For a positive integer $n$ denote $d(n)$ its greatest odd divisor. Find the value of the sum $$d(1008)+d(1009)+...+d(2015)$$

Consider the sequence $<{a_n}>$ of natural numbers such that {i} $a_n$ is a square numver for all $n$ ; (ii) $a_{n+1} - a_n$ is either a prime or a square of a prime for each $n$. Show that $<a_n>$ is a finite sequence. Determine the longest such sequence.

Let $a_d$ be the number of non-negative integer solutions $(a, b)$ to $a + b = d$ where $a \equiv b$ (mod $n$) for a fixed $n \in Z^+$. Consider the generating function $M(t) = a_0 + a_1t + a_2t^2 + ...$ Consider $$P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right).$$ Then $P(n)$, $n \in Z^+$ is a polynomial in $n$, so we can extend its domain to include all real numbers while having it remain a polynomial. Find $P(0)$.

Let $n$ be a natural number. Prove that there is a multiple of $n$ that can be written only with the digits $0$ and $1$.

The numbers of the set $\{1,2,\ldots,n\}$ are colored with $6$ colors. Let $$S:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have the same color}\}$$and $$D:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have three different colors}\}.$$Prove that $$|D|\le2|S|+\frac{n^2}2.$$

Does there exist an integer $z$ that can be written in two different ways as $z = x! + y!$, where $x, y$ are natural numbers with $x \le y$ ?

Let $\mathbb{Z}_{>0} = \{1, 2, 3, \ldots \}$ be the set of all positive integers. Find all strictly increasing functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(f(n)) = 3n$.

Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 | (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.

Let $A$ be the sum of the first $2k+1$ positive odd integers, and let $B$ be the sum of the first $2k+1$ positive even integers. Show that $A+B$ is a multiple of $4k+3$.

Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$

For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define \[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\] That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity. Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that \[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\] [i]Proposed by Rodrigo Sanches Angelo, Brazil[/i]

Does there exist a positive odd integer $n$ so that there are primes $p_1$, $p_2$ dividing $2^n-1$ with $p_1-p_2=2$?

Conside a positive odd integer $k$ and let $n_1<n_2<\ldots<n_k$ be $k$ positive odd integers. Prove that: \[n_1^2-n_2^2+n_3^2-n_4^2+\ldots+n_k^2\ge 2k^2-1\] [i]Titu Andreescu[/i]

International Mathematical Olympiad National Selection Test Malaysia 2021 Round 1 Primary Time: 2.5 hours [hide=Rules] $\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer. $\bullet$ No mark is deducted for a wrong answer. $\bullet$ The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.[/hide] [b]Part A[/b] (1 point each) p1. Faris has six cubes on his table. The cubes have a total volume of $2021$ cm$^3$. Five of the cubes have side lengths $5$ cm, $5$ cm, $6$ cm, $6$ cm, and $11$ cm. What is the side length of the sixth cube (in cm)? p2. What is the sum of the first $200$ even positive integers? p3. Anushri writes down five positive integers on a paper. The numbers are all different, and are all smaller than $10$. If we add any two of the numbers on the paper, then the result is never $10$. What is the number that Anushri writes down for certain? p4. If the time now is $10.00$ AM, what is the time $1,000$ hours from now? Note: Enter the answer in a $12$-hour system, without minutes and AM/PM. For example, if the answer is $9.00$ PM, just enter $9$. p5. Aminah owns a car worth $10,000$ RM. She sells it to Neesha at a $10\%$ profit. Neesha sells the car back to Aminah at a $10\%$ loss. How much money did Aminah make from the two transactions, in RM? [b]Part B[/b] (2 points each) p6. Alvin takes 250 small cubes of side length $1$ cm and glues them together to make a cuboid of size $5$ cm  $\times 5$ cm  $\times 10$ cm. He paints all the faces of the large cuboid with the color green. How many of the small cubes are painted by Alvin? p7. Cikgu Emma and Cikgu Tan select one integer each (the integers do not have to be positive). The product of the two integers they selected is $2021$. How many possible integers could have been selected by Cikgu Emma? p8. A three-digit number is called [i]superb[/i] if the first digit is equal to the sum of the other two digits. For example, $431$ and $909$ are superb numbers. How many superb numbers are there? p9. Given positive integers $a, b, c$, and $d$ that satisfy the equation $4a = 5b =6c = 7d$. What is the smallest possible value of $ b$? p10. Find the smallest positive integer n such that the digit sum of n is divisible by $5$, and the digit sum of $n + 1$ is also divisible by $5$. Note: The digit sum of $1440$ is $1 + 4 + 4 + 0 = 9$. [b]Part C[/b] (3 points each) p11. Adam draws $7$ circles on a paper, with radii $ 1$ cm, $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm, and $7$ cm. The circles do not intersect each other. He colors some circles completely red, and the rest of the circles completely blue. What is the minimum possible difference (in cm$^2$) between the total area of the red circles and the total area of the blue circles? p12. The number $2021$ has a special property that the sum of any two neighboring digits in the number is a prime number ($2 + 0 = 2$, $0 + 2 = 2$, and $2 + 1 = 3$ are all prime numbers). Among numbers from $2021$ to $2041$, how many of them have this property? p13. Clarissa opens a pet shop that sells three types of pets: goldshes, hamsters, and parrots. The pets inside the shop together have a total of $14$ wings, $24$ heads, and $62$ legs. How many goldshes are there inside Clarissa's shop? p14. A positive integer $n$ is called [i]special [/i] if $n$ is divisible by $4$, $n+1$ is divisible by $5$, and $n + 2$ is divisible by $6$. How many special integers smaller than $1000$ are there? p15. Suppose that this decade begins on $ 1$ January $2020$ (which is a Wednesday) and the next decade begins on $ 1$ January $2030$. How many Wednesdays are there in this decade? [b]Part D[/b] (4 points each) p16. Given an isosceles triangle $ABC$ with $AB = AC$. Let D be a point on $AB$ such that $CD$ is the bisector of $\angle ACB$. If $CB = CD$, what is $\angle ADC$, in degrees? p17. Determine the number of isosceles triangles with the following properties: all the sides have integer lengths (in cm), and the longest side has length $21$ cm. p18. Haz marks $k$ points on the circumference of a circle. He connects every point to every other point with straight lines. If there are $210$ lines formed, what is $k$? p19. What is the smallest positive multiple of $24$ that can be written using digits $4$ and $5$ only? p20. In a mathematical competition, there are $2021$ participants. Gold, silver, and bronze medals are awarded to the winners as follows: (i) the number of silver medals is at least twice the number of gold medals, (ii) the number of bronze medals is at least twice the number of silver medals, (iii) the number of all medals is not more than $40\%$ of the number of participants. The competition director wants to maximize the number of gold medals to be awarded based on the given conditions. In this case, what is the maximum number of bronze medals that can be awarded? PS. Problems 11-20 were also used in [url=https://artofproblemsolving.com/community/c4h2676837p23203256]Juniors [/url]as 1-10.

Find all pairs of integers $(x,y)$ such that \[ y^3=8x^6+2x^3y-y^2.\]

Let $a, b$ and $n$ be positive integers such that $ b$ is divisible by $a^n$. Prove that $(a+1)^b-1$ is divisible by $a^{n+1}$.

Let's call a positive integer $n$ special, if there exist two nonnegativ integers ($a, b$), such that $n=2^a\times 3^b$. Prove that if $k$ is a positive integer, then there are at most two special numbers greater then $k^2$ and less than $k^2+2k+1$.

Let $n$ be a positive integer. We call $C_n$ the number of positive integers $x$ less than $10^n$ such that the sum of the digits of $2x$ is less than the sum of the digits of $x$. Show that $C_n\geq\frac{4}{9}(10^{n}-1)$.

For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.

The sequence $(x_n)_{n\geqslant 0}$ is defined as such: $x_0=1, x_1=2$ and $x_{n+1}=4x_n-x_{n-1}$, for all $n\geqslant 1$. Determine all the terms of the sequence which are perfect squares. [i]George Stoica, Canada[/i]

A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$. [i]Proposed by Lewis Chen[/i]

Consider $f_1=1, f_2=1$ and $f_{n+1}=f_n+f_{n-1}$ for $n\geq 2$. Determine if exists $n\leq 1000001$ such that the last three digits of $f_n$ are zero.

Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$

Let $p,q$ be integers such that the polynomial $x^2+px+q+1$ has two positive integer roots. Show that $p^2+q^2$ is composite.