Let $a$ and $b$ be positive real numbers. Consider a regular hexagon of side $a$, and build externally on its sides six rectangles of sides $a$ and $b$. The new twelve vertices lie on a circle. Now repeat the same construction, but this time exchanging the roles of $a$ and $b$; namely; we start with a regular hexagon of side $b$ and we build externally on its sides six rectangles of sides $a$ and $b$. The new twelve vertices lie on another circle. Show that the two circles have the same radius.

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

What is the largest 2-digit prime factor of the integer $n = \binom{200}{100}$?

Which one is true for a quadrilateral $ABCD$ such that perpendicular bisectors of $[AB]$ and $[CD]$ meet on the diagonal $[AC]$? $\textbf{(A)}\ |BA| + |AD| \leq |BC| + |CD| \\ \textbf{(B)}\ |BD| \leq |AC| \\ \textbf{(C)}\ |AC| \leq |BD| \\ \textbf{(D)}\ |AD| + |DC| \leq |AB| + |BC| \\ \textbf{(E)}\ \text{None}$

There are nine points in the data square, of which no three are collinear. Prove that three of them are vertices of a triangle with an area not exceeding $ \frac{1}{8} $ the area of a square.

The faces of a cube are painted in six different colors: red (R), white (W), green (G), brown (B), aqua (A), and purple (P). Three views of the cube are shown below. What is the color of the face opposite the aqua face? $\text{\textbf{(A) }red}\qquad \text{\textbf{(B) } white}\qquad\text{\textbf{(C) }green}\qquad\text{\textbf{(D) }brown }\qquad\text{\textbf{(E)} purple}$ [asy] unitsize(2 cm); pair x, y, z, trans; int i; x = dir(-5); y = (0.6,0.5); z = (0,1); trans = (2,0); for (i = 0; i <= 2; ++i) { draw(shift(i*trans)*((0,0)--x--(x + y)--(x + y + z)--(y + z)--z--cycle)); draw(shift(i*trans)*((x + z)--x)); draw(shift(i*trans)*((x + z)--(x + y + z))); draw(shift(i*trans)*((x + z)--z)); } label(rotate(-3)*"$R$", (x + z)/2); label(rotate(-5)*slant(0.5)*"$B$", ((x + z) + (y + z))/2); label(rotate(35)*slant(0.5)*"$G$", ((x + z) + (x + y))/2); label(rotate(-3)*"$W$", (x + z)/2 + trans); label(rotate(50)*slant(-1)*"$B$", ((x + z) + (y + z))/2 + trans); label(rotate(35)*slant(0.5)*"$R$", ((x + z) + (x + y))/2 + trans); label(rotate(-3)*"$P$", (x + z)/2 + 2*trans); label(rotate(-5)*slant(0.5)*"$R$", ((x + z) + (y + z))/2 + 2*trans); label(rotate(-85)*slant(-1)*"$G$", ((x + z) + (x + y))/2 + 2*trans); [/asy]

We consider the set E of all fractions $\frac{1}{n}$, where $n$ is a natural number. A maximal arithmetic progression of length $k$ of the set E is an arithmetic progression of $k$ terms such that all its terms belong to the set E, and it is impossible to extend it to the right or to the left with another element of E. For example, $\frac{1}{20}, \frac{1}{8}, \frac{1}{5}$, is an arithmetic progression in E of length $3$, and it is maximal, since to extend it towards to the right you have to add $\frac{11}{40}$, which does not belong to E, and to extend it to the left you have to add $\frac{-1}{40}$ which does not belong to E either. Prove that for every integer $k&gt; 2$, there exists a maximal arithmetic progression of length $k$ of the set E.

Let $g$ be a real-valued function that is continuous on the closed interval $[0,1]$ and twice differentiable on the open interval $(0,1)$.  Suppose that for some real number $r>1$, \[ \lim_{x\to 0^+}\frac{g(x)}{x^r} = 0. \] Prove that either \[ \lim_{x\to 0^+}g'(x) = 0\qquad\text{or}\qquad \limsup_{x\to 0^+}x^r|g''(x)|= \infty. \]

Let there exist a configuration of [i]exactly[/i] 1 black king, $n$ black chess pieces (each of which can be a pawn, knight, bishop, rook, or queen), and a white [i]anti-king[/i] on a standard 8x8 board in which the white [i]anti-king[/i] is not under attack, but will be if it is moved. Compute the minimal value of $n$. *An [i]anti-king[/i] can move to any square is [b]not[/b] 1 square vertically, horizontally, or diagonally. It can also capture undefended pieces. [i]2022 CCA Math Bonanza Team Round #4[/i]

Let $a,b,$ and $c$ be positive integers with $a\ge b\ge c$ such that \begin{align*} a^2-b^2-c^2+ab&=2011\text{ and}\\ a^2+3b^2+3c^2-3ab-2ac-2bc&=-1997\end{align*} What is $a$? $ \textbf{(A)}\ 249 \qquad\textbf{(B)}\ 250 \qquad\textbf{(C)}\ 251 \qquad\textbf{(D)}\ 252 \qquad\textbf{(E)}\ 253 $

A function $f : R \to R$ satisfy the functional equation $f(x + 2y) + 2f(y - 2x) = 3x -4y + 6$ for all reals $x, y$. Compute $f(2548)$.

What is a centrally symmetric polygon of greatest area one can inscribe in a given triangle?

In the rectangular parallelepiped $ABCDA'B'C'D'$, the points $E$ and $F$ are the centers of the faces $ABCD$ and $ADD' A'$, respectively, and the planes $(BCF)$ and $(B'C'E)$ are perpendicular. Let $A'M \perp B'A$, $M \in B'A$ and $BN \perp B'C$, $N \in B'C$. Denote $n = \frac{C'D}{BN}$. a) Show that $n \ge \sqrt2$. . b) Express and in terms of $n$, the ratio between the volume of the tetrahedron $BB'M N$ and the volume of the parallelepiped $ABCDA'B'C'D'$.

In a tennis tournament, $n$ players want to make $2$ vs $2$ matches such that each player has each of the other players as opponents exactly once. Find all possible values of $n$.

If $a,b,c$ are positive reals such that $a+b+c=1$, prove that \[ 10(a^3+b^3+c^3)-9(a^5+b^5+c^5)\geq 1 . \]

What is the smallest value that the sum of the digits of the number $3n^2+n+1,$ $n\in\mathbb{N}$ can take?

For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.

Let $n$ be a positive integer. What is the smallest value of $m$ with $m > n$ such that the set $M = \{n, n + 1, ..., m\}$ can be partitioned into subsets so that in each subset, there is a number which equals to the sum of all other numbers of this subset?

Let $ABCD$ be a cyclic quadrilateral with $AB = 6$, $BC = 12$, $CD = 3$, and $DA = 6$. Let $E, F$ be the intersection of lines $AB$ and $CD$, lines $AD$ and $BC$, respectively. Find $EF$.

Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and \[T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).\] Show that for all $k$, \[1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],\] where $[x]$ denotes the greatest integer not exceeding $x.$

Let $f$ be a function defined on the positive integers with $f(n) \ge 0$ and $f(n) \le f(n+1)$ for all $n$. Prove that if \[\sum_{n = 1}^{\infty} \frac{f(n)}{n^2}\] diverges, there exists a sequence $a_1, a_2, \dots$ such that the sequence $\tfrac{a_n}{n}$ hits every natural number, while \[a_{n+m} \le a_n + a_m + f(n+m)\] holds for every pair $n$, $m$.

On this monthly calendar, the date behind one of the letters is added to the date behind $C$. If this sum equals the sum of the dates behind $A$ and $B$, then the letter is [asy] unitsize(12); draw((1,1)--(23,1)); draw((0,5)--(23,5)); draw((0,9)--(23,9)); draw((0,13)--(23,13)); for(int a=0; a<6; ++a) { draw((4a+2,0)--(4a+2,14)); } label("Tues.",(4,14),N); label("Wed.",(8,14),N); label("Thurs.",(12,14),N); label("Fri.",(16,14),N); label("Sat.",(20,14),N); label("C",(12,10.3),N); label("$\textbf{A}$",(16,10.3),N); label("Q",(12,6.3),N); label("S",(4,2.3),N); label("$\textbf{B}$",(8,2.3),N); label("P",(12,2.3),N); label("T",(16,2.3),N); label("R",(20,2.3),N);[/asy] $ \text{(A)}\ \text{P}\qquad\text{(B)}\ \text{Q}\qquad\text{(C)}\ \text{R}\qquad\text{(D)}\ \text{S}\qquad\text{(E)}\ \text{T} $

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Prove that when $ f, m, n $, are any non-negative integers, then the polynomial $$ P(x) = x^{3k+2} + x^{3m+1} + x^{3n}$$ is divisible by the polynomial $ x^2 + x + 1 $.

Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Nathan Xiong[/i]