Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$ $\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$

John scores $93$ on this year's AHSME. Had the old scoring system still been in effect, he would score only $84$ for the same answers. How many questions does he leave unanswered? (In the new scoring system one receives 5 points for correct answers, 0 points for wrong answers, and 2 points for unanswered questions. In the old system, one started with 30 points, received 4 more for each correct answer, lost one point for each wrong answer, and neither gained nor lost points for unanswered questions. There are 30 questions in the 1986 AHSME.) $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ \text{Not uniquely determined} $

[b]p1.[/b] Jose,, Luciano, and Placido enjoy playing cards after their performances, and you are invited to deal. They use just nine cards, numbered from $2$ through $10$, and each player is to receive three cards. You hope to hand out the cards so that the following three conditions hold: A) When Jose and Luciano pick cards randomly from their piles, Luciano most often picks a card higher than Jose, B) When Luciano and Placido pick cards randomly from their piles, Placido most often picks a card higher than Luciano, C) When Placido and Jose pick cards randomly from their piles, Jose most often picks a card higher than Placido. Explain why it is impossible to distribute the nine cards so as to satisfy these three conditions, or give an example of one such distribution. [b]p2.[/b] Is it possible to fill a rectangular box with a finite number of solid cubes (two or more), each with a different edge length? Justify your answer. [b]p3.[/b] Two parallel lines pass through the points $(0, 1)$ and $(-1, 0)$. Two other lines are drawn through $(1, 0)$ and $(0, 0)$, each perpendicular to the ¯rst two. The two sets of lines intersect in four points that are the vertices of a square. Find all possible equations for the first two lines. [b]p4.[/b] Suppose $a_1, a_2, a_3,...$ is a sequence of integers that represent data to be transmitted across a communication channel. Engineers use the quantity $$G(n) =(1 - \sqrt3)a_n -(3 - \sqrt3)a_{n+1} +(3 + \sqrt3)a_{n+2}-(1+ \sqrt3)a_{n+3}$$ to detect noise in the signal. a. Show that if the numbers $a_1, a_2, a_3,...$ are in arithmetic progression, then $G(n) = 0$ for all $n = 1, 2, 3, ...$. b. Show that if $G(n) = 0$ for all $n = 1, 2, 3, ...$, then $a_1, a_2, a_3,...$ is an arithmetic progression. [b]p5.[/b] The Olive View Airline in the remote country of Kuklafrania has decided to use the following rule to establish its air routes: If $A$ and $B$ are two distinct cities, then there is to be an air route connecting $A$ with $B$ either if there is no city closer to $A$ than $B$ or if there is no city closer to $B$ than $A$. No further routes will be permitted. Distances between Kuklafranian cities are never equal. Prove that no city will be connected by air routes to more than ¯ve other cities. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the board is written in decimal the integer positive number $N$. If it is not a single digit number, wipe its last digit $c$ and replace the number $m$ that remains on the board with a number $m -3c$. (For example, if $N = 1,204$ on the board, $120 - 3 \cdot 4 = 108$.) Find all the natural numbers $N$, by repeating the adjustment described eventually we get the number $0$.

Prove that $(a+b+c)^5 \ge 81 (a^2+b^2+c^2)abc$ for any positive real numbers $a,b,c$ I.Gorodnin

Let $a_1, a_2, \dots, a_n$ be positive real numbers, whose sum is $1$. Prove that \[ \frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+\dots+\frac{a_{n-1}^2}{a_{n-1}+a_n}+\frac{a_n^2}{a_n+a_1}\ge \frac{1}{2} \]

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 $EF$.