Let $ H$ denote the set of those natural numbers for which $ \tau(n)$ divides $ n$, where $ \tau(n)$ is the number of divisors of $ n$. Show that a) $ n! \in H$ for all sufficiently large $ n$, b)$ H$ has density $ 0$. [i]P. Erdos[/i]

Suppose that $ p$ is a prime number. Prove that the equation $ x^2\minus{}py^2\equal{}\minus{}1$ has a solution if and only if $ p\equiv1\pmod 4$.

Find all positive integers $n$ such that $n$ is equal to $100$ times the number of positive divisors of $n$.

Let $ n>1$ be an odd integer and define: \[ N\equal{}\{\minus{}n,\minus{}(n\minus{}1),\dots,\minus{}1,0,1,\dots,(n\minus{}1),n\}.\] A subset $ P$ of $ N$ is called [i]basis[/i] if we can express every element of $ N$ as the sum of $ n$ different elements of $ P$. Find the smallest positive integer $ k$ such that every $ k\minus{}$elements subset of $ N$ is basis.

Prove that for all $a, b$ and $x_0$ positive integers, in the sequence $x_1, x_2, x_3, \cdots$ defined by $$x_{n+1}=ax_n+b, n\geq 0$$ Exist an $x_i$ that is not prime for some $i\geq 1$

Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $23$, prove that $M$ contains a subset of $4$ elements whose product is the $4$th power of an integer.

Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?

$19.$ The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37 ?$

There are $n\geq2$ numbers $x_1, x_2, \ldots, x_n$ such that $x^2_i=1 (1\leq i\leq n)$ and $$x_1x_2+x_2x_3+\dots+x_{n-1}x_n+x_nx_1=0.$$ Prove that $n$ is divisible with $4$.

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

Let $P$ be a finite set of primes, $A$ an infinite set of positive integers, where every element of $A$ has a prime factor not in $P$. Prove that there exist an infinite subset $B$ of $A$, such that the sum of elements in any finite subset of $B$ has a prime factor not in $P$.

Let $d(n)$ denote the number of positive divisors of the positive integer $n$.Determine those numbers $n$ for which $d(n^3)=5d(n)$.

Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition \[0 < a_{n+1} - a_n \leq 2001\] for all $n \in \mathbb{N}^{*}$ Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.

Prove that it is impossible to arrange the numbers $1,2,\ldots,1997$ around a circle in such a way that, if $x$ and $y$ are any two neighboring numbers, then $499\le|x-y|\le997$.

Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$ has a solution in integers $a$ and $b$.

For every natural number $a$, consider the set $S(a)=\{a^n+a+1|n=2,3,\ldots\}$. Does there exist an infinite set $A\subset\mathbb N$ with the property that for any two distinct elements $x,y\in A$, $x$ and $y$ are coprime and $S(x)\cap S(y)=\emptyset$?

Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$: \[ f(3mn \plus{} m \plus{} n) \equal{} 4f(m)f(n) \plus{} f(m) \plus{} f(n). \]

How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$.

a) If positive integer $N$ is a perfect cube and is not divisible by $10$, then $N=(10m+n)^2$ where $m,n \in N$ with $1\le n\le 9$ b) Find all the positive integers $N$ which are perfect cubes, are not divisible by $10$, such that the number obtained by erasing the last three digits to be also also a perfect cube.

Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.

Prove that for each prime number distinct from $2$ and $5$ there exist infinitely many multiples of $p$ of the form $1111...1$.

Find all natural numbers $m$ having exactly three prime divisors $p,q,r$, such that $$p-1\mid m; \quad qr-1 \mid m; \quad q-1 \nmid m; \quad r-1 \nmid m; \quad 3 \nmid q+r.$$

Determine the number of positive odd and even factor of $ 5^6\minus{}1$.

Let $(f_n) _{n\ge 0}$ be the sequence defined by$ f_0 = 0, f_1 = 1, f_{n + 2 }= f_{n + 1} + f_n$ for $n> 0$ (Fibonacci string) and let $t_n =$ ${n+1}\choose{2}$ for $n \ge 1$ . Prove that: a) $f_1^2+f_2^2+...+f_n^2 = f_n \cdot f_{n+1}$ for $n \ge 1$ b) $\frac{1}{n^2} \cdot \Sigma_{k=1}^{n}\left( \frac{t_k}{f_k}\right)^2 \ge \frac{t_{n+1}^2}{9 f_n \cdot f_{n+1}}$

[i]20 problems for 20 minutes.[/i] [b]p1.[/b] Evaluate $=\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}$. [b]p2.[/b] A regular hexagon and a regular $n$-sided polygon have the same perimeter. If the ratio of the side length of the hexagon to the side length of the $n$-sided polygon is $2 : 1$, what is $n$? [b]p3.[/b] How many nonzero digits are there in the decimal representation of $2 \cdot 10\cdot 500 \cdot 2500$? [b]p4.[/b] When the numerator of a certain fraction is increased by $2012$, the value of the fraction increases by $2$. What is the denominator of the fraction? [b]p5.[/b] Sam did the computation $1 - 10 \cdot a + 22$, where $a$ is some real number, except he messed up his order of operations and computed the multiplication last; that is, he found the value of $(1 - 10) \cdot (a + 22)$ instead. Luckily, he still ended up with the right answer. What is $a$? [b]p6.[/b] Let $n! = n \cdot(n-1) \cdot\cdot\cdot 2 \cdot 1$. For how many integers $n$ between $1$ and $100$ inclusive is $n!$ divisible by $36$? [b]p7.[/b] Simplify the expression $\sqrt{\frac{3 \cdot 27^3}{27 \cdot 3^3}}$ [b]p8.[/b] Four points $A,B,C,D$ lie on a line in that order such that $\frac{AB}{CB}=\frac{AD}{CD}$ . Let $M$ be the midpoint of segment $AC$. If $AB = 6$, $BC = 2$, compute $MB \cdot MD$. [b]p9.[/b] Allan has a deck with $8$ cards, numbered $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$. He pulls out cards without replacement, until he pulls out an even numbered card, and then he stops. What is the probability that he pulls out exactly $2$ cards? [b]p10.[/b] Starting from the sequence $(3, 4, 5, 6, 7, 8, ... )$, one applies the following operation repeatedly. In each operation, we change the sequence $$(a_1, a_2, a_3, ... , a_{a_1-1}, a_{a_1} , a_{a_1+1},...)$$ to the sequence $$(a_2, a_3, ... , a_{a_1} , a_1, a_{a_1+1}, ...) .$$ (In other words, for a sequence starting with$ x$, we shift each of the next $x-1$ term to the left by one, and put x immediately to the right of these numbers, and keep the rest of the terms unchanged. For example, after one operation, the sequence is $(4, 5, 3, 6, 7, 8, ... )$, and after two operations, the sequence becomes $(5, 3, 6, 4, 7, 8,... )$. How many operations will it take to obtain a sequence of the form $(7, ... )$ (that is, a sequence starting with $7$)? [b]p11.[/b] How many ways are there to place $4$ balls into a $4\times 6$ grid such that no column or row has more than one ball in it? (Rotations and reflections are considered distinct.) [b]p12.[/b] Point $P$ lies inside triangle $ABC$ such that $\angle PBC = 30^o$ and $\angle PAC = 20^o$. If $\angle APB$ is a right angle, find the measure of $\angle BCA$ in degrees. [b]p13.[/b] What is the largest prime factor of $9^3 - 4^3$? [b]p14.[/b] Joey writes down the numbers $1$ through $10$ and crosses one number out. He then adds the remaining numbers. What is the probability that the sum is less than or equal to $47$? [b]p15.[/b] In the coordinate plane, a lattice point is a point whose coordinates are integers. There is a pile of grass at every lattice point in the coordinate plane. A certain cow can only eat piles of grass that are at most $3$ units away from the origin. How many piles of grass can she eat? [b]p16.[/b] A book has 1000 pages numbered $1$, $2$, $...$ , $1000$. The pages are numbered so that pages $1$ and $2$ are back to back on a single sheet, pages $3$ and $4$ are back to back on the next sheet, and so on, with pages $999$ and $1000$ being back to back on the last sheet. How many pairs of pages that are back to back (on a single sheet) share no digits in the same position? (For example, pages $9$ and $10$, and pages $89$ and $90$.) [b]p17.[/b] Find a pair of integers $(a, b)$ for which $\frac{10^a}{a!}=\frac{10^b}{b!}$ and $a < b$. [b]p18.[/b] Find all ordered pairs $(x, y)$ of real numbers satisfying $$\begin{cases} -x^2 + 3y^2 - 5x + 7y + 4 = 0 \\ 2x^2 - 2y^2 - x + y + 21 = 0 \end{cases}$$ [b]p19.[/b] There are six blank fish drawn in a line on a piece of paper. Lucy wants to color them either red or blue, but will not color two adjacent fish red. In how many ways can Lucy color the fish? [b]p20.[/b] There are sixteen $100$-gram balls and sixteen $99$-gram balls on a table (the balls are visibly indistinguishable). You are given a balance scale with two sides that reports which side is heavier or that the two sides have equal weights. A weighing is defined as reading the result of the balance scale: For example, if you place three balls on each side, look at the result, then add two more balls to each side, and look at the result again, then two weighings have been performed. You wish to pick out two different sets of balls (from the $32$ balls) with equal numbers of balls in them but different total weights. What is the minimal number of weighings needed to ensure this? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].