The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$

[b]7.[/b] Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that, with some positive number $C$, $\sum_{k=1}^{n}k\mid a_k \mid<n C$ ($n=1,2, \dots $) Putting $s_n= a_0 +a_1+\dots+a_n$, suppose that $\lim_{n \to \infty }(\frac{s_{0}+s_{1}+\dots+s_n}{n+1})= s$ exists. Prove that $\lim_{n \to \infty }(\frac{s_{0}^2+s_{1}^2+\dots+s_n^2}{n+1})= s^2$ [b](S. 7)[/b]

In the spherical shaped planet $X$ there are $2n$ gas stations. Every station is paired with one other station , and every two paired stations are diametrically opposite points on the planet. Each station has a given amount of gas. It is known that : if a car with empty (large enough) tank starting from any station it is always to reach the paired station with the initial station (it can get extra gas during the journey). Find all naturals $n$ such that for any placement of $2n$ stations for wich holds the above condotions, holds: there always a gas station wich the car can start with empty tank and go to all other stations on the planet.(Consider that the car consumes a constant amount of gas per unit length.)

Let $\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be two distinct points on line $AC$ ($A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ($A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C$ and $BQ_2 = P_2Q_2 = P_2C$. Find the distance between the circumcenters of $BP_1P_2$ and $CQ_1Q_2$.

Without using a calculator, determine which number is greater: $17^{24}$ or $31^{19}$

Find all positive reals $a,b,c$ which fulfill the following relation \[ 4(ab+bc+ca)-1 \geq a^2+b^2+c^2 \geq 3(a^3+b^3+c^3) . \] created by Panaitopol Laurentiu.

Given is a triangle $ABC$.On the extensions of the side $AB$ we consider points $A_1,B_1$ such that $AB_1=BA_1$ (with $A_1$ lying closer to $B$).On the extensions of the side $BC$ we consider points $B_4,C_4$ such that $CB_4=BC_4$ (with $B_4$ lying closer to $C$).On the extensions of the side $AC$ we consider points $C_1,A_4$ such that $AC_1=CA_4$ (with $C_1$ lying closer to $A$).On the segment $A_1A_4$ we consider points $A_2,A_3$ such that $A_1A_2=A_3A_4=mA_1A_4$ where $0<m<\frac{1}{2}$.Points $B_2,B_3$ and $C_2,C_3$ are defined similarly,on the segments $B_1B_4,C_1C_4$ respectively.If $D\equiv BB_2\cap CC_2 \ , \ E\equiv AA_3\cap CC_2 \ , \ F\equiv AA_3\cap BB_3$, $\ G\equiv BB_3\cap CC_3 \ , \ H\equiv AA_2\cap CC_3$ and $I\equiv AA_2\cap BB_2$,prove that the diagonals $DG,EH,FI$ of the hexagon $DEFGHI$ are concurrent. 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An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = $ \$ 1.60$, how much lire will the traveler receive in exchange for $ \$ 1.00$? $\text{(A)}\ 180 \qquad \text{(B)}\ 480 \qquad \text{(C)}\ 1800 \qquad \text{(D)}\ 1875 \qquad \text{(E)}\ 4875$

A bounded sequence $ (x_n)_{n\ge 1}$ of real numbers satisfies $ x_n \plus{} x_{n \plus{} 1} \ge 2x_{n \plus{} 2}$ for all $ n \ge 1$. Prove that this sequence has a finite limit.

Evaluate \[2\times (2\times (2\times (2\times (2\times (2\times 2-2)-2)-2)-2)-2)-2.\] [i]Proposed by Nathan Xiong[/i]

What is the value of \[1+2+3-4+5+6+7-8+\cdots+197+198+199-200?\] $\textbf{(A) } 9,800 \qquad \textbf{(B) } 9,900 \qquad \textbf{(C) } 10,000 \qquad \textbf{(D) } 10,100 \qquad \textbf{(E) } 10,200$

A professor organized five exams for a class consisting of at least two students. Before starting the first test, he deduced that there will be at least two students from that class that will have the same amount of passed exams. What is the minimum numer of students that class could have had such that the conclusion of the professor's reasoning was correct.

Let $ABCD$ be a rectangle and let $M, N$ and $P, Q$ be the points of intersections of some line $e$ with the sides $AB, CD$ and $AD, BC$, respectively (or their extensions). Given the points $M, N, P, Q$ and the length $p$ of side $AB$, construct the rectangle. Under what conditions can this problem be solved, and how many solutions does it have?

Let $a,b,c$ be positive real numbers such that \[ \cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1. \] Prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$. When does equality hold?

A sphere with center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through $A$, $B$ and $C$, and meets the edges $SA$, $SB$, $SC$ again at $A_1$, $B_1$, $C_1$, respectively. The planes through $A_1$, $B_1$, $C_1$ tangent to the sphere meet at $O$. Prove that $O$ is the circumcenter of the tetrahedron $SA_1B_1C_1$.

A given infinite geometric series with first term $a \ne 0$ and common ratio $2r$ sums to a value that is $6$ times the sum of an infinite geometric series with first term $2a$ and common ratio $r$. Then $r = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Suppose $a$, $b$, and $c$ are nonzero real numbers, and $a+b+c=0$. What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}$? $\textbf{(A) }0\qquad\textbf{(B) }1\text{ and }-1\qquad\textbf{(C) }2\text{ and }-2\qquad\textbf{(D) }0,2,\text{ and }-2\qquad\textbf{(E) }0,1,\text{ and }-1$

Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.

We throw a dice three times and the numbers are $x,y,z$ find out the possibility of $\binom{7}{x}<\binom{7}{y}<\binom{7}{z}$

In all triangles $ABC$ does it hold that: $$\sum\sqrt{\frac{a(h_a-2r)}{(3a+b+c)(h_a+2r)}}\le\frac34$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

$P(x)$ is a polynomial of degree $3n$ such that \begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*} Determine $n$.

A [i]coloring[/i] of the set of integers greater than or equal to $1$, must be done according to the following rule: Each number is colored blue or red, so that the sum of any two numbers (not necessarily different) of the same color is blue. Determine all the possible [i]colorings[/i] of the set of integers greater than or equal to $1$ that follow this rule.

Calculate the sum $\frac{5}{2.7}+\frac{5}{7.12}+...+\frac{5}{2002.2007}$

Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.

Five square carpets have been bought for a square hall with a side of $6$ m , two with the side $2$ m, one with the side $2.1$ m and two with the side $2.5$ m. Is it possible to place the five carpets so that they do not overlap in any way each other? The edges of the carpets do not have to be parallel to the cradles in the hall.