Suppose that $n$ consecutive positive integers were written on the board, where $n > 6$. Then some $5$ of the written numbers were erased, and it turned out that any two of the remaining numbers are coprime. Find the largest possible value of $n$.

For each letter in the English alphabet, William assigns an English word which contains that letter. His first document consists only of the word assigned to the letter $A$. In each subsequent document, he replaces each letter of the preceding document by its assigned word. The fortieth document begins with “Till whatsoever star that guides my moving.” Prove that this sentence reappears later in this document.

Let the sequence $(a_n)$ be defined as $a_1=\frac 13$ and $a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}$ for every $n\geq 1$. If $a_k$ is the largest term of the sequence satisfying $a_k < \frac 1{50}$, what is $k$? $ \textbf{(A)}\ 194 \qquad\textbf{(B)}\ 193 \qquad\textbf{(C)}\ 192 \qquad\textbf{(D)}\ 191 \qquad\textbf{(E)}\ \text{None of the above} $

By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$). For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$). Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.

Prove that there exist infinitely many natural numbers $n$, for which there $\exists \, f:\{0,1…n-1\}\rightarrow \{0,1…n-1\}$, satisfying the following conditions: 1) $f(x)\neq x$; 2) $f(f(x))=x$; 3) $f(f(f(x+1)+1)+1)=x$ for $\forall x\in \{0,1…n-1\}$.

[b]p1.[/b] We say that integers $a$ and $b$ are [i]friends [/i] if their product is a perfect square. Prove that if $a$ is a friend of $b$, then $a$ is a friend of $gcd (a, b)$. [b]p2.[/b] On the island of knights and liars, a traveler visited his friend, a knight, and saw him sitting at a round table with five guests. "I wonder how many knights are among you?" he asked. " Ask everyone a question and find out yourself" advised him one of the guests. "Okay. Tell me one: Who are your neighbors?" asked the traveler. This question was answered the same way by all the guests. "This information is not enough!" said the traveler. "But today is my birthday, do not forget it!" said one of the guests. "Yes, today is his birthday!" said his neighbor. Now the traveler was able to find out how many knights were at the table. Indeed, how many of them were there if [i]knights always tell the truth and liars always lie[/i]? [b]p3.[/b] A rope is folded in half, then in half again, then in half yet again. Then all the layers of the rope were cut in the same place. What is the length of the rope if you know that one of the pieces obtained has length of $9$ meters and another has length $4$ meters? [b]p4.[/b] The floor plan of the palace of the Shah is a square of dimensions $6 \times 6$, divided into rooms of dimensions $1 \times 1$. In the middle of each wall between rooms is a door. The Shah orders his architect to eliminate some of the walls so that all rooms have dimensions $2 \times 1$, no new doors are created, and a path between any two rooms has no more than $N$ doors. What is the smallest value of $N$ such that the order could be executed? [b]p5.[/b] There are $10$ consecutive positive integers written on a blackboard. One number is erased. The sum of remaining nine integers is $2011$. Which number was erased? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f: [0,1]\rightarrow(0,+\infty)$ be a continuous function. a) Show that for any integer $n\geq 1$, there is a unique division $0=a_{0}<a_{1}<\ldots<a_{n}=1$ such that $\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx$ holds for all $k=0,1,\ldots,n-1$. b) For each $n$, consider the $a_{i}$ above (that depend on $n$) and define $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$. Show that the sequence $(b_{n})$ is convergent and compute it's limit.

Let $ABCD$ be a unit square, and suppose that $E$ and $F$ are on $\overline{AD}$ and $\overline{AB}$ such that $AE = AF = \tfrac23$. Let $\overline{CE}$ and $\overline{DF}$ intersect at $G$. If the area of $\triangle CFG$ can be expressed as simplified fraction $\frac{p}{q}$, find $p + q$. Let $T = TNYWR$. A total of $2T$ students go on a road trip. They take two cars, each of which seats $T$ people. Call two students \textit{friendly} if they sat together in the same car going to the trip and in the same car going back home. What is the smallest possible number of friendly pairs of students on the trip?

The Fibonacci sequence is defined as follows: $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for all integers $n\ge 2$. Find the smallest positive integer $m$ such that $F_m\equiv 0 \pmod {127}$ and $F_{m+1}\equiv 1\pmod {127}$.

Suppose that $a_1,a_2,\cdots,a_{100}\in\mathbb{R}^+$ such that $a_i\geq a_{101-i}\,(i=1,2,\cdots,50).$ Let $x_k=\frac{ka_{k+1}}{a_1+a_2+\cdots+a_k}\,(k=1,2,\cdots,99).$ Prove that $$x_1x_2^2\cdots x_{99}^{99}\leq 1.$$

We define $I(n)$ as the result when the digits of $n$ are reversed. For example, $I(123)=321$, $I(2008)=8002$. Find all integers $n$, $1\leq{n}\leq10000$ for which $I(n)=\lceil{\frac{n}{2}}\rceil$. Note: $\lceil{x}\rceil$ denotes the smallest integer greater than or equal to $x$. For example, $\lceil{2.1}\rceil=3$, $\lceil{3.9}\rceil=4$, $\lceil{7}\rceil=7$.

Given a $ 4\times 4$ squares table. How many ways that we can fill the table with $ \{0,1\}$ such that two neighbor squares (have one common side) have product which is equal to $ 0$?

In a game of Fish, R2 and R3 are each holding a positive number of cards so that they are collectively holding a total of $24$ cards. Each player gives an integer estimate for the number of cards he is holding, such that each estimate is an integer between $80\%$ of his actual number of cards and $120\%$ of his actual number of cards, inclusive. Find the smallest possible sum of the two estimates.

Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}$. Susan makes a list as follows: for each two-element subset of $\mathcal{S}$, she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.

Rectangle $ABCD$ measures $70$ by $40$. Eighteen points (including $A$ and $C$) are marked on the diagonal $AC$ dividing the diagonal into $17$ congruent pieces. Twenty-two points (including A and B) are marked on the side $AB$ dividing the side into $21$ congruent pieces. Seventeen non-overlapping triangles are constructed as shown. Each triangle has two vertices that are two of these adjacent marked points on the side of the rectangle, and one vertex that is one of the marked points along the diagonal of the rectangle. Only the left $17$ of the $21$ congruent pieces along the side of the rectangle are used as bases of these triangles. Find the sum of the areas of these $17$ triangles. [asy] size(200); defaultpen(linewidth(0.8)); pair A=origin,B=(21,0),C=(21,12),D=(0,12); path P=origin; draw(A--B--C--D--cycle--C); for (int r = 1; r <= 17;++r) { P=P--(21*r/17,12*r/17)--(r,0); } P=P--cycle; filldraw(P,gray(0.7)); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); [/asy]

Find the number of elements that a set $B$ can have, contained in $(1, 2, ... , n)$, according to the following property: For any elements $a$ and $b$ on $B$ ($a \ne b$), $(a-b) \not| (a+b)$.

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.

Let $S$ be the set of the reciprocals of the first $2016$ positive integers and $T$ the set of all subsets of $S$ that form arithmetic progressions. What is the largest possible number of terms in a member of $T$? [i]2016 CCA Math Bonanza Lightning #3.4[/i]

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

A six digit number (base 10) is squarish if it satisfies the following conditions: (i) none of its digits are zero; (ii) it is a perfect square; and (iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers. How many squarish numbers are there? $\text{(A)} \ 0 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 8 \qquad \text{(E)} \ 9$

Consider the set $E$ of all natural numbers $n$ such that whenn divided by $11, 12, 13$, respectively, the remainders, int that order, are distinct prime numbers in an arithmetic progression. If $N$ is the largest number in $E$, find the sum of digits of $N$.

Let $ABCDEF$ be a hexagon inscribed in a circle of radius $r.$ Show that if $AB=CD=EF=r,$ then the midpoints of $BC, DE$ and $FA$ are the vertices of an equilateral triangle.

An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold: ($1$) no one stands between the two tallest players, ($2$) no one stands between the third and fourth tallest players, $\;\;\vdots$ ($N$) no one stands between the two shortest players. Show that this is always possible. [i]Proposed by Grigory Chelnokov, Russia[/i]

Three positive numbers are such that the sum of any one of them with the sum of squares of the remaining two numbers is the same. Is it true that all numbers are the same? (Author: L. Emelyanov)