The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Find all pairs $(m, n)$ of positive integers satsifying $m^6+5n^2=m+n^3$.

Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

Prove that if a natural number $ N $ can be represented in the form the sum of three squares of integers divisible by $3$, then it is also is represented as the sum of three squares of integers that are not divisible by $3$.

Consider a function f defined on the positive integers that meets the following conditions: $$f(1) = 1 \, , \,\, f(2n) = 2f(n) \, , \,\, nf(2n + 1) = (2n + 1)(f(n) + n) $$ for all $n \ge 1$. a) Prove that $f(n)$ is an integer for all $n$. b) Find all positive integers $m$ less than $2013$ that satisfy the equation $f(m) = 2m$.

[b]p1.[/b] A tennis net is made of strings tied up together which make a grid consisting of small squares as shown below. [img]https://cdn.artofproblemsolving.com/attachments/9/4/72077777d57408d9fff0ea5e79be5ecb6fe8c3.png[/img] The size of the net is $100\times 10$ small squares. What is the maximal number of sides of small squares which can be cut without breaking the net into two separate pieces? (The side is cut only in the middle, not at the ends). [b]p2.[/b] What number is bigger $2^{300}$ or $3^{200}$ ? [b]p3.[/b] All noble knights participating in a medieval tournament in Camelot used nicknames. In the tournament each knight had combats with all other knights. In each combat one knight won and the second one lost. At the end of tournament the losers reported their real names to the winners and to the winners of their winners. Was there a person who knew the real names of all knights? [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $10$ rocks in the first pile and $12$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] There is an interesting $5$-digit integer. With a $1$ after it, it is three times as large as with a $1$ before it. What is the number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n + F_{n-1}$, for all positive integer $n$, be the Fibonacci sequence. Prove that for any positive integer $m$ there exist infinitely many positive integers $n$ such that $F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2}$ mod $m$ .

Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.

Do there exist integers $x$ and $y$ such that $19^{19} = x^3 +y^4$ ? Justify your answer.

Let $a,b$ be positive real numbers. For any positive integer $n$, denote by $x_n$ the sum of digits of the number $[an+b]$ in it's decimal representation. Show that the sequence $(x_n)_{n\ge 1}$ contains a constant subsequence. [i]Laurentiu Panaitopol[/i]

How many integers $n$ are there such that $n^4 + 2n^3 + 2n^2 + 2n + 1$ is a prime number?

Dragon selects three positive real numbers with sum $100$, uniformly at random. He asks Cat to copy them down, but Cat gets lazy and rounds them all to the nearest tenth during transcription. If the probability the three new numbers still sum to $100$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m+n$. [i]Proposed by Aaron Lin[/i]

Find all pairs of positive integers $(m,n)$ such that $2^m-n^2$ is the square of an integer.

Let $r \geq 2023$ be a rational number. The real numbers $a, b$, and $c$ satisfy \[ 4a^2 + 4b^2 + 9c^2 = r. \] Does there exist a value of $r$ for which the number of rational triples $(a,b,c)$ that achieve the maximum possible value of $4ab + 6bc - 6ac$ is: a) zero b) finite, but non-zero?

Find all pairs of integers $(x,y)$ such that $x(x+2)=y^2(y^2+1)$

For a sequence $\{a_i\}_{i\ge1}$ consisting of only positive integers, prove that if for all different positive integers $i$ and $j$, we have $a_i\nmid a_j$, then $$\{p\mid p\text{ is a prime and }p\mid a_i\text{ for some }i\}$$is a infinite set.

An integer $n\geq 2$ having exactly $s$ positive divisors $1=d_1<d_2<\cdots<d_s=n$ is said to be [i]good[/i] if there exists an integer $k$, with $2\leq k\leq s$, such that $d_k>1+d_1+\cdots+d_{k-1}$. An integer $n\geq 2$ is said to be [i]bad[/i] if it is not good. (a) Show that there are infinitely many bad integers. (b) Prove that, among any seven consecutive integers all greater than $2$, there are always at least four good integers. (c) Show that there are infinitely many sequences of seven consecutive good integers.

Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$.

In Kacs Aladár street, houses are only found on one side of the road, so that only odd house numbers are found along the street. There are an odd number of allotments, as well. The middle three allotments belong to Scrooge McDuck, so he only put up the smallest of the three house numbers. The numbering of the other houses is standard, and the numbering begins with $1$. What is the largest number in the street if the sum of house numbers put up is $3133$?

The positive integers $a $ and $b $ satisfy the sistem $\begin {cases} a_{10} +b_{10} = a \\a_{11}+b_{11 }=b \end {cases} $ where $ a_1 <a_2 <\dots $ and $ b_1 <b_2 <\dots $ are the positive divisors of $a $ and $b$ . Find $a$ and $b $ .

Let $M$ be a positive real number. Determine the least positive real number $k$ with the following property: for each integer $n>M$, the interval $(n,kn]$ contains a power of $2$.

Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$.

Prove that there exist two infinite sequences $ \{a_n\}_{n\ge 1}$ and $ \{b_n\}_{n\ge 1}$ of positive integers such that the following conditions hold simultaneously: $ (i)$ $ 0 < a_1 < a_2 < a_3 < \cdots$; $ (ii)$ $ a_n < b_n < a_n^2$, for all $ n\ge 1$; $ (iii)$ $ a_n \minus{} 1$ divides $ b_n \minus{} 1$, for all $ n\ge 1$ $ (iv)$ $ a_n^2 \minus{} 1$ divides $ b_n^2 \minus{} 1$, for all $ n\ge 1$ [19 points out of 100 for the 6 problems]

Solve in N $1/x+2/y-4/z=1$

Prove that among any $1001$ numbers taken from the numbers $1,2,\ldots,1997$ there exist two with the difference $4$.