Determine all integers $n \ge 2$ such that $n \mid s_n-t_n$ where $s_n$ is the sum of all the integers in the interval $[1,n]$ that are mutually prime to $n$, and $t_n$ is the sum of the remaining integers in the same interval.

[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock? [b]p2.[/b] Compute $$\left( \frac{2012^{2012-2013} + 2013}{2013} \right) \times 2012.$$ Express your answer as a mixed number. [b]p3.[/b] What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$ [b]p4.[/b] Let $f(x)$ be a function such that $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$. If $f(2) = 3$ and $f(3) = 4$, find $f(12)$. [b]p5.[/b] Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$? [b]p6.[/b] Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $ 1$ Zergling day will the hour and minute hands be exactly opposite each other? [b]p7.[/b] Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies now than there were before, assuming every Sweet Candy is different? [b]p8.[/b] Ronny has a piece of paper in the shape of a right triangle $ABC$, where $\angle ABC = 90^o$, $\angle BAC = 30^o$, and $AC = 3$. Holding the paper fixed at $A$, Ronny folds the paper twice such that after the first fold, $\overline{BC}$ coincides with $\overline{AC}$, and after the second fold, $C$ coincides with $A$. If Ronny initially marked $P$ at the midpoint of $\overline{BC}$, and then marked $P'$ as the end location of $P$ after the two folds, find the length of $\overline{PP'}$ once Ronny unfolds the paper. [b]p9.[/b] How many positive integers have the same number of digits when expressed in base $3$ as when expressed in base $4$? [b]p10.[/b] On a $2 \times 4$ grid, a bug starts at the top left square and arbitrarily moves north, south, east, or west to an adjacent square that it has not already visited, with an equal probability of moving in any permitted direction. It continues to move in this way until there are no more places for it to go. Find the expected number of squares that it will travel on. Express your answer as a mixed number. PS. You had better use hide for answers.

Is there a positive integer divisible by the product of its digits such that this product is greater than $10^{2000}$?

There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 11$, has a remainder of $1$. What is the difference between the two smallest such integers? $\textbf{(A) }2310\qquad\textbf{(B) }2311\qquad\textbf{(C) }27,720\qquad\textbf{(D) }27,721\qquad \textbf{(E) }\text{none of these}$

Somebody incorrectly remembered Fermat's little theorem as saying that the congruence $a^{n+1} \equiv a \; \pmod{n}$ holds for all $a$ if $n$ is prime. Describe the set of integers $n$ for which this property is in fact true.

Let $Q_n$ be the product of the squares of even numbers less than or equal to $n$ and $K_n$ equal to the product of cubes of odd numbers less than or equal to $n$. What is the highest power of $98$, that [b]a)[/b]$Q_n$, [b]b)[/b] $K_n$ or [b]c)[/b] $Q_nK_n$ divides? If one divides $Q_{98}K_{98}$ by the highest power of $98$, then one get a number $N$. By which power-of-two number is $N$ still divisible?

For every non-negative integer $i$, define the number $M(i)$ as follows: write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$. (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ ) (a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $. Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ . (b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ . Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$. (A Kanel)

Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.

Determine the last digit of $5^5+6^6+7^7+8^8+9^9$.

Decide whether there exists a function $f : Z \rightarrow Z$ such that for each $k =0,1, ...,1996$ and for any integer $m$ the equation $f (x)+kx = m$ has at least one integral solution $x$.

Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$

Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$. [b](a)[/b] Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$. [b](b)[/b] What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

Given an integer $ n>6$. Consider those integers $ k\in (n(n\minus{}1),n^{2})$ which are coprime with $ n$. Prove that the greatest common divisor of the considered numbers is $ 1$.

Let $a, b, c, d$ be odd positive integers and pairwise coprime. For a positive integer $n$, let $$f(n) = \left[\frac{n}{a} \right]+\left[\frac{n}{b}\right]+\left[\frac{n}{c}\right]+\left[\frac{n}{d}\right]$$ Prove that $$\sum_{n=1}^{abcd}(-1)^{f(n)}=1$$

If $f:\mathbb N\to\mathbb R$ is a function such that $$\prod_{d\mid n}f(d)=2^n$$holds for all $n\in\mathbb N$, show that $f$ sends $\mathbb N$ to $\mathbb N$.

Given that $x$ and $y$ are positive integers such that $xy^{10}$ is perfect 33rd power of a positive integer, prove that $x^{10}y$ is also a perfect 33rd power!

Let $a$ and $b$ be positive integers. If $a^2 + b^2$ is divisible by $ab$, prove that $a = b$. (BR Frenkin)

For a fixed positive integer $k$, there are two sequences $A_n$ and $B_n$. They are defined inductively, by the following recurrences. $A_1 = k$, $A_2 = k$, $A_{n+2} = A_{n}A_{n+1}$ $B_1 = 1$, $B_2 = k$, $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$ Prove that for all positive integers $n$, $A_{2n}B_{n+3}$ is an integer.

Let $n$ be a fixed positive integer and let $x, y$ be positive integers such that $xy = nx+ny$. Determine the minimum and the maximum of $x$ in terms of $n$.

The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$. [b](a)[/b] Prove that $f_{2010} $ is divisible by $10$. [b](b)[/b] Is $f_{1005}$ divisible by $4$? Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.

$a$ and $b$ are fixed real numbers. With $x_n$ we denote the sum of the digits of $an+b$ in the decimal number system. Prove that the sequence $x_n$ contains an infinite constant subsequence.

Let $\{a_{n}\}_{n \geq 0}$ and $\{b_{n}\}_{n \geq 0}$ be two sequences of integer numbers such that: i. $a_{0}=0$, $b_{0}=8$. ii. For every $n \geq 0$, $a_{n+2}=2a_{n+1}-a_{n}+2$, $b_{n+2}=2b_{n+1}-b_{n}$. iii. $a_{n}^{2}+b_{n}^{2}$ is a perfect square for every $n \geq 0$. Find at least two values of the pair $(a_{1992},\, b_{1992})$.

There are $10$ consecutive 30-digit numbers. We write the biggest divisor for every number ( divisor is not equal number). Prove that some written numbers ends with same digit.

For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$? [i]David Yang.[/i]