For the nonnegative numbers $a, b, c$ prove the inequality: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge \frac52$$

Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.

Cut the non-equilateral triangle into four similar triangles, among which not all are the same.

Given triangle $ ABC$ with $ AB>AC$. $ l$ is tangent line of the circumcircle of triangle $ ABC$ at $ A$. A circle with center $ A$ and radius $ AC$, intersect $ AB$ at $ D$ and $ l$ at $ E$ and $ F$. Prove that the lines $ DE$ and $ DF$ pass through the incenter and excenter of triangle $ ABC$.

In $\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\angle ACB$ intersect $AB$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\triangle ABC$ at the two points $C$ and $D$. If $LI = 2$ and $LD = 3$, then $IC = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, derivable on $R\backslash\{x_0\}$, having finite side derivatives in $x_0$. Show that there exists a derivable function $g:\mathbb{R}\rightarrow\mathbb{R}$, a linear function $h:\mathbb{R}\rightarrow\mathbb{R}$ and $\alpha\in\{-1,0,1\}$ such that: \[ f(x)=g(x)+\alpha |h(x)|,\ \forall x\in\mathbb{R} \]

In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.

For every point on the plane, one of $ n$ colors are colored to it such that: $ (1)$ Every color is used infinitely many times. $ (2)$ There exists one line such that all points on this lines are colored exactly by one of two colors. Find the least value of $ n$ such that there exist four concyclic points with pairwise distinct colors.

Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying \[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\] Prove that $P(1983) = F_{1983} - 1.$

There are $k$ piles and there are $2019$ stones totally. In every move we split a pile into two or remove one pile. Using finite moves we can reach conclusion that there are $k$ piles left and all of them contain different number of stonws. Find the maximum of $k$.

King Arthur is placing $a + b + c$ knights around a table. $a$ knights are dressed in red, $b$ knights are dressed in brown, and $c$ knights are dressed in orange. Arthur wishes to arrange the knights so that no knight is seated next to someone dressed in the same colour as himself. Show that this is possible if, and only if, there exists a triangle whose sides have lengths $a +\frac12, b +\frac12$, and $c +\frac12$

$P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)$. $P_{n+4}$ is the midpoint of $P_nP_{n+1}$. $Q_n$ is the quadrilateral $P_{n}P_{n+1}P_{n+2}P_{n+3}$. $A_n$ is the interior of $Q_n$. Find $\cap_{n \geq 0}A_n$.

Let $\mathbb{K}$ be two-dimensional integer lattice. Is there a bijection $f:\mathbb{N} \rightarrow \mathbb{K}$, such that for every distinct $a,b,c \in \mathbb{N}$ we have: \[\gcd(a,b,c)>1 \Rightarrow f(a),f(b),f(c) \mbox{ are not colinear? }\]

Let $F_1, F_2, ...$ be Borel-measurable sets on the plane whose union is the whole plane. Prove that there is a natural number n and circle S for which the set $S \cap F_n$ is dense in S. Also show that the statement is not necessarily true if we omit the condition for the measurability of sets $F_j$.

Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether $$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$

A natural number $n$ is called breakable if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$. Find all breakable numbers.

Philip has $3$ triangles and $6$ pentagons. Let $S$ be the total number of sides of the shapes he has. Let $N$ be the number of shapes he has. What is $S+N$?

[b] Newton's Law of Gravity from Kepler's Laws?[/b] [list=1] [*] Planets in the solar system move in elliptic orbits with the sun at one of the foci. [/*] [*] The line joining the sun and the planet sweeps out equal areas in equal times. [/*] [*] The period of revolution ($T$) and the length of the semi-major axis $(a$) of the ellipse sit in the relation $T^2/a^3 = constant$. [/*] [/list] Now answer the following questions: [list] [*] Starting from Newton's Law of Gravitation and Kepler's first law, derive the second and third law. It is possible to derive the first law but that is beyond the scope of this exam. [/*] [*] For convenience work in the complex (Argand) plane and take the sun to be at the origin $(z=0)$. Show that points on the ellipse may be represented by, \[ z(\theta) = \frac{a(1-\epsilon^2)}{1+\epsilon\cos\theta}\exp(i\theta) = r(\theta) e^{i\theta}\] where $a$ is the length of the semi-major axis, $\epsilon$ is the eccentricity of the ellipse and $\theta$ is called the \emph{true anomaly} in celestial mechanics. [/*] [*] Show that Kepler's second law leads to, \[ \frac{1}{2}r^2 \dot{\theta} = constant\] where $r$ and $\theta$ are defined as in part (b) and a dot $(.)$ over a variable denotes its time derivative. What is this constant in terms of the other variables of the problem? [/*] [*] Using the results of parts (b) and (c) along with Kepler's third law obtain Newton's Law of Gravitation. [/*] [*] Can the above exercise truly be called a "derivation" of Newton's Law of Gravitation? State your reasons. [/*] [/list]

For any given non-negative integer $m$, let $f(m)$ be the number of $1$'s in the base $2$ representation of $m$. Let $n$ be a positive integer. Prove that the integer $$\sum^{2^n - 1}_{m = 0} \Big( (-1)^{f(m)} \cdot 2^m \Big)$$ contains at least $n!$ positive divisors.

Let the numbers x and y satisfy the conditions $\begin{cases} x^2 + y^2 - xy = 2 \\ x^4 + y^4 + x^2y^2 = 8 \end{cases}$ The value of $P = x^8 + y^8 + x^{2014}y^{2014}$ is: (A): $46$, (B): $48$, (C): $50$, (D): $52$, (E) None of the above.

Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number? $\textbf{(A) }8\qquad\textbf{(B) }11\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad\textbf{(E) }18$

Define a function $f(x)=\int_0^{\frac{\pi}{2}} \frac{\cos |t-x|}{1+\sin |t-x|}dt$ for $0\leq x\leq \pi$. Find the maximum and minimum value of $f(x)$ in $0\leq x\leq \pi$.

Let $ABC$ be a triangle and $E$ and $F$ lie on $AC$ and $AB$ such that $AE=AF$. $EF$ intersects $BC$ at $D$ and $(BDF)$ intersects $(CDE)$ at $X$. Let $O_1$ be the center of $(BDF)$ and $O_2$ be the center of $(CDE)$. Let $O$ be the center of $ABC$. Suppose that $XD$ intersects $(XO_1O_2)$ at $Z$. Show that $OZ\parallel BC$. [i](Proposed by Tan Rui Xuen and Yeoh Yi Shuen)[/i]