In an island, there exist 7 towns and a railway system which connected some of the towns. Every railway segment connects 2 towns, and in every town there exists at least 3 railway segments that connects the town to another towns. Prove that there exists a route that visits 4 different towns once and go back to the original town. (Example: $ A\minus{}B\minus{}C\minus{}D\minus{}A$)

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that a) points $T, \, \, X, \, \, B$ lie on one line; b) points $K, \, \, X, \, \, H$ lie on one line. (Vadym Mitrofanov)

Two circles, having their centers in A and B, intersect at points M and N. The radii AP and BQ are parallel and are in different semi-planes determined of the line AB. If the external common tangent intersect AB in D, and PQ intersects AB at C, prove that the <CND is right.

Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $\frac{ab}{a^5+b^5+ab} +\frac{bc}{b^5+c^5+bc}+\frac{ac}{c^5+a^5+ac}\leq 1$

Four circles are situated in the plane so that each is tangent to the other three. If three of the radii are $5$, $5$, and $8$, the largest possible radius of the fourth circle is $a/b$, where $a$ and $b$ are positive integers and gcd$(a, b) = 1$. Find $a + b$.

For positive integers $n$ consider the lattice points $S_n=\{(x,y,z):1\le x\le n, 1\le y\le n, 1\le z\le n, x,y,z\in \mathbb N\}.$ Is it possible to find a positive integer $n$ for which it is possible to choose more than $n\sqrt{n}$ lattice points from $S_n$ such that for any two chosen lattice points at least two of the coordinates of one is strictly greater than the corresponding coordinates of the other? [I]Proposed by Endre Csóka, Budapest[/i]

Given that $f(x)=(3+2x)^3(4-x)^4$ on the interval $\frac{-3}{2}<x<4$. Find the a. Maximum value of $f(x)$ b. The value of $x$ that gives the maximum in (a).

Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$. (Peter Cameron)

Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24$, $q(0)=30$, and \[p(q(x))=q(p(x))\] for all real numbers $x$. Find the ordered pair $(p(3),q(6))$.

There are three types of piece shown as below. Today Alice wants to cover a $100 \times 101$ board with these pieces without gaps and overlaps. Determine the minimum number of $1\times 1$ pieces should be used to cover the whole board and not exceed the board. (There are an infinite number of these three types of pieces.) [asy] size(9cm,0); defaultpen(fontsize(12pt)); draw((9,10) -- (59,10) -- (59,60) -- (9,60) -- cycle); draw((59,10) -- (109,10) -- (109,60) -- (59,60) -- cycle); draw((9,60) -- (59,60) -- (59,110) -- (9,110) -- cycle); draw((9,110) -- (59,110) -- (59,160) -- (9,160) -- cycle); draw((109,10) -- (159,10) -- (159,60) -- (109,60) -- cycle); draw((180,11) -- (230,11) -- (230,61) -- (180,61) -- cycle); draw((180,61) -- (230,61) -- (230,111) -- (180,111) -- cycle); draw((230,11) -- (280,11) -- (280,61) -- (230,61) -- cycle); draw((230,61) -- (280,61) -- (280,111) -- (230,111) -- cycle); draw((280,11) -- (330,11) -- (330,61) -- (280,61) -- cycle); draw((280,61) -- (330,61) -- (330,111) -- (280,111) -- cycle); draw((330,11) -- (380,11) -- (380,61) -- (330,61) -- cycle); draw((330,61) -- (380,61) -- (380,111) -- (330,111) -- cycle); draw((401,11) -- (451,11) -- (451,61) -- (401,61) -- cycle); [/asy] [i]Proposed by amano_hina[/i]

From point $A$ to point $B$, the car drove at a speed of $50$ km / h, and from $B$ to $A$ , at a speed of $30$ km / h. What was the average vehicle speed?

From the triangle $ABC$, are gicen only the incenter $I$, the touchpoint $K$ of the inscribed circle with the side $AB$, as well as the center $I_a$ of the exscribed circle, that touches the side $BC$ . Construct a triangle equal in size to triangle $ABC$. (Gryhoriy Filippovskyi)

Given an integer $n > 2$, consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.

Let $a,b,c$ be nonnegative real numbers.Prove that$$\frac{(a-bc)^2+(b-ca)^2+(c-ab)^2}{(a-b)^2+(b-c)^2+(c-a)^2}\geq\frac{1}{2}.$$

The non-negative integers $x,y$ satisfy $\sqrt{x}+\sqrt{x+60}=\sqrt{y}$. Find the largest possible value for $x$.

Through an internal point $O$ of $\Delta ABC$ one draws 3 lines, parallel to each of the sides, intersecting in the points shown on the picture. [img]https://cdn.artofproblemsolving.com/attachments/e/3/03d4d1bb61eda8b4a72ff84466d700de47c147.png[/img] Find the value of $\frac{|AF|}{|AB|}+\frac{|BE|}{|BC|}+\frac{|CN|}{|CA|}$.

Let $ ABC$ be an isoceles triangle with $ AC\equal{}BC$. A point $ P$ lies inside $ ABC$ such that \[ \angle PAB \equal{} \angle PBC, \angle PAC \equal{} \angle PCB.\] Let $ M$ be the midpoint of $ AB$ and $ K$ be the intersection of $ BP$ and $ AC$. Prove that $ AP$ and $ PK$ trisect $ \angle MPC$.

Given that $p$ and $p^4 + 34$ are both prime numbers, compute $p$.

Prove that if the number $A = 111 \cdots 1$ ($n$ digits) is prime, then $n$ is prime. Is the converse true?

For each $m\in \mathbb N$ we define $rad\ (m)=\prod p_i$, where $m=\prod p_i^{\alpha_i}$. [b]abc Conjecture[/b] Suppose $\epsilon >0$ is an arbitrary number, then there exist $K$ depinding on $\epsilon$ that for each 3 numbers $a,b,c\in\mathbb Z$ that $gcd (a,b)=1$ and $a+b=c$ then: \[ max\{|a|,|b|,|c|\}\leq K(rad\ (abc))^{1+\epsilon} \] Now prove each of the following statements by using the $abc$ conjecture : a) Fermat's last theorem for $n>N$ where $N$ is some natural number. b) We call $n=\prod p_i^{\alpha_i}$ strong if and only $\alpha_i\geq 2$. c) Prove that there are finitely many $n$ such that $n,\ n+1,\ n+2$ are strong. d) Prove that there are finitely many rational numbers $\frac pq$ such that: \[ \Big| \sqrt[3]{2}-\frac pq \Big|<\frac{2^ {1384}}{q^3} \]

Let D be an arbitrary point inside $\Delta ABC$. Let $\Gamma$ be the circumcircle of $\Delta BCD$. The external angle bisector of $\angle ABC$ meets $\Gamma$ again at $E$. The external angle bisector of $\angle ACB$ meets $\Gamma$ again at $F$. The line $EF$ meets the extension of $AB$ and $AC$ at $P$ and $Q$ respectively. Prove that the circumcircles of $\Delta BFP$ and $\Delta CEQ$ always pass through the same fixed point regardless of the position of $D$. (Assume all the labelled points are distinct.)

Given several nonnegative integers (repetitions allowed), the allowed operation is to choose a positive integer $a$ and replace each number $b$ greater than or equal to $a$ by $b-a$ (the numbers $a$ , if any, are replaced by $0$). Initially, the integers from $1$ are written on the blackboard until $2013$ inclusive. After a few operations the numbers on the board have a sum equal to $10$. Determine what the numbers that remained on the board could be. Find all the possibilities.

The area of polygon $ ABCDEF$ is 52 with $ AB\equal{}8$, $ BC\equal{}9$ and $ FA\equal{}5$. What is $ DE\plus{}EF$? [asy]defaultpen(linewidth(0.8));pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4); draw(a--b--c--d--e--f--cycle); draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a); draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b); draw(shift(-.25,0)*c--shift(-.25,.25)*c--shift(0,.25)*c); draw(shift(.25,0)*d--shift(.25,.25)*d--shift(0,.25)*d); draw(shift(.25,0)*f--shift(.25,.25)*f--shift(0,.25)*f); label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, SE); label("$D$", d, SW); label("$E$", e, SW); label("$F$", f, SW); label("5", (0,6.5), W); label("8", (4,9), N); label("9", (8, 4.5), E);[/asy] $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $

If $x^2-5x+6<0$ and $P=x^2+5x+6$ then $\textbf{(A)}\ P \; \text{can take any real value} \qquad \textbf{(B)}\ 20<P<30\\ \textbf{(C)}\ 0<P<20 \qquad \textbf{(D)}\ P<0 \qquad \textbf{(E)}\ P>30$

Let $\alpha,\beta,\gamma$ measures of angles of opposite to the sides of triangle with measures $a,b,c$ respectively.Prove that $$2(cos^2\alpha+cos^2\beta+cos^2\gamma)\geq \frac{a^2}{b^2+c^2}+\frac{b^2}{a^2+c^2}+\frac{c^2}{a^2+b^2}$$