Found problems: 85335
Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.
Find all positive integers $x,y,z$ such that $x^3 - (x + y + z)^2 = (y + z)^3 + 34$
The natural numbers from $100$ to $999$ are written on separate cards. They are gathered in one pile with their numbers down in arbitrary order. Let us open them in sequence and divide into $10$ piles according to the least significant digit. The first pile will contain cards with $0$ at the end, ... , the tenth -- with $9$. Then we shall gather $10$ piles in one pile, the first -- down, then the second, ... and the tenth -- up. Let us repeat the procedure twice more, but the next time we shall divide cards according to the second digit, and the last time -- to the most significant one. What will be the order of the cards in the obtained pile?
A circle with radius $r$ has area $505$. Compute the area of a circle with diameter $2r$.
[i]Proposed by Luke Robitaille & Yannick Yao[/i]
Let $a, b, c$ be reals satisfying $a^2+b^2+c^2=6$. Find the maximal values of the expressions
a) $(a-b)^2+(b-c)^2+(c-a)^2$;
b) $(a-b)^2 \cdot (b-c)^2 \cdot (c-a)^2$.
In both cases, describe all triples for which equality holds.
$30$ same balls are put into four boxes $ A$, $ B$, $ C$, $ D$ in such a way that sum of number of balls in $ A$ and $ B$ is greater than sum of in $ C$ and $ D$. How many possible ways are there?
$\textbf{(A)}\ 2472 \qquad\textbf{(B)}\ 2600 \qquad\textbf{(C)}\ 2728 \qquad\textbf{(D)}\ 2856 \qquad\textbf{(E)}\ \text{None}$
A positive integer is called [i]piola [/i] if the $9$ is the remainder obtained by dividing it by $2, 3, 4, 5, 6, 7, 8, 9$ and $10$ and it's digits are all different and nonzero. How many [i]piolas[/i] are there between $ 1$ and $100000$?
On the surface of a sphere, a non-intersecting closed curve is drawn. It divides the surface of the sphere in two regions, which are coloured red and blue. Prove that there exist two antipodes of different colours. [i]Note: the curve is colourless.[/i]
[i]Proposed by Vlad Spătaru[/i]
If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by:
\[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \]
Show that there exist a real number $a > 0$ such that:
[b]i.)[/b] for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$
[b]ii.)[/b] for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$
$100$ numbers $1$, $1/2$, $1/3$, $...$, $1/100$ are written on the blackboard. One may delete two arbitrary numbers $a$ and $b$ among them and replace them by the number $a + b + ab$. After $99$ such operations only one number is left. What is this final number?
(D. Fomin, Leningrad)
A number $k$ is [i]nice [/i] in base $b$ if there exists a $k$-digit number $n$ such that $n, 2n, . . . kn$ are each some cyclic shifts of the digits of $n$ in base $b$ (for example, $2$ is [i]nice [/i] in base $5$ because $2\cdot 135 = 315$). Determine all nice numbers in base $18$.
Let $G$ be a graph, not containing $K_4$ as a subgraph and $|V(G)|=3k$ (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in $G$?
Prove that it is not possible to write numbers $ 1,2,3,...,25$ on the squares of $ 5$x$ 5$ chessboard such that any neighboring numbers differ by at most $ 4$.
Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.
Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.
A number is called [i]winning[/i] if it can be expressed in the form $\frac{a}{20}+\frac{b}{23}$ where $a$ and $b$ are positive integers. How many [i]winning[/i] numbers are less than 1?
[i]Proposed by Andy Xu[/i]
Give examples of a function $f : R \to R$ that satisfies $0 < f(x) < f(x + f(x)) <\sqrt2 x$, for all positive $x$,
and show that there is no function $f : R \to R$ that satisfies $x < f(x + f(x)) <\sqrt2 f(x)$, for all positive $x$.
A convex quadrilateral $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Show that at least two circles have the same size.
Let $ A$ be a set of $ 2n$ points on the plane such that no three points are collinear. Prove that for any distinct two points $ a,b\in A$ there exists a line that partitions $ A$ into two subsets each containing $ n$ points and such that $ a,b$ lie on different sides of the line.
The right triangle below has legs of length $1$ and $2$. Find the sum of the areas of the shaded regions (of which there are infinitely many), given that the regions into which the triangle has been divided are all right triangles.
Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?
A rational number $x$ is given. Prove that there exists a sequence $x_0, x_1, x_2, \ldots$ of rational numbers with the following properties:
(a) $x_0=x$;
(b) for every $n\ge1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \textstyle\frac{1}{n}$;
(c) $x_n$ is an integer for some $n$.
If we add $1996$ to $1997$, we first add the unit digits $6$ and $7$. Obtaining $13$, we write down $3$ and “carry” $1$ to the next column. Thus we make a carry. Continuing, we see that we are to make three carries in total.
Does there exist a positive integer $k$ such that adding $1996\cdot k$ to $1997\cdot k$ no carry arises during the whole calculation?
Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.
Is it true that the center of the inscribed circle of the triangle lies inside the triangle formed by the lines of connecting it's midpoints?