Let $n$ and $a \leq n$ be two positive integers. There's $2n$ people sitting around a circle reqularly. Two people are friend iff one of their distance in the circle is $a$(that is , $a-1$ people are between them). Find all integers $a$ in terms of $n$ st we can choose $n$ of these people , no two of them positioned in front of each other(means they're not antipodes of each other in the circle) and the total friendship between them is an odd number.

The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?

Given any triangle $ABC$, show how to construct $A'$ on the side $AB$, $B'$ on the side $BC$ and $C'$ on the side $CA$, such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$) and $A'B'C'$ has the least possible area.

Let $g(k)$ be the number of partitions of a $k$-element set $M$, i.e., the number of families $\{ A_1,A_2,\ldots ,A_s\}$ of nonempty subsets of $M$ such that $A_i\cap A_j=\emptyset$ for $i\not= j$ and $\bigcup_{i=1}^n A_i=M$. Prove that, for every $n$, \[n^n\le g(2n)\le (2n)^{2n}\]

Alice guessed that there were $5$ skittles in a jar, while Kayla guessed $19$. It turns out that they were equally close to the true number of skittles. How many skittles are in the jar? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

The archipelago Barrantes - $n$ is a group of islands connected by bridges as follows: there are a main island (Humberto), in the first step I place an island below Humberto and one above from Humberto and I connect these 2 islands to Humberto. I put $2$ islands to the left of these $2$ new islands and I connect them with a bridge to the island that they have on their right. In the second step I take the last $2$ islands and I apply the same process that I applied to Humberto. In the third step I apply the same process to the $4$ new islands. We repeat this step n times we reflect the archipelago that we have on a vertical line to the right of Humberto. We connect Humberto with his reflection and so we have the archipelago Barrantes -$n$. However, the archipelago Barrantes -$n$ exists on a small planet cylindrical, so that the islands to the left of the archipelago are in fact the islands that are connected to the islands on the right. The figure shows the Barrantes archipelago -$2$, The islands at the edges are still numbered to show how the archipelago connects around the cylindrical world, the island numbered $1$ on the left is the same as the island numbered $1$ on the right. [img]https://cdn.artofproblemsolving.com/attachments/e/c/803d95ce742c2739729fdb4d74af59d4d0652f.png[/img] One day two bands of pirates arrive at the archipelago Barrantes - $n$: The pirates Black Beard and the Straw Hat Pirates. Blackbeard proposes a game to Straw Hat: The first player conquers an island, the next player must conquer an island connected to the island that was conquered in the previous turn (clearly not conquered on a previous shift). The one who cannot conquer any island in his turn loses. Straw Hat decides to give the first turn to Blackbeard. Prove that Straw Hat has a winning strategy for every $n$.

Consider the set $S= \{(a + b)^7 - a^7 - b^7 : a,b \in Z\}$. Find the greatest common divisor of all members in $S$.

A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $\mathcal{A}$ and $\mathcal{B}$. An $\mathcal{AB}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $\mathcal{A}$ and an endpoint in $\mathcal{B}$, and such that no segments form a closed polyline. An $\mathcal{AB}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$, and $A_2B_2$ in the $\mathcal{AB}$-tree such that $A_1$ is in $\mathcal{A}$ and $|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|$, and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{AB}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.

Find all the pairs of positive numbers such that the last digit of their sum is 3, their difference is a primer number and their product is a perfect square.

Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

The positive reals $ x,y$ and $ z$ are satisfying the relation $ x \plus{} y \plus{} z\geq 1$. Prove that: $ \frac {x\sqrt {x}}{y \plus{} z} \plus{} \frac {y\sqrt {y}}{z \plus{} x} \plus{} \frac {z\sqrt {z}}{x \plus{} y}\geq \frac {\sqrt {3}}{2}$ [i]Proposer[/i]:[b] Baltag Valeriu[/b]

Let $a,b,c$ be positive real numbers such that \[ a+b+c\geq abc. \] Prove that at least two of the inequalities \[ \frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq6,\;\;\;\;\;\frac{2}{b}+\frac{3}{c}+\frac{6}{a}\geq6,\;\;\;\;\;\frac{2}{c}+\frac{3}{a}+\frac{6}{b}\geq6 \] are true.

Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that \[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\] for all $x,y \in \mathbb{R} $.

For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. For how many three-digit positive integers $m$ is $m + t(t(m))$ odd? [i]Proposed by David Altizio[/i]

In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

Joe generates 4 independent random numbers in $(0,1)$ according to the uniform distribution. He shows one the numbers to Bill, who has to guess whether the number shown is one of the extremal numbers (that is, the smallest or the greatest) of the four numbers or not. Can Joe have a deterministic strategy such that no matter what Bill's method is, the probability of the right guess of Bill is at most $\frac12$?

Homer goes on the $100$-Donut Diet. A $100$-Donut Diet Plan specifies how many of $100$ total donuts Homer will eat each day. The diet requires that the number of donuts he eats does not increase from one day to the next. For example, one $5$-day Donut Diet Plan is $40$, $25$, $25$, $8$, $2$. Are there more $100$-Donut Diet Plans with an odd number of days or plans where Homer eats an odd number of donuts on the first day?

The numbers from 1 to 360 are partitioned into 9 subsets of consecutive integers and the sums of the numbers in each subset are arranged in the cells of a $3 \times 3$ square. Is it possible that the square turns out to be a magic square? Remark: A magic square is a square in which the sums of the numbers in each row, in each column and in both diagonals are all equal.

Given $12$ complex numbers $z_1,...,z_{12}$ st for each $1 \leq i \leq 12$: $$|z_i|=2 , |z_i - z_{i+1}| \geq 1$$ prove that : $$\sum_{1 \leq i \leq 12} \frac{1}{|z_i\overline{z_{i+1}}+1|^2} \geq \frac{1}{2}$$

10. Let $p=47$ be a prime. Call a function $f$ defined on the integers [i]lit[/i] if $f(x)$ is an integer from 1 to $p$ inclusive and $f(x+p)=f(x)$ for all integers $x$. How many [i]lit[/i] functions $g$ are there such that for all integers $x$, $p$ divides $g(x^2)-g(x)-x^8+x$? [i]Proposed by Monkey_king1[/i]

Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number $k$ such that there may be $k$ equal numbers between them.

When planning a trip from Temuco to the extreme north of the country, a truck driver notices that to cross the Atacama desert you must cross a distance of $800$ km between two stations consecutive service. Your truck can only store $50$ liters of benzene, and has a yield of $10$ km per liter. The trucker can leave gasoline stored in cans on the side of the road in different points along the way. For example, with an initial total charge of $50$ liters you can travel $100$ km, leave $30$ liters stored at the point you reached, and return to the starting point (with zero load) to refuel. The trucker decides to start the trip and arrives at the first service station with a zero load of fuel. a) Can the trucker cross the desert if at this service station the total supply is $140$ liters? b) Can the trucker cross the desert if the total supply of gasoline at the station is $180$ liters?

Let $ABC$ be an acute triangle with $AB<AC<BC$ and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$ respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $ respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide. by Evangelos Psychas, Greece

Prove that the inequality $x^2+y^2+1\ge 2(xy-x+y)$ is satisfied by any $x$, $y$ real numbers. Indicate when the equality is satisfied.

What is the greatest common divisor of the set of numbers \[\{{16}^{n}+10n-1 \; \vert \; n=1,2,\cdots \}?\]