Found problems: 85335
Let a $9$-digit number be balanced if it has all numerals $1$ to $9$. Let $S$ be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of $k$ such that any two subsequences of $S$ which has consecutive $k$ numerals are different from each other.
Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules:
[b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box.
[b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box.
Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.
Let $x$ be a positive real number and let $k$ be a positive integer. Assume that $x^k+\frac{1}{x^k}$ and $x^{k+1}+\frac{1}{x^{k+1}}$ are both rational numbers. Prove that $x+\frac{1}{x}$ is also a rational number.
Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of
\[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]
An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.
Show that
\[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\]
holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.
Suppose that $a,b$ are two odd positive integers such that $2ab+1 \mid a^2 + b^2 + 1$. Prove that $a=b$.
(15 points)
Let $ABCD$ be a cyclic quadrilateral in which internal angle bisectors $\angle ABC$ and $\angle ADC$ intersect on diagonal $AC$. Let $M$ be the midpoint of $AC$. Line parallel to $BC$ which passes through $D$ cuts $BM$ at $E$ and circle $ABCD$ in $F$ ($F \neq D$ ). Prove that $BCEF$ is parallelogram
[i]Proposed by Steve Dinh[/i]
We inscribed in triangle $ABC$ the rectangle $DEFG$ such that $D$ and $E$ fall on side $AB$, $F$ on side $BC$, and $G$ on side $AC$. We know that $AF$ bisects angle $\angle BAC$, and that $\frac{AD}{DE} = \frac12$. What is the measure of angle $\angle CAB$?
Find all functions $ f:\mathbb{Q}\to\mathbb{R}$ that satisfy $ f(x\plus{}y)\equal{}f(x)f(y)\minus{}f(xy)\plus{}1$ for every $x,y\in\mathbb{Q}$.
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
Let $ABCD$ be a cyclic quadrilateral with $AB<CD$, and let $P$ be the point of intersection of the lines $AD$ and $BC$.The circumcircle of the triangle $PCD$ intersects the line $AB$ at the points $Q$ and $R$. Let $S$ and $T$ be the points where the tangents from $P$ to the circumcircle of $ABCD$ touch that circle.
(a) Prove that $PQ=PR$.
(b) Prove that $QRST$ is a cyclic quadrilateral.
Is it possible to arrange $36$ distinct numbers in the cells of a $6 \times 6$ table, so that in each $1\times 5$ rectangle (both vertical and horizontal) the sum of the numbers equals $2022$ or $2023$?
Let $I = (0, 1]$ be the unit interval of the real line. For a given number $a \in (0, 1)$ we define a map $T : I \to I$ by the formula
if
\[ T (x, y) = \begin{cases} x + (1 - a),&\mbox{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \mbox{ if } a < x \leq 1.\end{cases} \]
Show that for every interval $J \subset I$ there exists an integer $n > 0$ such that $T^n(J) \cap J \neq \emptyset.$
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer.
[i]Laurentiu Panaitopol, Romania[/i]
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.
$\text{ }$
[asy]
unitsize(11);
for(int i=0; i<6; ++i)
{
if(i<5)
draw( (i, 0)--(i,5) );
else draw( (i, 0)--(i,2) );
if(i < 3)
draw((0,i)--(5,i));
else draw((0,i)--(4,i));
}
[/asy]
We are dividing the above figure into parts with shapes: [asy]
unitsize(11);
draw((0,0)--(0,2));
draw((1,0)--(1,2));
draw((2,1)--(2,2));
draw((0,0)--(1,0));
draw((0,1)--(2,1));
draw((0,2)--(2,2));
[/asy][asy]
unitsize(11);
draw((0,0)--(0,2));
draw((1,0)--(1,2));
draw((2,1)--(2,2));
draw((3,1)--(3,2));
draw((0,0)--(1,0));
draw((0,1)--(3,1));
draw((0,2)--(3,2));
[/asy]
After that division, find the number of
[asy]
unitsize(11);
draw((0,0)--(0,2));
draw((1,0)--(1,2));
draw((2,1)--(2,2));
draw((0,0)--(1,0));
draw((0,1)--(2,1));
draw((0,2)--(2,2));
[/asy]
shaped parts.
In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?
Solve in positive integers the following equation $$\left [\sqrt{1}\right]+\left [\sqrt{2}\right]+\left [\sqrt{3}\right]+\ldots+\left [\sqrt{x^2-2}\right]+\left [\sqrt{x^2-1}\right]=125,$$ where $[a]$ is the integer part of the real number $a$.
Consider an infinite strip of unit squares. The squares are numbered "1", "2", "3", ... A pawn starts on one of the squares and it can move according to the following rules:
(1) from the square numbered "$n$" to the square numbered "$2n$", and vice versa;
(2) from the square numbered "$n$" to the square numbered "$3n + 1$", and vice versa.
Show that the pawn can reach the square numbered "$1$" in a finite number of moves.
Let $ ABC$ be an acute-angled, non-isosceles triangle with orthocenter $H$, $M$ midpoint of side $AB$ and $w$ bisector of angle $\angle ACB$. Let $S$ be the point of intersection of the perpendicular bisector of side $AB$ with $w$ and $F$ the foot of the perpendicular from $H$ on $w$. Prove that the segments $MS$ and $MF$ are equal.
[i](Karl Czakler)[/i]
$ABC$ is a triangle. Points $D, E, F$ are chosen on $BC, CA, AB$ such that $B$ is equidistant from $D$ and $F$, and $C$ is equidistant from $D$ and $E$. Show that the circumcenter of $AEF$ lies on the bisector of $EDF$.
Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously:
(1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$;
(2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$;
(3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$.
Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.
Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$
$A$ -is $20$-digit number. We write $101$ numbers $A$ then erase last $11$ digits. Prove that this $2009$-digit number can not be degree of $2$