Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even? $\text{(A)}\ \frac{1}{6} \qquad \text{(B)}\ \frac{3}{7} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{2}{3} \qquad \text{(E)}\ \frac{5}{7}$ [asy] unitsize(36); draw(circle((-3,0),1)); draw(circle((0,0),1)); draw((0,0)--dir(30)); draw((0,0)--(0,-1)); draw((0,0)--dir(150)); draw((-2.293,.707)--(-3.707,-.707)); draw((-2.293,-.707)--(-3.707,.707)); fill((-2.9,1)--(-2.65,1.25)--(-2.65,1.6)--(-3.35,1.6)--(-3.35,1.25)--(-3.1,1)--cycle,black); fill((.1,1)--(.35,1.25)--(.35,1.6)--(-.35,1.6)--(-.35,1.25)--(-.1,1)--cycle,black); label("$5$",(-3,.2),N); label("$3$",(-3.2,0),W); label("$4$",(-3,-.2),S); label("$8$",(-2.8,0),E); label("$6$",(0,.2),N); label("$9$",(-.2,.1),SW); label("$7$",(.2,.1),SE); [/asy]

Let $f:[0,\infty)\to\mathbb R$ be differentiable and satisfy $$f'(x)=-3f(x)+6f(2x)$$for $x>0$. Assume that $|f(x)|\le e^{-\sqrt x}$ for $x\ge0$. For $n\in\mathbb N$, define $$\mu_n=\int^\infty_0x^nf(x)dx.$$ $a.$ Express $\mu_n$ in terms of $\mu_0$. $b.$ Prove that the sequence $\frac{3^n\mu_n}{n!}$ always converges, and the the limit is $0$ only if $\mu_0$.

In $R^d$ , all $n^d$ points of an n × n × ··· × n cube grid are contained in 2n - 3 hyperplanes. Prove that n ($n\geq3$) hyperplanes can be chosen from these so that they contain all points of the grid.

Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$. Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$. [i]N. Agakhanov[/i]

An octahedron (a solid with 8 triangular faces) has a volume of $1040$. Two of the spatial diagonals intersect, and their plane of intersection contains four edges that form a cyclic quadrilateral. The third spatial diagonal is perpendicularly bisected by this plane and intersects the plane at the circumcenter of the cyclic quadrilateral. Given that the side lengths of the cyclic quadrilateral are $7, 15, 24, 20$, in counterclockwise order, the sum of the edge lengths of the entire octahedron can be written in simplest form as $a/b$. Find $a + b$.

Let $\triangle ABC$ be a triangle with a right angle $\angle ABC$. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AC}$, and let $F$ be the midpoint of $\overline{AB}$. Let $G$ be the midpoint of $\overline{EC}$. One of the angles of $\triangle DFG$ is a right angle. What is the least possible value of $\frac{BC}{AG}$?

Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$. (1) Find the maximum value of $ y$. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis. (3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.

Given that $a_1, a_2, . . . , a_n$ are nonnegative real numbers with $a_1 + \cdots + a_n = 1$, prove that the expression $$ \frac{a_1}{1+a_2 +a_3 +\cdots +a_n }\; +\; \frac{a_2}{1+a_1 +a_3 +\cdots +a_n }\; +\; \cdots \; +\, \frac{a_n }{1+a_1 +a_2+\cdots +a_{n-1} }$$ attains its minimum, and determine this minimum.

Consider an odd prime number $p$ and $p$ consecutive positive integers $m_1,m_2,…,m_p$. Choose a permutation $\sigma$ of $1,2,…,p$ . Show that there exist two different numbers $k,l\in{(1,2,…,p)}$ such that $p\mid{m_k.m_{\sigma(k)}-m_l.m_{\sigma(l)}}$

Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$

Prove that there are infinitely many integers $n$ such that both the arithmetic mean of its divisors and the geometric mean of its divisors are integers. (Recall that for $k$ positive real numbers, $a_1, a_2, \dotsc, a_k$, the arithmetic mean is $\frac{a_1 +a_2 +\dotsb +a_k}{k}$, and the geometric mean is $\sqrt[k]{a_1 a_2\dotsb a_k}$.)

Let $p$ and $q$ be odd prime numbers and $a$ a positive integer so that $p|a^q+1$ and $q|a^p+1$. Show that $p|a+1$ or $q|a+1$. [i]Authored by Nikola Velov[/i]

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove: \[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3} \] In what case does equality hold?

Find all positive integers $ a, b, c, d $ so that $ a^2+b^2+c^2+d^2=13 \cdot 4^n $

Three distinct positive integers are chosen at random from $\{1,2,3...,12\}$. The probability that no two elements of the set have an absolute difference less than or equal to $2$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$.

Let $p(x)$ be a polynomial of degree $4$ with leading coefficient $1$. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$ and $p(4)=4$. Then $p(5)=$? $\textbf{(A)}~5$ $\textbf{(B)}~\frac{25}6$ $\textbf{(C)}~29$ $\textbf{(D)}~35$

Let $k_1$ and $k_2$ be two circles that are externally tangent at point $C$. We have a point $A$ on $k_1$ and a point $B$ on $k_2$ such that $C$ is an interior point of segment $AB$. Let $k_3$ be a circle that passes through points $A$ and $B$ and intersects circles $k_1$ and $k_2$ another time at points $M$ and $N$ respectively. Let $k_4$ be the circumscribed circle of triangle $CMN$. Prove that the centres of circles $k_1, k_2, k_3$ and $k_4$ all lie on the same circle.

A convex $1993$-gon is divided into convex $7$-gons. Prove that there are $3$ neighbouring sides of the $1993$-gon belonging to one such $7$-gon. (A vertex of a $7$-gon may not be positioned on the interior of a side of the $1993$-gon, and two $7$-gons either have no common points, exactly one common vertex or a complete common side.) (A Kanel-Belov)

For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exacly $2004$ of other lines? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 1 $

When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions: (i) $n$ divides $A_m$, (ii) $n$ divides $m$, (iii) $n$ divides the sum of the digits of $A_m$.

Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has \[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]

Let $ABC$ be an arbitrary triangle,$P$ is the intersection point of the altitude from $C$ and the tangent line from $A$ to the circumcircle. The bisector of angle $A$ intersects $BC$ at $D$ . $PD$ intersects $AB$ at $K$, if $H$ is the orthocenter then prove : $HK\perp AD$

Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that $ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$ Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.