Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.

Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers? [i]Boris Frenkin[/i]

How many 4-digit numbers $ABCD$ are there with the property that $|A-B|= |B-C|= |C-D|$? Note that the first digit $A$ of a four-digit number cannot be zero.

[u]Set 2[/u] [b]p4.[/b] $\vartriangle ABC$ is an isosceles triangle with $AB = BC$. Additionally, there is $D$ on $BC$ with $AC = DA = BD = 1$. Find the perimeter of $\vartriangle ABC$. [b]p5[/b]. Let $r$ be the positive solution to the equation $100r^2 + 2r - 1 = 0$. For an appropriate $A$, the infinite series $Ar + Ar^2 + Ar^3 + Ar^4 +...$ has sum $1$. Find $A$. [b]p6.[/b] Let $N(k)$ denote the number of real solutions to the equation $x^4 -x^2 = k$. As $k$ ranges from $-\infty$ to $\infty$, the value of $N(k)$ changes only a finite number of times. Write the sequence of values of $N(k)$ as an ordered tuple (i.e. if $N(k)$ went from $1$ to $3$ to $2$, you would write $(1, 3, 2)$). PS. You should use hide for answers.

Find the greatest integer $n$ such that $10^n$ divides $$\frac{2^{10^5} 5^{2^{10}}}{10^{5^2}}$$

Does the equation $$z(y-x)(x+y)=x^3$$ have finitely many solutions in the set of positive integers? [i]Proposed by Nikola Velov[/i]

Let $d(n)$ denote the number of positive divisors of $n$ and let $e(n)=\left[2000\over n\right]$ for positive integer $n$. Prove that \[d(1)+d(2)+\dots+d(2000)=e(1)+e(2)+\dots+e(2000).\]

How many of the following statements are false? a. $2005$ distinct positive integers exist such that the sum of their squares is a cube and the sum of their cubes is a square. b. There are $2$ integral solutions to $x^2 + y^2 + z^2 = x^2y^2$. c. If the vertices of a triangle are lattice points in a plane, the diameter of the triangle’s circumcircle will never exceed the product of the triangle’s side lengths.

Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy \[(ab + 1)(bc + 1)(ca + 1) = 84.\]

Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.

Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

Does there exist two infinite sets $A,B$ such that every number can be written uniquely as a sum of an element of $A$ and an element of $B$?

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Let $p$ be an odd prime and let $$F(n)=1+2n+3n^2+\ldots+(p-1)n^{p-2}.$$Prove that if $a$ and $b$ are distinct integers in $\{0,1,2,\ldots,p-1\}$ then $F(a)$ and $F(b)$ are not congruent modulo $p$.

Prove that every natural number is the difference of two natural numbers that have the same number of prime factors. (Each prime divisor is counted once, for example, the number $12$ has two prime factors: $2$ and $3$.)

Show that the equation $x^2+y^5=z^3$ has infinitely many solutions in integers $x, y,z$ for which $xyz \neq 0$.

A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i]. a) Prove that there are infinite [i]non-charrua[/i] pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].

We say that a positive integer $n$ is [i]good[/i] if the result of multiplying the first $n$ positive odd integers consists only of the digits $1$, $3$, $5$ and $9$. For example, $n = 3$ is good because $1 \times 3 \times 5 = 15$, but $n = 4$ is not good because $1 \times 3 \times 5 \times 7 = 105$. Determine all the good numbers.

Find all $n \in \mathbb N$ such that $n = \varphi (n) + 402$, where $\varphi$ denotes the Euler phi function.

The positive integers $a, b, c, d$ satisfy (i) $a + b + c + d = 2023$ (ii) $2023 \text{ } | \text{ } ab - cd$ (iii) $2023 \text{ } | \text{ } a^2 + b^2 + c^2 + d^2.$ Assuming that each of the numbers $a, b, c, d$ is divisible by $7$, prove that each of the numbers $a, b, c, d$ is divisible by $17$.

Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

[b]p1.[/b] Let $D(n)$ denote the number of positive factors of the integer $n$. For example, $D(6) = 4$ , since the factors of $6$ are $1, 2, 3$ , and $6$ . Note that $D(n) = 2$ if and only if $n$ is a prime number. (a) Describe the set of all solutions to the equation $D(n) = 5$ . (b) Describe the set of all solutions to the equation $D(n) = 6$ . (c) Find the smallest $n$ such that $D(n) = 21$ . [b]p2.[/b] At a party with $n$ married couples present (and no one else), various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening Mr. Jones asked everyone else, including his wife, how many hands he or she had shaken. To his surprise, he got a different answer from each person. Determine the number of hands that Mr. Jones shook that evening, (a) if $n = 2$ . (b) if $n = 3$ . (c) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [b]p3.[/b] Let $n$ be a positive integer. A square is divided into triangles in the following way. A line is drawn from one corner of the square to each of $n$ points along each of the opposite two sides, forming $2n + 2$ nonoverlapping triangles, one of which has a vertex at the opposite corner and the other $2n + 1$ of which have a vertex at the original corner. The figure shows the situation for $n = 2$ . Assume that each of the $2n + 1$ triangles with a vertex in the original corner has area $1$. Determine the area of the square, (a) if $n = 1$ . (b) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [img]https://cdn.artofproblemsolving.com/attachments/1/1/62a54011163cc76cc8d74c73ac9f74420e1b37.png[/img] [b]p4.[/b] Arthur and Betty play a game with the following rules. Initially there are one or more piles of stones, each pile containing one or more stones. A legal move consists either of removing one or more stones from one of the piles, or, if there are at least two piles, combining two piles into one (but not removing any stones). Arthur goes first, and play alternates until a player cannot make a legal move; the player who cannot move loses. (a) Determine who will win the game if initially there are two piles, each with one stone, assuming that both players play optimally. (b) Determine who will win the game if initially there are two piles, each with $n$ stones, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . (c) Determine who will win the game if initially there are $n$ piles, each with one stone, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . [b]p5.[/b] Suppose $x$ and $y$ are real numbers such that $0 < x < y$. Define a sequence$ A_0 , A_1 , A_2, A_3, ...$ by-setting $A_0 = x$ , $A_1 = y$ , and then $A_n= |A_{n-1}| - A_{n-2}$ for each $n \ge 2$ (recall that $|A_{n-1}|$ means the absolute value of $A_{n-1}$ ). (a) Find all possible values for $A_6$ in terms of $x$ and $y$ . (b) Find values of $x$ and $y$ so that $A_{1987} = 1987$ and $A_{1988} = -1988$ (simultaneously). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].