Assume every $ 7$-digit whole number is a possible telephone number except those which begin with $ 0$ or $ 1$. What fraction of telephone numbers begin with $ 9$ and end with $ 0$? \[ \textbf{(A)}\ \frac{1}{63} \qquad \textbf{(B)}\ \frac{1}{80} \qquad \textbf{(C)}\ \frac{1}{81} \qquad \textbf{(D)}\ \frac{1}{90} \qquad \textbf{(E)}\ \frac{1}{100} \]

A treasure is buried under a square of an $8\times 8$ board. Under each other square is a message which indicates the minimum number of steps needed to reach the square with the treasure. Each step takes one from a square to another square sharing a common side. What is the minmum number of squares we must dig up in order to bring up the treasure for sure?

In triangle $ABC$, point $I$ is the center of the inscribed circle points, points $I_A$ and $I_C$ are the centers of the excircles, tangent to sides $BC$ and $AB$, respectively. Point $O$ is the center of the circumscribed circle of triangle $II_AI_C$. Prove that $OI \perp AC$

Let $ABC$ be a triangle with $AB < AC < BC$. Let the incenter and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ intersect the circumcircle of triangle $ABC$ at $P \ne A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively. Prove that $\angle KIL + \angle YPX = 180^{\circ}$. [i]Proposed by Dominik Burek, Poland[/i]

For all pairs $(m, n)$ of positive integers that have the same number $k$ of divisors we define the operation $\circ$. Write all their divisors in an ascending order: $1=m_1<\ldots<m_k=m$, $1=n_1<\ldots<n_k=n$ and set $$ m\circ n= m_1\cdot n_1+\ldots+m_k\cdot n_k. $$ Find all pairs of numbers $(m, n)$, $m\geqslant n$, such that $m\circ n=497$.

Does there exist a strictly increasing sequence $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for $\forall$ $c\in \mathbb{Z}$ the sequence $c+a_1,c+a_2,...,c+a_n...$ has finite number of primes? Explain your answer.

Find all pairs of functions $f ,g : Z \rightarrow Z $ that satisfy $f (g(x)+y) = g( f (y)+x) $ for all integers $ x,y$ and such that $g(x) = g(y)$ only if $x = y$.

Let $I$ be the incenter of a scalene triangle $\vartriangle ABC$. In other words, $\overline{AB},\overline{BC},\overline{CA}$ are distinct. Prove that if $D,E$ are two points on rays $\overrightarrow{BA},\overrightarrow{CA}$, satisfying $\overline{BD}=\overline{CA},\overline{CE}=\overline{BA}$ then line $DE$ pass through the orthocenter of $\vartriangle BIC$. [img]http://2.bp.blogspot.com/-aHCD5tL0FuA/XnYM1LoZjWI/AAAAAAAALeE/C6hO9W9FGhcuUP3MQ9aD7SNq5q7g_cY9QCK4BGAYYCw/s1600/imoc2019g1.png[/img]

The pentagon $A_1A_2A_3A_4A_5$ contains bisectors $\ell_1$, $\ell_2$, $...$, $\ell_5$ of angles $\angle A_1$, $\angle A_2$, $ ...$ , $\angle A_5$ respectively. Bisectors $\ell_1$ and $\ell_2$ intersect at point $B_1$, $\ell_2$ and $\ell_3$ - at point $B_2$, etc., $\ell_5$ and $\ell_1$ intersect at point $B_5$. Can the pentagon $B_1B_2B_3B_4B_5$ be convex?

A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $ 2/\pi $, as shown. What is the perimeter of this region? [asy] size(90); defaultpen(linewidth(0.7)); filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle,gray(0.5)); filldraw(arc((1,0),1,180,0, CCW)--cycle,gray(0.7)); filldraw(arc((0,1),1,90,270)--cycle,gray(0.7)); filldraw(arc((1,2),1,0,180)--cycle,gray(0.7)); filldraw(arc((2,1),1,270,90, CCW)--cycle,gray(0.7));[/asy] $ \textbf{(A) }\frac {4}\pi\qquad\textbf{(B) }2\qquad\textbf{(C) }\frac {8}\pi\qquad\textbf{(D) }4\qquad\textbf{(E) }\frac{16}{\pi} $

Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $ 1$ Joule of energy to hop one step north or one step south, and $ 1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $ 100$ Joules of energy, and hops till he falls asleep with $ 0$ energy. How many different places could he have gone to sleep?

How many even integers between 4000 and 7000 have four different digits?

Let $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A+B=AB+BA$. Prove that: a) if $n$ is odd, then $\det(AB-BA)=0$; b) if $\text{tr}(A)\neq \text{tr}(B)$, then $\det(AB-BA)=0$.

The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is: $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ -1\qquad\textbf{(E)}\ -19 $

In $\triangle ABC$, $P$ is a point on $BC$. $F\in AB,E\in AC,PF//CA,PE//BA$. If $S_{\triangle ABC}=1$, prove that at least one of $S_{\triangle BPF},S_{\triangle PCE},S_{PEAF}$ is not less than $\frac{4}{9}$.

Let $a, b, c$ and $d$ be non-negative integers. Prove that the numbers $2^a7^b$ and $2^c7^d$ give the same remainder when divided by $15$ iff the numbers $3^a5^b$ and $3^c5^d$ give the same remainder when divided by $16$.

Prove that there do not exist natural numbers $ n\ge 10$ having all digits different from zero, and such that all numbers which are obtained by permutations of its digits are perfect squares.

Compute the number of non-empty subsets $S$ of $\{-3, -2, -1, 0, 1, 2, 3\}$ with the following property: for any $k \ge 1$ distinct elements $a_1, \dots, a_k \in S$ we have $a_1 + \dots + a_k \neq 0$. [i]Proposed by Evan Chen[/i]

Suppose $a$ is a complex number such that \[a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0\] If $m$ is a positive integer, find the value of \[a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}\]

For every positive $a, b, c, d$ such that $a + c\le ac$ and $b + d \le bd$ prove that $ab + cd \ge 8$.

Given that $0 < a, b, c, d < 1$ and $abcd = (1 - a)(1 - b)(1 - c)(1 - d)$, prove that $(a + b + c + d) -(a + c)(b + d) \ge 1$

[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$ [i] Ivan Landgev[/i]

At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by $4$. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She think she has a $\frac{1}{6}$ chance of getting an A in English, and a $\frac{1}{4}$ chance of getting a B. In History, she has a $\frac{1}{4}$ chance of getting an A, and a $\frac{1}{3}$ chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5? $\textbf{(A) }\frac{11}{72}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{3}{16}\qquad\textbf{(D) }\frac{11}{24}\qquad\textbf{(E) }\frac{1}{2}$

Does there exist a convex pentagon, all of whose vertices are lattice points in the plane, with no lattice point in the interior?

[u]Round 1 [/u] [b]p1.[/b] In the Hundred Acre Wood, all the animals are either knights or liars. Knights always tell the truth and liars always lie. One day in the Wood, Winnie-the-Pooh, a knight, decides to visit his friend Rabbit, also a noble knight. Upon arrival, Pooh finds his friend sitting at a round table with $5$ other guests. One-by-one, Pooh asks each person at the table how many of his two neighbors are knights. Surprisingly, he gets the same answer from everybody! "Oh bother!" proclaims Pooh. "I still don't have enough information to figure out how many knights are at this table." "But it's my birthday," adds one of the guests. "Yes, it's his birthday!" agrees his neighbor. Now Pooh can tell how many knights are at the table. Can you? [b]p2.[/b] Harry has an $8 \times 8$ board filled with the numbers $1$ and $-1$, and the sum of all $64$ numbers is $0$. A magical cut of this board is a way of cutting it into two pieces so that the sum of the numbers in each piece is also $0$. The pieces should not have any holes. Prove that Harry will always be able to find a magical cut of his board. (The picture shows an example of a proper cut.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/98dec239cfc757e6f2996eef7876cbfd79d202.png[/img] [b]p3.[/b] Several girls participate in a tennis tournament in which each player plays each other player exactly once. At the end of the tournament, it turns out that each player has lost at least one of her games. Prove that it is possible to find three players $A$, $B$, and $C$ such that $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. [b]p4.[/b] $120$ bands are participating in this year's Northwest Grunge Rock Festival, and they have $119$ fans in total. Each fan belongs to exactly one fan club. A fan club is called crowded if it has at least $15$ members. Every morning, all the members of one of the crowded fan clubs start arguing over who loves their favorite band the most. As a result of the fighting, each of them leaves the club to join another club, but no two of them join the same one. Is it true that, no matter how the clubs are originally arranged, all these arguments will eventually stop? [b]p5.[/b] In Infinite City, the streets form a grid of squares extending infinitely in all directions. Bonnie and Clyde have just robbed the Infinite City Bank, located at the busiest intersection downtown. Bonnie sets off heading north on her bike, and, $30$ seconds later, Clyde bikes after her in the same direction. They each bike at a constant speed of $1$ block per minute. In order to throw off any authorities, each of them must turn either left or right at every intersection. If they continue biking in this manner, will they ever be able to meet? [u]Round 2 [/u] [b]p6.[/b] In a certain herd of $33$ cows, each cow weighs a whole number of pounds. Farmer Dan notices that if he removes any one of the cows from the herd, it is possible to split the remaining $32$ cows into two groups of equal total weight, $16$ cows in each group. Show that all $33$ cows must have the same weight. [b]p7.[/b] Katniss is thinking of a positive integer less than $100$: call it $x$. Peeta is allowed to pick any two positive integers $N$ and $M$, both less than $100$, and Katniss will give him the greatest common divisor of $x+M$ and $N$ . Peeta can do this up to seven times, after which he must name Katniss' number $x$, or he will die. Can Peeta ensure his survival? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].