Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$ [i]Note: A primitive root of $r$ of a prime $p$ is an integer $1 \le r < p$ such that the smallest positive integer $k$ such that $r^k \equiv 1 \pmod{p}$ is $k = p-1$.[/i]

[b]p1.[/b] Find the smallest positive integer $N$ such that $N!$ is a multiple of $10^{2010}$. [b]p2.[/b] An equilateral triangle $T$ is externally tangent to three mutually tangent unit circles, as shown in the diagram. Find the area of $T$. [b]p3. [/b]The polynomial $p(x) = x^3 + ax^2 + bx + c$ has the property that the average of its roots, the product of its roots, and the sum of its coefficients are all equal. If $p(0) = 2$, find $b$. [b]p4.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs AiBj , for $1 \le i \le 5$ and $1 \le j \le 4$. Find the maximum of $f(P, S)$ over all pairs of shapes. [b]p5.[/b] Let $ a, b, c$ be three three-digit perfect squares that together contain each nonzero digit exactly once. Find the value of $a + b + c$. [b]p6. [/b]There is a big circle $P$ of radius $2$. Two smaller circles $Q$ and $R$ are drawn tangent to the big circle $P$ and tangent to each other at the center of the big circle $P$. A fourth circle $S$ is drawn externally tangent to the smaller circles $Q$ and $R$ and internally tangent to the big circle $P$. Finally, a tiny fifth circle $T$ is drawn externally tangent to the $3$ smaller circles $Q, R, S$. What is the radius of the tiny circle $T$? [b]p7.[/b] Let $P(x) = (1 +x)(1 +x^2)(1 +x^4)(1 +x^8)(...)$. This infinite product converges when $|x| < 1$. Find $P\left( \frac{1}{2010}\right)$. [b]p8.[/b] $P(x)$ is a polynomial of degree four with integer coefficients that satisfies $P(0) = 1$ and $P(\sqrt2 + \sqrt3) = 0$. Find $P(5)$. [b]p9.[/b] Find all positive integers $n \ge 3$ such that both roots of the equation $$(n - 2)x^2 + (2n^2 - 13n + 38)x + 12n - 12 = 0$$ are integers. [b]p10.[/b] Let $a, b, c, d, e, f$ be positive integers (not necessarily distinct) such that $$a^4 + b^4 + c^4 + d^4 + e^4 = f^4.$$ Find the largest positive integer $n$ such that $n$ is guaranteed to divide at least one of $a, b, c, d, e, f$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there exists a real number $C > 1$ with the following property. Whenever $n > 1$ and $a_0 < a_1 < a_2 <\cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1} \cdots \frac{1}{a_n}$ form an arithmetic progression, then $a_0 > C^n$.

Determine all the pairs $(a,b)$ of positive integers, such that all the following three conditions are satisfied: 1- $b>a$ and $b-a$ is a prime number 2- The last digit of the number $a+b$ is $3$ 3- The number $ab$ is a square of an integer.

Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $2000$ digits?

Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers \[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \] are integers. Prove that $p=q=r $. [i]Nazar Agakhanov[/i]

Prove that the product of four consecutive natural numbers cannot be the square of an integer.

Find all pairs of $ (a, n) $ natural numbers such that $ \varphi (a ^ n + n) = 2 ^ n. $ ($ \varphi (n) $ is the Euler function, that is, the number of integers from $1$ up to $ n $, relative prime to $ n $)

How many integers less than $400$ have exactly $3$ factors that are perfect squares?

Let $0 \leq b \leq c \leq d \leq a$ and $a>14$ are integers. Prove, that there is such natural $n$ that can not be represented as $$n=x(ax+b)+y(ay+c)+z(az+d)$$ where $x,y,z$ are some integers. [i]K. Kohas[/i]

[b]p1.[/b] $2011$ distinct points are arranged along the perimeter of a circle. We choose without replacement four points $P$, $Q$, $R$, $S$. What is the probability that no two of the segments $P Q$, $QR$, $RS$, $SP$ intersect (disregarding the endpoints)? [b]p2.[/b] In Soviet Russia, all phone numbers are between three and six digits and contain only the digits $1$, $2$, and $3$. No phone number may be the prefix of another phone number, so, for example, we cannot have the phone numbers $123$ and $12332$. If the Soviet bureaucracy has preassigned $10$ phone numbers of length $3$, $20$ numbers of length $4$, and $77$ phone numbers of length $6$, what is the maximum number of phone numbers of length $5$ that the authorities can allocate? [b]p3.[/b] The sequence $\{a_n\}_{n\ge 1}$ is defined as follows: we have $a_1 = 1$, $a_2 = 0$, and for $n \ge 3$ we have $$a_n = \frac12 \sum\limits_{\substack{1\le i,j\\ i+j+k=n}} a_ia_ja_k.$$ Find $$\sum^{\infty}_{n=1} \frac{a_n}{2^n}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,\cdots,a_{2000}$ be distinct positive integers such that $1 \leq a_1 < a_2 < \cdots < a_{2000} < 4000$ such that the LCM (least common multiple) of any two of them is $\geq 4000$. Show that $a_1 \geq 1334$

Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.

Sara wrote on the board an integer with less than thirty digits and ending in $2$. Celia erases the $2$ from the end and writes it at the beginning. The number that remains written is equal to twice the number that Sara had written. What number did Sara write?

Given a natural number $k$, find the smallest natural number $C$ such that $$\frac C{n+k+1}\binom{2n}{n+k}$$is an integer for every integer $n\ge k$.

Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.

For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation \[1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}\] Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 7)[/i]

[u]Round 1[/u] [b]p1. [/b]The Queen of Bees invented a new language for her hive. The alphabet has only $6$ letters: A, C, E, N, R, T; however, the alphabetic order is different than in English. A word is any sequence of $6$ different letters. In the dictionary for this language, the word TRANCE immediately follows NECTAR. What is the last word in the dictionary? [b]p2.[/b] Is it possible to solve the equation $\frac{1}{x}= \frac{1}{y} +\frac{1}{z}$ with $x,y,z$ integers (positive or negative) such that one of the numbers $x,y,z$ has one digit, another has two digits, and the remaining one has three digits? [b]p3.[/b] The $10,000$ dots in a $100\times 100$ square grid are all colored blue. Rekha can paint some of them red, but there must always be a blue dot on the line segment between any two red dots. What is the largest number of dots she can color red? The picture shows a possible coloring for a $5\times 7$ grid. [img]https://cdn.artofproblemsolving.com/attachments/0/6/795f5ab879938ed2a4c8844092b873fb8589f8.jpg[/img] [b]p4.[/b] Six flies rest on a table. You have a swatter with a checkerboard pattern, much larger than the table. Show that there is always a way to position and orient the swatter to kill at least five of the flies. Each fly is much smaller than a swatter square and is killed if any portion of a black square hits any part of the fly. [b]p5.[/b] Maryam writes all the numbers $1-81$ in the cells of a $9\times 9$ table. Tian calculates the product of the numbers in each of the nine rows, and Olga calculates the product of the numbers in every column. Could Tian's and Olga's lists of nine products be identical? [u]Round 2[/u] [b]p6.[/b] A set of points in the plane is epic if, for every way of coloring the points red or blue, it is possible to draw two lines such that each blue point is on a line, but none of the red points are. The figure shows a particular set of $4$ points and demonstrates that it is epic. What is the maximum possible size of an epic set? [img]https://cdn.artofproblemsolving.com/attachments/e/f/44fd1679c520bdc55c78603190409222d0b721.jpg[/img] [b]p7.[/b] Froggy Chess is a game played on a pond with lily pads. First Judit places a frog on a pad of her choice, then Magnus places a frog on a different pad of his choice. After that, they alternate turns, with Judit moving first. Each player, on his or her turn, selects either of the two frogs and another lily pad where that frog must jump. The jump must reduce the distance between the frogs (all distances between the lily pads are different), but both frogs cannot end up on the same lily pad. Whoever cannot make a move loses. The picture below shows the jumps permitted in a particular situation. Who wins the game if there are $2017$ lily pads? [img]https://cdn.artofproblemsolving.com/attachments/a/9/1a26e046a2a614a663f9d317363aac61654684.jpg[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For each positive integer $ n$,define $ f(n)\equal{}lcm(1,2,...,n)$. (a)Prove that for every $ k$ there exist $ k$ consecutive positive integers on which $ f$ is constant. (b)Find the maximum possible cardinality of a set of consecutive positive integers on which $ f$ is strictly increasing and find all sets for which this maximum is attained.

Let $p$ be a prime number congruent to $3$ modulo $4$, and consider the equation $(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1$. Prove that this equation has infinitely many solutions in positive integers, and show that if $(x,y) = (x_{0}, y_{0})$ is a solution of the equation in positive integers, then $p | x_{0}$.

Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$. (Folklore)

Prove that there is a natural number $n > 1$ such that the product of some $n$ consecutive natural numbers is equal to the product of some $n + 100$ consecutive natural numbers.

Let $ a$ be a fixed integer. Find all integer solutions $ x,y,z$ of the system: $ 5x\plus{}(a\plus{}2)y\plus{}(a\plus{}2)z\equal{}a,$ $ (2a\plus{}4)x\plus{}(a^2\plus{}3)y\plus{}(2a\plus{}2)z\equal{}3a\minus{}1,$ $ (2a\plus{}4)x\plus{}(2a\plus{}2)y\plus{}(a^2\plus{}3)z\equal{}a\plus{}1.$

Given that $m$ and $n$ are positive integers with property: \[(mn)\mid(m^{2010}+n^{2010}+n)\] Show that there exists a positive integer $k$ such that $n=k^{2010}$ [i]Nanang Susyanto, Yogyakarta[/i]