Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada? [i]Proposed by Oleksii Masalitin[/i]

Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$ [i]Proposed by Mihai Baluna, Romania[/i]

Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$

[b]p1.[/b] Find all real solutions of the system $$x^5 + y^5 = 1$$ $$x^6 + y^6 = 1$$ [b]p2.[/b] The centers of three circles of the radius $R$ are located in the vertexes of equilateral triangle. The length of the sides of the triangle is $a$ and $\frac{a}{2}< R < a$. Find the distances between the intersection points of the circles, which are outside of the triangle. [b]p3.[/b] Prove that no positive integer power of $2$ ends with four equal digits. [b]p4.[/b] A circle is divided in $10$ sectors. $90$ coins are located in these sectors, $9$ coins in each sector. At every move you can move a coin from a sector to one of two neighbor sectors. (Two sectors are called neighbor if they are adjoined along a segment.) Is it possible to move all coins into one sector in exactly$ 2004$ moves? [b]p5.[/b] Inside a convex polygon several points are arbitrary chosen. Is it possible to divide the polygon into smaller convex polygons such that every one contains exactly one given point? Justify your answer. [b]p6.[/b] A troll tried to spoil a white and red $8\times 8$ chessboard. The area of every square of the chessboard is one square foot. He randomly painted $1.5\%$ of the area of every square with black ink. A grasshopper jumped on the spoiled chessboard. The length of the jump of the grasshopper is exactly one foot and at every jump only one point of the chessboard is touched. Is it possible for the grasshopper to visit every square of the chessboard without touching any black point? Justify your answer. PS. You should use hide for answers.

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 01[/b] Find all $(x,y,z)$ positive integers, such that: $\sqrt{\frac{2006}{x+y}} + \sqrt{\frac{2006}{y+z}} + \sqrt{\frac{2006}{z+x}},$ is an integer. --- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88509]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

Find all positive integers $n$, for which there exist positive integers $a>b$, satisfying $n=\frac{4ab}{a-b}$.

Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

For positive integers $m, n$ define the function $f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}$. Prove that there are only finitely many pairs of positive integers $(a, b)$ such that $f_n(a) + f_n(b)$ is a prime number. [i]Proposed by Nazar Serdyuk[/i]

We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$. A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial. Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that $$P(n)|f(n)$$ for all positive integers $n$.

Prove that there exists an infinite number of relatively prime pairs $(m, n)$ of positive integers such that the equation \[x^3-nx+mn=0\] has three distint integer roots.

Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$.

Let $k$ be a positive integer. A ring $(A,+,\cdot)$ has property $P_k$ if for any $a,b\in A$ there exists $c\in A$ such that $a^k=b^k+c^k.$[list=a] [*]Give an example of a finite ring $(A,+,\cdot)$ which [i]does not[/i] have $P_k$ for any $k\geqslant 2.$ [*]Let $n\geqslant 3$ be an integer and $M_n=\{m\in\mathbb{N}:(\mathbb{Z}_n,+,\cdot)\text{ has }P_m\}.$ Prove that all the elements of $M_n$ are odd integers and that $(M_n,\cdot)$ is a monoid. [/list]

The cells of the $11 \times 111 \times11$ cube contain the numbers $ 1, 2, , . .. . . 1331$, once each number. Two worms are sent from one corner cube to the opposite corner. Each of them can crawl into a cube adjacent to the edge, while the first can crawl if the number in the adjacent cube differs by $8$, the second - if they differ by $ 9$. Is there such an arrangement of numbers that both worms can get to the opposite corner cube?

Does there exist a positive integer, $x$, such that $(x+2)^{2023}-x^{2023}$ has exactly $2023^{2023}$ factors? [i]Proposed by Wong Jer Ren[/i]

Determine all pairs of integers $(x, y)$ satisfying the equation \[y(x + y) = x^3- 7x^2 + 11x - 3.\]

Let $\lambda$ be a positive real number satisfying $\lambda=\lambda^{2/3}+1$. Show that there exists a positive integer $M$ such that $|M-\lambda^{300}|<4^{-100}$. [i]Proposed by Evan Chen[/i]

Prove that an integer $n$ can be expressed as the sum of two squares if and only if $2n$ can be expressed as the sum of two squares.

Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$ a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple. b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple. [i]There are exactly $n$ factors in the product on the left hand side.[/i]

For which positive integer couples $(k,n)$, the equality $\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}$ holds?

Determine the smallest positive integer \( n \) with the following property: for every triple of positive integers \( x, y, z \), with \( x \) dividing \( y^3 \), \( y \) dividing \( z^3 \), and \( z \) dividing \( x^3 \), it also holds that \( (xyz) \) divides \( (x + y + z)^n \).

For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only? Alexey Glebov

Determine the set of all real numbers $\alpha$ with the following property: For each positive $c$ there exists a rational number $\frac mn~(m\in\mathbb Z,n\in\mathbb N)$ different than $\alpha$ such that $$\left|\alpha-\frac mn\right|<\frac cn.$$

Let $a_1=0$, $a_2=1$, and $a_{n+2}=a_{n+1}+a_n$ for all positive integers $n$. Show that there exists an increasing infinite arithmetic progression of integers, which has no number in common in the sequence $\{a_n\}_{n \ge 0}$.

$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$

[b]p1.[/b] Let $x_1 = 0$, $x_2 = 1/2$ and for $n >2$, let $x_n$ be the average of $x_{n-1}$ and $x_{n-2}$. Find a formula for $a_n = x_{n+1} - x_{n}$, $n = 1, 2, 3, \dots$. Justify your answer. [b]p2.[/b] Given a triangle $ABC$. Let $h_a, h_b, h_c$ be the altitudes to its sides $a, b, c,$ respectively. Prove: $\frac{1}{h_a}+\frac{1}{h_b}>\frac{1}{h_c}$ Is it possible to construct a triangle with altitudes $7$, $11$, and $20$? Justify your answer. [b]p3.[/b] Does there exist a polynomial $P(x)$ with integer coefficients such that $P(0) = 1$, $P(2) = 3$ and $P(4) = 9$? Justify your answer. [b]p4.[/b] Prove that if $\cos \theta$ is rational and $n$ is an integer, then $\cos n\theta$ is rational. Let $\alpha=\frac{1}{2010}$. Is $\cos \alpha $ rational ? Justify your answer. [b]p5.[/b] Let function $f(x)$ be defined as $f(x) = x^2 + bx + c$, where $b, c$ are real numbers. (A) Evaluate $f(1) -2f(5) + f(9)$ . (B) Determine all pairs $(b, c)$ such that $|f(x)| \le 8$ for all $x$ in the interval $[1, 9]$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].