There are $n>2$ lines on the plane in general position; Meaning any two of them meet, but no three are concurrent. All their intersection points are marked, and then all the lines are removed, but the marked points are remained. It is not known which marked point belongs to which two lines. Is it possible to know which line belongs where, and restore them all? [i]Proposed by Boris Frenkin - Russia[/i]

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

In the right triangle $ABC$ with hypotenuse $AB$, the incircle touches $BC$ and $AC$ at points ${{A}_{1}}$ and ${{B}_{1}}$ respectively. The straight line containing the midline of $\Delta ABC$ , parallel to $AB$, intersects its circumcircle at points $P$ and $T$. Prove that points $P,T,{{A}_{1}}$ and ${{B}_{1}}$ lie on one circle.

Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let $m$ be an integer, with $1 \leq m \leq 720$. At precisely $m$ minutes after 12:00, the angle made by the hour hand and minute hand is exactly $1^\circ$. Determine all possible values of $m$.

A river flows between two houses $ A$ and $ B$, the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from $ A$ to $ B$, using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks.

In a blackboard there are $K$ circles in a row such that one of the numbers $1,...,K$ is assigned to each circle from the left to the right. Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle. For every positive divisor $d$ of $K$ ,$1\leq d\leq K$ we change the situation of the circles in which their assigned numbers are divisible by $d$,performing for each divisor $d$ $K$ changes of situation. Determine the value of $K$ for which the following holds;when this procedure is applied once for all positive divisors of $K$ ,then all numbers $1,2,3,...,K$ are written in the circles they were assigned in.

On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches. $\textbf{(A) }.33\qquad\textbf{(B) }.34\qquad\textbf{(C) }.35\qquad\textbf{(D) }.38\qquad \textbf{(E) }.66$

Let $a_1, a_2, . . ., a_n$ be a finite sequence of $0$ and $1$. Under any two consecutive terms of this sequence $0$ is written if the digits are equal and $1$ is written otherwise. This way a new sequence of length $n -1$ is obtained. By repeating this procedure $n - 1$ times one obtains a triangular table of $0$ and $1$. Find the maximum possible number of ones that can appear on this table

Let $ABCD$ be a convex quadrilateral and $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$,and $O$ their intersection point.Point $M$ is the midpoint of $AB$ and $N$ is the midpoint of $CD$ such that $ MN \cap AC ={E},MN \cap BD={F}$.Prove that $OE \cdot QF= OF\cdot PE $

Prove that for any positive integer $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}>\frac{1}{2}$$

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c} \ge 2\sqrt2 \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)$$

Solve for $x$ if $4x + 1 = 37$. $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x), f(x)g(x), f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3 - 3x^2 + 5$ and $x^2 - 4x$ are written on the blackboard. Can we write a nonzero polynomial of the form $x^n - 1$ after a finite number of steps? Justify your answer.

Find the smallest positive integer $N$ such that each of the $101$ intervals $$[N^2, (N+1)^2), [(N+1)^2, (N+2)^2), \cdots, [(N+100)^2, (N+101)^2)$$ contains at least one multiple of $1001.$ [i]Proposed by Kyle Lee[/i]

Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.

Prove that an arbitrary triangle can be cut into three polygons, one of which must be an obtuse triangle, so that they can then be folded into a rectangle. (Turning over parts is possible).

Let $C$ and $C'$ be two concentric circles of radii $r$ and $r'$ respectively. Determine how much the quotient $r'/r$ must be worth so that in the limited crown (annulus) through $C$ and $C'$ there are eight circles $C_i$ , $i = 1, . . . , 8$, which are tangent to $C$ and to $C'$ , and also that $C_i$ is tangent to $C_{i+1}$ for $i = 1, . . . ,7$ and $C_8$ tangent to $C_1$ .