Found problems: 85335
Prove that there exists a $k_0\in\mathbb{N}$ such that for every $k\in\mathbb{N},k>k_0$, there exists a finite number of lines in the plane not all parallel to one of them, that divide the plane exactly in $k$ regions. Find $k_0$.
Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.
A quadrilateral and a pentagon (both not self-intersecting) intersect each other at $N$ distinct points, where $N$ is a positive integer. What is the maximal possible value of $N$?
[i]Proposed by James Lin
Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers.
Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer.
(Walther Janous)
For each positive integer $m$, denote by $d(m)$ the number of positive divisors of $m$. We say that a positive integer $n$ is [i]Burapha[/i] integer if it satisfy the following condition
[list]
[*] $d(n)$ is an odd integer.
[*] $d(k) \leqslant d(\ell)$ holds for every positive divisor $k, \ell$ of $n$, such that $k < \ell$
[/list]
Find all Burapha integer.
If $ p(x)$ is a polynomial, denote by $ p^n(x)$ the polynomial $ p(p(...(p(x))..)$, where $ p$ is iterated $ n$ times. Prove that the polynomial $ p^{2003}(x)\minus{}2p^{2002}(x)\plus{}p^{2001}(x)$ is divisible by $ p(x)\minus{}x$
Consider the sequence $p_{1},p_{2},...$ of primes such that for each $i\geq2$, either $p_{i}=2p_{i-1}-1$ or $p_{i}=2p_{i-1}+1$. Show that any such sequence has a finite number of terms.
[u]Round 1 [/u]
[b]p1.[/b] Pirate Jim had $8$ boxes with gun powder weighing $1, 2, 3, 4, 5, 6, 7$, and $8$ pounds (the weight is printed on top of every box). Pirate Bob hid a $1$-pound gold bar in one of these boxes. Pirate Jim has a balance scale that he can use, but he cannot open any of the boxes. Help him find the box with the gold bar using two weighings on the balance scale.
[b]p2.[/b] James Bond will spend one day at Dr. Evil's mansion to try to determine the answers to two questions:
a) Is Dr. Evil at home?
b) Does Dr. Evil have an army of ninjas?
The parlor in Dr. Evil's mansion has three windows. At noon, Mr. Bond will sneak into the parlor and use open or closed windows to signal his answers. When he enters the parlor, some windows may already be opened, and Mr. Bond will only have time to open or close one window (or leave them all as they are).
Help Mr. Bond and Moneypenny design a code that will tell Moneypenny the answers to both questions when she drives by later that night and looks at the windows. Note that Moneypenny will not have any way to know which window Mr. Bond opened or closed.
[b]p3.[/b] Suppose that you have a triangle in which all three side lengths and all three heights are integers. Prove that if these six lengths are all different, there cannot be four prime numbers among them.
p4. Fred and George have designed the Amazing Maze, a $5\times 5$ grid of rooms, with Adorable Doors in each wall between rooms. If you pass through a door in one direction, you gain a gold coin. If you pass through the same door in the opposite direction, you lose a gold coin. The brothers designed the maze so that if you ever come back to the room in which you started, you will find that your money has not changed.
Ron entered the northwest corner of the maze with no money. After walking through the maze for a while, he had $8$ shiny gold coins in his pocket, at which point he magically teleported himself out of the maze. Knowing this, can you determine whether you will gain or lose a coin when you leave the central room through the north door?
[b]p5.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him?
[u]Round 2 [/u]
[b]p6.[/b] $1000$ non-zero numbers are written around a circle and every other number is underlined. It happens that each underlined number is equal to the sum of its two neighbors and that each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be?
[b]p7.[/b] A grasshopper is sitting at the edge of a circle of radius $3$ inches. He can hop exactly $4$ inches in any direction, as long as he stays within the circle. Which points inside the circle can the grasshopper reach if he can make as many jumps as he likes?
[img]https://cdn.artofproblemsolving.com/attachments/1/d/39b34b2b4afe607c1232f4ce9dec040a34b0c8.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Define a sequence $\{a_n\}$ by\[a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),\] find integer $k$ such that $a_{k}<1<a_{k+1}.$
(September 29, 2012, Hohhot)
For what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8?$
[i]Proposed by Alex Li[/i]
Find all positive integer $n$ satifying $$2n+3|n!-1$$
[i]Proposed by ltf0501[/i]
Let the function $f:N^*\to N^*$ such that
[b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;
[b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$
Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.
(High School Affiliated to Nanjing Normal University )
Let $S$ be a subset of $\{1,2,3,\dots,30\}$ with the property that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the largest possible size of $S$?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 $
Find the smallest positive $\alpha$ (in degrees) for which all the numbers \[\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots\] are negative.
Triangle $\vartriangle ABC$ has side lengths $AB = AC = 27$ and $BC = 18$. Point $D$ is on $\overline{AB}$ and point $E$ is on $\overline{AC}$ such that $\angle BCD = \angle CBE = \angle BAC$. Compute $DE$.
For every positive integer $n$, define
$$f(n)=\sum_{p\mid n}{p^{k_p}},$$where the sum is taken over all positive prime divisors $p$ of $n$, and $k_p$ is the unique integer satisfying $$p^{k_p}\leqslant n<p^{k_p+1}.$$Find$$\limsup_{n\to \infty} \frac{f(n)\log \log n}{n\log n} .$$
Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]
Given is an acute angled triangle $ABC$. A circle going through $B$ and the triangle's circumcenter, $O$, intersects $BC$ and $BA$ at points $P$ and $Q$ respectively. Prove that the intersection of the heights of the triangle $POQ$ lies on line $AC$.
For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.
Numbers $1,2,3,...,2n$ are divided onto two equal groups. Let $a_1,a_2,...,a_n$ be the first group numbers in the increasing order, and $b_1,b_2,...,b_n$ -- the second group numbers in the decreasing order. Prove that $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2$$
The tangent at $C$ to $\Omega$, the circumcircle of scalene triangle $ABC$ intersects $AB$ at $D$. Through point $D$, a line is drawn that intersects segments $AC$ and $BC$ at $K$ and $L$ respectively. On the segment $AB$ points $M$ and $N$ are marked such that $AC \parallel NL$ and $BC \parallel KM$. Lines $NL$ and $KM$ intersect at point $P$ lying inside the triangle $ABC$. Let $\omega$ be the circumcircle of $MNP$. Suppose $CP$ intersects $\omega$ again at $Q$. Show that $DQ$ is tangent to $\omega$.
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
Find all $ x,y$ and $ z$ in positive integer: $ z \plus{} y^{2} \plus{} x^{3} \equal{} xyz$ and $ x \equal{} \gcd(y,z)$.
Polynomial $w(x)$ of second degree with integer coefficients takes for integer arguments values, which are squares of integers. Prove that polynomial $w(x)$ is a square of a polynomial.
A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$.
Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$.
a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$.
b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.