Find all positive real numbers $x$, that verify $x+\left[\frac{x}{3}\right]=\left[\frac{2x}{3}\right]+\left[\frac{3x}{5}\right]$.

In the first $1999$ cells of the computer are written numbers in the specified order:: $1$, $2$, $4$,$... $, $2^{1998}$. Two programmers take turns reducing in one move per unit number in five different cells. If a negative number appears in one of the cells, then the computer breaks down and the broken repairs are paid for. Which programmer can protect himself from financial losses, regardless of his partner’s moves, and how should he do this act?

Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. The ratio of the area of square $EFGH$ to the area of square $ABCD$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers and $m<n$. Find $10n+m$.

Let be a natural number $ n\ge 3 $ with the property that $ 1+3n $ is a perfect square. Show that there are three natural numbers $ a,b,c, $ such that the number $$ 1+\frac{3n+3}{a^2+b^2+c^2} $$ is a perfect square.

A positive integer $n$ is called [i]smooth[/i] if there exist integers $a_1,a_2,\dotsc,a_n$ satisfying \[a_1+a_2+\dotsc+a_n=a_1 \cdot a_2 \cdot \dotsc \cdot a_n=n.\] Find all smooth numbers.

Find a function $f(n)$ on the positive integers with positive integer values such that $f( f(n) ) = 1993 n^{1945}$ for all $n$.

In a Pythagorean triangle all sides are longer than 5. Is it possible that (a) all three sides are prime numbers, (b) exactly two sides are prime numbers. (Note: We call a triangle "Pythagorean", if it is a right-angled triangle where all sides are positive integers.)

A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

Show that it is possible to color the points of $\mathbb Q\times\mathbb Q$ in two colors in such a way that any two points having distance $1$ have distinct colors.

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

Find all functions $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ such that the conditions $\quad a) \quad a-b \mid f(a)-f(b)$ for all $a\neq b$ and $a,b \in \mathbb{Z}^{+}$ $\quad b) \quad f(\varphi(a))=\varphi(f(a))$ for all $a \in \mathbb{Z}^{+}$ where $\varphi$ is the Euler's totient function. holds

Let $A$ be a set of $N$ residues $\pmod{N^2}$. Prove that there exists a set $B$ of $N$ residues $\pmod{N^2}$ such that the set $A+B=\{a+b \vert a \in A, b \in B \}$ contains at least half of all the residues $\pmod{N^2}$.

What is the largest possible value of the expression $$gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )$$ for naturals $n$? [hide]original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n? [/hide]

Let $ f (n) $ be a function that fulfills the following properties: $\bullet$ For each natural $ n $, $ f (n) $ is an integer greater than or equal to $ 0 $. $\bullet$ $f (n) = 2010 $, if $ n $ ends in $ 7 $. For example, $ f (137) = 2010 $. $\bullet$ If $ a $ is a divisor of $ b $, then: $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $. Find $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $ and justify your answer.

Arnaldo selects a nonnegative integer $a$ and Bernaldo selects a nonnegative integer $b$. Both of them secretly tell their number to Cernaldo, who writes the numbers $5$, $8$, and $15$ on the board, one of them being the sum $a+b$. Cernaldo rings a bell and Arnaldo and Bernaldo, individually, write on different slips of paper whether they know or not which of the numbers on the board is the sum $a+b$ and they turn them in to Cernaldo. If both of the papers say NO, Cernaldo rings the bell again and the process is repeated. It is known that both Arnaldo and Bernaldo are honest and intelligent. What is the maximum number of times that the bell can be rung until one of them knows the sum? Personal note: They really phoned it in with the names there…

For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

For which positive integers $n$ is $3^{2n+1} - 2^{2n+1} - 6^n$ composite?

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

Find last three digits of the number $\frac{2019!}{2^{1009}}$ .

The letters $A,B,C$ and $D$ each represent a different digit, so each of the four-digit numbers $ABCD$, $BCDA$, $CDAB$ and $DABC$ satisfy that its least prime divisor is equal to $11$. Determine all possible values of the sum $$ABCD +BCDA+CDAB+DABC$$ and for each possible value of said sum, give an example of a choice of digits $A,B,C$ and $D$ with which to obtain that value and which satisfies the conditions established above.

Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.

One hundred concentric circles with radii $1, 2, 3, \dots, 100$ are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Pete wrote down $21$ pairwise distinct positive integers, each not greater than $1,000,000$. For every pair $(a, b)$ of numbers written down by Pete, Nick wrote the number $$F(a;b)=a+b -\gcd(a;b)$$ on his piece of paper. Prove that one of Nick’s numbers differs from all of Pete’s numbers.

Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$. For instance $d(6)=12$ and $\phi(7)=6$. Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.

Define the sequnce ${(a_n)}_{n\ge1}$ by $a_1=1$ and $a_n=5a_{n-1}+3^{n-1}$ for $n\ge2$. Find the greatest power of $2$ that divides $a_{2^{2019}}$.