An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold: ($1$) no one stands between the two tallest players, ($2$) no one stands between the third and fourth tallest players, $\;\;\vdots$ ($N$) no one stands between the two shortest players. Show that this is always possible. [i]Proposed by Grigory Chelnokov, Russia[/i]

Let $M>1$ be a natural number. Tom and Jerry play a game. Jerry wins if he can produce a function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying [list] [*]$f(M) \ne M$ [/*] [*] $f(k)<2k$ for all $k \in \mathbb{N}$[/*] [*] $f^{f(n)}(n)=n$ for all $n \in \mathbb{N}$. For each $\ell>0$ we define $f^{\ell}(n)=f\left(f^{\ell-1}(n)\right)$ and $f^0(n)=n$[/*] [/list] Tom wins otherwise. Prove that for infinitely many $M$, Tom wins, and for infinitely many $M$, Jerry wins. [i]Proposed by Anant Mudgal[/i]

A point $P$ was chosen on the smaller arc $BC$ of the circumcircle of the acute-angled triangle $ABC$. Points $R$ and $S$ on the sides$ AB$ and $AC$ are respectively selected so that $CPRS$ is a parallelogram. Point $T$ on the arc $AC$ of the circumscribed circle of $\vartriangle ABC$ such that $BT \parallel CP$. Prove that $\angle TSC = \angle BAC$. (Anton Trygub)

Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

For real numbers $a, b, c, d \in [0, 1]$, find the largest possible value of the following expression: $$a^2+b^2+c^2+d^2-ab-bc-cd-da$$ [i]Proposed by Mykhailo Shtandenko[/i]

Given two circles $(O_1)$ and $(O_2)$ intersect at $A$ and $B$. Let $d_1$ and $d_2$ be two lines through $A$ and be symmetric with respect to $AB$. The line $d_1$ cuts $(O_1)$ and $(O_2)$ at $G, E$ ($\ne A$), respectively, the line $d_2$ cuts $(O_1)$ and $(O_2)$ at $F, H$ ($\ne A$), respectively, such that $E$ is between $A, G$ and $F$ is between $A, H$. Let $J$ be the intersection of $EH$ and $FG$. The line $BJ$ cuts $(O_1), (O_2)$ at $K, L$ ($\ne B$), respectively. Let $N$ be the intersection of $O_1K$ and $O_2L$. Prove that the circle $(NLK)$ is tangent to $AB$.

An ant starts at the bottom left corner of a $5\times5$ grid of dots and walks to the top right corner. It can walk from one dot to any dot that is horizontally or vertically adjacent to it. If it never walks between the same pair of dots twice, what is the length of the longest path the ant can take? $\text{(A) }30\qquad\text{(B) }31\qquad\text{(C) }32\qquad\text{(D) }33\qquad\text{(E) }34$

Let $\mathcal{S}$ be a set of integers. Given a real positive $r$, we say that $\mathcal{S}$ is a $r$-discerning, if for any pair $m, n > 1$ of distinct integers such that $\left| \frac{m-n}{m+n} \right|  < r$, there exists $a \in \mathcal{S}$ and $k \geq 1$ such that $a^k$ divides $m$ but not $n$, or $a^k$ divides $n$ but not $m$ 1. Show that for every $r > 0$ every $r$-discerning set contains an infinite number of primes. 2. For every $r > 0$ determine the maximal possible cardinality of $\mathcal{P} \backslash \mathcal{S}$ where $\mathcal{P}$ is the set of primes and $\mathcal{S} \subseteq \mathcal{P}$ is a $r$-discerning set.

Let $ABCD$ be a parallelogram, and let $P$ be a point on the side $AB$. Let the line through $P$ parallel to $BC$ intersect the diagonal $AC$ at point $Q$. Prove that $$|DAQ|^2 = |PAQ| \times |BCD| ,$$ where $|XY Z|$ denotes the area of triangle $XY Z$.

Luis' friends played a prank on him in his geometry homework. They erased the entire triangle but left traces equivalent to two sides measuring $a$ and $b$, with $b > a$, and the height $h$ falling on the side measuring $b$, with $h < a$. Help Luis reconstruct the original triangle using only a straightedge and compass. Since Luis' method does not involve measurements, prove that his method results in a triangle longer than its given sides and height.

Given a positive integer $m$, Prove that there exists a positive integers $n_0$ such that all first digits after the decimal points of $\sqrt{n^2+817n+m}$ in decimal representation are equal, for all integers $n>n_0$.

Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) $ is a perfect square.

Find all integers $N$ with the following property: for $10$ but not $11$ consecutive positive integers, each one is a divisor of $N$.

Let $X$ be the collection of all functions $f: \{0,1,\dots, 2016\} \rightarrow \{0,1,\dots, 2016\}$. Compute the number of functions $f \in X$ such that \[ \max_{g \in X} \left( \min_{0 \le i \le 2016} \big( \max (f(i), g(i)) \big) - \max_{0 \le i \le 2016} \big( \min (f(i),g(i)) \big) \right) = 2015. \]

$ A$ and $ B$ play the following game with a polynomial of degree at least 4: \[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0 \] $ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy?

For real $\theta_i$, $i = 1, 2, \dots, 2011$, where $\theta_1 = \theta_{2012}$, find the maximum value of the expression \[ \sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}. \] [i]Proposed by Lewis Chen [/i]

Prove there do exist infinitely many positive integers $n$ such that if a prime $p$ divides $n(n+1)$ then $p^2$ also divides it (all primes dividing $n(n+1)$ bear exponent at least two). Exhibit (at least) two values, one even and one odd, for such numbers $n>8$. (Pál Erdös & Kurt Mahler)

p1. It is known that $a_1=2$ , $a_2=3$ . For $k > 2$, define $a_k=\frac{1}{2}a_{k-2}+\frac{1}{3}a_{k-1}$. Find the infinite sum of of $a_1+a_2+a_3+...$ p2. The [i]multiplied [/i] number is a natural number in two-digit form followed by the result time. For example, $7\times 8 = 56$, then $7856$ and $8756$ are multiplied numbers . $2\times 3 = 6$, then $236$ and $326$ are multiplied. $2\times 0 = 0$, then $200$ is the multiplied. For the record, the first digit of the number times can't be $0$. a. What is the difference between the largest and the smallest multiplied number? b. Find all the multiplied numbers that consist of three digits and each digit is square number. c. Given the following "boxes" that must be filled with multiple numbers. [img]https://cdn.artofproblemsolving.com/attachments/b/6/ac086a3d1a0549fae909c072224605430daf1d.png[/img] Determine the contents of the shaded box. Is this content the only one? d. Complete all the empty boxes above with multiplied numbers. p3. Look at the picture of the arrangement of three squares below. [img]https://cdn.artofproblemsolving.com/attachments/1/3/c0200abae77cc73260b083117bf4bafc707eea.png[/img]Prove that $\angle BAX + \angle CAX = 45^o$ p4. Prove that $(n-1)n (n^3 + 1)$ is always divisible by $6$ for all natural number $n$.

The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?

Let $A_1A_2A_3...A_{72}$ be a regurar $72$-gon with center $O$. Calculate an extenral angle of that polygon and the angles $\angle A_{45} OA_{46}$, $\angle A_{44} A_{45}A_{46}$. How many diagonals does this polygon have?

Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$: $$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$ [i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]

The cubic polynomials $p(x)$ and $q(x)$ satisfy • $p(1) = q(2)$ • $p(3) = q(4)$ • $p(5) = q(6)$ • $p(7) = q(8) + 13$. Find $p(9)-q(10)$.

Let $m, n > 1$ be coprime odd integers. Show that $$\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor$$ is an even integer, where $\phi$ is Euler’s totient function.

Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.

A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M,$ where $r$ and $M$ are positive integers and $M$ is not divisible by $3.$ What is $r?$ $\textbf{(A) }5 \qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$