Let $n\geq 2$ be an integer. Consider the solutions of the system $$\begin{cases} n=a+b-c \\ n=a^2+b^2-c^2 \end{cases}$$ where $a,b,c$ are integers. Show that there is at least one solution and that the solutions are finitely many.

Find all integer solutions of the equation $xy+5x-3y=27$.

A wobbly number is a positive integer whose digits are alternately zero and non-zero with the last digit non-zero (for example, 201). Find all positive integers which do not divide any wobbly number.

[u]Round 1[/u] [b]p1.[/b] If this mathathon has $7$ rounds of $3$ problems each, how many problems does it have in total? (Not a trick!) [b]p2.[/b] Five people, named $A, B, C, D,$ and $E$, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away from where they started? [b]p3.[/b] At Barrios’s absurdly priced fish and chip shop, one fish is worth $\$13$, one chip is worth $\$5$. What is the largest integer dollar amount of money a customer can enter with, and not be able to spend it all on fish and chips? [u]Round 2[/u] [b]p4.[/b] If there are $15$ points in $4$-dimensional space, what is the maximum number of hyperplanes that these points determine? [b]p5.[/b] Consider all possible values of $\frac{z_1 - z_2}{z_2 - z_3} \cdot \frac{z_1 - z_4}{z_2 - z_4}$ for any distinct complex numbers $z_1$, $z_2$, $z_3$, and $z_4$. How many complex numbers cannot be achieved? [b]p6.[/b] For each positive integer $n$, let $S(n)$ denote the number of positive integers $k \le n$ such that $gcd(k, n) = gcd(k + 1, n) = 1$. Find $S(2015)$. [u]Round 3 [/u] [b]p7.[/b] Let $P_1$, $P_2$,$...$, $P_{2015}$ be $2015$ distinct points in the plane. For any $i, j \in \{1, 2, ...., 2015\}$, connect $P_i$ and $P_j$ with a line segment if and only if $gcd(i - j, 2015) = 1$. Define a clique to be a set of points such that any two points in the clique are connected with a line segment. Let $\omega$ be the unique positive integer such that there exists a clique with $\omega$ elements and such that there does not exist a clique with $\omega + 1$ elements. Find $\omega$. [b]p8.[/b] A Chinese restaurant has many boxes of food. The manager notices that $\bullet$ He can divide the boxes into groups of $M$ where $M$ is $19$, $20$, or $21$. $\bullet$ There are exactly $3$ integers $x$ less than $16$ such that grouping the boxes into groups of $x$ leaves $3$ boxes left over. Find the smallest possible number of boxes of food. [b]p9.[/b] If $f(x) = x|x| + 2$, then compute $\sum^{1000}_{k=-1000} f^{-1}(f(k) + f(-k) + f^{-1}(k))$. [u]Round 4 [/u] [b]p10.[/b] Let $ABC$ be a triangle with $AB = 13$, $BC = 20$, $CA = 21$. Let $ABDE$, $BCFG$, and $CAHI$ be squares built on sides $AB$, $BC$, and $CA$, respectively such that these squares are outside of $ABC$. Find the area of $DEHIFG$. [b]p11.[/b] What is the sum of all of the distinct prime factors of $7783 = 6^5 + 6 + 1$? [b]p12.[/b] Consider polyhedron $ABCDE$, where $ABCD$ is a regular tetrahedron and $BCDE$ is a regular tetrahedron. An ant starts at point $A$. Every time the ant moves, it walks from its current point to an adjacent point. The ant has an equal probability of moving to each adjacent point. After $6$ moves, what is the probability the ant is back at point $A$? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782011p24434676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p>3$ be a prime number. Prove that the product of all primitive roots between 1 and $p-1$ is congruent 1 modulo $p$.

Let a fixed natural number m be given. Call a positive integer n to be an MTRP-number iff [list] [*] $n \equiv 1\ (mod\ m)$ [*] Sum of digits in decimal representation of $n^2$ is greater than equal to sum of digits in decimal representation of $n$ [/list] How many MTRP-numbers are there ?

Find all triplets of positive integers $ (a,m,n)$ such that $ a^m \plus{} 1 \mid (a \plus{} 1)^n$.

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

Define the [i]distance [/i] between two $5$-digit numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ to be the largest integer $j$ such that $a_j \ne b_j$ . (Example: the distance between $16523$ and $16452$ is $5$.) Suppose all $5$-digit numbers are written in a line in some order. What is the minimal possible sum of the distances of adjacent numbers in that written order?

Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$. (A Galochkin)

[b]p1.[/b] The numerator of the fraction was increased by 20%. By what percentage should its denominator be reduced so that the resulting fraction doubles? [b]p2.[/b] From point $O$ on the plane there are four rays $OA$, $OB$, $OC$ and $OD$ (not necessarily in that order). It is known that $\angle AOB =40^o$, $\angle BOC = 70^o$, $\angle COD = 80^o$. What values can the angle between rays $OA$ and $OD$ take? (The angle between the rays is from $0^o$ to $180^o$.) [b]p3.[/b] Three equal circles have a common interior point. Prove that there is a circle of the same radius containing the centers of these three circles. [b]p4.[/b] Two non-leap years are consecutive. The first one has more Mondays than Wednesdays. Which of the seven days of the week will occur most often in the second year? [b]p5.[/b] The difference between two four-digit numbers is $7$. How much can the sums of their digits differ? [b]p6.[/b] The numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written on the board. In one move you can increase any of the numbers by $3$ or $5$. What is the minimum number of moves you need to make for all the numbers to become equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$. [i]Victor Wang[/i]

There are integers $v,w,x,y,z$ and real numbers $0\le \theta < \theta' \le \pi$ such that $$\cos 3\theta = \cos 3\theta' = v^{-1}, \qquad w+x\cos \theta + y\cos 2\theta = z\cos \theta'.$$ Given that $z\ne 0$ and $v$ is positive, find the sum of the $4$ smallest possible values of $v$. [i]Proposed by Vijay Srinivasan[/i]

Given a set $S_n = \{1, 2, 3, \ldots, n\}$, we define a [i]preference list[/i] to be an ordered subset of $S_n$. Let $P_n$ be the number of preference lists of $S_n$. Show that for positive integers $n > m$, $P_n - P_m$ is divisible by $n - m$. [i]Note: the empty set and $S_n$ are subsets of $S_n$.[/i]

Let $a, m$ be two positive integers, $a \ne 10^k$, for all non-negative integers $k$ and $d_1, d_2, ... , d_m$ random decimal$^1$ digits with $d_1 > 0$. Prove that there exists some positive integer $n$ for which the representation in the decimal base of the number $a^n$ begins with the digits $d_1, d_2, ... , d_m$ in this order. $^1$ lesser or equal with $9$

What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is [i]palindromic[/i] if the sequence of decimal digits are the same when read backwards.)

Prove that if a prime is the sum of four perfect squares then the product of two of these is equal to the product of the other two. [i]Gherghe Stoica[/i]

Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$. Find all almost perfect numbers. [i]Paulius Ašvydis[/i]

Prove that there are $100$ distinct positive integers $n_1,n_2,\dots,n_{99},n_{100}$ such that $\frac{n_1^3+n_2 ^3+\dots +n_{100}^3}{100}$ is a perfect cube.

Let the sequence ($a_n$) be defined by $a_n = n^6 +5n^4 -12n^2 -36, n \ge 2$. (a) Prove that any prime number divides some term in this sequence. (b) Prove that there is a positive integer not dividing any term in the sequence. (c) Determine the least $n \ge 2$ for which $1989 | a_n$.

Let $S$ be an infinite set of positive integers and define: $T=\{ x+y|x,y \in S , x \neq y \} $ Suppose that there are only finite primes $p$ so that: 1.$p \equiv 1 \pmod 4$ 2.There exists a positive integer $s$ so that $p|s,s \in T$. Prove that there are infinity many primes that divide at least one term of $S$.

Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that: $$a^3 + b^3 + c^3 = 2016$$

Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that: (i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and (ii) each $b_i$ is a product of one or more terms of $(a_i)$.

p1. From the measurement of the height of nine trees obtained data as following. a) There are three different measurement results (in meters) b) All data are positive numbers c) Mean$ =$ median $=$ mode $= 3$ d) The sum of the squares of all data is $87.$ Determine all possible heights of the nine trees. p2. If $x$ and $y$ are integers, find the number of pairs $(x,y)$ that satisfy $|x|+|y|\le 50$. p3. The plane figure $ABCD$ on the side is a trapezoid with $AB$ parallel to $CD$. Points $E$ and $F$ lie on $CD$ so that $AD$ is parallel to $BE$ and $AF$ is parallel to $BC$. Point $H$ is the intersection of $AF$ with $BE$ and point $G$ is the intersection of $AC$ with $BE$. If the length of $AB$ is $4$ cm and the length of $CD$ is $10$ cm, calculate the ratio of the area of ​​the triangle $AGH$ to the area of ​​the trapezoid $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/c/7/e751fa791bce62f091024932c73672a518a240.png[/img] p4. A prospective doctor is required to intern in a hospital for five days in July $2011$. The hospital leadership gave the following rules: a) Internships may not be conducted on two consecutive days. b) The fifth day of internship can only be done after four days counted since the fourth day of internship. Suppose the fourth day of internship is the date $20$, then the fifth day of internship can only be carried out at least the date $24$. Determine the many possible schedule options for the prospective doctor. p5. Consider the following sequences of natural numbers: $5$, $55$, $555$, $5555$, $55555$, $...$ ,$\underbrace{\hbox{5555...555555...}}_{\hbox{n\,\,numbers}}$ . The above sequence has a rule: the $n$th term consists of $n$ numbers (digits) $5$. Show that any of the terms of the sequence is divisible by $2011$.

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.