Let $K$ be a closed subset of the closed unit ball in $\mathbb{R}^3$. Suppose there exists a family of chords $\Omega$ of the unit sphere $S^2$, with the following property: for every $X,Y\in S^2$, there exist $X',Y'\in S^2$, as close to $X$ and $Y$ correspondingly, as we want, such that $X'Y'\in \Omega$ and $X'Y'$ is disjoint from $K$. Verify that there exists a set $H\subset S^2$, such that $H$ is dense in the unit sphere $S^2$, and the chords connecting any two points of $H$ are disjoint from $K$. EDIT: The statement fixed. See post #4

If $ a$ and $ b$ are two unequal positive numbers, then: $ \textbf{(A)}\ \frac {2ab}{a \plus{} b} > \sqrt {ab} > \frac {a \plus{} b}{2} \qquad\textbf{(B)}\ \sqrt {ab} > \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2}$ $ \textbf{(C)}\ \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2} > \sqrt {ab} \qquad\textbf{(D)}\ \frac {a \plus{} b}{2} > \frac {2ab}{a \plus{} b} > \sqrt {ab}$ $ \textbf{(E)}\ \frac {a \plus{} b}{2} > \sqrt {ab} > \frac {2ab}{a \plus{} b}$

Let $ S(t)$ be the area of the traingle $ OAB$ with $ O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t)$. Evaluate $ \int_1^ e S(t)^2\ln t\ dt$.

Call a real number [i]amiable[/i] if it can be expressed in the form $a - b\sqrt{2}$, where $1 \le a, b \le 100$ are integers. Find the amiable number $x$ that minimizes $\left|x - \frac{1}{3}\right|$.

The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $$\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor$$ where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )$? $\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}$

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

The normals to a surface all intersect a fixed straight line. Show that the surface is a portion of a surface of revolution.

We say that a sequence $a_1,a_2,\cdots$ is [i]expansive[/i] if for all positive integers $j,\; i<j$ implies $|a_i-a_j|\ge \tfrac 1j$. Find all positive real numbers $C$ for which one can find an expansive sequence in the interval $[0,C]$.

Six cities are located at the vertices of a convex hexagon, all angles of which are equal. Three sides of this hexagon have length $a$, and the remaining three have length $b$ ($a \le b$). It is necessary to connect these cities with a network of roads so that from each city you can drive to any other (possibly through other cities). Find the shortest length of such a road network.

$ A',B',C' $ are the feet of the heights of an acute-angled triangle $ ABC. $ Calculate $$ \frac{\text{area} (ABC)}{\text{area}\left( A'B'C'\right)} , $$ knowing that $ ABC $ and $ A'B'C' $ have the same center of mass. [i]Carmen[/i] and [i]Viorel Botea[/i]

The real numbers $a, b, c$ are such that $a^2 + b^2 = 2c^2$, and also such that $a \ne b, c \ne -a, c \ne -b$. Show that \[\frac{(a+b+2c)(2a^2-b^2-c^2)}{(a-b)(a+c)(b+c)}\] is an integer.

Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.

Kevin is at the point $(19,12)$. He wants to walk to a point on the ellipse $9x^2 + 25y^2 = 8100$, and then walk to $(-24, 0)$. Find the shortest length that he has to walk. [i]Proposed by Kevin Zhao[/i]

A quadrilateral is given, in which the lengths of some two sides are equal to $1$ and $4$. Also, the diagonal of length $2$ divides it into two isosceles triangles. Find the perimeter of this quadrilateral.

Find all real numbers $ x$ such that $ 4x^5 \minus{} 7$ and $ 4x^{13} \minus{} 7$ are both perfect squares.

Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.

Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$ inequality for all $x,y\in I$ and $t\in [0,1]$. Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$-Lipschitz property, that is $$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$

Let $ABC$ be a non-equilateral triangle and let $M_a$, $M_b$, $M_c$ be the midpoints of the sides $BC$, $CA$, $AB$, respectively. Let $S$ be a point lying on the Euler line. Denote by $X$, $Y$, $Z$ the second intersections of $M_aS$, $M_bS$, $M_cS$ with the nine-point circle. Prove that $AX$, $BY$, $CZ$ are concurrent.

Find all triples (m,n,N) of positive integers numbers m,n and N such that $m^N-n^N=2^{100}$ with N>1

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect at point $O$. Let a line through $O$ intersect segment $AB$ at $M$ and segment $CD$ at $N$. Prove that the segment $MN$ is not longer than at least one of the segments $AC$ and $BD$.

There are $2014$ airports in the faraway land of Artinia. Each pair of airports is connected by a nonstop flight in one or both directions. Show that there is some airport from which it is possible to reach every other airport in at most two flights.

If $n\ge2$ is an integer, prove the equality $$\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.$$

Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

A four-digit number has the property that the units digit equals the tens digit increased by $1$, the hundreds digit equals twice the tens digit, and the thousands digit is at least twice the units. Determine this four-digit number, knowing that it is twice a prime number.

A $100 \times 100$ table is given. At the beginning, every unit square has number $"0"$ written in them. Two players playing a game and the game stops after $200$ steps (each player plays $100$ steps). In every step, one can choose a row or a column and add $1$ to the written number in all of it's squares $\pmod 3.$ First player is the winner if more than half of the squares ($5000$ squares) have the number $"1"$ written in them, Second player is the winner if more than half of the squares ($5000$ squares) have the number $"0"$ written in them. Otherwise, the game is draw. Assume that both players play at their best. What will be the result of the game ? [i]Proposed by Mahyar Sefidgaran[/i]