Prove that $$ \sin^2 \alpha + \sin^2 \beta \geq \sin \alpha \sin \beta + \sin \alpha + \sin \beta - 1.$$

Denote by $d(n)$ be the biggest prime divisor of $|n|>1$. Find all polynomials with integer coefficients satisfy; $$P(n+d(n))=n+d(P(n)) $$ for the all $|n|>1$ integers such that $P(n)>1$ and $d(P(n))$ can be defined.

Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that \[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \] Give an example of two such progressions having at least five terms. [i]Mihai Baluna[/i]

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

Determine the real numbers $x$, $y$, $z > 0$ for which $xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$

Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$, \[f(a+b)=f(a)g(b)+g(a)f(b)\\ g(a+b)=g(a)g(b)-f(a)f(b).\]

Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all real numbers $a$ for which there exists a function $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $3(f(x))^{2}=2f(f(x))+ax^{4}$, for all $x \in \mathbb{R}^{+}$.

The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$. Determine the value of $P(1)/P(-1)$.

i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is: Find all positive integers m for which for all $\alpha,\beta \in \mathbb{Z}-\{0\}$ \[ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} \]

Let us call a polynomial [i]admissible[/i] if all it's coefficients are $0, 1, 2$ or $3$. For given $n$ find the number of all the [i]admissible [/i] polynomials $P$ such, that $P(2) = n$.

Let $a,b,c,d$ be positive irrational numbers with $a+b = 1$. Show that $c+d = 1$ if and only if $[na]+[nb] = [nc]+[nd]$ for all positive integers $n$.

Suppose that $a,b,c$ are complex numbers with $a+b+c = 0$, $|abc| = 1$, $|b| = |c|$, and $$\frac{9-\sqrt{33}}{48} \le \cos^2 \left( arg \left( \frac{b}{a} \right) \right)\le \frac{9+\sqrt{33}}{48} .$$ Find the maximum possible value of $|-a^6+b^6+c^6|$.

Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?

[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$. (b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$. Proposed by Karl Czakler

Real numbers $a,b,c$ with $a\neq b$ verify $$a^2(b+c)=b^2(c+a)=2023.$$ Find the numerical value of $E=c^2(a+b)$.

Let $f: \mathbb R \to\mathbb R$ be a function such that $|f(x)| \leq 1$ for all $x \in \mathbb R$ and \[f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right), \quad \forall x \in \mathbb R.\] Show that $f$ is a periodic function.

Positive $x,y,z$ satisfy a system: $\begin{cases} x^2 + xy + y^2/3= 25 \\ y^2/ 3 + z^2 = 9 \\ z^2 + zx + x^2 = 16 \end{cases}$ Find the value of expression $xy + 2yz + 3zx$.

Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.

Let \( f \) and \( g \) be two nonzero polynomials with integer coefficients such that \( \deg(f) > \deg(g) \). Suppose that for infinitely many prime numbers \( p \), the polynomial \( pf + g \) has a rational root. Prove that \( f \) has a rational root. Clarification: A rational root of a polynomial \( f \) is a number \( q \in \mathbb{Q} \) such that \( f(q) = 0 \).

Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system \[a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.\]

Let $a, b, c$ be positive real numbers such that $abc = \dfrac {1} {8}$. Prove the inequality:$$a ^ 2 + b ^ 2 + c ^ 2 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ 2a ^ 2 \geq \dfrac {15} {16}$$ When the equality holds?