ into four regions $\mathcal{R}_i ~ , ~i=1,2,3,4$. If $\mid \mathcal{R}_i \mid$ denotes the area of the region $\mathcal{R}_i$ and if $\mid \mathcal{R}_1 \mid >$ $\mid \mathcal{R}_2 \mid >$ $\mid \mathcal{R}_3 \mid > $ $\mid \mathcal{R}_4 \mid $, determine $\mid \mathcal{R}_1 \mid $ $-$ $\mid \mathcal{R}_2 \mid $ $-$ $\mid \mathcal{R}_3 \mid $ $+$ $\mid \mathcal{R}_4 \mid $.

Find the greatest real number $M$ for which \[ a^2+b^2+c^2+3abc \geq M(ab+bc+ca) \] for all non-negative real numbers $a,b,c$ satisfying $a+b+c=4.$

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

The famous conjecture of Goldbach is the assertion that every even integer greater than $2$ is the sum of two primes. Except $2$, $4$, and $6$, every even integer is a sum of two positive composite integers: $n=4+(n-4)$. What is the largest positive even integer that is not a sum of two odd composite integers?

Alice and Bob take $a$ and $b$ candies respectively, where $0\leq a,b\leq3$, from a pile of $6$ identical candies. They draw the candies one at a time, but one person may draw multiple candies in a row. For example, if $a=2$ and $b=3$, a possible order of drawing could be Alice, Bob, Bob, Alice, Bob. In how many ways (considering order of drawing and values of $a$ and $b$) can this happen? [i]2017 CCA Math Bonanza Team Round #6[/i]

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

Find all functions $f : \mathbb R \to \mathbb R$ such that the equality \[y^2f(x) + x^2f(y) + xy = xyf(x + y) + x^2 + y^2\] holds for all $x, y \in \Bbb R$, where $\Bbb R$ is the set of real numbers.

Let $a,b,c\in\mathbb R^{n}, a+b+c=0$ and $\lambda>0$. Prove that \[\prod_{cycle}\frac{|a|+|b|+(2\lambda+1)|c|}{|a|+|b|+|c|}\geq(2\lambda+3)^{3}\]

Penni Precisely buys $\$100$ worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up $20\%$, BB was down $25\%$, and CC was unchanged. For the second year, AA was down $20\%$ from the previous year, BB was up $25\%$ from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then $\textbf{(A)}\ A=B=C \qquad \textbf{(B)}\ A=B<C \qquad \textbf{(C)}\ C<B=A$ $\textbf{(D)}\ A<B<C \qquad \textbf{(E)}\ B<A<C$

Consider the power series expansion \[\dfrac{1}{1-2x-x^2}=\sum_{n=0}^\infty a_nx^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2+a_{n+1}^2=a_m.\]

Let $S$ be the sum of the greatest odd divisors of the natural numbers $1$ through $2^n$. Prove that $3S = 4^n + 2$.

Let $S$ be the set of all those 2007 place decimal integers $\overline{2s_1a_2a_3 \ldots a_{2006}}$ which contain odd number of digit $9$ in each sequence $a_1, a_2, a_3, \ldots, a_{2006}$. The cardinal number of $S$ is ${ \textbf{(A)}\ \frac{1}{2}(10^{2006}+8^{2006})\qquad\textbf{(B)}\ \frac{1}{2}(10^{2006}-8^{2006})\qquad\textbf{(C)}\ 10^{2006}+8^{2006}\qquad \textbf{(D)}}\ 10^{2006}-8^{2006}\qquad $

The integers between $1$ and $9$ inclusive are distributed in the units of a $3$ x $3$ table. You sum six numbers of three digits: three that are read in the rows from left to right, and three that are read in the columns from top to bottom. Is there any such distribution for which the value of this sum is equal to $2001$?

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?

A sequence of polynomial $f_n(x)\ (n=0,1,2,\cdots)$ satisfies $f_0(x)=2,f_1(x)=x$, \[f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots)\] Let $x_n\ (n\geqq 2)$ be the maximum real root of the equation $f_n(x)=0\ (|x|\leqq 2)$ Evaluate \[\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx\]

Let $ABC$ be a triangle. A point $P$ lies on the segment $BC$ such that the circle with diameter $BP$ passes through the incenter of $ABC$. Show that $\frac{BP}{PC}=\frac{c}{s-c}$ where $c$ is the length of segment $AB$ and $2s$ is the perimeter of $ABC$.

Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. See the figure below. The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$. What is $a+b$? [asy] unitsize(12); pair A = (0, 8/sqrt(3)), B = rotate(-120)*A, C = rotate(120)*A; real theta = 41.5; pair P1 = rotate(theta)*(2+2*sqrt(7/3), 0), P2 = rotate(-120)*P1, P3 = rotate(120)*P1; filldraw(P1--P2--P3--cycle, gray(0.9)); draw(Circle(A, 4)); draw(Circle(B, 4)); draw(Circle(C, 4)); dot(P1); dot(P2); dot(P3); defaultpen(fontsize(10pt)); label("$P_1$", P1, E*1.5); label("$P_2$", P2, SW*1.5); label("$P_3$", P3, N); label("$\omega_1$", A, W*17); label("$\omega_2$", B, E*17); label("$\omega_3$", C, W*17); [/asy] $\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554$

Jenn is competing in a puzzle hunt with six regular puzzles and one additional meta-puzzle. Jenn can solve any puzzle regularly. Additionally, if she has already solved the meta-puzzle, Jenn can also back-solve a puzzle. A back-solve is distinguishable from a regular solve. The meta puzzle cannot be the first puzzle solved. How many possible solve orders for the seven puzzles are possible? For example, Jenn may solve #3, solve #5, solve #6, solve the meta-puzzle, solve #2, solve #1, and then solve #4. However, she may not solve #2, solve #4, solve #6, back-solve #1, solve #3, solve #5, and then solve the meta-puzzle.

A piece of paper is the square $ABCD$. We fold it by placing the vertex $D$ on the point $D' $ of the side $BC$. We assume that $AD$ moves on the segment $A' D'$ and that $A' D' $ intersects $AB$ at $E$. Prove that the perimeter of the triangle $EBD' $ is one half of the perimeter of the square.

We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$. Compute the greatest possible weight of a positive integer less than 100.

Let $n > 1$ be an integer and let $a_1, a_2, \ldots, a_n$ be integers such that $n \mid a_i-i$ for all integers $1 \leq i \leq n$. Prove there exists an infinite sequence $b_1,b_2, \ldots$ such that [list] [*] $b_k\in\{a_1,a_2,\ldots, a_n\}$ for all positive integers $k$, and [*] $\sum\limits_{k=1}^{\infty}\frac{b_k}{n^k}$ is an integer. [/list]

Sam writes three $3$-digit positive integers (that don't end in $0$) on the board and adds them together. Jessica reverses each of the integers, and adds the reversals together. (For example, $\overline{XYZ}$ becomes $\overline{ZYX}$.) What is the smallest possible positive three-digit difference between Sam's sum and Jessica's sum?

When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even? $\textbf{(A) }\dfrac38\qquad\textbf{(B) }\dfrac49\qquad\textbf{(C) }\dfrac59\qquad\textbf{(D) }\dfrac9{16}\qquad\textbf{(E) }\dfrac58$

Point $P$ is inside triangle $\triangle{ABC}$ such that $\angle{ABP}=\angle{ACP}.$ Given that $AB=6, AC=8, BC=7,$ and $\tfrac{BP}{PC}=\tfrac{1}{2},$ compute $\tfrac{[BPC]}{[ABC]}.$ (Here, $[XYZ]$ denotes the area of $\triangle{XYZ}.$)

The parliament of the country Ar consists of two houses, upper and lower, both have the same number of people. The law says that each member must vote "Yes" or "No". One day, when all members of both houses were present and voted on an important issue, the speaker informed the press that the number of members voted "Yes" was greater by $23$ than the number of members voted "No". Prove that he made a mistake.