Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that \[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]

Let $n$ be the $200$th smallest positive real solution to the equation $x- \frac{\pi}{2} =\ tan x$. Find the greatest integer that does not exceed $\frac{n}{2}$.

Prove that the ten's digit of any power of 3 is even.

Define $f(x, y)$ to be $\frac{|x|}{|y|}$ if that value is a positive integer, $\frac{|y|}{|x|}$ if that value is a positive integer, and zero otherwise. We say that a sequence of integers $\ell_1$ through $\ell_n$ is [i]good [/i] if $f(\ell_i, \ell_{i+1})$ is nonzero for all $i$ where $1 \le i \le n - 1$, and the score of the sequence is $\sum^{n-1}_{i=1} f(\ell_i, \ell_{i+1})$

All four angles of quadrilateral are greater than $60^o$. Prove that we can choose three sides to make a triangle.

Let $n$ be an integer bigger than $0$. Let $\mathbb{A}= ( a_1,a_2,...,a_n )$ be a set of real numbers. Find the number of functions $f:A \rightarrow A$ such that $f(f(x))-f(f(y)) \ge x-y$ for any $x,y \in \mathbb{A}$, with $x>y$.

The diagram below shows nine points on a circle where $AB=BC=CD=DE=EF=FG=GH$. Given that $\angle GHJ=117^\circ$ and $\overline{BH}$ is perpendicular to $\overline{EJ}$, there are relatively prime positive integers $m$ and $n$ so that the degree measure of $\angle AJB$ is $\textstyle\frac mn$. Find $m+n$. [asy] size(175); defaultpen(linewidth(0.6)); draw(unitcircle,linewidth(0.9)); string labels[] = {"A","B","C","D","E","F","G"}; int start=110,increment=20; pair J=dir(210),x[],H=dir(start-7*increment); for(int i=0;i<=6;i=i+1) { x[i]=dir(start-increment*i); draw(J--x[i]--H); dot(x[i]); label("$"+labels[i]+"$",x[i],dir(origin--x[i])); } draw(J--H); dot(H^^J); label("$H$",H,dir(origin--H)); label("$J$",J,dir(origin--J)); [/asy]

Show that $\tan \left( \frac{\pi}{m} \right)$ is irrational for all positive integers $m \ge 5$.

Find the maximum number of queens you could put on $2017 \times 2017$ chess table such that each queen attacks at most $1$ other queen.

Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.

Does there exist an infinite number of sets $C$ consisting of $1983$ consecutive natural numbers such that each of the numbers is divisible by some number of the form $a^{1983}$, with $a \in \mathbb N, a \neq 1?$

Let $a,b,c $ be real positive numbers such that $abc(a+b+c)=3$ Prove that $(a+b)(b+c)(c+a) \geq 8$

Prove that, for any natural numbers $k,m,n$: $[k,m] \cdot [m,n] \cdot [n,k] \ge [k,m,n]^2$

Let \( M \) and \( N \) be the midpoints of sides \( BC \) and \( AC \), respectively, in an acute-angled triangle \( ABC \). Suppose there exists a point \( P \) on the line segment \( AM \) such that \( \angle NPC = \angle MPC \). Let \( D \) be the intersection point of the line \( NP \) and the line parallel to \( CP \) passing through \( B \). Prove that \( AD = AB \). Proposed by Soroush Behroozifar

$n$ people attend a party. There are no more than $n$ pairs of friends among them. Two people shake hands if and only if they have at least $1$ common friend. Given integer $m\ge 3$ such that $n\leq m^3$. Prove that there exists a person $A$, the number of people that shake hands with $A$ is no more than $m-1$ times of the number of $A$‘S friends.

Let $n\geq 4$ and $y_1,\dots, y_n$ real with $$\sum_{k=1}^n y_k=\sum_{k=1}^n k y_k=\sum_{k=1}^n k^2y_k=0$$ and $$y_{k+3}-3y_{k+2}+3y_{k+1}-y_k=0$$ for $1\leq k\leq n-3$. Prove that $$\sum_{k=1}^n k^3y_k=0$$

Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition \[\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.\] Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $.

For how many integers $n$, does the equation system \[\begin{array}{rcl} 2x+3y &=& 7\\ 5x + ny &=& n^2 \end{array}\] have a solution over integers? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{None of the preceding} $

A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$

Prove that any continuos function $ f: \mathbb{R}\rightarrow \mathbb{R}$ with \[ f(x)\equal{}\left\{ \begin{aligned} a_1x\plus{}b_1\ ,\ \text{for } x\le 1 \\ a_2x\plus{}b_2\ ,\ \text{for } x>1 \end{aligned} \right.\] where $ a_1,a_2,b_1,b_2\in \mathbb{R}$, can be written as: \[ f(x)\equal{}m_1x\plus{}n_1\plus{}\epsilon|m_2x\plus{}n_2|\ ,\ \text{for } x\in \mathbb{R}\] where $ m_1,m_2,n_1,n_2\in \mathbb{R}$ and $ \epsilon\in \{\minus{}1,\plus{}1\}$.

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$? $ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $

An angle $< 45^o$ is given in the plane of the drawing. Furthermore, the projection $P_1$ of a point $P$ lying above the plane of the drawing and the distance from $P$ to $P_1$ are given. $P_1$ lies within the given angle. On the legs of the given angle, construct points $A$ and $B$, respectively, such that the triangle $PAB$ has a minimal perimeter.

Given a positive integer $n$ with prime factorization $p_1^{e_1}p_2^{e_2}... p_k^{e_k}$ , we define $f(n)$ to be $\sum^k_{i=1}p_ie_i$. In other words, $f(n)$ is the sum of the prime divisors of $n$, counted with multiplicities. Let $M$ be the largest odd integer such that $f(M) = 2023$, and $m$ the smallest odd integer so that $f(m) = 2023$. Suppose that $\frac{M}{m}$ equals $p_1^{e_1}p_2^{e_2}... p_l^{e_l}$ , where the $e_i$ are all nonzero integers and the $p_i$ are primes. Find $\left| \sum^l_{i=1} (p_i + e_i) \right|$.

Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. See the figure below. The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$. What is $a+b$? [asy] unitsize(12); pair A = (0, 8/sqrt(3)), B = rotate(-120)*A, C = rotate(120)*A; real theta = 41.5; pair P1 = rotate(theta)*(2+2*sqrt(7/3), 0), P2 = rotate(-120)*P1, P3 = rotate(120)*P1; filldraw(P1--P2--P3--cycle, gray(0.9)); draw(Circle(A, 4)); draw(Circle(B, 4)); draw(Circle(C, 4)); dot(P1); dot(P2); dot(P3); defaultpen(fontsize(10pt)); label("$P_1$", P1, E*1.5); label("$P_2$", P2, SW*1.5); label("$P_3$", P3, N); label("$\omega_1$", A, W*17); label("$\omega_2$", B, E*17); label("$\omega_3$", C, W*17); [/asy] $\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554$