Find the smallest positive integer $n$ for which there exist integers $x_{1} < x_{2} <...< x_{n}$ such that every integer from $1000$ to $2000$ can be written as a sum of some of the integers from $x_1,x_2,..,x_n$ without repetition.

Prove that for any positive even integer $n$ larger than $38$, $n$ can be written as $a\times b+c\times d$ where $a, b, c, d$ are odd integers larger than $1$.

How many positive integers satisfy the following three conditions: a) All digits of the number are from the set $\{1,2,3,4,5\}$; b) The absolute value of the difference between any two consecutive digits is $1$; c) The integer has $1994$ digits?

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Find all positive integers $n$ such that $n^{2}+2^{n}$ is square of an integer.

We consider the following figure: [See attachment] We are looking for labellings of the nine fields with the numbers 1, 2, ..., 9. Each of these numbers has to be used exactly once. Moreover, the six sums of three resp. four numbers along the drawn lines have to be be equal. Give one such labelling. Show that all such labellings have the same number in the top field. How many such labellings do there exist? (Two labellings are considered different, if they disagree in at least one field.) (Walther Janous)

Determine all $3$-digit numbers which are equal to cube of the sum of all its digits.

The decimal notation of three natural numbers consists of equal digits: $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$. For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$

Let $a,b,c,d,$ be real and $ad-bc = 1$. Show that $Q = a^2 + b^2 + c^2 + d^2 + ac +bd$ $\not= 0, 1, -1$

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) $\frac{1}{n^2} \cdot \Sigma_{k=1}^{n}\left( \frac{t_k}{f_k}\right)^2 \ge \frac{t_{n+1}^2}{9 f_n \cdot f_{n+1}}$

Nirajan is trapped in a magical dungeon. He has infinitely many magical cards with arbitrary MPs(Mana Points) which is always an integer $\mathbb{Z}$. To escape, he must give the dungeon keeper some magical cards whose MPs add up to an integer with at least $2024$ divisors. Can Nirajan always escape? [i]( Proposed by Vlad Spǎtaru, Romania)[/i]

The natural numbers $x, y > 1$, are such that $x^2 + xy -y$ is the square of a natural number. Prove that $x + y + 1$ is a composite number.

Which is the leastest natural number $n$ such that $n!$ has, at least, $2013$ divisors?

Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.

Let $(A_i)_{i\ge 1}$ be sequence of sets of two integer numbers, such that no integer is contained in more than one $A_i$ and for every $A_i$ the sum of its elements is $i$. Prove that there are infinitely many values of $k$ for which one of the elements of $A_k$ is greater than $13k/7$.

Let be a nonnegative integer $ n $ and three real numbers $ a,b,c $ satisfying $$ a^n+c=b^n+a=c^n+b=a+b+c. $$ Show that $ a=b=c. $ [i]Gheorghe Iurea[/i]

Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

[b]7.[/b] Prove that for any odd prime number $p$, the polynomial $2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})$ is congruent mod $p$ to the square of a polynomial with integer coefficients. [b](N. 21)[/b] *This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594

Prove that $ N\geq 2n \minus{} 2$ integers, of absolute value not higher than $ n > 2$, and of absolute value of their sum $ S$ less than $ n \minus{} 1,$ there exist some of sum $ 0.$ Show that for $ |S| \equal{} n \minus{} 1$ this is not anymore true, and neither for $ N \equal{} 2n \minus{} 3$ (when even for $ |S| \equal{} 1$ this is not anymore true).

Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take. [i]Budi Surodjo, Jogjakarta[/i]

All empty white triangles in figure are to be filled with integers such that for each gray triangle the three numbers in the white neighboring triangles sum to a multiple of $5$. The lower left and the lower right white triangle are already filled with the numbers $12$ and $3$, respectively. Find all integers that can occur in the uppermost white triangle. (G. Woeginger, Eindhoven, The Netherlands) [img]https://cdn.artofproblemsolving.com/attachments/8/a/764732f5debbd58a147e9067e83ba8d31f7ee9.png[/img]

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

Find the smallest positive integer that ends in $56$, is a multiple of $56$, and has the sum of its digits equal to $56$.

Prove that there are no perfect squares $a,b,c$ such that $ab-bc = a$.