Found problems: 85335
Let $n, r$ be positive integers. Find the smallest positive integer $m$ satisfying the following condition. For each partition of the set $\{1, 2, \ldots ,m \}$ into $r$ subsets $A_1,A_2, \ldots ,A_r$, there exist two numbers $a$ and $b$ in some $A_i, 1 \leq i \leq r$, such that
\[ 1 < \frac ab < 1 +\frac 1n.\]
Call a positive integer [i]monotonous[/i] if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there?
$\textbf{(A)} \text{ 1024} \qquad \textbf{(B)} \text{ 1524} \qquad \textbf{(C)} \text{ 1533} \qquad \textbf{(D)} \text{ 1536} \qquad \textbf{(E)} \text{ 2048}$
Let $D$ and $E$ be the midpoints of the sides $BC$ and $AC$ of a right triangle $ABC$. Prove that if $\angle CAD=\angle ABE$, then $$\frac{5}{6} \le \frac{AD}{AB}\le \frac{\sqrt{73}}{10}.$$
Let $\phi=\tfrac{1+\sqrt 5}{2}$. Find
\[\left(4+\phi^{\frac12}\right)\left(4-\phi^{\frac12}\right)\left(4+i\phi^{-\frac12}\right)\left(4-i\phi^{-\frac12}\right).\]
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
Let $A$ be a set of positive integers such that
a) if $a\in A$, the all the positive divisors of $a$ are also in $A$;
b) if $a,b\in A$, with $1<a<b$, then $1+ab \in A$.
Prove that if $A$ has at least 3 elements, then $A$ is the set of all positive integers.
Show that for any integer $n \ge 2$ the sum of the fractions $\frac{1}{ab}$, where $a$ and $b$ are relatively prime positive integers such that $a < b \le n$ and $a+b > n$, equals $\frac{1}{2}$.
(Integers $a$ and $b$ are called relatively prime if the greatest common divisor of $a$ and $b$ is $1$.)
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
Inside a square $ABCD$ point $P$ is marked, and on the sides $AB$, $BC$, $CD$ and $DA$ points $K,L,M$ and $N$ are chosen respectively. Lines $KP,LP,MP$ and $NP$ intersect sides $CD,DA,AB$ and $BC$ at points $K_1, L_1, M_1$ and $N_1$ respectively. It turned out that
$$\frac{KP}{PK_1}+\frac{LP}{PL_1}+\frac{MP}{PM_1}+\frac{NP}{PN_1}=4$$
Prove that $KP+LP+MP+NP=K_1P+L_1P+M_1P+N_1P$.
Let $n$ be a positive integer. There is a collection of cards that meets the following properties:
$\bullet$Each card has a number written in the form $m!$, where $m$ is a positive integer.
$\bullet$For every positive integer $t\le n!$, it is possible to choose one or more cards from the collection in such a way
$\text{ }$that the sum of the numbers of those cards is $t$.
Determine, based on $n$, the smallest number of cards that this collection can have.
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that:
$ \text{(i)}\quad f(0)=0 $
$ \text{(ii)}\quad f'(x)\neq 0,\quad\forall x\in\mathbb{R} $
$ \text{(iii)}\quad \left. f''\right|_{\mathbb{R}}\text{ exists and it's continuous} $
Demonstrate that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ g(x)=\left\{\begin{matrix}\cos\frac{1}{f(x)},\quad x\neq 0\\ 0,\quad x=0\end{matrix}\right. $$
is primitivable.
Determine all positive integers $k$ for which there exist positive integers $r$ and $s$ that satisfy the equation $$(k^2-6k+11)^{r-1}=(2k-7)^{s}.$$
Let $N,K,L$ be points on the sides $\overline{AB}, \overline{BC}, \overline{CA}$ respectively. Suppose $AL=BK$ and $\overline{CN}$ is the internal bisector of angle $ACB$. Let $P$ be the intersection of lines $\overline{AK}$ and $\overline{BL}$ and let $I,J$ be the incenters of triangles $APL$ and $BPK$ respectively. Let $Q$ be the intersection of lines $\overline{IJ}$ and $\overline{CN}$. Prove that $IP=JQ$.
A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?
[b]3.[/b] Let $k$ and $K$ be concentric circles on the plane, and let $k$ be contained inside $K$. Assume that $k$ is covered by a finite system of convex angular domains with vertices on $K$. Prove that the sum of the angles of the domains is not less than the angle under which $k$ can be seen from a point of $K$. ([b]G.38[/b])
[Zs.. Páles]
A set of $2003$ positive numbers is such that for any two numbers $a$ and $b$ included in it ($a > b$) at least one of the numbers $a + b$ or $a - b$ also included in the set. Prove that if these numbers are ordered by increasing, then the differences between adjacent numbers will be the same.
Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.
Find the remainder when
\[\sum_{k=1}^{2^{16}}\binom{2k}{k}(3\cdot 2^{14}+1)^k (k-1)^{2^{16}-1}\]is divided by $2^{16}+1$. ([i]Note:[/i] It is well-known that $2^{16}+1=65537$ is prime.)
[i]Victor Wang.[/i]
Given are positive integers $n>20$ and $k>1$, such that $k^2$ divides $n$. Prove that there exist positive integers $a, b, c$, such that $n=ab+bc+ca$.
Consider a complex number whose affix in the complex plane is situated on the first quadrant of the unit circle centered at origin. Then, the following inequality holds.
$$ \sqrt{2} +\sqrt{2+\sqrt{2}} \le |1+z|+|1+z^2|+|1+z^4|\le 6 $$
[i]Costică Ambrinoc[/i]
Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits.
Let $ABC$ be a triangle with circumcircle $w_1$ and incenter $I$. Suppose $w_2$ is a circle tangent to $AB,AC$ at $X,Y$, and internally tangent to $w$ at $D$. Let the parallel to the exterior angle bisector of $A$ through $D$ meet $w_2$ at $P$. Show that $AP, DI$ intersect on $w_2$.
Calculate the limit of the following sequences:
[b]a)[/b] n^{n!}/(n!)^n
[b]b)[/b] n^{ln n}/n!
[i]Adrian Troie[/i]
Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
Let $ABC$ be a triangle with circumradius $17$, inradius $4$, circumcircle $\Gamma$ and $A$-excircle $\Omega$. Suppose the reflection of $\Omega$ over line $BC$ is internally tangent to $\Gamma$. Compute the area of $\triangle ABC$.