Numbers from $1$ to $49$ are randomly placed in a $35 \times 35$ table such that number $i$ is used exactly $i$ times. Some random cells of the table are removed so that table falls apart into several connected (by sides) polygons. Among them, the one with the largest area is chosen (if there are several of the same largest areas, a random one of them is chosen). What is the largest number of cells that can be removed that guarantees that in the chosen polygon there is a number which occurs at least $15$ times?

For a positive integer $a$, $a'$ is the integer obtained by the following method: the decimal writing of $a'$ is the inverse of the decimal writing of $a$ (the decimal writing of $a'$ can begin by zeros, but not the one of $a$); for instance if $a=2370$, $a'=0732$, that is $732$. Let $a_{1}$ be a positive integer, and $(a_{n})_{n \geq 1}$ the sequence defined by $a_{1}$ and the following formula for $n \geq 1$: \[a_{n+1}=a_{n}+a'_{n}. \] Can $a_{7}$ be prime?

We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.

A triangle $ABC$ is inscribed in a circle $G$. Points $M$ and $N$ are the midpoints of the arcs $BC$ and $AC$ respectively, and $D$ is an arbitrary point on the arc $AB$ (not containing $C$). Points $I_1$ and $I_2$ are the incenters of the triangles $ADC$ and $BDC$, respectively. If the circumcircle of triangle $DI_1I_2$ meets $G$ again at $P$, prove that triangles $PNI_1$ and $PMI_2$ are similar.

Find the largest possible number of rooks that can be placed on a $3n \times 3n$ chessboard so that each rook is attacked by at most one rook.

There are n cups arranged on the circle. Under one of cups is hiden a coin. For every move, it is allowed to choose 4 cups and verify if the coin lies under these cups. After that, the cups are returned into its former places and the coin moves to one of two neigbor cups. What is the minimal number of moves we need in order to eventually find where the coin is?

Let $ n$ be an integer, $ n\geq 2$. Find all sets $ A$ with $ n$ integer elements such that the sum of any nonempty subset of $ A$ is not divisible by $ n\plus{}1$.

Two equal rectangles are intersecting in $8$ points. Prove that the common part area is greater than the half of the rectangle's area.

In the center of a square is a rabbit and at each vertex of this even square, a wolf. The wolves only move along the sides of the square and the rabbit moves freely in the plane. Knowing that the rabbit move at a speed of $10$ km / h and that the wolves move to a maximum speed of $14$ km / h, determine if there is a strategy for the rabbit to leave the square without being caught by the wolves.

[list=a] [*] Prove that for any positive integers $a,b,c$, there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)$$ is a perfect square. [*] Prove that there exist five distinct positive integers $a,b,c,d,e$ for which there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)(N+d^2)(N+e^2)$$ is a perfect square. [/list] [i]Luminița Popescu[/i]

The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$? $\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$

Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.

Segments $AA'$, $BB'$, and $CC'$ are the bisectrices of triangle $ABC$. It is known that these lines are also the bisectrices of triangle $A'B'C'$. Is it true that triangle $ABC$ is regular?

For any finite set $S$, let $f(S)$ be the sum of the elements of $S$ (if $S$ is empty then $f(S)=0$). Find the sum over all subsets $E$ of $S$ of $\dfrac{f(E)}{f(S)}$ for $S=\{1,2,\cdots,1999\}$.

[u]Round 9[/u] [b]p25.[/b] Let $S$ be the set of the first $2017$ positive integers. Find the number of elements $n \in S$ such that $\sum^n_{i=1} \left\lfloor \frac{n}{i} \right\rfloor$ is even. [b]p26.[/b] Let $\{x_n\}_{n \ge 0}$ be a sequence with $x_0 = 0$,$x_1 = \frac{1}{20}$ ,$x_2 = \frac{1}{17}$ ,$x_3 = \frac{1}{10}$ , and $x_n = \frac12 ((x_{n-2} +x_{n-4})$ for $n\ge 4$. Compute $$ \left\lfloor \frac{1}{x_{2017!} -x_{2017!-1}} \right\rfloor.$$ [b]p27.[/b] Let $ABCDE$ be be a cyclic pentagon. Given that $\angle CEB = 17^o$, find $\angle CDE + \angle EAB$, in degrees. [u]Round 10[/u] [b]p28.[/b] Let $S = \{1,2,4, ... ,2^{2016},2^{2017}\}$. For each $0 \le i \le 2017$, let $x_i$ be chosen uniformly at random from the subset of $S$ consisting of the divisors of $2^i$ . What is the expected number of distinct values in the set $\{x_0,x_1,x_2,... ,x_{2016},x_{2017}\}$? [b]p29.[/b] For positive real numbers $a$ and $b$, the points $(a, 0)$, $(20,17)$ and $(0,b)$ are collinear. Find the minimum possible value of $a+b$. [b]p30.[/b] Find the sum of the distinct prime factors of $2^{36}-1$. [u]Round 11[/u] [b]p31.[/b] There exist two angle bisectors of the lines $y = 20x$ and $y = 17x$ with slopes $m_1$ and $m_2$. Find the unordered pair $(m_1,m_2)$. [b]p32.[/b] Triangle 4ABC has sidelengths $AB = 13$, $BC = 14$, $C A =15$ and orthocenter $H$. Let $\Omega_1$ be the circle through $B$ and $H$, tangent to $BC$, and let $\Omega_2$ be the circle through $C$ and $H$, tangent to $BC$. Finally, let $R \ne H$ denote the second intersection of $\Omega_1$ and $\Omega_2$. Find the length $AR$. [b]p33.[/b] For a positive integer $n$, let $S_n = \{1,2,3, ...,n\}$ be the set of positive integers less than or equal to $n$. Additionally, let $$f (n) = |\{x \in S_n : x^{2017}\equiv x \,\, (mod \,\, n)\}|.$$ Find $f (2016)- f (2015)+ f (2014)- f (2013)$. [u]Round 12[/u] [b]p34. [/b] Estimate the value of $\sum^{2017}_{n=1} \phi (n)$, where $\phi (n)$ is the number of numbers less than or equal $n$ that are relatively prime to n. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be max $\max \left(0,\lfloor 15 - 75 \frac{|A-E|}{A} \rceil \right).$ [b]p35.[/b] An up-down permutation of order $n$ is a permutation $\sigma$ of $(1,2,3, ..., n)$ such that $\sigma(i ) <\sigma (i +1)$ if and only if $i$ is odd. Denote by $P_n$ the number of up-down permutations of order $n$. Estimate the value of $P_{20} +P_{17}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, 16 -\lceil \max \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$ [b]p36.[/b] For positive integers $n$, superfactorial of $n$, denoted $n\$ $, is defined as the product of the first $n$ factorials. In other words, we have $n\$ = \prod^n_{i=1}(i !)$. Estimate the number of digits in the product $(20\$)\cdot (17\$)$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the sequence $a_n = 3^n- 2^n$ contains no three numbers in geometric progression.