Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60$. ${{ \textbf{(A)}\ \frac{60}{13} \qquad\textbf{(B)}\ \frac{13}{5} \qquad\textbf{(C)}\ \frac{13}{12} \qquad\textbf{(D)}\ 1}\qquad\textbf{(E)}\ 0 } $

a)Does there exist for any rational number $\frac{a}{b}$ some rational numbers $x_1,x_2,....x_n$ such that $x_1*x_2*....*x_n=1$ and $x_1+x_2+....+x_n=\frac{a}{b}$ a)Does there exist for any rational number $\frac{a}{b}$ some rational numbers $x_1,x_2,....x_n$ such that $x_1*x_2*....*x_n=\frac{a}{b}$ and $x_1+x_2+....+x_n=1$

Statement 1: In a cuboid, we can always find a point, its distances to eight vertexes are equal. Statement 2: In a cuboid, we can always find a point, its distances to twelve edges are equal. Statement 3: In a cuboid, we can always find a point, its distances to six surfaces are equal. In the statements above, how many of them are true? $\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}3$

The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? $\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$

The vertices of two regular octagons are numbered from $1$ to $8$, in some order, which may vary between both octagons (each octagon must have all numbers from $1$ to $8$). After this, one octagon is placed on top of the other so that every vertex from one octagon touches a vertex from the other. Then, the numbers of the vertices which are in contact are multiplied (i.e., if vertex $A$ has a number $x$ and is on top of vertex $A'$ that has a number $y$, then $x$ and $y$ are multiplied), and the $8$ products are then added. Prove that, for any order in which the vertices may have been numbered, it is always possible to place one octagon on top of the other so that the final sum is at least $162$. Note: the octagons can be rotated.

Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.

Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$ [i]Author: Hojoo Lee, South Korea[/i]

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that \[x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.\] Prove that $f$ is a constant function.

A triangle with perimeter $2p$ is inscribed in a circle of radius $R$ and also circumscribed on a circle of radius $r$. Prove that $p < 2(R+r)$.

A trapezoid with bases $AB$ and $CD$ is inscribed into a circle centered at $O$. Let $AP$ and $AQ$ be the tangents from $A$ to the circumcircle of triangle $CDO$. Prove that the circumcircle of triangle $APQ$ passes through the midpoint of $AB$.

Given a real number $a> 0$. How many positive real solutions of the equation is $ a^{x}=x^{a} $

Let $\Gamma$ be the circumcircle of isosceles triangle $ABC$ with vertex $C$. An arbitrary point $M$ is chosen on the segment $BC$ and point $N$ lies on the ray $AM$ with $M$ between $A,N$ such that $AN=AC$. The circumcircle of $CMN$ cuts $\Gamma$ in $P$ other than $C$ and $AB,CP$ intersect at $Q$. Prove that $\angle BMQ = \angle QMN.$

Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$

Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear. [i]Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand[/i]

The expression $ a^3 \minus{} a^{ \minus{} 3}$ equals: $ \textbf{(A)}\ \left(a \minus{} \frac {1}{a}\right)\left(a^2 \plus{} 1 \plus{} \frac {1}{a^2}\right) \qquad\textbf{(B)}\ \left(\frac {1}{a} \minus{} a\right)\left(a^2 \minus{} 1 \plus{} \frac {1}{a^2}\right)$ $ \textbf{(C)}\ \left(a \minus{} \frac {1}{a}\right)\left(a^2 \minus{} 2 \plus{} \frac {1}{a^2}\right) \qquad\textbf{(D)}\ \left(\frac {1}{a} \minus{} a\right)\left(a^2 \plus{} 1 \plus{} \frac {1}{a^2}\right)$ $ \textbf{(E)}\ \text{none of these}$

Two triangular pyramids have common base. One pyramid contains the other. Can the sum of the lengths of the edges of the inner pyramid be longer than that of the outer one?

Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.

5. $A$ and $B$ play alternating turns on a $2012 \times 2013$ board with enough pieces of the following types: Type $1$: Piece like Type $2$ but with one square at the right of the bottom square. Type $2$: Piece of $2$ consecutive squares, one over another. Type $3$: Piece of $1$ square. At his turn, $A$ must put a piece of the type $1$ on available squares of the board. $B$, at his turn, must put exactly one piece of each type on available squares of the board. The player that cannot do more movements loses. If $A$ starts playing, decide who has a winning strategy. Note: The pieces can be rotated but cannot overlap; they cannot be out of the board. The pieces of the types $1$, $2$ and $3$ can be put on exactly $3$, $2$ and $1$ squares of the board respectively.

An engineer is given a fixed volume $V_m$ of metal with which to construct a spherical pressure vessel. Interestingly, assuming the vessel has thin walls and is always pressurized to near its bursting point, the amount of gas the vessel can contain, $n$ (measured in moles), does not depend on the radius $r$ of the vessel; instead it depends only on $V_m$ (measured in $\text{m}^3$), the temperature $T$ (measured in $\text{K}$), the ideal gas constant $R$ (measured in $\text{J/(K} \cdot \text{mol})$), and the tensile strength of the metal $\sigma$ (measured in $\text{N/m}^2$). Which of the following gives $n$ in terms of these parameters? (A) $n=\frac{2}{3}\frac{V_m\sigma}{RT}$ (B) $n=\frac{2}{3}\frac{\sqrt[3]{V_m\sigma}}{RT}$ (C) $n=\frac{2}{3}\frac{\sqrt[3]{V_m\sigma^2}}{RT}$ (D) $n=\frac{2}{3}\frac{\sqrt[3]{V_m^2\sigma}}{RT}$ (E) $n=\frac{2}{3}\sqrt[3]{\frac{V_m\sigma^2}{RT}}$

For $n \geq 3$, let $(b_0, b_1,..., b_{n-1}) = (1, 1, 1, 0, ..., 0).$ Let $C_n = (c_{i, j})$ the $n \times n$ matrix defined by $c_{i, j} = b _{(j -i) \mod n}$. Show that $\det (C_n) = 3$ if $n$ is not a multiple of 3 and $\det (C_n) = 0$ if $n$ is a multiple of 3.

Let $a(x)$ and $b(x)$ be continuous functions on $[0,1]$ and let $0 \leq a(x) \leq a <1$ on that range. Under what other conditions (if any) is the solution of the equation for $u,$ $$ u= \max_{0 \leq x \leq 1} b(x) +a(x)u$$ given by $$u = \max_{0 \leq x \leq 1} \frac{b(x)}{1-a(x)}.$$

Fourty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat? $\textbf{(A) }27\qquad\textbf{(B) }37\qquad\textbf{(C) }47\qquad\textbf{(D) }57\qquad\textbf{(E) }67$

Let $f(a)$ be the largest integer less than or equal to the fourth root of " $a$". Calculate $$f(1)+f(2)+...+f(2005).$$

Let $G$ be a finite abelian group and $f :\mathbb{Z}^+ \to G$ a completely multiplicative function (i.e. $f(mn) = f(m)f(n)$ for any positive integers $m, n$). Prove that there are infinitely many positive integers $k$ such that $f(k) = f(k + 1).$