Positive real numbers $a_1>a_2>\ldots>a_n$ with sum $s$ are such that the equation $nx^2-sx+1=0$ has a positive root $a_{n+1}$ smaller than $a_n$. Prove that there exists a positive integer $r \leq n$ such that the inequality $a_ra_{r+1} \geq \frac{1}{r}$ holds. [i]M. Zorka[/i]

Let $a$ be a real number in the open interval $(0,1)$. Let $n\geq 2$ be a positive integer and let $f_n\colon\mathbb{R}\to\mathbb{R}$ be defined by $f_n(x) = x+\frac{x^2}{n}$. Show that \[\frac{a(1-a)n^2+2a^2n+a^3}{(1-a)^2n^2+a(2-a)n+a^2}<(f_n \circ\ \cdots\ \circ f_n)(a)<\frac{an+a^2}{(1-a)n+a}\] where there are $n$ functions in the composition.

In how many ways can we form three teams of four players each from a group of $12$ participants?

[u]Linear? What's The Problem?[/u] A function $f(x_1, x_2,..., x_n)$ is said to be linear in each of its variables if it is a polynomial such that no variable appears with power higher than one in any term. For example, $1 + x + xy$ is linear in $x$ and $y$, but $1 + x^2$ is not. Similarly, $2x + 3yz$ is linear in $x$, $y$, and $z$, but $xyz^2$ is not. [b]p8.[/b] A function $f(x,y)$ is linear in $x$ and in $y$. $f(x,y) =\frac{1}{xy}$ for $x,y \in \{3, 4\}$. What is $f(5,5)$? [b]p9.[/b] A function $f(x, y,z)$ is linear in $x$, $y$, and $z$ such that $f(x,y, z) = \frac{1}{xyz}$ for $x,y,z \in \{3,4\}$. What is $f(5, 5, 5)$? [b]p10.[/b] A function $f(x_1, x_2,..., x_n)$ is linear in each of the $x_i$ and $f(x_1, x_2,..., x_n)= \frac{1}{x_1x_2...x_n}$ when $x_i \in \{3,4\}$ for all $ i$. In terms of $n$, what is $f(5,5,...,5)$?

Let be two sequences $ \left( a_n \right)_{n\ge 0} , \left( b_n \right)_{n\ge 0} $ satisfying the following system: $$ \left\{ \begin{matrix} a_0>0,& \quad a_{n+1} =a_ne^{-a_n} , &\quad\forall n\ge 0 \\ b_{0}\in (0,1) ,& \quad b_{n+1} =b_n\cos \sqrt{b_n} ,& \quad\forall n\ge 0 \end{matrix} \right. $$ Calculate $ \lim_{n\to\infty} \frac{a_n}{b_n} . $ [i]Florian Dumitrel[/i]

Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$, for all $x,y \in \mathbb{R}$. Prove that if for any real $x$, the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that $f(x)=x+a$, for all $x \in \mathbb R$.

Show that there are no positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ such that $$(1+a_1 \omega)(1+a_2 \omega)(1+a_3 \omega)(1+a_4 \omega)(1+a_5 \omega)(1+a_6 \omega)$$ is an integer where $\omega$ is an imaginary $5$th root of unity.

Let $ABCD$ be a convex quadrilateral such that $m \left (\widehat{DAB} \right )=m \left (\widehat{CBD} \right )=120^{\circ}$, $|AB|=2$, $|AD|=4$ and $|BC|=|BD|$. If the line through $C$ which is parallel to $AB$ meets $AD$ at $E$, what is $|CE|$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None of the preceding} $

Some cities of Russia are connected with some cities of Ukraine with international airlines. The Interstate Council for the Promotion of Migration intends to introduce a one-way traffic on each airline so that, by taking off from the city, it could no longer be returned in this city (using other one-way airlines). Prove that the number of ways to do this is not divided by $3$.

Alphonse and Beryl play a game involving $n$ safes. Each safe can be opened by a unique key and each key opens a unique safe. Beryl randomly shuffles the $n$ keys, and after placing one key inside each safe, she locks all of the safes with her master key. Alphonse then selects $m$ of the safes (where $m < n$), and Beryl uses her master key to open just the safes that Alphonse selected. Alphonse collects all of the keys inside these $m$ safes and tries to use these keys to open up the other $n - m$ safes. If he can open a safe with one of the $m$ keys, he can then use the key in that safe to try to open any of the remaining safes, repeating the process until Alphonse successfully opens all of the safes, or cannot open any more. Let $P_m(n)$ be the probability that Alphonse can eventually open all $n$ safes starting from his initial selection of $m$ keys. (a) Show that $P_2(3) = \frac23$. (b) Prove that $P_1(n) = \frac1n$. (c) For all integers $n \geq 2$, prove that $$P_2(n) = \frac2n \cdot P_1(n-1) + \frac{n-2}{n} \cdot P_2(n-1).$$ (d) Determine a formula for $P_2 (n)$.

Let $k$ be a positive integer. There are $2k$ pieces arranged in a row. A [i]move[/i] consists of swapping two adjacent pieces. Several moves must be made so that each piece passes through both the first and the last position. What is the minimum number of moves required to achieve this?

Let $ABC$ be a triangle with altitude $\overline{AF}.$ Let $AB=5, AC=8, BC=7.$ Let $P$ be on $\overline{AF}$ such that it lies between $A$ and $F.$ Let $\omega_1, \omega_2$ be the circumcircles of $APB, APC$ respectively. Let $\overline{BC}$ intersect $\omega_1$ at $B' \neq B.$ Also, let $\overline{BC}$ intersect $\omega_2$ at $C' \neq C.$ Let $X \neq A$ be on $\omega_1$ such that $B'X=B'A.$ Let $Y \neq A$ be on $\omega_2$ such that $C'A=C'Y.$ Let $X, Y, A$ all lie on one line $h.$ Find the length of $PA.$

Octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle $\Omega$ with center $O$. It is known that $A_1A_2\|A_5A_6$, $A_3A_4\|A_7A_8$ and $A_2A_3\|A_5A_8$. The circle $\omega_{12}$ passes through $A_1$, $A_2$ and touches $A_1A_6$; circle $\omega_{34}$ passes through $A_3$, $A_4$ and touches $A_3A_8$; the circle $\omega_{56}$ passes through $A_5$, $A_6$ and touches $A_5A_2$; the circle $\omega_{78}$ passes through $A_7$, $A_8$ and touches $A_7A_4$. The common external tangent to $\omega_{12}$ and $\omega_{34}$ cross the line passing through ${A_1A_6}\cap{A_3A_8}$ and ${A_5A_2}\cap{A_7A_4}$ at the point $X$. Prove that one of the common tangents to $\omega_{56}$ and $\omega_{78}$ passes through $X$.

If the length of a diagonal of a square is $ a \plus{} b$, then the area of the square is: $ \textbf{(A)}\ (a \plus{} b)^2 \qquad\textbf{(B)}\ \frac {1}{2}(a \plus{} b)^2 \qquad\textbf{(C)}\ a^2 \plus{} b^2$ $ \textbf{(D)}\ \frac {1}{2}(a^2 \plus{} b^2) \qquad\textbf{(E)}\ \text{none of these}$

Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$ $$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$

Find all natural numbers $x,y>1$and primes $p$ that satisfy $$\frac{x^2-1}{y^2-1}=(p+1)^2. $$

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Compute the value of the sum \[ \sum_{k = 1}^{11} \frac{\sin(2^{k + 4} \pi / 89)}{\sin(2^k \pi / 89)} \, . \]

$k$ is a positive integer. $A$ company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself $;$ it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least $k$ customers, this person gets a present. Prove that, if $n$ persons have bought clocks, then at most $\frac{n}{k+2}$ presents have been accepted.

An $\textit{abundant number}$ is a natural number, the sum of whose proper divisors is greater than the number itself. For instance, $12$ is an abundant number: \[1+2+3+4+6=16>12.\] However, $8$ is not an abundant number: \[1+2+4=7<8.\] Which one of the following natural numbers is an abundant number? $\begin{array}{c@{\hspace{14em}}c@{\hspace{14em}}c} \textbf{(A) }14&\textbf{(B) }28&\textbf{(C) }56\end{array}$

Let $\alpha\in\mathbb R\backslash \{ 0 \}$ and suppose that $F$ and $G$ are linear maps (operators) from $\mathbb R^n$ into $\mathbb R^n$ satisfying $F\circ G - G\circ F=\alpha F$. a) Show that for all $k\in\mathbb N$ one has $F^k\circ G-G\circ F^k=\alpha kF^k$. b) Show that there exists $k\geq 1$ such that $F^k=0$.

Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$

Source: 1976 Euclid Part A Problem 4 ----- The points $(1,y_1)$ and $(-1,y_2)$ lie on the curve $y=px^2+qx+5$. If $y_1+y_2=14$, then the value of $p$ is $\textbf{(A) } 2 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 2-q \qquad \textbf{(E) }\text{none of these}$

The incircle of triangle $ABC$ is tangent to $BC,CA,AB$ at $D,E,F$ respectively. Consider the triangle formed by the line joining the midpoints of $AE,AF$, the line joining the midpoints of $BF,BD$, and the line joining the midpoints of $CD,CE$. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle $ABC$.

Let $ABCD$ be a rectangle with $BC=24.$ Point $X$ lies inside the rectangle such that $\angle{AXB}=90^\circ.$ Given that triangles $\triangle{AXD}$ and $\triangle{BXC}$ are both acute and have circumradii $13$ and $15,$ respectively, compute $AB.$