Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that the stack of crates is exactly $ 41$ ft tall, where $ m$ and $ n$ are relatively prime positive integers. Find $ m$.

Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.

Is it true that for any positive integer $n>1$, there exists an infinite arithmetic progression $M_n$ of positive integers, such that for any $m \in M_n$, the number $n^m-1$ is not a perfect power (a positive integer is a perfect power if it is of the form $a^b$ for positive integers $a, b>1$)?

Let $n$ be a positive integer. Show that the equation \[\sqrt{x}+\sqrt{y}=\sqrt{n}\] have solution of pairs of positive integers $(x,y)$ if and only if $n$ is divisible by some perfect square greater than $1$. [i]Proposer: Nanang Susyanto[/i]

Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying $$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$ for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$. [i]Ankan Bhattacharya[/i]

Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$ [i]Proposed by Oleksii Masalitin (Ukraine)[/i]

The following sentece is written on a board: [center]The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.[/center] Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?

[list=a] [*]Prove that for every real number $t$ such that $0 < t < \tfrac{1}{2}$ there exists a positive integer $n$ with the following property: for every set $S$ of $n$ positive integers there exist two different elements $x$ and $y$ of $S$, and a non-negative integer $m$ (i.e. $m \ge 0 $), such that \[ |x-my|\leq ty.\] [*]Determine whether for every real number $t$ such that $0 < t < \tfrac{1}{2} $ there exists an infinite set $S$ of positive integers such that \[|x-my| > ty\] for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m > 0$).

Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation $$ a^2 + b^2 = c^2,$$ then: 1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $, 2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $, 3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.

Find all pairs of prime numbers $p$ and $q$ such that: \[ p(p + q) = q^p+ 1. \]

A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of $360^\circ$. That is, if the center of the smaller disk has moved to the point $D$, and the point on the smaller disk that began at $A$ has now moved to point $B$, then $\overline{AC}$ is parallel to $\overline{BD}$. Then $\sin^2(\angle BEA)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

(a) Two identical cogwheels with $14$ teeth each are given . One is laid horizontally on top of the other in such a way that their teeth coincide (thus the projections of the teeth on the horizontal plane are identical ) . Four pairs of coinciding teeth are cut off. Is it always possible to rotate the two cogwheels with respect to each other so that their common projection looks like that of an entire cogwheel? (The cogwheels may be rotated about their common axis, but not turned over.) (b) Answer the same question , but with two $13$-tooth cogwheels and four pairs of cut-off teeth.

Let $ 1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $ \sum_{i\equal{}1}^n 1/a_i \leq 1$. Let $ y$ denote the number of positive integers smaller that $ x$ not divisible by any of the $ a_i$. Prove that \[ y > \frac{cx}{\log x}\] with a suitable positive constant $ c$ (independent of $ x$ and the numbers $ a_i$). [i]I. Z. Ruzsa[/i]

Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

Bored of waiting for his plane to travel to the International Mathematics Olympiad, Daniel began to write powers of $2$ in a list in his notebook as follows: $\bullet$ Starting with the number $1$, Daniel writes the next power of $2$ at the end of his list and reverses the order of the numbers in the list. Let us call such a modification of the list, including the first step, a [i]move[/i]. The list in each of the first $4$ moves it looks like this: $$1 \,\,\,\, \to 2, 1 \,\,\,\, \,\,\,\, \to 4, 1, 2 \,\,\,\, \,\,\,\, \to 8, 2, 1, 4$$ Daniel plans to carry out operations until his plane arrives, but he is worried let the list grow too. After $2020$ moves, what is the sum of the first $1010$ numbers?

For a positive integer $n$, denote $A_n=\{(x,y)\in\mathbb Z^2|x^2+xy+y^2=n\}$. (a) Prove that the set $A_n$ is always finite. (b) Prove that the number of elements of $A_n$ is divisible by $6$ for all $n$. (c) For which $n$ is the number of elements of $A_n$ divisible by $12$?

Given are positive integers $a, b$ satisfying $a \geq 2b$. Does there exist a polynomial $P(x)$ of degree at least $1$ with coefficients from the set $\{0, 1, 2, \ldots, b-1 \}$ such that $P(b) \mid P(a)$?

The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r,$ where $m, n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0.$ Find $m+n+r.$

Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that: $\bullet$ it is a $4$-digit number whose first digit is $5$. $\bullet$ it is a multiple of $9$; $\bullet$ The larger number is more likely to be a password than a smaller number. Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed: Alice: ”I have no idea what the number is.” Bob: ”I have no idea too.” After that both of them knew which number they should try first. Identify this number

Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

In one country there are coins of value $1,2,3,4$ or $5$. Nisse wants to buy a pair of shoes. While paying, he tells the seller that he has $100$ coins in the bag, but that he does not know the exact number of coins of each value. ”Fine, then you will have the exact amount”, the seller responds. What is the price of the shoes, and how did the seller conclude that Nisse would have the exact amount?

Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}\plus{}4y^{4}\plus{}5z^{3}\minus{}y^{4}z \equiv 0$ is $ p^2$

Consider the sequence $ \{a_n\}_{n\geq1}$ defined as follows: $ a_1 \equal{} 1$, $ a_{2k} \equal{} 1 \plus{} a_k$ and $ a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}}$ for every $ k\geq 1$. Prove that every positive rational number appears on the sequence $ \{a_n\}$ exactly once.

Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.