Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

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

In a triangle $ABC$, $\angle A = 2 \cdot \angle B$. Prove that $a^2 = b (b+c)$.

The price of a diamond is proportional to the square of its weight. Show that, breaking it into two parts, there is a depreciation of its value. When is it the maximum depreciation?

Pit and Bill play the following game. First Pit writes down a number $a$, then Bill writes a number $b$, then Pit writes a number $c$. Can Pit always play so that the three equations $$x^3+ax^2+bx+c, x^3+bx^2+cx+a, x^3+cx^2+ax+b$$ have (a) a common real root; (b) a common negative root?

Let $a$, $b$, $c$ and $d$ positive real numbers with $a > c$ and $b < d$. Assume that \[a + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d\] Prove that $a + b + c + d > 1$. [i]Proposed by Victor Domínguez[/i]

Problem 2. Let $P(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers. If $$P(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab$$ then prove that $$(a - b)(b - c)(c - a)(a + b + c) = 0.$$

An infinite sequence of distinct real numbers is given. Prove that it contains a subsequence of $1999$ terms which is either monotonically increasing or monotonically decreasing.

For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always: \[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

The polynomial $f(x)=1-x+x^2-x^3+\cdots-x^{19}+x^{20}$ is written into the form $g(y)=a_0+a_1y+a_2y^2+\cdots+a_{20}y^{20}$, where $y=x-4$, then $a_0+a_1+\cdots+a_{20}=$________.

Find all real solutions to the equation $(x^2-9x+19)^{x^2-3x+2} = 1$.

Given is a graph $G$ of $n+1$ vertices, which is constructed as follows: initially there is only one vertex $v$, and one a move we can add a vertex and connect it to exactly one among the previous vertices. The vertices have non-negative real weights such that $v$ has weight $0$ and each other vertex has a weight not exceeding the avarage weight of its neighbors, increased by $1$. Prove that no weight can exceed $n^2$.

In an infinite sequence $\{\alpha\}, \{\alpha^2\}, \{\alpha^3\}, \cdots $ there are finitely many distinct values$.$ Show that $\alpha$ is an integer$. (\{x\}$ denotes the fractional part of$ x.)$ [i](Golovanov A.S.)[/i]

Find all positive integers $d$ for which there exist polynomials $P(x)$ and $Q(x)$ with real coefficients such that degree of $P$ equals $d$ and $$P(x)^2+1=(x^2+1)Q(x)^2.$$

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality $$ \left ( \frac{1+a}{b}+2 \right ) \left ( \frac{1+b}{c}+2 \right ) \left ( \frac{1+c}{a}+2 \right )\geq 216.$$ When does equality hold? [i]Authored by Anastasija Trajanova[/i]

Let $a$, $b$, $c$, $d$ be positive numbers. Prove that $$(a^2 + b^2 + c^2 + d^2)^2 \ge (a+b)(b+c)(c+d)(d+a)$$ When does equality hold? (Georg Anegg)

[b]7.1[/b] Construct a trapezoid given four sides. [b]7.2[/b] Prove that $$(1 + x + x^2 + ...+ x^{100})(1 + x^{102}) - 102x^{101} \ge 0 .$$ [b]7.3 [/b] In a quadrilateral $ABCD$, $M$ is the midpoint of AB, $N$ is the midpoint of $CD$. Lines $AD$ and BC intersect $MN$ at points $P$ and $Q$, respectively. Prove that if $\angle BQM = \angle APM$ , then $BC=AD$. [img]https://cdn.artofproblemsolving.com/attachments/a/2/1c3cbc62ee570a823b5f3f8d046da9fbb4b0f2.png[/img] [b]7.4 / 6.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]7.5 / 8.4[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. [img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png[/img] [b]7.6 / 6.5 [/b]The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

For real numbers $B,M,$ and $T,$ we have $B^2+M^2+T^2 =2022$ and $B+M+T =72.$ Compute the sum of the minimum and maximum possible values of $T.$

Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0,1,2,\cdots, 12$ such that the value of the resulting expression is divisible by 5. [i]Proposed by Matthew Lerner-Brecher[/i]

A sequence $\{z_n\}$ satisfies that for any positive integer $i,$ $z_i\in\{0,1,\cdots,9\}$ and $z_i\equiv i-1 \pmod {10}.$ Suppose there is $2021$ non-negative reals $x_1,x_2,\cdots,x_{2021}$ such that for $k=1,2,\cdots,2021,$ $$\sum_{i=1}^kx_i\geq\sum_{i=1}^kz_i,\sum_{i=1}^kx_i\leq\sum_{i=1}^kz_i+\sum_{j=1}^{10}\dfrac{10-j}{50}z_{k+j}.$$ Determine the least possible value of $\sum_{i=1}^{2021}x_i^2.$

Let $a$ be a given positive integer. We consider the sequence $(x_n)_{n \ge 1}$ defined by $x_n=\frac{1}{1+na},$ for every positive integer $n.$ Prove that for any integer $k \ge 3,$ there exist positive integers $n_1<n_2<\ldots<n_k$ such that the numbers $x_{n_1},x_{n_2},\ldots,x_{n_k}$ are consecutive terms in an arithmetic progression.

Let $a,b,c$ be distinct nonzero real numbers. If the equations $ax^3+bx+c=0$, $bx^3+cx+a=0,$ and $cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real roots(not necessarily distinct).

Find the largest constant $k$ such that the inequality $$a^2+b^2+c^2-ab-bc-ca \ge k \left|\frac{a^3-b^3}{a+b}+\frac{b^3-c^3}{b+c}+\frac{c^3-a^3}{c+a}\right|$$ holds for any for non negative real numbers $a,b,c$ with $(a+b)(b+c)(c+a)>0$.

Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$ \[ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . \] [i]Proposed by F. Petrov[/i]