Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.

A snowman is built on a level plane by placing a ball radius $6$ on top of a ball radius $8$ on top of a ball radius $10$ as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$. [asy] size(150); draw(circle((0,0),24)); draw(ellipse((0,0),24,9)); draw(circle((0,-56),32)); draw(ellipse((0,-56),32,12)); draw(circle((0,-128),40)); draw(ellipse((0,-128),40,15)); [/asy]

Let $m$ and $n$ be positive integers. Let $S$ be the set of all points $(x, y)$ with integer coordinates such that $1 \leq x,y \leq m+n-1$ and $m+1 \leq x +y \leq 2m+n-1.$ Let $L$ be the set of the $3m+3n-3$ lines parallel to one of $x = 0, y = 0,$ or $x + y = 0$ and passing through at least one point in $S$. For which pairs $(m, n)$ does there exist a subset $T$ of $S$ such that every line in $L$ intersects an odd number of elements of $T$?

Prove that there are infinitely many positive integers $n$ such that $n \times n \times n$ can not be filled completely with 2 x 2 x 2 and 3 x 3 x 3 solid cubes.

Find all functions $f : \mathbb N \to \mathbb N$ such that: i) $f^{2000}(m)=f(m)$ for all $m \in \mathbb N$, ii) $f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}$, for all $m,n\in \mathbb N$, and iii) $f(m)=1$ if and only if $m=1$.

For each positive integer $n$, define $V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor$. Prove that, in the sequence $V_1,V_2,\ldots,$ there are infinitely many odd integers, as well as infinitely many even integers. [i]Remark.[/i] $\lfloor x\rfloor$ is the largest integer that does not exceed the real number $x$.

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

Let $k, n$ be odd positve integers greater than $1$. Prove that if there a exists natural number $a$ such that $k|2^a+1, \ n|2^a-1$, then there is no natural number $b$ satisfying $k|2^b-1, \ n|2^b+1$.

$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that $$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$ [i]Proposed by Ali Partofard[/i]

$\boxed{\text{A5}}$ Determine all positive integers $a,b$ such that $a^{2}b^{2}+208=4([a,b]+(a,b))^2$ where $[a,b]$-lcm of $a,b$ and $(a,b)$-gcd of $a,b$.

Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that \[\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.\]

Let $a$ be a positive integer and $(a_n)_{n\geqslant 1}$ be a sequence of positive integers satisfying $a_n<a_{n+1}\leqslant a_n+a$ for all $n\geqslant 1$. Prove that there are infinitely many primes which divide at least one term of the sequence. [i]Moldavia Olympiad, 1994[/i]

Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $m/n$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$ Calculate $S_n=\sum_{k=1}^{2^n}T(K)$.

Determine the value of $(101 \times 99)$ - $(102 \times 98)$ + $(103 \times 97)$ − $(104 \times 96)$ + ... ... + $(149 \times 51)$ − $(150 \times 50)$.

Determine the quantity of positive integers $N$ of $10$ digits with the following properties: I- All the digits of $N$ are non-zero. II- $11|N$. III- $N$ and all the permutation(s) of the digits of $N$ are divisible by $12$.

Natural numbers are written in the cells of of a $2014\times2014$ regular square grid such that every number is the average of the numbers in the adjacent cells. Describe and prove how the number distribution in the grid can be.

Find integers $a_1,a_2,\cdots,a_{18}$, s.t. $a_1=1,a_2=2,a_{18}=2019$, and for all $3\le k\le 18$, there exists $1\le i<j<k$ with $a_k=a_i+a_j$.

Let $\Phi$ denote the Euler totient function. Prove that for infinitely many $k$ we have $\Phi (2^k+1) < 2^{k-1}$ and that for infinitely many $m$ one has $\Phi (2^m+1) > 2^{m-1}$

Let $p$ be a prime number. We know that each natural number can be written in the form \[\sum_{i=0}^{t}a_ip^i (t,a_i \in \mathbb N\cup \{0\},0\le a_i\le p-1)\] Uniquely. Now let $T$ be the set of all the sums of the form \[\sum_{i=0}^{\infty}a_ip^i (0\le a_i \le p-1).\] (This means to allow numbers with an infinite base $p$ representation). So numbers that for some $N\in \mathbb N$ all the coefficients $a_i, i\ge N$ are zero are natural numbers. (In fact we can consider members of $T$ as sequences $(a_0,a_1,a_2,...)$ for which $\forall_{i\in \mathbb N}: 0\le a_i \le p-1$.) Now we generalize addition and multiplication of natural numbers to this set so that it becomes a ring (it's not necessary to prove this fact). For example: $1+(\sum_{i=0}^{\infty} (p-1)p^i)=1+(p-1)+(p-1)p+(p-1)p^2+...$ $=p+(p-1)p+(p-1)p^2+...=p^2+(p-1)p^2+(p-1)p^3+...$ $=p^3+(p-1)p^3+...=...$ So in this sum, coefficients of all the numbers $p^k, k\in \mathbb N$ are zero, so this sum is zero and thus we can conclude that $\sum_{i=0}^{\infty}(p-1)p^i$ is playing the role of $-1$ (the additive inverse of $1$) in this ring. As an example of multiplication consider \[(1+p)(1+p+p^2+p^3+...)=1+2p+2p^2+\cdots\] Suppose $p$ is $1$ modulo $4$. Prove that there exists $x\in T$ such that $x^2+1=0$. [i]Proposed by Masoud Shafaei[/i]

Find the number of ordered palindromic partitions of an integer $n$.

Suppose that $a$ is an integer and that $n! + a$ divides $(2n)!$ for infinitely many positive integers $n$. Prove that $a = 0$.

Compute the product of all positive integers $n$ for which $3(n!+1)$ is divisible by $2n - 5$.

Let $f:\mathbb N\to \mathbb N$. Show that $f(m)+n\mid f(n)+m$ for all positive integers $m\le n$ if and only if $f(m)+n\mid f(n)+m$ for all positive integers $m\ge n$. [i]Proposed by Carl Schildkraut[/i]

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.