Let $a,b,c \in \mathbb{C}$ such that $a|bc| + b|ca| + c|ab| = 0$. Prove that $|(a-b)(b-c)(c-a)| \ge 3\sqrt{3}|abc|$.

A TV station holds a math talent competition, where each participant will be scored by 8 people. The scores are F (failed), G (good), or E (exceptional). The competition is participated by three people, A, B, and C. In the competition, A and B get the same score from exactly 4 people. C states that he has differing scores with A from at least 4 people, and also differing scores with B from at least 4 people. Assuming C tells the truth, how many scoring schemes can occur?

Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

A cycling group that has $4n$ members will have several cycling events, such that: a) Two cycling events are done every week; once on Saturday and once on Sunday. b) Exactly $2n$ members participate in any cycling event. c) No member may participate in both cycling events of a week. d) After all cycling events are completed, the number of events where each pair of members meet is the same for all pairs of members. Prove that after all cycling events are completed, the number of events where each group of three members meet is the same value $t$ for all groups of three members, and that for $n \ge 2$, $t$ is divisible by $n-1$.

Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that \[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\] and \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\] Prove that \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]

Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that: a) $d(f(x)) = x$ for all $x \in \mathbb{N}$ b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that \[f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y)\] for all $x,y \in \mathbb{R}$.

An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.

Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.

The Fibonacci sequence $\{F_n\}$ is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$. Determine all triplets of positive integers $(k,m,n)$ such that $F_n = F_m^k$.

Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that \[f(x+t) - f(x) = P(x)\] for all $x \in \mathbb{R}$.

Let $\omega$ be a circle with center $O$, and let $l$ be a line not intersecting $\omega$. $E$ is a point on $l$ such that $OE$ is perpendicular with $l$. Let $M$ be an arbitrary point on $M$ different from $E$. Let $A$ and $B$ be distinct points on the circle $\omega$ such that $MA$ and $MB$ are tangents to $\omega$. Let $C$ and $D$ be the foot of perpendiculars from $E$ to $MA$ and $MB$ respectively. Let $F$ be the intersection of $CD$ and $OE$. As $M$ moves, determine the locus of $F$.

Let $S$ be a subset of $\{1,2,3,4,5,6,7,8,9,10\}$. If $S$ has the property that the sums of three elements of $S$ are all different, find the maximum number of elements of $S$.

Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$.

Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$. (A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)

Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.

The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.

Determine all integer $n > 1$ such that \[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\] for all integer $1 \le m < n$.

Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.

The positive integers are colored with black and white such that: - There exists a bijection from the black numbers to the white numbers, - The sum of three black numbers is a black number, and - The sum of three white numbers is a white number. Find the number of possible colorings that satisfies the above conditions.

The cross of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.

The sequence $a_i$ is defined as $a_1 = 1$ and \[a_n = a_{\left\lfloor \dfrac{n}{2} \right\rfloor} + a_{\left\lfloor \dfrac{n}{3} \right\rfloor} + a_{\left\lfloor \dfrac{n}{4} \right\rfloor} + \cdots + a_{\left\lfloor \dfrac{n}{n} \right\rfloor} + 1\] for every positive integer $n > 1$. Prove that there are infinitely many values of $n$ such that $a_n \equiv n \mod 2012$.

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

Let $P_1, P_2, \ldots, P_n$ be distinct $2$-element subsets of $\{1, 2, \ldots, n\}$. Suppose that for every $1 \le i < j \le n$, if $P_i \cap P_j \neq \emptyset$, then there is some $k$ such that $P_k = \{i, j\}$. Prove that if $a \in P_i$ for some $i$, then $a \in P_j$ for exactly one value of $j$ not equal to $i$.

Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$. Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle. Should the first sentence read: Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ with respect to $M$. ? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.

Given a non-zero integer $y$ and a positive integer $n$. If $x_1, x_2, \ldots, x_n \in \mathbb{Z} - \{0, 1\}$ and $z \in \mathbb{Z}^+$ satisfy $(x_1x_2 \ldots x_n)^2y \le 2^{2(n+1)}$ and $x_1x_2 \ldots x_ny = z + 1$, prove that there is a prime among $x_1, x_2, \ldots, x_n, z$. It appears that the problem statement is incorrect; suppose $y = 5, n = 2$, then $x_1 = x_2 = -1$ and $z = 4$. They all satisfy the problem's conditions, but none of $x_1, x_2, z$ is a prime. What should the problem be, or did I misinterpret the problem badly?

Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.

Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.

Let $P_1P_2\ldots P_n$ be an $n$-gon such that for all $i,j \in \{1,2,\ldots,n\}$ where $i \neq j$, there exists $k \neq i,j$ such that $\angle P_iP_kP_j = 60^\circ$. Prove that $n=3$.

Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.