Let $a,b,c\in R^+$ with $abc=1$. Prove that $$\frac{a^3b^3}{a+b}+\frac{b^3c^3}{b+c}+\frac{c^3c^3}{c+a} \ge \frac12 \left(\frac{1}{a}+ \frac{1}{b}+\frac{1}{c}\right)$$ [i](Zhuge Liang)[/i]

For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers? $ \textbf{(A)}\ 794\qquad \textbf{(B)}\ 796\qquad \textbf{(C)}\ 798\qquad \textbf{(D)}\ 800\qquad \textbf{(E)}\ 802$

[b]a)[/b] Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $3$-element subset of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $3$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $3$-element subsets of it are green. [b]b)[/b] Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $k$-element subset ($k>3$) of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $k$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $k$-element subsets of it are green.

Let $P$ be a point inside a triangle $ABC$. A line through $P$ parallel to $AB$ meets $BC$ and $CA$ at points $L$ and $F$, respectively. A line through $P$ parallel to $BC$ meets $CA$ and $BA$ at points $M$ and $D$ respectively, and a line through $P$ parallel to $CA$ meets $AB$ and $BC$ at points $N$ and $E$ respectively. Prove \begin{align*} [PDBL] \cdot [PECM] \cdot [PFAN]=8\cdot [PFM] \cdot [PEL] \cdot
[PDN] \\ \end{align*} [i]Proposed by Steve Dinh[/i]

Given a positive integer $N$, define $u(N)$ as the number obtained by making the ones digit the left-most digit of $N$, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of $N$ until it becomes the first (left-most) digit; for example, $u(2023) = 3202$. [b](1)[/b] Find a $6$-digit positive integer $N$ such that \[\frac{u(N)}{N} = \frac{23}{35}.\] [b](2)[/b] Prove that there is no positive integer $N$ with less than $6$ digits such that \[\frac{u(N)}{N} = \frac{23}{35}.\]

Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.) [i]Proposed by Sean Li[/i]

Let $ABC$ be a triangle and $D$ a point on its side $BC.$ Points $E, F$ lie on the lines $AB, AC$ beyond vertices $B, C,$ respectively, such that $BE = BD$ and $CF = CD.$ Let $P$ be a point such that $D$ is the incenter of triangle $P EF.$ Prove that $P$ lies inside the circumcircle $\Omega$ of triangle $ABC$ or on it.

Let $a_1,a_2,...,a_m(m\geq 1)$ be all the positive divisors of $n$. If there exist $m$ integers $b_1,b_2,...b_m$ such that $n=\sum_{i=1}^m (-1)^{b_i} a_i$, then $n$ is a $\textit{good}$ number. Prove that there exist a good number with exactly $2013$ distinct prime factors.

If $\theta$ is an acute angle, and $\sin 2\theta=a$, then $\sin\theta+\cos\theta$ equals $\textbf{(A) }\sqrt{a+1}\qquad\textbf{(B) }(\sqrt{2}-1)a+1\qquad\textbf{(C) }\sqrt{a+1}-\sqrt{a^2-a}\qquad$ $\textbf{(D) }\sqrt{a+1}+\sqrt{a^2-a}\qquad \textbf{(E) }\sqrt{a+1}+a^2-a$

Let $P(x), Q(x), $ and $R(x)$ be three monic quadratic polynomials with only real roots, satisfying $$P(Q(x))=(x-1)(x-3)(x-5)(x-7)$$$$Q(R(x))=(x-2)(x-4)(x-6)(x-8)$$ for all real numbers $x.$ What is $P(0)+Q(0)+R(0)?$ [i]Proposed by Kyle Lee[/i]

The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner? $\textbf{(A)} 4 \qquad\textbf{(B)} 7 \qquad\textbf{(C)} 8 \qquad\textbf{(D)} 15 \qquad\textbf{(E)} 16$

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

For which positive integers $n$ is it possible to cover a $(2n+1) \times (2n+1)$ chessboard which has one of its corner squares cut out with tiles shown in the figure (each tile covers exactly $4$ squares, tiles can be rotated and turned around)? [img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]

Find all possible triples $(a, b, c)$ of positive integers with the following properties: • $gcd(a, b) = gcd(a, c) = gcd(b, c) = 1$, • $a$ is a divisor of $a + b + c$, • $b$ is a divisor of $a + b + c$, • $c$ is a divisor of $a + b + c$. (Here $gcd(x,y)$ is the greatest common divisor of $x$ and $y$.)

$\triangle ABC$ is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are $AD$, $BE$ and $CF$. The line parallel to $ DF$ through $E$ meets $BC$ at $Y$ and $BA$ at $X$. $DF$ and $CA$ meet at $Z$. Circumcircle of $XYZ$ meets $AC$ at $S$. Given, $\angle B=33 ^\circ.$ find the angle $\angle FSD $ with proof.

Given two positive integers $n$ and $m$ and a function $f : \mathbb{Z} \times \mathbb{Z} \to \left\{0,1\right\}$ with the property that \begin{align*} f\left(i, j\right) = f\left(i+n, j\right) = f\left(i, j+m\right) \qquad \text{for all } \left(i, j\right) \in \mathbb{Z} \times \mathbb{Z} . \end{align*} Let $\left[k\right] = \left\{1,2,\ldots,k\right\}$ for each positive integer $k$. Let $a$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i+1, j\right) = f\left(i, j+1\right) . \end{align*} Let $b$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i-1, j\right) = f\left(i, j-1\right) . \end{align*} Prove that $a = b$.

Find all functions satisfying the equality $$(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x$$ for all $x \ne 1$.

Let $P(x)$ be a nonzero quadratic polynomial such that $P(1) = P(2) = 0$. Given that $P(3)^2 = P(4)+P(5)$, find $P(6)$. [i]Proposed by Andy Xu[/i]

There are $10$ seats in each of $10$ rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats? $ \textbf{a)}\ \dfrac{1}{55} \qquad\textbf{b)}\ \dfrac{1}{50} \qquad\textbf{c)}\ \dfrac{2}{55} \qquad\textbf{d)}\ \dfrac{1}{25} \qquad\textbf{e)}\ \text{None of above} $

Let $d$ be any positive integer not equal to 2, 5, or 13. Show that one can find distinct $a$ and $b$ in the set $\{2,5,13,d\}$ such that $ab - 1$ is not a perfect square.

Find the real number coefficient $c$ of polynomial $x^2+x+c$, if his roots $x_1$ and $x_2$ satisfy following: $$\frac{2x_1^3}{2+x_2}+\frac{2x_2^3}{2+x_1}=-1$$

Let $a$ and $n>0$ be integers. Define $a_{n} = 1+a+a^2...+a^{n-1}$. Show that if $p|a^p-1$ for all prime divisors of $n_{2}-n_{1}$, then the number $\frac{a_{n_{2}}-a_{n_{1}}}{n_{2}-n_{1}}$ is an integer.

Let $BH$ be an altitude of right angled triangle $ABC$($\angle B = 90^o$). An excircle of triangle $ABH$ opposite to $B$ touches $AB$ at point $A_1$; a point $C_1$ is defined similarly. Prove that $AC // A_1C_1$.

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

What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$? $\textbf{(A) } \text{It is a negative number ending with a 1.}$ $\textbf{(B) } \text{It is a positive number ending with a 1.}$ $\textbf{(C) } \text{It is a negative number ending with a 5.}$ $\textbf{(D) } \text{It is a positive number ending with a 5.}$ $\textbf{(E) } \text{It is a negative number ending with a 0.}$