Suppose that $4^{n}+2^{n}+1$ is prime for some positive integer $n$. Show that $n$ must be a power of $3$.

Find out wich of the following polynomials are irreducible. a) $t^4+1$ over $\mathbb{R}$; b) $t^4+1$ over $\mathbb{Q}$; c) $t^3-7t^2+3t+3$ over $\mathbb{Q}$; d) $t^4+7$ over $\mathbb{Z}_{17}$; e) $t^3-5$ over $\mathbb{Z}_{11}$; f) $t^6+7$ over $\mathbb{Q}(i)$.

If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$. [i]Author: Alex Zhu[/i]

Prove that if $1 < a < b < c$, then $$\log_a(\log_a b) + \log_b(\log_b c) + \log_c(\log_c a) > 0.$$

Let $ a$, $ b$, $ c$ be positive real numbers for which $ a \plus{} b \plus{} c \equal{} 1$. Prove that \[ {{a\minus{}bc}\over{a\plus{}bc}} \plus{} {{b\minus{}ca}\over{b\plus{}ca}} \plus{} {{c\minus{}ab}\over{c\plus{}ab}} \leq {3 \over 2}.\]

We're given the functions $f(x)=|x-1|-|x-2|$ and $g(x)=|x-3|$. a) Draw the graph of the function $f(x)$. b) Determine the area of the section enclosed by the functions $f(x)$ and $g(x)$.

p1. Given a set of $n$ the first natural number. If one of the numbers is removed, then the average number remaining is $21\frac14$ . Specify the number which is deleted. p2. Ipin and Upin play a game of Tic Tac Toe with a board measuring $3 \times 3$. Ipin gets first turn by playing $X$. Upin plays $O$. They must fill in the $X$ or $O$ mark on the board chess in turn. The winner of this game was the first person to successfully compose a sign horizontally, vertically, or diagonally. Determine as many final positions as possible, if Ipin wins in the $4$th step. For example, one of the positions the end is like the picture on the side. [img]https://cdn.artofproblemsolving.com/attachments/6/a/a8946f24f583ca5e7c3d4ce32c9aa347c7e083.png[/img] p3. Numbers $ 1$ to $10$ are arranged in pentagons so that the sum of three numbers on each side is the same. For example, in the picture next to the number the three numbers are $16$. For all possible arrangements, determine the largest and smallest values ​​of the sum of the three numbers. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3dd629361715b4edebc7803e2734e4f91ca3dc.png[/img] p4. Define $$S(n)=\sum_{k=1}^{n}(-1)^{k+1}\,\, , \,\, k=(-1)^{1+1}1+(-1)^{2+1}2+...+(-1)^{n+1}n$$ Investigate whether there are positive integers $m$ and $n$ that satisfy $S(m) + S(n) + S(m + n) = 2011$ p5. Consider the cube $ABCD.EFGH$ with side length $2$ units. Point $A, B, C$, and $D$ lie in the lower side plane. Point $I$ is intersection point of the diagonal lines on the plane of the upper side. Next, make a pyramid $I.ABCD$. If the pyramid $I.ABCD$ is cut by a diagonal plane connecting the points $A, B, G$, and $H$, determine the volume of the truncated pyramid low part.

If $a,b,c>0,a+b+c=1$,then: $\frac{1}{abc}+\frac{4}{a^{2}+b^{2}+c^{2}}\geq\frac{13}{ab+bc+ca}$

A frog is jumping on $N$ stones which are numbered from $1$ to $N$ from left to right. The frog is jumping to the previous stone (to the left) with probability $p$ and is jumping to the next stone (to the right) with probability $1-p$. If the frog has jumped to the left from the leftmost stone or to the right from the rightmost stone, it will fall into the water. The frog is initially on the leftmost stone. If $p< \tfrac 13$, show that the frog will fall into the water from the rightmost stone with a probability higher than $\tfrac 12$.

Find the smallest positive period of the function $f(x)=\sin (1998x)+ \sin (2000x)$

Let $R$ be the set of real numbers and $f : R \to R$ be a function that satisfies: $$f(xy) + y + f(x + f(y)) = (y + 1)f(x),$$ for all real numbers $x, y$. a) Determine the value of $f(0)$. b) Prove that $f(x) = 2-x$ for every real number $x$.

A function $f$ satisfies the following condition for each $n\in N$: $f (1)+ f (2)+...+ f (n) = n^2 f (n)$. Find $f (1997)$ if $f (1) = 999$.

[b]p1. [/b] If $f(x) = 3x - 1$, what is $f^6(2) = (f \circ f \circ f \circ f \circ f \circ f)(2)$? [b]p2.[/b] A frog starts at the origin of the $(x, y)$ plane and wants to go to $(6, 6)$. It can either jump to the right one unit or jump up one unit. How many ways are there for the frog to jump from the origin to $(6, 6)$ without passing through point $(2, 3)$? [b]p3.[/b] Alfred, Bob, and Carl plan to meet at a café between noon and $2$ pm. Alfred and Bob will arrive at a random time between noon and $2$ pm. They will wait for $20$ minutes or until $2$ pm for all $3$ people to show up after which they will leave. Carl will arrive at the café at noon and leave at $1:30$ pm. What is the probability that all three will meet together? [b]p4.[/b] Let triangle $ABC$ be isosceles with $AB = AC$. Let $BD$ be the altitude from $ B$ to $AC$, $E$ be the midpoint of $AB$, and $AF$ be the altitude from $ A$ to $BC$. If $AF = 8$ and the area of triangle $ACE$ is $ 8$, find the length of $CD$. [b]p5.[/b] Find the sum of the unique prime factors of $(2018^2 - 121) \cdot (2018^2 - 9)$. [b]p6.[/b] Compute the remainder when $3^{102} + 3^{101} + ... + 3^0$ is divided by $101$. [b]p7.[/b] Take regular heptagon $DUKMATH$ with side length $ 3$. Find the value of $$\frac{1}{DK}+\frac{1}{DM}.$$ [b]p8.[/b] RJ’s favorite number is a positive integer less than $1000$. It has final digit of $3$ when written in base $5$ and final digit $4$ when written in base $6$. How many guesses do you need to be certain that you can guess RJ’s favorite number? [b]p9.[/b] Let $f(a, b) = \frac{a^2+b^2}{ab-1}$ , where $a$ and $b$ are positive integers, $ab \ne 1$. Let $x$ be the maximum positive integer value of $f$, and let $y$ be the minimum positive integer value of f. What is $x - y$ ? [b]p10.[/b] Haoyang has a circular cylinder container with height $50$ and radius $5$ that contains $5$ tennis balls, each with outer-radius $5$ and thickness $1$. Since Haoyang is very smart, he figures out that he can fit in more balls if he cuts each of the balls in half, then puts them in the container, so he is ”stacking” the halves. How many balls would he have to cut up to fill up the container? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$. Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$ [i]Proposed by Nikola Velov[/i]

prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.

Denote by $F_0(x)$, $F_1(x)$, $\ldots$ the sequence of Fibonacci polynomials, which satisfy the recurrence $F_0(x)=1$, $F_1(x)=x$, and $F_n(x)=xF_{n-1}(x)+F_{n-2}(x)$ for all $n\geq 2$. It is given that there exist unique integers $\lambda_0$, $\lambda_1$, $\ldots$, $\lambda_{1000}$ such that \[x^{1000}=\sum_{i=0}^{1000}\lambda_iF_i(x)\] for all real $x$. For which integer $k$ is $|\lambda_k|$ maximized?

Numbers from $1$ to $8$ are placed on the vertices of a cube, one on each of the eight vertices, so that the sum of the numbers on any three vertices of the same face is greater than $9$. Determines the minimum value that the sum of the numbers on one side can have.

Let $(a_i)_{i\in \mathbb{N}}$ be a sequence with $a_1=\frac{3}2$ such that $$a_{n+1}=1+\frac{n}{a_n}$$ Find $n$ such that $2020\le a_n <2021$

Suppose $x>1$ is a real number such that $x+\tfrac 1x = \sqrt{22}$. What is $x^2-\tfrac1{x^2}$?

Let $ 9 < n_{1} < n_{2} < \ldots < n_{s} < 1992$ be positive integers and \[ P(x) \equal{} 1 \plus{} x^{2} \plus{} x^{9} \plus{} x^{n_{1}} \plus{} \cdots \plus{} x^{n_{s}} \plus{} x^{1992}.\] Prove that if $ x_{0}$ is real root of $ P(x)$ then $ x_{0}\leq\frac {1 \minus{} \sqrt {5}}{2}$.

Suppose $a$ and $b$ are real numbers such that the polynomial $x^3+ax^2+bx+15$ has a factor of $x^2-2$. Find the value of $a^2b^2$.

Let $p(z)$ be a polynomial of degree $n,$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=\frac{p(z)}{z^{n/2}}.$ Show that all zeros of $g'(z)=0$ have absolute value $1.$

Let $ X: \equal{} \{x_1,x_2,\ldots,x_{29}\}$ be a set of $ 29$ boys: they play with each other in a tournament of Pro Evolution Soccer 2009, in respect of the following rules: [list]i) every boy play one and only one time against each other boy (so we can assume that every match has the form $ (x_i \text{ Vs } x_j)$ for some $ i \neq j$); ii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the win of the boy $ x_i$, then $ x_i$ gains $ 1$ point, and $ x_j$ doesn’t gain any point; iii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the parity of the two boys, then $ \frac {1}{2}$ point is assigned to both boys. [/list] (We assume for simplicity that in the imaginary match $ (x_i \text{ Vs } x_i)$ the boy $ x_i$ doesn’t gain any point). Show that for some positive integer $ k \le 29$ there exist a set of boys $ \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X$ such that, for all choice of the positive integer $ i \le 29$, the boy $ x_i$ gains always a integer number of points in the total of the matches $ \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}$. [i](Paolo Leonetti)[/i]

Let $n$ be a positive integer greater than 1. Determine all the collections of real numbers $x_1,\ x_2,\dots,\ x_n\geq1\mbox{ and }x_{n+1}\leq0$ such that the next two conditions hold: (i) $x_1^{\frac12}+x_2^{\frac32}+\cdots+x_n^{n-\frac12}= nx_{n+1}^\frac12$ (ii) $\frac{x_1+x_2+\cdots+x_n}{n}=x_{n+1}$

Let $k$ be an integer number and $P(X)$ be the polynomial $$P(X) = X^{1997}-X^{1995} +X^2-3kX+3k+1$$ Prove that: a) the polynomial has no integer root; β) the numbers $P(n)$ and $P(n) + 3$ are relatively prime, for every integer $n$.