Let $(a_n)_1^{\infty}$ be a sequence such that $a_n \le a_{n+m} \le a_n + a_m$ for all positive integers $n$ and $m$. Prove that $\frac{a_n}{n}$ has a limit as $n$ approaches infinity.

Prove that there is no real function $ f(x)$ satisfying $ f\left(f(x)\right) \equal{} x^2 \minus{} 2$ for all real number $ x$.

[b]6.1[/b] The student bought a briefcase, a fountain pen and a book. If the briefcase cost 5 times cheaper, the fountain pen was 2 times cheaper, and the book was 2 1/2 times cheaper cheaper, then the entire purchase would cost 2 rubles. If the briefcase was worth 2 times cheaper, a fountain pen is 4 times cheaper, and a book is 3 times cheaper, then the whole the purchase would cost 3 rubles. How much does it really cost? ´ [b]6.2.[/b] Which number is greater: $$\underbrace{888...88}_{19 \, digits} \cdot \underbrace{333...33}_{68 \, digits} \,\,\, or \,\,\, \underbrace{444...44}_{19 \, digits} \cdot \underbrace{666...67}_{68 \, digits} \, ?$$ [b]6.3[/b] Distance between Luga and Volkhov 194 km, between Volkhov and Lodeynoye Pole 116 km, between Lodeynoye Pole and Pskov 451 km, between Pskov and Luga 141 km. What is the distance between Pskov and Volkhov? [b]6.4 [/b] There are $4$ objects in pairs of different weights. How to use a pan scale without weights Using five weighings, arrange all these objects in order of increasing weights? [b]6.5 [/b]. Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams. [b]6.6 [/b] In task 6.1, determine what is more expensive: a briefcase or a fountain pen. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

The polynomial $ P(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}d$ has three distinct real roots. The polynomial $ P(Q(x))$, where $ Q(x)\equal{}x^2\plus{}x\plus{}2001$, has no real roots. Prove that $ P(2001)>\frac{1}{64}$.

Prove that there is no function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(f(n))=n+1.$ Here $\mathbb{N}$ is the positive integers $\{1,2,3,\dots\}.$

Assume that the polynomial $\left(x+1\right)^n-1$ is divisible by some polynomial $P\left(x\right)=x^k+c_{k-1}x^{k-1}+c_{k-2}x^{k-2}+...+c_1x+c_0$, whose degree $k$ is even and whose coefficients $c_{k-1}$, $c_{k-2}$, ..., $c_1$, $c_0$ all are odd integers. Show that $k+1\mid n$.

$x \geq 0$ and $y$ are real numbers for which $y^2 \geq x(x + 1)$. Prove that: $(y - 1)^2 \geq x(x-1)$.

Does there exist a triangle with length $a,b,c$ such that: [b]a)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=28 \end{cases}$ [b]b)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=30 \end{cases}$

Real numbers $a_1,a_2,...,a_n$ ($n \ge 3$) satisfy the conditions $a_1 = a_n = 0$ and $$a_{k-1}+a_{k+1} \ge 2a_k$$ for $k = 2$,$3$$,...,$$n -1$. Prove that none of the numbers $a_1$,$...$,$a_n$ is positive.

Suppose $P$ is a cubic polynomial satisfying $P(0) = 3$ and \[(x^3 - 2x + 1 - P(x))(2x^3 - 5x^2 + 4 - P(x))\leq 0\] for all $x\in\mathbb R$. Determine all possible values of $P(-1)$.

Let $a,b,c \in \mathbb{R}^+$ where $a^2+b^2+c^2=1$. Find the minimum value of . $$a+b+c+\frac{3}{ab+bc+ca}$$ [i](PP-nine)[/i]

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

Let $Q(x)$ be a quadratic trinomial. Given that the function $P(x)=x^{2}Q(x)$ is increasing in the interval $(0,\infty )$, prove that: \[P(x) + P(y) + P(z) > 0\] for all real numbers $x,y,z$ such that $x+y+z>0$ and $xyz>0$.

Given unequal integers $x, y, z$ prove that $(x-y)^5 + (y-z)^5 + (z-x)^5$ is divisible by $5(x-y)(y- z)(z-x)$.

Suppose $ a, b,$ and $ c$ are positive integers with $ a \plus{} b \plus{} c \equal{} 2006$, and $ a!b!c! \equal{} m\cdot10^n$, where $ m$ and $ n$ are integers and $ m$ is not divisible by 10. What is the smallest possible value of $ n$? $ \textbf{(A) } 489 \qquad \textbf{(B) } 492 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 498 \qquad \textbf{(E) } 501$

On a circle $4$ points are chosen and for each point we wrote the multiple of its distances to the rest. Could the written numbers be $1,2,3, 4$ in some order?

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

For any real number$m$, the equation $$x^2+(m-2)x- (m+3)=0$$ has two solutions, denoted $x_1 $and $ x_2$. Determine $m$ such that $x_1^2+x_2^2$ is the minimum possible.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying \[ f(xy)(f(x) - f(y)) = (x-y)f(x)f(y) \] for all $x,y$.

Given a positive integer $k$, let $||k||$ denote the absolute difference between $k$ and the nearest perfect square. For example, $||13||=3$ since the nearest perfect square to $13$ is $16$. Compute the smallest positive integer $n$ such that $\frac{||1|| + ||2|| + ...+ ||n||}{n}=100$.

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

The polynomial $-400x^5+2660x^4-3602x^3+1510x^2+18x-90$ has five rational roots. Suppose you guess a rational number which could possibly be a root (according to the rational root theorem). What is the probability that it actually is a root?

The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.

Let $f(x) =\frac{x+a}{x+b}$ satisfy $f(f(f(x))) = x$ for real numbers $a, b$. If the maximum value of a is $p/q$, where $p, q$ are relatively prime integers, what is $|p| + |q|$?

a) Let $a$ and $b$ be two non-negative real numbers. Show that $a+b \ge \sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab}$ b) Let $a$ and $b$ be two real numbers in $[0, 3]$. Show that $\sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab} \ge \frac{(a+b)^2}{2}$