$P$ is an monic integer coefficient polynomial which has no integer roots. deg$P=n$ and define $A$ $:=${$v_2(P(m))|m\in Z, v_2(P(m)) \ge 1$}. If $|A|=n$, show that all of the elements of $A$ is smaller than $\frac{3}{2}n^2$.

Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$. Suppose that $\lambda$ ($|\lambda|>1$) is a complex root of the equation, prove that $\lambda^{n+1}=1$.

Find all primes $p, q$ and natural numbers $n$ such that: $p(p+1)+q(q+1)=n(n+1)$

If x,y<0 prove that $\left(x+\frac{2}{y} \right) \left(\frac{y}{x}+2 \right)\geq 8$. When do we have equality?

Find all functions $f : R \to R$ that satisfy $f(x^2 + f(y) - y) = (f(x))^2$ for all $x,y \in R$.

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

Let $ a_1, $ $ a_2, $ $ \ldots, $ $ a_ {99} $ be positive real numbers such that $ ia_j + ja_i \ge i + j $ for all $ 1 \le i <j \le 99. $ Prove , that $ (a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!$ .

[b]p1.[/b] At a certain point in time, $20\%$ of seniors, $30\%$ of juniors, and $50\%$ of sophomores at a school had a cold. If the number of sick students was the same for each grade, the fraction of sick students across all three grades can be written as $\frac{a}{b}$ , where a and b are relatively prime positive integers. Find $a + b$. [b]p2.[/b] The average score on Mr. Feng’s recent test is a $63$ out of $100$. After two students drop out of the class, the average score of the remaining students on that test is now a $72$. What is the maximum number of students that could initially have been in Mr. Feng’s class? (All of the scores on the test are integers between $0$ and $100$, inclusive.) [b]p3.[/b] Madeline is climbing Celeste Mountain. She starts at $(0, 0)$ on the coordinate plane and wants to reach the summit at $(7, 4)$. Every hour, she moves either $1$ unit up or $1$ unit to the right. A strawberry is located at each of $(1, 1)$ and $(4, 3)$. How many paths can Madeline take so that she encounters exactly one strawberry? [b]p4.[/b] Let $E$ be a point on side $AD$ of rectangle $ABCD$. Given that $AB = 3$, $AE = 4$, and $\angle BEC = \angle CED$, the length of segment $CE$ can be written as $\sqrt{a}$ for some positive integer $a$. Find $a$. [b]p5.[/b] Lucy has some spare change. If she were to convert it into quarters and pennies, the minimum number of coins she would need is $66$. If she were to convert it into dimes and pennies, the minimum number of coins she would need is $147$. How much money, in cents, does Lucy have? [b]p6.[/b] For how many positive integers $x$ does there exist a triangle with altitudes of length $20$, $22$, and $x$? [b]p7.[/b] Compute the number of positive integers $x$ for which $\frac{x^{20}}{x+22}$ is an integer. [b]p8.[/b] Vincent the Bug is crawling along an octagonal prism. He starts on a fixed vertex $A$, visits all other vertices exactly once by traveling along the edges, and returns to $A$. Find the number of paths Vincent could have taken. [b]p9.[/b] Point $U$ is chosen inside square $ALEX$ so that $\angle AUL = 90^o$. Given that $UL = 56$ and $UE = 65$, what is the sum of all possible values for the area of square $ALEX$? [b]p10.[/b] Miranda has prepared $8$ outfits, no two of which are the same quality. She asks her intern Andrea to order these outfits for the new runway show. Andrea first randomly orders the outfits in a list. She then starts removing outfits according to the following method: she chooses a random outfit which is both immediately preceded and immediately succeeded by a better outfit and then removes it. Andrea repeats this process until there are no outfits that can be removed. Given that the expected number of outfits in the final routine can be written as $\frac{a}{b}$ for some relatively prime positive integers $a$ and $b$, find $a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

a) prove that every discrete subgroup of $(\mathbb R^2,+)$ is in one of these forms: i-$\{0\}$. ii-$\{mv|m\in \mathbb Z\}$ for a vector $v$ in $\mathbb R^2$. iii-$\{mv+nw|m,n\in \mathbb Z\}$ for tho linearly independent vectors $v$ and $w$ in $\mathbb R^2$.(lattice $L$) b) prove that every finite group of symmetries that fixes the origin and the lattice $L$ is in one of these forms: $\mathcal C_i$ or $\mathcal D_i$ that $i=1,2,3,4,6$ ($\mathcal C_i$ is the cyclic group of order $i$ and $\mathcal D_i$ is the dyhedral group of order $i$).(20 points)

Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$ \[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \] Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.

Let be a finite set $ A $ of real numbers, and define the sets $ S_{\pm }=\{ x\pm y| x,y\in A \} . $ Show that $ \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 . $

Prove that exists a infinity of triplets $a, b, c\in\mathbb{R}$ satisfying simultaneously the relations $a+b+c=0$ and $a^4+b^4+c^4=50$. Moldova National Math Olympiad 2010, 12th grade

Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$

Given a positive integer $N$. Prove that: there are infinitely elements of the [i]Fibonacci[/i] sequence that are divisible by $N$.

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of integers defined recursively as $ x_{n+2}=5x_{n+1}-x_n. $ Prove that $ \left( x_n\right)_{n\ge 1} $ has a subsequence whose terms are multiples of $ 22 $ if $ \left( x_n\right)_{n\ge 1} $ has a term that is multiple of $ 22. $

Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$ Prove that the system of equations \[\sqrt{y-a}+\sqrt{z-a}=1\] \[\sqrt{z-b}+\sqrt{x-b}=1\] \[\sqrt{x-c}+\sqrt{y-c}=1\] has exactly one solution $(x,y,z)$ in real numbers. It was proposed by Poland. Have fun! :lol:

Find all polynomials $P(x)$ , such that $$P(x^3+1)=\left(P (x+1)\right)^3$$

For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality $$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | \geq 2$$ and determine all cases of equality. Prove that if we also impose $a,b,c \geq 0$, then $$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | > 3,$$ with the value $3$ being the best constant possible. ([i]Dan Schwarz[/i])

Find all $3$-tuples $(a, b, c)$ of positive integers, with $a \geq b \geq c$, such that $a^2 + 3b$, $b^2 + 3c$, and $c^2 + 3a$ are all squares.

In the sequence of the natural (i.e. positive integers) numbers every member from the third equals the absolute value of the difference of the two previous. What is the maximal possible length of such a sequence, if every member is less or equal to $1967$?

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying $$f(x^2 + f(x)f(y)) = xf(x + y)$$ for all real numbers $x$ and $y$.

It is known that the number $\pi$ is transcendental, that is, it is not a root of any polynomial with integer coefficients. Using this fact, prove that the same is true for the number $\pi + \sqrt2$.

Consider integers $m\ge 2$ and $n\ge 1$. Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .

Prove that : $\cos36-\sin18=\frac{1}{2}$

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]