Nickolas and Peter divide $2n+1$ nuts amongst each other. Both of them want to get as many as possible. Three methods are suggested to them for doing so, each consisting of three stages. The first two stages are the same in all three methods: [i]Stage 1:[/i] Peter divides the nuts into 2 heaps, each containing at least 2 nuts. [i]Stage 2:[/i] Nickolas divides both heaps into 2 heaps, each containing at least 1 nut. Finally, stage 3 varies among the three methods as follows: [i]Method 1:[/i] Nickolas takes the smallest and largest of the heaps. [i]Method 2:[/i] Nickolas takes the two middle size heaps. [i]Method 3:[/i] Nickolas chooses between taking the biggest and the smallest heap or the two middle size heaps, but gives one nut to Peter for the right of choice. Determine the most and the least profitable method for Nickolas.

Two circles $\omega_1(O)$ and $\omega_2$ intersect each other at $A,B$ ,and $O$ lies on $\omega_2$. Let $S$ be a point on $AB$ such that $OS\perp AB$. Line $OS$ intersects $\omega_2$  at $P$ (other than $O$). The bisector of $\hat{ASP}$ intersects  $\omega_1$ at $L$ ($A$ and $L$ are on the same side of the line $OP$). Let $K$ be a point on $\omega_2$ such that $PS=PK$ ($A$ and $K$ are on the same side of the line $OP$). Prove that $SL=KL$. [i]Proposed by Ali Zamani [/i]

(a) Prove that the number of all digits in the sequence $1, 2, 3,... , 10^8$ is equal to the number of all zeros in the sequence $1, 2, 3, ... , 10^9$. (b) Prove that the number of all digits in the sequence $1, 2, 3, ... , 10^k$ is equal to the number of all zeros in the sequence $1, 2, 3, ... , 10^{k+1}$.

1. If five fair coins are flipped simultaneously, what is the probability that at least three of them show heads? 2. How many perfect squares divide $10^{10}$? 3. Evaluate $\frac{2016!^2}{2015!2017!}$ . Here $n!$ denotes $1 \times 2 \times \ldots \times n$.

Inside triangle $ABC$ there is a chosen point $P$. The rays $AP$, $BP$, $CP$ intersect the boundary of the triangle in the points $A'$, $B'$, $C'$ respectively. Set $u = |AP| : |PA'|, v = |BP| : |PB'|, w = |CP| : |PC'|$. Express the product $uvw$ in terms of the sum $u + v + w$.

Let $P$ be a non-selfintersecting closed polygon with $n$ sides. Let its vertices be $P_1 , P_2 ,\ldots, P_n .$ Let $m$ other points,$Q_1 , Q_2 ,\ldots, Q_m $ , interior to $P$, be given. Let the figure be triangulated. This means that certain pairs of the $(n+m)$ points $P_1 ,\ldots , Q_m$ are connected by line segments such that (i) the resulting figure consists exclusively of a set $T$ of triangles, (ii) if two different triangles in $T$ have more than a vertex in common then they have exactly a side in common, and (iii) the set of vertices of the triangles in $T$ is precisely the set of the $(n+m)$ points $P_1 ,\ldots , Q_m.$ How many triangles are in $T$?

Consider an acute non-isosceles triangle. In a single step it is allowed to cut any one of the available triangles into two triangles along its median. Is it possible that after a finite number of cuttings all triangles will be isosceles? [i]Proposed by E. Bakaev[/i]

Prove by the method of induction that: [b]a)[/b] $ a!b! $ divides $ (a+b)! , $ for any natural numbers $ a,b. $ [b]b)[/b] $ p $ divides $ (-1)^{k+1} +\binom{p-1}{k} , $ for any odd primes $ p $ and $ k\in\{ 0,1,\ldots ,p-1\} . $

In the $xy$-plane, how many lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer pass through the point $(4,3)$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

There are exactly two four-digit numbers that are multiples of three where their first digit is double their second digit, their third digit is three more than their fourth digit, and their second digit is $2$ less than their fourth digit. Find the difference of these two numbers.

Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? $\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

In each cell of a $4 \times 4$ grid, one of the two diagonals is drawn uniformly at random. Compute the probability that the resulting $32$ triangular regions can be colored red and blue so that any two regions sharing an edge have different colors.

Let $C_1$ and $C_2$ be two concentric circles with center $O$, in such a way that the radius of $C_1$ is smaller than the radius of $C_2$. Let $P$ be a point other than $O$ that is in the interior of $C_1$, and $L$ a line through $P$ and intersects $C_1$ at $A$ and $B$. Ray $\overrightarrow{OB}$ intersects $C_2$ at $C$. Determine the locus that determines the circumcenter of triangle $ABC$ as $L$ varies.

Define \[f(h,t) = \begin{cases} 8h & h = t \\ (h-t)^2 & h \neq t. \end{cases}\] Cody plays a game with a fair coin, where he begins by flipping it once. At each turn in the game, if he has flipped $h$ heads and $t$ tails and $h + t < 6$, he can choose either to stop and receive $f(h,t)$ dollars or he can flip the coin again; if $h + t = 6$ then the game ends and he receives $f(h,t)$ dollars. If Cody plays to maximize expectancy, how much money, in dollars, can he expect to win from this game?

Five points are chosen on the sphere, no three of them lying on a great circle (a great circle is the intersection of the sphere with some plane passing through the sphere’s centre). Two great circles not containing any of the chosen points are called equivalent if one of them can be moved to the other without passing through any chosen points. (a) How many nonequivalent great circles not containing any chosen points can be drawn on the sphere? (b) Answer the same problem, but with $n$ chosen points.

At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for \$5. This week they are on sale at 5 boxes for \$4. The percent decrease in the price per box during the sale was closest to $\textbf{(A)}\ 30\% \qquad \textbf{(B)}\ 35\% \qquad \textbf{(C)}\ 40\% \qquad \textbf{(D)}\ 45\% \qquad \textbf{(E)}\ 65\%$

For every positive integer$ n$, let $P(n)$ be the greatest prime divisor of $n^2+1$. Show that there are infinitely many quadruples $(a, b, c, d)$ of positive integers that satisfy $a < b < c < d$ and $P(a) = P(b) = P(c) = P(d)$.

In each cell of the table $ 3 \times 3 $ there is one of the numbers $1, 2$ and $3$. Dima counted the sum of the numbers in each row and in each column. What is the greatest number of different sums he could get?

Define an [i]upno[/i] to be a positive integer of $2$ or more digits where the digits are strictly increasing moving left to right. Similarly, define a [i]downno[/i] to be a positive integer of $2$ or more digits where the digits are strictly decreasing moving left to right. For instance, the number $258$ is an [i]upno[/i] and $8620$ is a [i]downno[/i]. Let $U$ equal the total number of [i]upno[/i]s and let $D$ equal the total number of [i]downno[/i]s. What is $|U-D|$? $\textbf{(A)}~512\qquad\textbf{(B)}~10\qquad\textbf{(C)}~0\qquad\textbf{(D)}~9\qquad\textbf{(E)}~511$

Let $a$, $b$, $c$ be real numbers, not all of them are equal. Prove that $a+b+c=0$ if and only if $a^2+ab+b^2=b^2+bc+c^2=c^2+ca+a^2$.

The $48$ faces of $8$ unit cubes are painted white. What is the smallest number of these faces that can be repainted black so that it becomes impossible to arrange the $8$ unit cubes into a two by two by two cube, each of whose $6$ faces is totally white?

For how many postive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$? $ \textbf{(A)}\ 212 \qquad\textbf{(B)}\ 206 \qquad\textbf{(C)}\ 191 \qquad\textbf{(D)}\ 185 \qquad\textbf{(E)}\ 173 $

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

In a planar coordinate system, the points have non-negative integer coordinates numbered according to the figure. E.g. the point $(3,1)$ has the number $12$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/28725d75f281ac4618129067037d751c8d8f83.png[/img] What is the number of the point$(x,y)$?

Six tennis players gather to play in a tournament where each pair of persons play one game, with one person declared the winner and the other person the loser. A triplet of three players $\{\mathit{A}, \mathit{B}, \mathit{C}\}$ is said to be [i]cyclic[/i] if $\mathit{A}$ wins against $\mathit{B}$, $\mathit{B}$ wins against $\mathit{C}$ and $\mathit{C}$ wins against $\mathit{A}$. [list] [*] (a) After the tournament, the six people are to be separated in two rooms such that none of the two rooms contains a cyclic triplet. Prove that this is always possible. [*] (b) Suppose there are instead seven people in the tournament. Is it always possible that the seven people can be separated in two rooms such that none of the two rooms contains a cyclic triplet?[/list]