Let $a_1, a_2, \dots, a_{14}$ be some positive real numbers. Prove that
$$
\frac{a_1}{a_2+a_3} + \frac{a_2}{a_3+a_4} + \dots + \frac{a_{14}}{a_1+a_2} \ge \frac{a_1}{a_{14}+a_1} + \frac{a_2}{a_1+a_2} + \dots + \frac{a_{14}}{a_{13}+a_{14}}.
$$
When does the equality occur?
Let $x$, $y$ and $z$ be positive real numbers. Find the maximum value of the following expression
$$
\frac{x^3 y^4 z^3}{(x^4 + y^4)(xy + z^2)^3} + \frac{y^3 z^4 x^3}{(y^4 + z^4)(yz + x^2)^3} + \frac{z^3 x^4 y^3}{(z^4 + x^4)(zx + y^2)^3}
$$
Let $(a_n)$ be the sequence such that $a_1 = \frac{3}{2}$ and
$$
a_{n+1} = a_n - \frac{3n+2}{2n(n+1)(2n+1)},\ n \ge 1.
$$
Find the limit $\lim_{n \to +\infty} a_n$.
Points $M, N, K$ lie on the sides $B C, C A, A B$ respectively, of a triangle $A B C$, and are different from its vertices. The triangle $M N K$ is called beautiful if the triangles $M N K$ and $A B C$ are similar (with the vertices respectively in this order). Show that if in the triangle $A B C$ there are two beautiful triangles with a common vertex, then $\triangle A B C$ is right-angled.
Given an acute, scalene triangle $ABC$ ($AB \lt AC$) with the circumcircle $(O)$. Let $I$ be the midpoint of arc $BC$ that does not contain $A$. Take $K$ on $AC$ ($K \neq C$) such that $IK = IC$. Line $BK$ intersects $(O)$ again at $D$ and intersects $AI$ at $E$. $DI$ meets $AC$ at $F$.
a) Prove that $EF = \frac{BC}{2}$.
b) Let $M$ be the point on $DI$ such that $CM$ is parallel to $AD$. Line $KM$ meets $BC$ at $N$ and the circumcircle of triangle $BKN$ meets $(O)$ again at $P$. Prove that $PK$ passes through the midpoint of the segment $AD$.
Given a regular 103-sided polygon with 79 vertices are colored red and the remaining vertices are colored blue. Denote $A$ to be the number of pairs of adjacent red vertices and $B$ to be the number of pairs of adjacent blue vertices.
a) Find all possible values of $(A, B)$.
b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B = 14$. Note that two ways of colorings are considered similar if one of them can be obtained from another with a rotation at the circumcircle of the polygon.
Find all sets of 2014 rational numbers not necessarily distinct such that: If an arbitrary number in the set is removed, one always can divide the remaining 2013 numbers into three sets such that each set has exactly 671 elements and the products of all elements in each set are equal.
Given that $x$ and $y$ are nonzero real numbers such that $x + \frac{1}{y} = 10$ and $y + \frac{1}{x} = \frac{5}{12}$, find all possible values of $x$.
What is the smallest positive integer $n$ which cannot be written in any of the following forms?
- $n=1+2+\cdots+k$ for a positive integer $k$.
- $n=p^{k}$ for a prime number $p$ and integer $k$.
- $n=p+1$ for a prime number $p$.
- $n=p q$ for some distinct prime numbers $p$ and $q$
For any positive integer $x$, define $\operatorname{Accident}(x)$ to be the set of ordered pairs $(s, t)$ with $s \in \{0,2,4,5,7,9,11\}$ and $t \in \{1,3,6,8,10\}$ such that $x+s-t$ is divisible by $12$. For any nonnegative integer $i$, let $a_{i}$ denote the number of $x \in \{0,1, \ldots, 11\}$ for which $|\operatorname{Accident}(x)|=i$. Find
$$
a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}
$$
Let $A=\{a_{1}, a_{2}, \ldots, a_{7}\}$ be a set of distinct positive integers such that the mean of the elements of any nonempty subset of $A$ is an integer. Find the smallest possible value of the sum of the elements in $A$.
Seven little children sit in a circle. The teacher distributes pieces of candy to the children in such a way that the following conditions hold.
- Every little child gets at least one piece of candy.
- No two little children have the same number of pieces of candy.
- The numbers of candy pieces given to any two adjacent little children have a common factor other than $1$.
- There is no prime dividing every little child's number of candy pieces.
What is the smallest number of pieces of candy that the teacher must have ready for the little children?
Let $x_{1}, \ldots, x_{100}$ be defined so that for each $i$, $x_{i}$ is a (uniformly) random integer between $1$ and $6$ inclusive. Find the expected number of integers in the set $\{x_{1}, x_{1}+x_{2}, \ldots, x_{1}+x_{2}+\ldots+x_{100}\}$ that are multiples of $6$.
There are given 100 distinct positive integers. We call a pair of integers among them good if the ratio of its elements is either $2$ or $3$. What is the maximum number $g$ of good pairs that these 100 numbers can form? (A same number can be used in several pairs.)
There are 100 students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly 20 students. The 100 students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?
Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\frac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x, y\}$ is said to be *amicable* if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2^{\binom{n/2}{2}}$ unordered pairs of vertices which are amicable.
Let a sequence $\{a_{n}\}_{n=0}^{\infty}$ be defined by $a_{0}=\sqrt{2}$, $a_{1}=2$, and $a_{n+1}=a_{n} a_{n-1}^{2}$ for $n \geq 1$. The sequence of remainders when $a_{0}, a_{1}, a_{2}, \cdots$ are divided by $2014$ is eventually periodic with some minimal period $p$ (meaning that $a_{m}=a_{m+p}$ for all sufficiently large integers $m$, and $p$ is the smallest such positive integer). Find $p$.