The George Washington Bridge is $2016$ meters long. Sally is standing on the George Washington Bridge, $1010$ meters from its left end. Each step, she either moves $1$ meter to the left or $1$ meter to the right, each with probability $\dfrac{1}{2}$. What is the expected number of steps she will take to reach an end of the bridge?

A sequence $(a_n)$ of positive integers is defined by $a_0=m$ and $a_{n+1}= a_n^5 +487$ for all $n\ge 0$. Find all positive integers $m$ such that the sequence contains the maximum possible number of perfect squares.

For a positive integer, $m$, answer the following questions. 1) Show that $2^{m+1}+1$ is a prime number, when $2^{m+1}+1$ is a factor of $3^{2^m}+1$. 2) Is converse of 1) true?

Set $T$ consists of $66$ points in plane, and $P$ consists of $16$ lines in plane. Pair $(A,l)$ is [i]good[/i] if $A \in T$, $l \in P$ and $A \in l$. Prove that maximum number of good pairs is no greater than $159$, and prove that there exits configuration with exactly $159$ good pairs.

Prove that there exist integers $a$, $b$, $c$ with $1 \le a < b < c \le 25$ and \[ S(a^6+2014) = S(b^6+2014) = S(c^6+2014) \] where $S(n)$ denotes the sum of the decimal digits of $n$. [i]Proposed by Evan Chen[/i]

An arbitrary point $D$ was marked on the median $BM$ of the triangle $ABC$. It is known that the point $DE\parallel AB$ and $CE \parallel BM$. Prove that $BE = AD$

There are 100 students taking an exam. The professor calls them one by one and asks each student a single person question: “How many of 100 students will have a “passed” mark by the end of this exam?” The students answer must be an integer. Upon receiving the answer, the professor immediately publicly announces the student’s mark which is either “passed” or “failed.” After all the students have got their marks, an inspector comes and checks if there is any student who gave the correct answer but got a “failed” mark. If at least one such student exists, then the professor is suspended and all the marks are replaced with “passed.” Otherwise no changes are made. Can the students come up with a strategy that guarantees a “passed” mark to each of them? [i] Denis Afrizonov [/i]

There were $n> 1$ aborigines living on an island, each of them telling only the truth or only lying, and each having at least one friend among the others. The new governor asked each aborigine whether there are more truthful aborigines or liars among his friends, or an equal number of both. Each aborigine answered that there are more liars than truthful aborigines among his friends. The governor then ordered one of the aborigines to be executed for being a liar and asked each of the remaining $n- 1$ aborigines the same question again. This time each aborigine answered that there are more truthful aborigines than liars among his friends. Determine whether the executed aborigine was truthful or a liar, and whether there are more truthful aborigines or liars remaining on the island.

Given is a triangle $ABC$, $\angle C = 60^o$, $R$ the midpoint of side $AB$. There exist a point $P$ on the line $BC$ and a point $Q$ on the line $AC$ such that the perimeter of the triangle $PQR$ is minimal. a) Prove that and also indicate how the points $P$ and $Q$ can be constructed. b) If $AB = c$, $AC = b$, $BC = a$, then prove that the perimeter of the triangle $PQR$ equals $\frac12\sqrt{3c^2+6ab}$ .

Each positive integer shall be coloured red or green such that it satisfies the following properties: - The sum of three not necessarily distinct red numbers is a red number. - The sum of three not necessarily distinct green numbers is a green number. - There are red and green numbers. Find all such colorations!

The sequence $a = (a_0, a_1,a_2,...)$ is defined by $a_0 = 0, a_1 =2$ and $$a_{n+2} = 2a_{n+1} + 41a_n$$Prove that $a_{2016}$ is divisible by $2017.$

Let $m$ and $n$ be positive integers. A circular necklace contains $mn$ beads, each either red or blue. It turned out that no matter how the necklace was cut into $m$ blocks of $n$ consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair $(m, n)$. [i]Proposed by Rishabh Das[/i]

Let $(a_n)\subset (\frac{1}{2},1)$. Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$. Is this sequence convergent? If yes find the limit.

Compute the number of positive integers $n \le 1890$ such that n leaves an odd remainder when divided by all of $2, 3, 5$, and $7$.

Let $ABC$ be a triangle. Points $D$ and $E$ are placed on $\overline{AC}$ in the order $A$, $D$, $E$, and $C$, and point $F$ lies on $\overline{AB}$ with $EF\parallel BC$. Line segments $\overline{BD}$ and $\overline{EF}$ meet at $X$. If $AD = 1$, $DE = 3$, $EC = 5$, and $EF = 4$, compute $FX$.

Let $ABC$ be scalene and acute triangle with $CA>CB$ and let $P$ be an internal point, satisfying $\angle APB=180^{\circ}-\angle ACB$; the lines $AP, BP$ meet $BC, CA$ at $A_1, B_1$. If $M$ is the midpoint of $A_1B_1$ and $(A_1B_1C)$ meets $(ABC)$ at $Q$, show that $\angle PQM=\angle BQA_1$.

A real number $x$ satisfies $9^x + 3^x = 6$. Compute the value of $16^{1/x} + 4^{1/x} $.

Three integers $A, B, C$ are written on a whiteboard. Every move, Mr. Doba can either subtract $1$ from all numbers on the board, or choose two numbers on the board and subtract $1$ from both of them whilst leaving the third untouched. For how many ordered triples $(A, B, C)$ with $1 \le A < B < C\le 20$ is it possible for Mr. Doba to turn all three of the numbers on the board to $0$?

Kyle found the sum of the digits of $2014^{2014}$. Then, Shannon found the sum of the digits of Kyle's result. Finally, James found the sum of the digits of Shannon's result. What number did James find? $\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }11\qquad\text{(D) }16\qquad\text{(E) }18$

Find all positive integers $m,n$ with $m \leq 2n$ that solve the equation \[ m \cdot \binom{2n}{n} = \binom{m^2}{2}. \] [i](German MO 2016 - Problem 4)[/i]

A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? $\textbf{(A) } 200 \qquad\textbf{(B) } 200\sqrt{2} \qquad\textbf{(C) } 200\sqrt{3} \qquad\textbf{(D) } 300\sqrt{2} \qquad\textbf{(E) } 500$

In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played? $\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$

There are three cities each of which has exactly the same number of citizens, say $n$. Every citizen in each city has exactly a total of $(n+1)$ friends in the other two cities. Show that there exist three people, one from each city, such that they are friends. We assume that friendship is mutual (that is, a symmetric relation).

Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that \[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]