Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$. [i]Nanang Susyanto, Jogjakarta[/i]

Let $n = p_1p_2 \dots p_k$ be the product of distinct primes $p_1, p_2, \dots , p_k$, with $k > 1$. Find all $n$ such that $n$ is multiple of $p_1 - 1, p_2 - 1, \dots , p_k - 1$.

Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Ali and Mohammad Play the following game. At each step, Ali chooses $b_1,b_2,\dots,b_m \in \mathbb{N}$ and then Mohammad chosses a positive integers $s$ and obtains a new sequence $\{c_i=a_i+b_{i+s}\}_{i=1}^m$, where $$b_{m+1}=b_1,\ b_{m+2}=b_2, \dots,\ b_{m+s}=b_s$$ The goal of Ali is to make all the numbers divisible by $n$ in a finite number of steps. FInd all positive integers $m$ and $n$ such that Ali has a winning strategy, no matter how the initial values $a_1, a_2,\dots,a_m$ are. [hide=clarification] after we create the $c_i$ s, this sequence becomes the sequence that we continue playing on, as in it is our 'new' $a_i$[/hide] Proposed by Shayan Gholami

(a) Prove that \[ [5x]\plus{}[5y] \ge [3x\plus{}y] \plus{} [3y\plus{}x],\] where $ x,y \ge 0$ and $ [u]$ denotes the greatest integer $ \le u$ (e.g., $ [\sqrt{2}]\equal{}1$). (b) Using (a) or otherwise, prove that \[ \frac{(5m)!(5n)!}{m!n!(3m\plus{}n)!(3n\plus{}m)!}\] is integral for all positive integral $ m$ and $ n$.

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

Find all solution $(p,r)$ of the "Pythagorean-Euler Theorem" $$p^p+(p+1)^p+\cdots+(p+r)^p=(p+r+1)^p$$Where $p$ is a prime and $r$ is a positive integer. [i]Proposed by Li4 and Untro368[/i]

Let $a$ be number of $n$ digits ($ n > 1$). A number $b$ of $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values values of $k$.

Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$

Is it possible to find $2005$ different positive square numbers such that their sum is also a square number ?

If the natural numbers $a, b, c, d$ verify the relationships: $$(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2$$ $$(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2$$ and the $gcd(a, b, c, d) = 1$, prove that $a + b + c + d$ is a perfect square.

Find if there are solutions : $ a,b \in\mathbb{N} $ , $a^2+b^2=2018 $ , $ 7|a+b $ .

Find all positive integer $n$ such that $$\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1$$ holds for all $\theta$ which are not integral multiples of $\frac{\pi}{2}$

For any positive integer $m$, denote by $P(m)$ the product of positive divisors of $m$ (e.g $P(6)=36$). For every positive integer $n$ define the sequence $$a_1(n)=n,\qquad a_{k+1}(n)=P(a_k(n))\quad (k=1,2,\dots,2016)$$ Determine whether for every set $S\subset\{1,2,\dots,2017\}$, there exists a positive integer $n$ such that the following condition is satisfied: For every $k$ with $1\leq k\leq 2017$, the number $a_k(n)$ is a perfect square if and only if $k\in S$.

Find all pairs $(b,c)$ of positive integers, such that the sequence defined by $a_1=b$, $a_2=c$ and $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$ for $n \geq 1$ has only finite number of composite terms. [i]Proposed by Oleg Mushkarov and Nikolai Nikolov[/i]

Let $N$ be a positive integer. Consider the sequence $a_1, a_2, ..., a_N$ of positive integers, none of which is a multiple of $2^{N+1}$. For $n \ge N +1$, the number $a_n$ is defined as follows: choose $k$ to be the number among $1, 2, ..., n - 1$ for which the remainder obtained when $a_k$ is divided by $2^n$ is the smallest, and define $a_n = 2a_k$ (if there are more than one such $k$, choose the largest such $k$). Prove that there exist $M$ for which $a_n = a_M$ holds for every $n \ge M$.

Each of six fruit baskets contains pears, plums and apples. The number of plums in each basket equals the total number of apples in all other baskets combined while the number of apples in each basket equals the total number of pears in all other baskets combined. Prove that the total number of fruits is a multiple of $31$.

For a positive integer $n$, denote $A_n=\{(x,y)\in\mathbb Z^2|x^2+xy+y^2=n\}$. (a) Prove that the set $A_n$ is always finite. (b) Prove that the number of elements of $A_n$ is divisible by $6$ for all $n$. (c) For which $n$ is the number of elements of $A_n$ divisible by $12$?

What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is [i]palindromic[/i] if the sequence of decimal digits are the same when read backwards.)

Let $f(x)$ be a monic polynomial of degree $2023$ with positive integer coefficients. Show that for any sufficiently large integer $N$ and any prime number $p>2023N$, the product \[f(1)f(2)\dots f(N)\] is at most $\binom{2023}{2}$ times divisible by $p$. [i]Proposed by Ashwin Sah[/i]

Show that for any positive integers $a$ and $b$, $(36a+b)(a+36b)$ cannot be a power of $2$.

Let $a_1,a_2,a_3, \cdots $ be distinct positive integers, and $0<c<\frac{3}{2}$ . Prove that : There exist infinitely many positive integers $k$, such that $[a_k,a_{k+1}]>ck $.

Find all pairs of integers $(p,q)$ for which all roots of the trinomials $x^2+px+q$ and $x^2+qx+p$ are integers.

[b]p1.[/b] An airplane travels between two cities. The first half of the distance between the cities is traveled at a constant speed of $600$ mi/hour, and the second half of the distance is traveled at a a constant speed of $900$ mi/hour. Find the average speed of the plane. [b]p2.[/b] The figure below shows two egg cartons, $A$ and $B$. Carton $A$ has $6$ spaces (cell) and has $3$ eggs. Carton $B$ has $12$ cells and $3$ eggs. Tow cells from the total of $18$ cells are selected at random and the contents of the selected cells are interchanged. (Not that one or both of the selected cells may be empty.) [img]https://cdn.artofproblemsolving.com/attachments/6/7/2f7f9089aed4d636dab31a0885bfd7952f4a06.png[/img] (a) Find the number of selections/interchanges that produce a decrease in the number of eggs in cartoon $A$- leaving carton $A$ with $2$ eggs. (b) Assume that the total number of eggs in cartons $A$ and $B$ is $6$. How many eggs must initially be in carton $A$ and in carton $B$ so that the number of selections/interchanges that lead to an increase in the number of eggs in $A$ equals the number of selections/interchanges that lead to an increase in the number of eggs in $B$. $\bullet$ In other words, find the initial distribution of $6$ eggs between $A$ and $B$ so that the likelihood of an increase in A equals the likelihood of an increase in $B$ as the result of a selection/interchange. Prove your answer. [b]p3.[/b] Divide the following figure into four equal parts (parts should be of the same shape and of the same size, they may be rotated by different angles however they may not be disjoint and reconnected). [img]https://cdn.artofproblemsolving.com/attachments/f/b/faf0adbf6b09b5aaec04c4cfd7ab1d6397ad5d.png[/img] [b]p4.[/b] Find the exact numerical value of $\sqrt[3]{5\sqrt2 + 7}- \sqrt[3]{5\sqrt2 - 7}$ (do not use a calculator and do not use approximations). [b]p5.[/b] The medians of a triangle have length $9$, $12$ and $15$ cm respectively. Find the area of the triangle. [b]p6. [/b]The numbers $1, 2, 3, . . . , 82$ are written in an arbitrary order. Prove that it is possible to cross out $72$ numbers in such a sway the remaining number will be either in increasing order or in decreasing order. PS. You should use hide for answers.

Given a positive integer whose base-$10$ representation is $\overline{d_k\ldots d_0}$ for some integer $k \geq 0$, where $d_k \neq 0$, a move consists of selecting some integers $0 \leq i \leq j \leq k$, such that the digits $d_j,\ldots,d_i$ are not all $0$, erasing them from $n$, and replacing them with a divisor of $\overline{d_j\ldots d_i}$ (this divisor need not have the same number of digits as $\overline{d_j\ldots d_i}$). Prove that for all sufficiently large even integers $n$, we may apply some sequence of moves to $n$ to transform it into $2024$. [i]Allen Wang[/i]

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.