Let Pascal triangle be an equilateral triangular array of number, consists of $2019$ rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it. How many ways to assign each of numbers $a_0, a_1,...,a_{2018}$ (from left to right) in the bottom row by $0$ or $1$ such that the number $S$ on the top is divisible by $1019$.

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

Determine whether or not there exists a natural number $N$ which satisfies the following three criteria: 1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$, 2. $N$ only has three different digits, and none of them are zero, 3. Exactly 99.9% of the digits of $N$ are odd.

Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$.

Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.

A positive integer n is called [i]primary divisor [/i] if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$, $2$, $4$, and $8$ each differ by $1$ from a prime number ($2$, $3$, $5$, and $7$, respectively), while $9$ is not divisor primary, because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor number.

Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

A sequence of integers $(a_0,...,a_k)$ is said to be [i]medaled[/i] if, for each $i = 0,...,k$, there are exactly $a_i$ elements of the sequence equal to $i$. For example, $(1,2,1,0)$ is a [i]medaled [/i] seqence. Indicates all [i]medaled [/i] sequences $(a_0,...,a_{2015})$.

Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.

A quadratic polynomial with positive integer coefficients and rational roots can be written as $196x^2+Bx + 135$ for some integer $B.$ What is the sum of all possible values of $B$ such that $\gcd(B, 196 \cdot 135) = 1$?

Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$. Compute the sum of all bases and exponents in the prime factorization of $A$. For example, if $A=7\cdot 11^5$, the answer would be $7+11+5=23$.

Given a prime $p$, show that there exists a positive integer $n$ such that the decimal representation of $p^n$ has a block of $2002$ consecutive zeros.

The set $S = \{1, 2, \dots, 2022\}$ is to be partitioned into $n$ disjoint subsets $S_1, S_2, \dots, S_n$ such that for each $i \in \{1, 2, \dots, n\}$, exactly one of the following statements is true: (a) For all $x, y \in S_i$, with $x \neq y, \gcd(x, y) > 1.$ (b) For all $x, y \in S_i$, with $x \neq y, \gcd(x, y) = 1.$ Find the smallest value of $n$ for which this is possible.

Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.

Let $p>q$ be primes, such that $240 \nmid p^4-q^4$. Find the maximal value of $\frac{q} {p}$.

Define the sequence of positive prime numbers. $p_1,p_2,p_3,...$. Let set $A$ be the infinite set of positive integers whose prime divisor does not exceed $p_n$. How many at least members must be selected from the set $A$ , such that we ensures that there are $2$ numbers whose products are perfect squares? [i](PP-nine)[/i]

Determine all solutions of the diophantine equation $a^2 = b \cdot (b + 7)$ in integers $a\ge 0$ and $b \ge 0$. (W. Janous, Innsbruck)

If $p$ is a prime number and $x, y$ are positive integers, find in terms of $p$, all pairs $(x, y)$ that satisfy the equation: $$p(x -2) = x(y -1).$$ If $x+y = 21$, find all triples $(x, y, p)$ that satisfy this equation.

p1. Let $A = \{(x, y)|3x + 5y\ge 15, x + y^2\le 25, x\ge 0, x, y$ integer numbers $\}$. Find all pairs of $(x, zx)\in A$ provided that $z$ is non-zero integer. p2. A shop owner wants to be able to weigh various kinds of weight objects (in natural numbers) with only $4$ different weights. (For example, if he has weights $ 1$, $2$, $5$ and $10$. He can weighing $ 1$ kg, $2$ kg, $3$ kg $(1 + 2)$, $44$ kg $(5 - 1)$, $5$ kg, $6$ kg, $7$ kg, $ 8$ kg, $9$ kg $(10 - 1)$, $10$ kg, $11$ kg, $12$ kg, $13$ kg $(10 + 1 + 2)$, $14$ kg $(10 + 5 -1)$, $15$ kg, $16$ kg, $17$ kg and $18$ kg). If he wants to be able to weigh all the weight from $ 1$ kg to $40$ kg, determine the four weights that he must have. Explain that your answer is correct. p3. Given the following table. [img]https://cdn.artofproblemsolving.com/attachments/d/8/4622407a72656efe77ccaf02cf353ef1bcfa28.png[/img] Table $4\times 4$ ​​is a combination of four smaller table sections of size $2\times 2$. This table will be filled with four consecutive integers such that: $\bullet$ The horizontal sum of the numbers in each row is $10$ . $\bullet$ The vertical sum of the numbers in each column is $10$ $\bullet$ The sum of the four numbers in each part of $2\times 2$ which is delimited by the line thickness is also equal to $10$. Determine how many arrangements are possible. p4. A sequence of real numbers is defined as following: $U_n=ar^{n-1}$, if $n = 4m -3$ or $n = 4m - 2$ $U_n=- ar^{n-1}$, if $n = 4m - 1$ or $n = 4m$, where $a > 0$, $r > 0$, and $m$ is a positive integer. Prove that the sum of all the $ 1$st to $2009$th terms is $\frac{a(1+r-r^{2009}+r^{2010})}{1+r^2}$ 5. Cube $ABCD.EFGH$ is cut into four parts by two planes. The first plane is parallel to side $ABCD$ and passes through the midpoint of edge $BF$. The sceond plane passes through the midpoints $AB$, $AD$, $GH$, and $FG$. Determine the ratio of the volumes of the smallest part to the largest part.

Let $f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor$ and $g(n) = f(n) -f(n - 1)$. For how many $n$ from $1$ to $100$ inclusive is $g(n)$ even?

Rational numbers $a, b$ are such that $a+b$ and $a^2+b^2$ are integers. Prove that $a, b$ are integers.

For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \] Find all natural numbers $ n $ such that $ s(n) = 2010 $

Find all pairs of positive integers $m$, $n$ such that the $(m+n)$-digit number \[\underbrace{33\ldots3}_{m}\underbrace{66\ldots 6}_{n}\] is a perfect square.

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

Let $p$ be a prime number greater than $2$. Patricia wants $7$ not-necessarily different numbers from $\{1, 2, . . . , p\}$ to the black dots in the figure below, on such a way that the product of three numbers on a line or circle always has the same remainder when divided by $p$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/ef0d63b8ff5341ffc340de0cc75b24c7229e23.png[/img] (a) Suppose Patricia uses the number $p$ at least once. How many times does she have the number $p$ then a minimum sum needed? (b) Suppose Patricia does not use the number $p$. In how many ways can she assign numbers? (Two ways are different if there is at least one black one dot different numbers are assigned. The figure is not rotated or mirrored.)