Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds: $$(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)$$

Suppose that a council consists of five members and that decisions in this council are made according to a method based on the positive or negative vote of its members. The method used by this council has the following two properties: $\bullet$ [b]Ascension:[/b]If the presumptive final decision is favorable and one of the opposing members changes his/her vote, the final decision will still be favorable. $\bullet$ [b]Symmetry:[/b] If all of the members change their vote, the final decision will change too. Prove that the council uses a weighted decision-making method ; that is , nonnegative weights $\omega _1 , \omega _2 , \cdots ,\omega _5$ can be assigned to members of the council such that the final decision is favorable if and only if sum of the weights of those in favor is greater than sum of the weights of the rest. Remark. The statement isn't true at all if you replace $5$ with arbitrary $n$ . In fact , finding a counter example for $n=6$ , was appeared in the same year's [url=https://artofproblemsolving.com/community/c6h1459567p8417532]Iran MO 2nd round P6[/url]

Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained. [i]Brazitikos Silouanos, Greece[/i]

The set of positive integers is partitioned into $n$ disjoint infinite arithmetic progressions $S_1, S_2, \ldots, S_n$ with common differences $d_1, d_2, \ldots, d_n$, respectively. Prove that there exists exactly one index $1\leq i \leq n$ such that\[ \frac{1}{d_i}\prod_{j=1}^n d_j \in S_i.\]

Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

The angles $\alpha, \beta, \gamma$ of a triangle are in arithmetic progression. If $\sin 20\alpha$, $\sin 20\beta$, and $\sin 20\gamma$ are in arithmetic progression, how many different values can $\alpha$ take? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None of the above} $

In a convex quadrilateral $ABCD$ with $\angle ABC = \angle ADC = 135^\circ$, points $M$ and $N$ are taken on the rays $AB$ and $AD$ respectively such that $\angle MCD = \angle NCB = 90^\circ$. The circumcircles of triangles $AMN$ and $ABD$ intersect at $A$ and $K$. Prove that $AK \perp KC.$

Natural numbers have been divided in groups as follow: $(1), (2, 4), (3, 5, 7), (6, 8, 10, 12), (9, 11, 13, 15, 17), \ldots$. Let $S_n$ be the sum of the elements of the $n$th group. Prove that $\frac{S_{2n+1}}{2n+1}-\frac{S_{2n}}{2n}$ is even.

The numbers $a, b, c, d$ and $e$ satisfy $$a + b < c + d < e + a < b + c < d + e .$$ Which of the numbers is the smallest, and which is the largest?

Prove that there exists a surjective function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ having the property that for all natural numbers $ n\ge 2, $ there exists an infinite set $ A_n $ such that $ f(x)=n, $ for all $ x\in A_n. $

Find all real numbers $x$ such that $-1 < x \le 2 $ and $$\sqrt{2 - x}+\sqrt{2 + 2x} =\sqrt{\frac{x^4 + 1}{x^2 + 1}}+ \frac{x + 3}{x + 1}.$$ .

If five boys and three girls are randomly divided into two four-person teams, what is the probability that all three girls will end up on the same team? $\text{(A) }\frac{1}{7}\qquad\text{(B) }\frac{2}{7}\qquad\text{(C) }\frac{1}{10}\qquad\text{(D) }\frac{1}{14}\qquad\text{(E) }\frac{1}{28}$

The Fermat-numbers are defined by $F_n = 2^{2^n}+1$ for $n\in N$. (a) Prove that $F_n = F_{n-1}F_{n-2}....F_1F_0 +2$ for $n > 0$. (b) Prove that any two different Fermat numbers are coprime

The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.

Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1 .$ Determine the largest real number $u$ such that $$u \leq \int_{0}^{1} |f'(x) -f(x) | \, dx $$ for all $f$ in $C.$

$ 1994$ girls are seated at a round table. Initially one girl holds $ n$ tokens. Each turn a girl who is holding more than one token passes one token to each of her neighbours. a.) Show that if $ n < 1994$, the game must terminate. b.) Show that if $ n \equal{} 1994$ it cannot terminate.

How many ways are there to place the integers from $1$ to $8$ on the vertices of a regular octagon such that the sum of the numbers on any $4$ vertices forming a rectangle is even? Rotations and reflections of the same arrangement are considered distinct

Let $ O$ be an interior point of the triangle $ ABC$. Prove that there exist positive integers $ p,q$ and $ r$ such that \[ |p\cdot\overrightarrow{OA} \plus{} q\cdot\overrightarrow{OB} \plus{} r\cdot\overrightarrow{OC}|<\frac{1}{2007}\]

If $x_{1}, x_{2},\ldots ,x_{n}$ are positive real numbers with $x_{1}^2+x_2^{2}+\ldots +x_{n}^{2}=1$, find the minimum value of $\sum_{i=1}^{n}\frac{x_{i}^{5}}{x_{1}+x_{2}+\ldots +x_{n}-x_{i}}$.

On the plane are placed two triangles exterior to each other. Show that there always exists a line passing through two vertices of one triangle and separating the third vertex from all vertices of the other triangle.

Maryssa, Stephen, and Cynthia played a game. Each of them independently privately chose one of Rock, Paper, and Scissors at random, with all three choices being equally likely. Given that at least one of them chose Rock and at most one of them chose Paper, the probability that exactly one of them chose Scissors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Yannick Yao[/i]

Show that there are no functions $f : R \to R$ satisfying $f(x + f(y)) = f(x) + y^2$ for all real numbers $x$ and $y$

Elmer's new car gives $50\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip? $\textbf{(A) } 20\% \qquad \textbf{(B) } 26\tfrac23\% \qquad \textbf{(C) } 27\tfrac79\% \qquad \textbf{(D) } 33\tfrac13\% \qquad \textbf{(E) } 66\tfrac23\%$

a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$ b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$