Let the sequence ($a_n$) be defined by $a_n = n^6 +5n^4 -12n^2 -36, n \ge 2$. (a) Prove that any prime number divides some term in this sequence. (b) Prove that there is a positive integer not dividing any term in the sequence. (c) Determine the least $n \ge 2$ for which $1989 | a_n$.

Let $S$ be an infinite set of positive integers and define: $T=\{ x+y|x,y \in S , x \neq y \} $ Suppose that there are only finite primes $p$ so that: 1.$p \equiv 1 \pmod 4$ 2.There exists a positive integer $s$ so that $p|s,s \in T$. Prove that there are infinity many primes that divide at least one term of $S$.

100 couples are invited to a traditional Modolvan dance. The $200$ people stand in a line, and then in a $\textit{step}$, (not necessarily adjacent) many swap positions. Find the least $C$ such that whatever the initial order, they can arrive at an ordering where everyone is dancing next to their partner in at most $C$ steps.

Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that: $$a^3 + b^3 + c^3 = 2016$$

What is the number of $4$ digit natural numbers such that the sum of digits is even? (A)$4999$ (B)$5000$ (C)$5050$ (D)$4500$

Given the point $O$ in the space and $1979$ straight lines $l_1, l_2, ... , l_{1979}$ containing it. Not a pair of lines is orthogonal. Given a point $A_1$ on $l_1$ that doesn't coincide with $O$. Prove that it is possible to choose the points $A_i$ on $l_i$ ($i = 2, 3, ... , 1979$) in so that $1979$ pairs will be orthogonal: $A_1A_3$ and $l_2$, $A_2A_4$ and $l_3$,$ ...$ , $A_{i-1}A_{i+1}$ and $l_i$,$ ...$ , $A_{1977}A_{1979}$ and $l_{1978}$, $A_{1978}A_1$ and $l_{1979}$, $A_{1979}A_2$ and $l_1$

Nanna and Sofie move in the same direction along two parallel paths, which are $200$ meters apart. Nanna's speed is $3$ meters per second, Sofie's only $1$ meter per second. A tall, cylindrical building with a diameter of $100$ meters is located in the middle between the two paths. Since the building first once the line of sight breaks between the girls, their distance between them is $200$ metres. How long will it be before the two girls see each other again?

Let $x_1,x_2,\ldots,x_n\ge0$, $\alpha,\beta>0$, $\beta\ge\alpha$, $t\in\mathbb R$, such that $x_1^{x_2^t}\cdot x_2^{x_3^t}\cdots x_n^{x_1^t}=1$. Then prove that $$x_1^\beta x_2^t+x_2^\beta x_3^t+\ldots+x_n^\beta x_1^t\ge x_1^\alpha x_2^t+x_2^\alpha x_3^t+\ldots+x_n^\alpha x_1^t.$$ [i]Proposed by Marius Drăgan[/i]

Prove that for $a, b, c \in [0;1]$, $$(1-a)(1+ab)(1+ac)(1-abc) \leq (1+a)(1-ab)(1-ac)(1+abc).$$

Let $P$ be a right prism whose two bases are equilateral triangles with side length 2. The height of $P$ is $2\sqrt{3}$. Let $l$ be the line connecting the centroids of the bases. Remove the solid, keeping only the bases. Rotate one of the bases $180^\circ$ about $l$. Let $T$ be the convex hull of the two current triangles. What is the volume of $T$?

Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.

Let $0!!=1!!=1$ and $n!!=n\cdot (n-2)!!$ for all integers $n\geq 2$. Find all positive integers $n$ such that \[\dfrac{(2^n+1)!!-1}{2^{n+1}}\] is an integer.

In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.

Define $a*b=\frac{a-b}{1-ab}$. What is $(1*(2*(3*\cdots (n*(n+1))\cdots )))$?

Does there exist a field such that its multiplicative group is isomorphism to its additive group? [i]Proposed by Alexandre Chapovalov, New York University, Abu Dhabi[/i]

Right triangle ABC with a right angle at A has AB = 20 and AC = 15. Point D is on AB with BD = 2. Points E and F are placed on ray CA and ray CB, respectively, such that CD is a median of $\triangle$ CEF. Find the area of $\triangle$CEF.

We call a polynomial $P(x)=a_dx^d+...+a_0$ of degree $d$ [i]nice[/i] if $$\frac{2021(|a_d|+...+|a_0|)}{2022}<max_{0 \le i \le d}|a_i|$$ Initially Shayan has a sequence of $d$ distinct real numbers; $r_1,...,r_d \neq \pm 1$. At each step he choose a positive integer $N>1$ and raises the $d$ numbers he has to the exponent of $N$, then delete the previous $d$ numbers and constructs a monic polynomial of degree $d$ with these number as roots, then examine whether it is nice or not. Prove that after some steps, all the polynomials that shayan produces would be nice polynomials Proposed by [i]Navid Safaei[/i]

Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that: (i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and (ii) each $b_i$ is a product of one or more terms of $(a_i)$.

Given a sequence $(x_1,x_2,\ldots, x_n)$ of 0's and 1's, let $A$ be the number of triples $(x_i,x_j,x_k)$ with $i<j<k$ such that $(x_i,x_j,x_k)$ equals $(0,1,0)$ or $(1,0,1)$. For $1\leq i \leq n$, let $d_i$ denote the number of $j$ for which either $j < i$ and $x_j = x_i$ or else $j > i$ and $x_j\neq x_i$. (a) Prove that \[A = \binom n3 - \sum_{i=1}^n\binom{d_i}2.\] (Of course, $\textstyle\binom ab = \tfrac{a!}{b!(a-b)!}$.) [5 points] (b) Given an odd number $n$, what is the maximum possible value of $A$? [15 points]

Reconstruct a triangle, given the incenter, the midpoint of some side and the foot of the altitude drawn on this side.

I have a $2$ by $4$ grid of squares; how many ways can I shade at least one of the squares such that no two shaded squares share an edge?

Let $ABCDEF$ be a hexagon inscribed in a circle in which $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines $AD, BE$ and $CF$ intersect at one point.

p1. From the measurement of the height of nine trees obtained data as following. a) There are three different measurement results (in meters) b) All data are positive numbers c) Mean$ =$ median $=$ mode $= 3$ d) The sum of the squares of all data is $87.$ Determine all possible heights of the nine trees. p2. If $x$ and $y$ are integers, find the number of pairs $(x,y)$ that satisfy $|x|+|y|\le 50$. p3. The plane figure $ABCD$ on the side is a trapezoid with $AB$ parallel to $CD$. Points $E$ and $F$ lie on $CD$ so that $AD$ is parallel to $BE$ and $AF$ is parallel to $BC$. Point $H$ is the intersection of $AF$ with $BE$ and point $G$ is the intersection of $AC$ with $BE$. If the length of $AB$ is $4$ cm and the length of $CD$ is $10$ cm, calculate the ratio of the area of ​​the triangle $AGH$ to the area of ​​the trapezoid $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/c/7/e751fa791bce62f091024932c73672a518a240.png[/img] p4. A prospective doctor is required to intern in a hospital for five days in July $2011$. The hospital leadership gave the following rules: a) Internships may not be conducted on two consecutive days. b) The fifth day of internship can only be done after four days counted since the fourth day of internship. Suppose the fourth day of internship is the date $20$, then the fifth day of internship can only be carried out at least the date $24$. Determine the many possible schedule options for the prospective doctor. p5. Consider the following sequences of natural numbers: $5$, $55$, $555$, $5555$, $55555$, $...$ ,$\underbrace{\hbox{5555...555555...}}_{\hbox{n\,\,numbers}}$ . The above sequence has a rule: the $n$th term consists of $n$ numbers (digits) $5$. Show that any of the terms of the sequence is divisible by $2011$.

Let us say that a subset $M$ of the plane contains a hole if there exists a disc not contained in $M$, but contained inside some polygon with the boundary lying in $M$. Can the plane be presented as a union of $n$ convex sets such that the union of any $n-1$ from them contains a hole? Proposed by: N.Spivak

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.