Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.

In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$. [i]Ray Li.[/i]

Find, with proof, a positive integer $n$ such that \[\frac{(n + 1)(n + 2)
\cdots
(n + 500)}{500!}\] is an integer with no prime factors less than $500$.

Prove the following two inequalities: 1) If $n > 49$, then exist positive integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}<1$$ 2) If $n > 4$, then exist integer integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}>1$$

Let $ABC$ is acute angled triangle and $K,L$ is points on $AC,BC$ respectively such that $\angle{AKB}=\angle{ALB}$. $P$ is intersection of $AL$ and $BK$ and $Q$ is the midpoint of segment $KL$. Let $T,S$ are the intersection $AL,BK$ with $(ABC)$ respectively. Prove that $TK,SL,PQ$ are concurrent.

Let $ S$ be a semigroup without proper two-sided ideals and suppose that for every $ a,b \in S$ at least one of the products $ ab$ and $ ba$ is equal to one of the elements $ a,b$. Prove that either $ ab\equal{}a$ for all $ a,b \in S$ or $ ab\equal{}b$ for all $ a,b \in S$. [i]L. Megyesi[/i]

For a real number $x$, the minimum value of the expression $$\frac{2x^2 + x - 3}{x^2 - 2x + 3}$$ can be written in the form $\frac{a-\sqrt{b}}{c}$, where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$

Find the maximal possible value of $n$ such that there exist points $P_1,P_2,P_3,\ldots,P_n$ in the plane and real numbers $r_1,r_2,\ldots,r_n$ such that the distance between any two different points $P_i$ and $P_j$ is $r_i+r_j$.

In convex hexagon $ABCDEF$, $\angle A \cong \angle B$, $\angle C \cong \angle D$, and $\angle E \cong \angle F$. Prove that the perpendicular bisectors of $\overline{AB}$, $\overline{CD}$, and $\overline{EF}$ pass through a common point. [i]Proposed by Lewis Chen[/i]

Determine all triples of real numbers that satisfy the following system of equations: \[x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1\]

Let $A > 0$ and $B = (3 + 2\sqrt{2})A$. Prove that in the finite sequence $a_k = \lfloor k / \sqrt{2} \rfloor$ for $k \in (A, B) \cap \mathbb{Z}$, the number of even and odd terms differs by at most $2$.

In a non-equilateral acute-angled triangle $ABC$ with $\angle C = 60^\circ$, $U$ is the circumcenter, $H$ the orthocenter and $D$ the intersection of $AH$ and $BC$. Prove that the Euler line $HU$ bisects the angle $BHD$.

Find the maximum value of $c$ such that \begin{align*} 1&=-cx+y\\ -7&=x^2+y^2+8y \end{align*} has a unique real solution $(x,y)$.

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$

For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi^{\prime}$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi^{\prime}(x) = \pi^{\prime}(y)$.

Let $ A$, $ M$, and $ C$ be nonnegative integers such that $ A \plus{} M \plus{} C \equal{} 12$. What is the maximum value of $ A \cdot M \cdot C \plus{} A\cdot M \plus{} M \cdot C \plus{} C\cdot A$? $ \textbf{(A)}\ 62 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 112$

Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$: \[ f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}. \] Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.

Let $q$ be the sum of the expressions $a_1^{-a_2^{a_3^{a_4}}}$ over all permutations $(a_1, a_2, a_3, a_4)$ of $(1,2,3,4).$ Determine $\lfloor q \rfloor.$

Find the least positive integer $n$ such that for every prime number $p, p^2 + n$ is never prime.

For any $n\in\mathbb N$, denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$, where the last summand consists of $n$ digits of $2$. Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$.

Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)

A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is: $\textbf{(A)}\ 800 \qquad \textbf{(B)}\ 640\qquad \textbf{(C)}\ 400 \qquad \textbf{(D)}\ 320 \qquad \textbf{(E)}\ 160$

Let $n$ a positive integer and let $f\colon [0,1] \to \mathbb{R}$ an increasing function. Find the value of : \[ \max_{0\leq x_1\leq\cdots\leq x_n\leq 1}\sum_{k=1}^{n}f\left ( \left | x_k-\frac{2k-1}{2n} \right | \right )\]

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $ 1$ place to its right in the alphabet (assuming that the letter $ A$ is one place to the right of the letter $ Z$). The second time this same letter appears in the given message, it is replaced by the letter that is $ 1\plus{}2$ places to the right, the third time it is replaced by the letter that is $ 1 \plus{} 2 \plus{} 3$ places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter $ \text{s}$ in the message "Lee's sis is a Mississippi miss, Chriss!"? $ \textbf{(A)}\ \text{g}\qquad \textbf{(B)}\ \text{h}\qquad \textbf{(C)}\ \text{o}\qquad \textbf{(D)}\ \text{s}\qquad \textbf{(E)}\ \text{t}$