For how many positive integers $n \le 1000$ is $$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$$ not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) $\textbf{(A) } 22 \qquad\textbf{(B) } 23 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 26$

Determine all $n \geq 3$ for which there are $n$ positive integers $a_1, \cdots , a_n$ any two of which have a common divisor greater than $1$, but any three of which are coprime. Assuming that, moreover, the numbers $a_i$ are less than $5000$, find the greatest possible $n$.

Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$, it holds that $a \nmid b$ and $b \nmid a$. Show that \[ \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}. \]

The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$). Prove that there are two consecutive positive integers such that the largest prime divisor of one of them is $p$ and that of the other is $q$.

For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$ for all $n\ge 1$ [i]Cyprus[/i]

Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.

Find all ordered pairs of integers $(x, y)$ with $y \ge 0$ such that $x^2 + 2xy + y! = 131$.

Find all primes $p$ such that $\dfrac{2^{p+1}-4}{p}$ is a perfect square.

There are two $3$-digit numbers which end in $99$. These two numbers are also the product of two integers which differ by $2$. What is the sum of these two numbers?

[b]8.1[/b] Find all primes $p,q$ and $r$ such that $$pqr= 5(p + q + r).$$ [b]8.2 [/b] Prove that if $\overline{ab}/\overline{bc} = a/c$, then $$\overline{abb...bb}/\overline{bb...bbc} = a/c$$ (each number has $n$ digits). [b]8.3 / 9.1[/b] Construct a triangle with perimeter, altitude and angle at the base. [b]8.4. / 9.4[/b] Prove that the square of the sum of N distinct non-zero squares of integers is also the sum of $N $squares of non-zero integers. [b]8.5.[/b] In the quadrilateral $ABCD$ the diagonals $AC$ and $BD$ are drawn. Prove that if the circles inscribed in $ABC$ and $ ADC$ touch each other each other, then the circles inscribed in $BAD$ and in $BCD$ also touch each other. [b]8.6 / 9.6[/b] If the numbers $A$ and $n$ are coprime, then there are integers $X$ and $Y$ such that $ |X| <\sqrt{n}$, $|Y| <\sqrt{n} $ and $AX-Y$ is divided by $n$. Prove it. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

Let $a\geq 2$ a fixed natural number, and let $a_{n}$ be the sequence $a_{n}=a^{a^{.^{.^{a}}}}$ (e.g $a_{1}=a$, $a_{2}=a^a$, etc.). Prove that $(a_{n+1}-a_{n})|(a_{n+2}-a_{n+1})$ for every natural number $n$.

Find all polynomials $ f\in\mathbb Z[x]$ such that for each $ a,b,x\in\mathbb N$ \[ a\plus{}b\plus{}c|f(a)\plus{}f(b)\plus{}f(c)\]

We call [i]Pythagorean triple[/i] a triple $(x,y,z)$ of positive integers such that $x<y<z$ and $x^2+y^2=z^2$. Prove that for all $n \in \mathbb{N}$ number $2^{n+1}$ is in exactly $n$ [i]Pythagorean triples[/i]

Given a $101$-digit number $a$ and an arbitrary positive integer $b$. Prove that there is at most a $102$-digit positive integer $c$ such that any number of the form $\overline{caaa \dots ab}$ is composite.

Given that $a_{n}= \binom{200}{n} \cdot 6^{\frac{200-n}{3}} \cdot (\dfrac{1}{\sqrt{2}})^n$ ($ 1 \leq n \leq 95$). How many integers are there in the sequence $\{a_n\}$?

Let $m$ be an integer where $|m|\ge 2$. Let $a_1,a_2,\cdots$ be a sequence of integers such that $a_1,a_2$ are not both zero, and for any positive integer $n$, $a_{n+2}=a_{n+1}-ma_n$. Prove that if positive integers $r>s\ge 2$ satisfy $a_r=a_s=a_1$, then $r-s\ge |m|$.

A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

If $x,y,K,m \in N$, let us define: $a_m= \underset{k \, twos}{2^{2^{,,,{^{2}}}}}$, $A_{km} (x)= \underset{k \, twos}{ 2^{2^{,,,^{x^{a_m}}}}}$, $B_k(y)= \underset{m \, fours}{4^{4^{4^{,,,^{4^y}}}}}$, Determine all pairs $(x,y)$ of non-negative integers, dependent on $k>0$, such that $A_{km} (x)=B_k(y)$

[b]p1.[/b] Consider the system of simultaneous equations $$(x+y)(x+z)=a^2$$ $$(x+y)(y+z)=b^2$$ $$(x+z)(y+z)=c^2$$ , where $abc \ne 0$. Find all solutions $(x,y,z)$ in terms of $a$,$b$, and $c$. [b]p2.[/b] Shown in the figure is a triangle $PQR$ upon whose sides squares of areas $13$, $25$, and $36$ sq. units have been constructed. Find the area of the hexagon $ABCDEF$ . [img]https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png[/img] [b]p3.[/b] Suppose $p,q$, and $r$ are positive integers no two of which have a common factor larger than $1$. Suppose $P,Q$, and $R$ are positive integers such that $\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}$ is an integer. Prove that each of $P/p$, $Q/q$, and $R/r$ is an integer. [b]p4.[/b] An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles. [img]https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png[/img] [b]p5.[/b] Suppose that $p_1$, $p_2$, $p_3$ and $p_4$ are the centers of four non-overlapping circles of radius $1$ in a plane and that, $p$ is any point in that plane. Prove that $$\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for any prime $p,$ there exists a positive integer $n$ such that \[1^n+2^{n-1}+3^{n-2}+\cdots+n^1\equiv 2020\pmod{p}.\] [i]Robin Son[/i]

Prove that if natural numbers $a,b,c,d$ satisfy the equality $ab = cd$, then $\frac{gcd(a,c)gcd(a,d)}{gcd(a,b,c,d)}= a$

The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $

How many distinct four-tuples $ (a,b,c,d)$ of rational numbers are there with $ a \log_{10} 2 \plus{} b \log_{10} 3 \plus{} c \log_{10} 5 \plus{} d \log_{10} 7 \equal{} 2005$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 2004\qquad \textbf{(E)}\ \text{infinitely many}$

Let $a_1,a_2,...,a_{11}$ be integers. Prove that there are numbers $b_1,b_2,...,b_{11}$, each $b_i$ equal $-1,0$ or $1$, but not all being $0$, such that the number $$N=a_1b_1+a_2b_2+...+a_{11}b_{11}$$ is divisible by $2015$.

Let $a,b$ be two different positive integers. Suppose that $a,b$ are relatively prime. Prove that $\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is not an integer.